diff --git a/app/src/data/questions/comp2804/exam-fall-2014/exam-fall-2014.json b/app/src/data/questions/comp2804/exam-fall-2014/exam-fall-2014.json new file mode 100644 index 00000000..1b851e8c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/exam-fall-2014.json @@ -0,0 +1,32 @@ +{ + "id": "exam-fall-2014", + "title": "Fall 2014 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-fall-2014/f14f01.xml", + "comp2804/exam-fall-2014/f14f02.xml", + "comp2804/exam-fall-2014/f14f03.xml", + "comp2804/exam-fall-2014/f14f04.xml", + "comp2804/exam-fall-2014/f14f05.xml", + "comp2804/exam-fall-2014/f14f06.xml", + "comp2804/exam-fall-2014/f14f07.xml", + "comp2804/exam-fall-2014/f14f08.xml", + "comp2804/exam-fall-2014/f14f09.xml", + "comp2804/exam-fall-2014/f14f10.xml", + "comp2804/exam-fall-2014/f14f11.xml", + "comp2804/exam-fall-2014/f14f12.xml", + "comp2804/exam-fall-2014/f14f13.xml", + "comp2804/exam-fall-2014/f14f14.xml", + "comp2804/exam-fall-2014/f14f15.xml", + "comp2804/exam-fall-2014/f14f16.xml", + "comp2804/exam-fall-2014/f14f17.xml", + "comp2804/exam-fall-2014/f14f18.xml", + "comp2804/exam-fall-2014/f14f19.xml", + "comp2804/exam-fall-2014/f14f20.xml", + "comp2804/exam-fall-2014/f14f21.xml", + "comp2804/exam-fall-2014/f14f22.xml", + "comp2804/exam-fall-2014/f14f23.xml", + "comp2804/exam-fall-2014/f14f24.xml", + "comp2804/exam-fall-2014/f14f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f01.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f01.xml new file mode 100644 index 00000000..4bc50e7e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f01.xml @@ -0,0 +1,12 @@ + + + + A password must contain exactly one odd digit. How many passwords are there? + ]]> + + $13 \cdot 5^{12}$ + $13 \cdot 5^{13}$ + $13 \cdot 9^{12}$ + $13 \cdot 5 \cdot 9^{12}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f02.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f02.xml new file mode 100644 index 00000000..96f03605 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f02.xml @@ -0,0 +1,13 @@ + + + + A password must contain at least one odd digit and at most two even digits. How many passwords are + there? + ]]> + + $5^{12} + 13 \cdot 5^{12} + {13 \choose 2}5^{12}$ + $5^{13} + 13 \cdot 9^{12} + {13 \choose 2}9^{12}$ + $5^{13} + 13 \cdot 5^{13} + {13 \choose 2}5^{13}$ + $5 \cdot 9^{12} + 13 \cdot 5 \cdot 9^{12} + {13 \choose 2}5 \cdot 9^{12}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f03.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f03.xml new file mode 100644 index 00000000..a6f8dbc7 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f03.xml @@ -0,0 +1,14 @@ + + + + + $(-2)^{45}$ + $4^{45}$ + $(-8)^{45}$ + $10^{45}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f04.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f04.xml new file mode 100644 index 00000000..b34652b7 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f04.xml @@ -0,0 +1,11 @@ + + + + + $2^{10} + 2^{11}$ + $3 \cdot 2^{10} - 2^{8}$ + $2^{13} - (2^{10} + 2^{11})$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f05.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f05.xml new file mode 100644 index 00000000..7fc4efe2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f05.xml @@ -0,0 +1,12 @@ + + + + + $S_n = S_{n-1} + S_{n-2}\ \mathrm{for}\ n \geq 3.$ + $S_n = S_{n-1} + S_{n-3}\ \mathrm{for}\ n \geq 3.$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3}\ \mathrm{for}\ n \geq 3.$ + $S_n = 2 \cdot S_{n-1}\ \mathrm{for}\ n \geq 3.$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f06.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f06.xml new file mode 100644 index 00000000..4080934a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f06.xml @@ -0,0 +1,23 @@ + + + +
  • + 40 like 8:30am classes, +
    • +
  • + 30 like the course COMP 2804, +
    • +
  • + 50 do not like 8:30am classes and do not like the course COMP 2804. +
    • + + How many students in this group like 8:30am classes and like the course COMP 2804? + ]]> +     + 10 + 20 + 30 + 40 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f07.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f07.xml new file mode 100644 index 00000000..b8353a12 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f07.xml @@ -0,0 +1,13 @@ + + + + + $7!{m \choose 7} + 7!{n \choose 7}$ + $m!{m \choose 7} + n!{n \choose 7}$ + $7!{m + n \choose 7}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f08.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f08.xml new file mode 100644 index 00000000..ca69cbd1 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f08.xml @@ -0,0 +1,11 @@ + + + + + $10!$ + $\frac{10!}{2!3!5!}$ + ${10 \choose 3}{10 \choose 2}{10 \choose 5}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f09.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f09.xml new file mode 100644 index 00000000..3c5e8138 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f09.xml @@ -0,0 +1,12 @@ + + + + + ${13 \choose 2}$ + ${14 \choose 2}$ + ${15 \choose 2}$ + ${16 \choose 2}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f10.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f10.xml new file mode 100644 index 00000000..7f2bb29a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f10.xml @@ -0,0 +1,24 @@ + + + + + + $f(0) = $ + $-17,$ + + + $f(n) = $ + $f(n - 1) + 8n - 2\; \ \text{for all}$ $\text{integers}\ n \geq 1.$ + + + + Which of the following is true? + ]]> + + for all $n \geq 0$: $f(n) = 3n^{2} - n - 17$ + for all $n \geq 0$: $f(n) = 3n^{2} + n - 17$ + for all $n \geq 0$: $f(n) = 4n^{2} - 2n - 17$ + for all $n \geq 0$: $f(n) = 4n^{2} + 2n - 17$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f11.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f11.xml new file mode 100644 index 00000000..de7cf924 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f11.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + If we run algorithm $\Fib(20)$, how many calls are there to $\Fib(16)$? + ]]> +     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f12.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f12.xml new file mode 100644 index 00000000..68224de0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f12.xml @@ -0,0 +1,12 @@ + + + + + $\frac{{5 \choose 2} + {4 \choose 2} + {6 \choose 2}}{{15 \choose 2}}$ + $\frac{{15 \choose 2}}{{5 \choose 2} + {4 \choose 2} + {6 \choose 2}}$ + $\frac{{5 \choose 2}{4 \choose 2}{6 \choose 2}}{{15 \choose 2}}$ + $\frac{{15 \choose 2}}{{5 \choose 2}{4 \choose 2}{6 \choose 2}}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f13.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f13.xml new file mode 100644 index 00000000..bdf100cd --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f13.xml @@ -0,0 +1,12 @@ + + + + + $\frac{1}{364 \cdot 365}$ + $\frac{1}{365^2}$ + $\frac{365}{364^{2}}$ + $\frac{1}{365^{3}}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f14.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f14.xml new file mode 100644 index 00000000..f8c2f394 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f14.xml @@ -0,0 +1,13 @@ + + + + + $1 - (1/2)^{7}$ + $\sum_{k=0}^{7} (1/2)^{k}$ + $\sum_{k=0}^{7} (1/2)^{k+1}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f15.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f15.xml new file mode 100644 index 00000000..9378cdea --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f15.xml @@ -0,0 +1,17 @@ + + + + + $\frac{{4 \choose 2}}{{5 \choose 2} + {4 \choose 2}}$ + $\frac{{5 \choose 2} + {4 \choose 2}}{{4 \choose 2}}$ + $\frac{{4 \choose 2}}{{9 \choose 2}}$ + $\frac{{4 \choose 2}}{{5 \choose 2}{4 \choose 2}}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f16.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f16.xml new file mode 100644 index 00000000..78bdc0a0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f16.xml @@ -0,0 +1,16 @@ + + + + + 3/10 + 1/3 + 1/2 + 2/3 + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f17.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f17.xml new file mode 100644 index 00000000..be64a0f4 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f17.xml @@ -0,0 +1,18 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f18.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f18.xml new file mode 100644 index 00000000..2b910a44 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f18.xml @@ -0,0 +1,18 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f19.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f19.xml new file mode 100644 index 00000000..b0fd3107 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f19.xml @@ -0,0 +1,20 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f20.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f20.xml new file mode 100644 index 00000000..b6c80a7e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f20.xml @@ -0,0 +1,13 @@ + + + + + $\frac{1}{3}$ + $\frac{1}{4}$ + $\frac{1}{4} \cdot \frac{1}{3} + \frac{3}{4} \cdot \frac{1}{4}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f21.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f21.xml new file mode 100644 index 00000000..bc6a0fc0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f21.xml @@ -0,0 +1,14 @@ + + + + + 2 + 1 + 1/4 + 1/2 + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f22.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f22.xml new file mode 100644 index 00000000..dcb9061d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f22.xml @@ -0,0 +1,18 @@ + + + + + 2/5 + 2 + 5/2 + 5 + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f23.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f23.xml new file mode 100644 index 00000000..c152c5ba --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f23.xml @@ -0,0 +1,37 @@ + + + + $\newcommand{\ThreeHeadsOrThreeTails}{{\rm T {\scriptsize HREE} H {\scriptsize EADS} O {\scriptsize R} T {\scriptsize HREE} T {\scriptsize AILS}}}$ + + + Consider the following recursive algorithm $\ThreeHeadsOrThreeTails$, which takes as input a positive integer $k$: + +
    + $\mathbf{Algorithm}$ $\ThreeHeadsOrThreeTails(k)\mathrm{:}$
    + + + + + + + +
    $//$ + $\text{all coin flips made}$ $\text{are mutually independent}$ +
    + $\text{flip a fair coin three times};$
    + $\mathbf{if}\ \text{the result is \(HHH\) or \(TTT\)}$
    + $\mathbf{then}\ \mathrm{return}\ k$
    + $\mathbf{else}\ \ThreeHeadsOrThreeTails(k + 1)$
    + $\mathbf{endif}$ +
    + + You run algorithm $\ThreeHeadsOrThreeTails(1)$, i.e., with $k = 1$. Define the random variable $X$ to be the value of the output + of this algorithm. What is the expected value of $X$? + ]]> +     + 2 + 3 + 4 + 5 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f24.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f24.xml new file mode 100644 index 00000000..7b1c6f10 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f24.xml @@ -0,0 +1,15 @@ + + + + + $-n/2$ + $n/2$ + $-n$ + $n$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2014/f14f25.xml b/app/src/data/questions/comp2804/exam-fall-2014/f14f25.xml new file mode 100644 index 00000000..d6789cfe --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2014/f14f25.xml @@ -0,0 +1,7 @@ + + Who invented the Fibonacci numbers? + Justin Bieber + Britney Spears + Fibonacci + Carl Friedrich Gauss + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/exam-fall-2015.json b/app/src/data/questions/comp2804/exam-fall-2015/exam-fall-2015.json new file mode 100644 index 00000000..dc8802d9 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/exam-fall-2015.json @@ -0,0 +1,32 @@ +{ + "id": "exam-fall-2015", + "title": "Fall 2015 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-fall-2015/f15f01.xml", + "comp2804/exam-fall-2015/f15f02.xml", + "comp2804/exam-fall-2015/f15f03.xml", + "comp2804/exam-fall-2015/f15f04.xml", + "comp2804/exam-fall-2015/f15f05.xml", + "comp2804/exam-fall-2015/f15f06.xml", + "comp2804/exam-fall-2015/f15f07.xml", + "comp2804/exam-fall-2015/f15f08.xml", + "comp2804/exam-fall-2015/f15f09.xml", + "comp2804/exam-fall-2015/f15f10.xml", + "comp2804/exam-fall-2015/f15f11.xml", + "comp2804/exam-fall-2015/f15f12.xml", + "comp2804/exam-fall-2015/f15f13.xml", + "comp2804/exam-fall-2015/f15f14.xml", + "comp2804/exam-fall-2015/f15f15.xml", + "comp2804/exam-fall-2015/f15f16.xml", + "comp2804/exam-fall-2015/f15f17.xml", + "comp2804/exam-fall-2015/f15f18.xml", + "comp2804/exam-fall-2015/f15f19.xml", + "comp2804/exam-fall-2015/f15f20.xml", + "comp2804/exam-fall-2015/f15f21.xml", + "comp2804/exam-fall-2015/f15f22.xml", + "comp2804/exam-fall-2015/f15f23.xml", + "comp2804/exam-fall-2015/f15f24.xml", + "comp2804/exam-fall-2015/f15f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f01.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f01.xml new file mode 100644 index 00000000..b3e9617b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f01.xml @@ -0,0 +1,13 @@ + + + + + ${20 \choose 3}$ + ${15 \choose 3}$ + ${15 \choose 3} + 15 \cdot {5 \choose 2}$ + ${15 \choose 3} + {15 \choose 2} \cdot 5 + 15 \cdot {5 \choose 2}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f02.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f02.xml new file mode 100644 index 00000000..78ec8b0e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f02.xml @@ -0,0 +1,14 @@ + + + + + ${19 \choose 6}$ + ${19 \choose 7}$ + ${20 \choose 6}$ + ${20}\choose{7}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f03.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f03.xml new file mode 100644 index 00000000..71638a4f --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f03.xml @@ -0,0 +1,12 @@ + + + + + $2^{17} - 2^{15}$ + $2^{17} - 2^{16}$ + $2^{18} - 2^{16}$ + $2^{18} - 2^{17}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f04.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f04.xml new file mode 100644 index 00000000..a721dce3 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f04.xml @@ -0,0 +1,11 @@ + + + + + ${20 \choose 5} \cdot 3^{15} \cdot 5^5$ + $- {20 \choose 5} \cdot 3^{15} \cdot 5^5$ + ${20 \choose 5} \cdot 5^{15} \cdot 3^5$ + $- {20 \choose 5} \cdot 5^{15} \cdot 3^5$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f05.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f05.xml new file mode 100644 index 00000000..1780940c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f05.xml @@ -0,0 +1,23 @@ + + + +
  • + 10 are blond, +
    • +
  • + 20 have green eyes, +
    • +
  • + 9 are not blond and do not have green eyes. +
    • + + How many people in this group are blond and have green eyes? + ]]> +     + 8 + 9 + 10 + 11 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f06.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f06.xml new file mode 100644 index 00000000..536c7931 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f06.xml @@ -0,0 +1,12 @@ + + + + + $S_n = n \cdot 2^{n-1}$ + $S_n = n \cdot 3^{n-1}$ + $S_n = 3^{n} - n$ + $S_n = 3^{n} - 2^{n}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f07.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f07.xml new file mode 100644 index 00000000..11a49afc --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f07.xml @@ -0,0 +1,12 @@ + + + + + $S_n = 2 \cdot S_{n-1} + 3^{n-1}$ + $S_n = 2 \cdot S_{n-1} + 2 \cdot S_{n-2}$ + $S_n = 3 \cdot S_{n-1}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f08.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f08.xml new file mode 100644 index 00000000..b4b6c317 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f08.xml @@ -0,0 +1,24 @@ + + + + + + $f(0) = $ + $7,$ + + + $f(n) = $ + $2 \cdot f(n - 1) + 1\; \ \text{for all}$ $\text{integers}\ n \geq 1.$ + + + + Which of the following is true? + ]]> + + $f(n) = 8n + 7$ + $f(n) = 4n^{2} + 4n + 7$ + $f(n) = 2^{n+3} - 1$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f09.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f09.xml new file mode 100644 index 00000000..e8ed5645 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f09.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the following recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + If we run algorithm $\Fib(18)$, how many calls are there to $\Fib(14)$? + ]]> +     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f10.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f10.xml new file mode 100644 index 00000000..67019f89 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f10.xml @@ -0,0 +1,11 @@ + + + + + $2 \cdot (n-1) \cdot 2^{n-2}$ + $2 \cdot n \cdot 2^{n-1}$ + $2^n - 4$ + $2^n - 2$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f11.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f11.xml new file mode 100644 index 00000000..bd73e2aa --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f11.xml @@ -0,0 +1,15 @@ + + + + + ${79 \choose 4}$ + ${79 \choose 5}$ + ${80 \choose 4}$ + ${80 \choose 5}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f12.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f12.xml new file mode 100644 index 00000000..d33fc4da --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f12.xml @@ -0,0 +1,15 @@ + + + + + $\frac{{5 \choose 2}}{{12 \choose 3}}$ + $\frac{5 \cdot {7 \choose 2}}{{12 \choose 3}}$ + $\frac{{5 \choose 2} \cdot 7}{{12 \choose 3}}$ + $\frac{{7 \choose 2}}{{12 \choose 3}}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f13.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f13.xml new file mode 100644 index 00000000..80a5c9ab --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f13.xml @@ -0,0 +1,15 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f14.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f14.xml new file mode 100644 index 00000000..e4afd9c2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f14.xml @@ -0,0 +1,17 @@ + + + + + $n \cdot (1/365) \cdot (364/365)^{n-1}$ + $365 \cdot n \cdot (364/365)^{n-1}$ + $1 - (1/365)^{n}$ + $1 - (364/365)^{n}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f15.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f15.xml new file mode 100644 index 00000000..4d545f44 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f15.xml @@ -0,0 +1,31 @@ + + + +
  • + The course COMP 9999 runs over a period of one year, starting on January 1 and ending on December + 31. There is one lecture every day; thus, the total number of lectures is 365. +
    • +
  • + Dania and Nick take this course. Dania's birthday is on November 19. Nick's birthday is on December + 3. +
    • +
  • + Professor G. Ruesome teaches the course. Professor Ruesome decides to have 20 quizzes during the + year. For this, he chooses a uniformly random subset of 20 days; the quizzes will be on the 20 + chosen days. (It is possible that there is a quiz on January 1.) +
    • + + + Determine $\Pr(A)$, where $A$ is the event + [defs] + A = "There is a quiz on Dania's birthday and there is a quiz on Nick's birthday". + [/defs] + ]]> +     + $1 - \left. {363 \choose 20} \middle/ {365 \choose 20} \right.$ + $\left. {363 \choose 18} \middle/ {365 \choose 20} \right.$ + $\left. {365 \choose 18} \middle/ {365 \choose 20} \right.$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f16.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f16.xml new file mode 100644 index 00000000..e13e1467 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f16.xml @@ -0,0 +1,32 @@ + + + +
  • + The course COMP 9999 runs over a period of one year, starting on January 1 and ending on December + 31. There is one lecture every day; thus, the total number of lectures is 365. +
    • +
  • + Dania and Nick take this course. Dania's birthday is on November 19. Nick's birthday is on December + 3. +
    • +
  • + Professor G. Ruesome teaches the course. Professor Ruesome decides to have 20 quizzes during the + year. For this, he chooses a uniformly random subset of 20 days; the quizzes will be on the 20 + chosen days. (It is possible that there is a quiz on January 1.) +
    • + + + Determine the conditional probability $\Pr(B|C)$, where $B$ and $C$ are the events + [defs] + B = "there is a quiz on Nick's birthday", + C = "there are exactly 5 quizzes in December". + [/defs] + ]]> +     + 4/31 + 5/31 + 4/32 + 5/32 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f17.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f17.xml new file mode 100644 index 00000000..491a50e3 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f17.xml @@ -0,0 +1,26 @@ + + + +
  • + The course COMP 9999 runs over a period of one year, starting on January 1 and ending on December + 31. There is one lecture every day; thus, the total number of lectures is 365. +
    • +
  • + At the beginning of each of the 365 lectures, Nick flips a fair and independent coin twice. + If the coin comes up heads twice, then Nick eats 3 bananas during the lecture; otherwise, + Nick eats 5 bananas during the lecture. +
    • + + + Let $X$ be the total number of bananas that Nick eats during the 365 lectures of the course COMP + 9999. What is the expected value $\mathbb{E}(X)$ of $X$?
    + (n.b., you may find it useful to apply Linearity of Expectation) + ]]> +     + $\frac{9 \cdot 365}{2}$ + $\frac{7 \cdot 365}{2}$ + $\frac{5 \cdot 365}{2}$ + $4 \cdot 365$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f18.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f18.xml new file mode 100644 index 00000000..f3e3a6e0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f18.xml @@ -0,0 +1,19 @@ + + + + Let $X$ be the number of students who give their gift to themselves. What is the expected value + $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use an indicator random variable for each student. +

    + ]]> +     + $1$ + $2$ + $1 + 1/n$ + $2 + 1/n$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f19.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f19.xml new file mode 100644 index 00000000..5695abc1 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f19.xml @@ -0,0 +1,19 @@ + + + + For each $i = 1,2,\dots,n$, let $v_i$ be the value (in dollars) of the gift that student $S_i$ + buys. Let $Y$ be the value of the gift that student $S_1$ receives, and let $Z$ be the value of the + gift that student $S_2$ receives. What is $\mathbb{E}(2 \cdot Y - Z)$? +

    + ]]> +     + $2v_1 - v_2$ + $\sum_{i=3}^{n} v_i/n$ + $\sum_{i=1}^{n} v_i/n$ + $2 \sum_{i=2}^{n} v_i/n - (v_1/n + \sum_{i=3}^{n} v_i/n)$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f20.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f20.xml new file mode 100644 index 00000000..f8330fdc --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f20.xml @@ -0,0 +1,19 @@ + + + + + What is the expected value $\mathbb{E}(X)$ of $X$? + ]]> + + 3/8 + 8/3 + 8 + 12 + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f21.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f21.xml new file mode 100644 index 00000000..fc9c103b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f21.xml @@ -0,0 +1,20 @@ + + + + + What is the expected value $\mathbb{E}(X)$ of $X$?
    + Hint: Use indicator random variables. + ]]> +     + $2n/27$ + $2(n-2)/27$ + $(n-2)/8$ + $n/8$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f22.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f22.xml new file mode 100644 index 00000000..6425598b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f22.xml @@ -0,0 +1,22 @@ + + + + + Let $X$ be the size of the set $T'$. + What is the expected value $\mathbb{E}(X)$ of $X$?
    + Hint: Use indicator random variables. + ]]> +     + $n/9$ + $m/9$ + $4n/9$ + $4m/9$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f23.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f23.xml new file mode 100644 index 00000000..9bac8de7 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f23.xml @@ -0,0 +1,12 @@ + + + + + 1 + 3/2 + 2 + 5/2 + diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f24.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f24.xml new file mode 100644 index 00000000..ce225164 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f24.xml @@ -0,0 +1,38 @@ + + + + $\newcommand{\Heads}{{\rm H \scriptsize EADS}}$ + + + Consider a coin that comes up heads with probability 1/3 and, thus, tails with probability 2/3. + Consider the following recursive algorithm $\Heads$, which takes as input a positive integer $k$: + +
    + $\mathbf{Algorithm}\ \Heads(k)\mathrm{:}$
    + + + + + + + +
    $//$ + $\text{all coin flips made}$ $\text{are mutually independent}$ +
    + $\text{flip the coin};$
    + $\mathbf{if}\ \text{the coin came up heads}$
    + $\mathbf{then}\ \mathrm{return}\ k + 1$
    + $\mathbf{else}\ \Heads(k + 1)$
    + $\mathbf{endif}$ +
    + + You run algorithm $\Heads(1)$, i.e., with $k = 1$. Define the random variable $X$ to be the value of + the output of this algorithm. Let $m \geq 1$ be an integer. What is $\Pr(X = m + 1)$? + ]]> +     + $2^{m-1}/3^{m}$ + $(2/3)^{m}$ + $2^{m}/3^{m+1}$ + $(2/3)^{m+1}$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2015/f15f25.xml b/app/src/data/questions/comp2804/exam-fall-2015/f15f25.xml new file mode 100644 index 00000000..d92a6220 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2015/f15f25.xml @@ -0,0 +1,9 @@ + + + Who discovered Newton's Binomial Theorem? + + Professor Binomial. + Justin Bieber. + Isaac Newton. + Professor G. Ruesome (the guy teaching COMP 9999). + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/exam-fall-2016.json b/app/src/data/questions/comp2804/exam-fall-2016/exam-fall-2016.json new file mode 100644 index 00000000..ee8ddc68 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/exam-fall-2016.json @@ -0,0 +1,32 @@ +{ + "id": "exam-fall-2016", + "title": "Fall 2016 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-fall-2016/f16f01.xml", + "comp2804/exam-fall-2016/f16f02.xml", + "comp2804/exam-fall-2016/f16f03.xml", + "comp2804/exam-fall-2016/f16f04.xml", + "comp2804/exam-fall-2016/f16f05.xml", + "comp2804/exam-fall-2016/f16f06.xml", + "comp2804/exam-fall-2016/f16f07.xml", + "comp2804/exam-fall-2016/f16f08.xml", + "comp2804/exam-fall-2016/f16f09.xml", + "comp2804/exam-fall-2016/f16f10.xml", + "comp2804/exam-fall-2016/f16f11.xml", + "comp2804/exam-fall-2016/f16f12.xml", + "comp2804/exam-fall-2016/f16f13.xml", + "comp2804/exam-fall-2016/f16f14.xml", + "comp2804/exam-fall-2016/f16f15.xml", + "comp2804/exam-fall-2016/f16f16.xml", + "comp2804/exam-fall-2016/f16f17.xml", + "comp2804/exam-fall-2016/f16f18.xml", + "comp2804/exam-fall-2016/f16f19.xml", + "comp2804/exam-fall-2016/f16f20.xml", + "comp2804/exam-fall-2016/f16f21.xml", + "comp2804/exam-fall-2016/f16f22.xml", + "comp2804/exam-fall-2016/f16f23.xml", + "comp2804/exam-fall-2016/f16f24.xml", + "comp2804/exam-fall-2016/f16f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f01.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f01.xml new file mode 100644 index 00000000..2b153042 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f01.xml @@ -0,0 +1,12 @@ + + + + + $7^9$ + $9^7$ + $7!$ + $9!$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f02.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f02.xml new file mode 100644 index 00000000..02e46142 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f02.xml @@ -0,0 +1,12 @@ + + + + + $\frac{7!}{2}$ + $\frac{7!}{3}$ + $\frac{9!}{2}$ + $\frac{9!}{3}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f03.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f03.xml new file mode 100644 index 00000000..de6a6637 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f03.xml @@ -0,0 +1,11 @@ + + + + + $2^{55} - 2^{48}$ + $2^{51} + 2^{52}$ + $2^{51} + 2^{52} - 2^{48}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f04.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f04.xml new file mode 100644 index 00000000..e2e51ce7 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f04.xml @@ -0,0 +1,11 @@ + + + + + The number of bitstrings of length $n$ that have at least two 1's. + The number of bitstrings of length $n$ that have at most two 1's. + The number of subsets of a set of size $n$ that have size at least two. + The number of subsets of a set of size $n$ that have size at least three. + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f05.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f05.xml new file mode 100644 index 00000000..1bd5e5c4 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f05.xml @@ -0,0 +1,23 @@ + + + +
  • + 25 drink cider, +
    • +
  • + 50 drink beer, +
    • +
  • + 33 do not drink cider and do not drink beer. +
    • + + How many people in this group drink both cider and beer? + ]]> +     + 8 + 9 + 10 + 11 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f06.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f06.xml new file mode 100644 index 00000000..443a686a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f06.xml @@ -0,0 +1,13 @@ + + + + + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ + $S_n = S_{n-1} + 2 \cdot S_{n-2}$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-3}$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-2}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f07.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f07.xml new file mode 100644 index 00000000..e8fa1ec8 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f07.xml @@ -0,0 +1,13 @@ + + + + + $S_n = {n \choose 2}$ + $S_n = {n \choose 2} - n - 1$ + $S_n = {n \choose 2} - n$ + $S_n = {n \choose 2} - n + 1$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f08.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f08.xml new file mode 100644 index 00000000..fa76de8c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f08.xml @@ -0,0 +1,16 @@ + + + + + $f(n) = \frac{5}{n!}$ + $f(n) = \frac{5^{n}}{n!}$ + $f(n) = \frac{5^{n}}{(n+1)!}$ + $f(n) = \frac{5^{n+1}}{n!}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f09.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f09.xml new file mode 100644 index 00000000..0c05a838 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f09.xml @@ -0,0 +1,30 @@ + + + + $\newcommand{\Hello}{{\rm H {\scriptsize ELLO}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the recursive algorithm $\Hello$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Hello(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \text{print ``hello''}$
    + $\mathbf{else}\ \mathbf{if}\ n\ \mathrm{is}\ \mathrm{even}$
    + $\elsesp \mathbf{then}\ \Hello(n / 2)$
    + $\elsesp \mathbf{else}\ \Hello(n - 1);$
    + $\elsesp \elsesp \Hello(n - 2)$
    + $\elsesp \mathbf{endif};$
    + $\mathbf{endif}$ +

    + + If we run algorithm $\Hello(7)$, how many times is the word "hello" printed? + ]]> +     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f10.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f10.xml new file mode 100644 index 00000000..e06f4c11 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f10.xml @@ -0,0 +1,13 @@ + + + + + $S_n = 5 \cdot 4^{n-1}$ + $S_n = 5 \cdot 4^{n-2}$ + $S_n = 5^{n} - 5(n-1) \cdot 4^{n-2}$ + $S_n = 5^{n} - 5(n-1) \cdot 4^{n-1}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f11.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f11.xml new file mode 100644 index 00000000..d6452817 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f11.xml @@ -0,0 +1,16 @@ + + + $x_5 \geq 0$ are integers? + ]]> + + ${33 \choose 4}$ + ${33 \choose 5}$ + ${32 \choose 4}$ + ${32 \choose 5}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f12.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f12.xml new file mode 100644 index 00000000..759a82c2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f12.xml @@ -0,0 +1,17 @@ + + + + + 1/3 + 1/4 + 1/5 + 1/6 + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f13.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f13.xml new file mode 100644 index 00000000..5f6a5b6a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f13.xml @@ -0,0 +1,16 @@ + + + + + $\frac{{12 \choose 4}+{8 \choose 3}}{20 \choose 7}$ + $\frac{{12 \choose 4}{8 \choose 3}}{20 \choose 7}$ + $\frac{{20 \choose 3}{17 \choose 4}}{20 \choose 7}$ + $\frac{{20 \choose 4}{16 \choose 3}}{20 \choose 7}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f14.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f14.xml new file mode 100644 index 00000000..9041d246 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f14.xml @@ -0,0 +1,17 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f15.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f15.xml new file mode 100644 index 00000000..5a29de62 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f15.xml @@ -0,0 +1,16 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f16.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f16.xml new file mode 100644 index 00000000..89712421 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f16.xml @@ -0,0 +1,17 @@ + + + + + ${n \choose 2} \cdot \left( \frac{1}{365} \right)^2 \cdot \left( \frac{364}{365} \right)^{n-2}$ + $1 - {n \choose 2} \cdot \left( \frac{1}{365} \right)^2 \cdot \left( \frac{364}{365} \right)^{n-2}$ + $1 - \left( \frac{364}{365} \right)^n - n \cdot \frac{1}{365} \cdot \left( \frac{364}{365} \right)^{n-1}$ + ${\sum_{k=2}^{n}} {n \choose k} \cdot \left( \frac{1}{365} \right)^k$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f17.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f17.xml new file mode 100644 index 00000000..f4644b82 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f17.xml @@ -0,0 +1,15 @@ + + + + + $1 - \frac{{98}\choose{15}}{{100}\choose{17}}$ + $1 - \frac{{100 \choose 2} \cdot {98 \choose 15}}{100 \choose 17}$ + $\frac{2 \cdot {99 \choose 16}}{100 \choose 17}$ + $\frac{2 \cdot {99 \choose 16} - {98 \choose 15}}{100 \choose 17}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f18.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f18.xml new file mode 100644 index 00000000..7606eadc --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f18.xml @@ -0,0 +1,15 @@ + + + + + 1/2 + 1/3 + 1/4 + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f19.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f19.xml new file mode 100644 index 00000000..bb9ef31c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f19.xml @@ -0,0 +1,19 @@ + + + + + What is the expected value $\mathbb{E}(X)$ of $X$?
    + Hint: Use indicator random variables. + ]]> +     + $\frac{n-1}{4}$ + $\frac{n}{4}$ + $\frac{n-1}{2}$ + $\frac{n}{2}$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f20.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f20.xml new file mode 100644 index 00000000..69b9555e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f20.xml @@ -0,0 +1,19 @@ + + + + + Define the random variable $X$ to be the number of unordered pairs $\{P_i,P_j\}$ of people that have + the same birthday.
    + + What is the expected value $\mathbb{E}(X)$ of $X$?
    + Hint: Use indicator random variables. + ]]> +     + $\frac{1}{365} \cdot {n \choose 2}$ + $\left( \frac{1}{365} \right)^2 \cdot {n \choose 2}$ + $\frac{1}{365} \cdot n^2$ + $\left( \frac{1}{365} \right)^2 \cdot n^2$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f21.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f21.xml new file mode 100644 index 00000000..8d3ea6c6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f21.xml @@ -0,0 +1,14 @@ + + + + + What is the expected value $\mathbb{E}(X)$ of $X$? + ]]> + + 7/2 + 32/7 + 17/7 + 7/17 + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f22.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f22.xml new file mode 100644 index 00000000..bd04621b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f22.xml @@ -0,0 +1,19 @@ + + + + + The random variables $X$ and $Y$ are independent. + The random variables $X$ and $Y$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f23.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f23.xml new file mode 100644 index 00000000..eff64287 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f23.xml @@ -0,0 +1,28 @@ + + + + + The random variables $X$ and $Y$ are independent. + The random variables $X$ and $Y$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f24.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f24.xml new file mode 100644 index 00000000..aed29a19 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f24.xml @@ -0,0 +1,15 @@ + + + + + What is the expected value $\mathbb{E}(X)$ of $X$? + ]]> + + 2 + 3 + 4 + 5 + diff --git a/app/src/data/questions/comp2804/exam-fall-2016/f16f25.xml b/app/src/data/questions/comp2804/exam-fall-2016/f16f25.xml new file mode 100644 index 00000000..883b2797 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2016/f16f25.xml @@ -0,0 +1,11 @@ + + + + + Beer + Cider + Hot chocolate + Poutine + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/exam-fall-2017.json b/app/src/data/questions/comp2804/exam-fall-2017/exam-fall-2017.json new file mode 100644 index 00000000..f53ad739 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/exam-fall-2017.json @@ -0,0 +1,32 @@ +{ + "id": "exam-fall-2017", + "title": "Fall 2017 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-fall-2017/f17f01.xml", + "comp2804/exam-fall-2017/f17f02.xml", + "comp2804/exam-fall-2017/f17f03.xml", + "comp2804/exam-fall-2017/f17f04.xml", + "comp2804/exam-fall-2017/f17f05.xml", + "comp2804/exam-fall-2017/f17f06.xml", + "comp2804/exam-fall-2017/f17f07.xml", + "comp2804/exam-fall-2017/f17f08.xml", + "comp2804/exam-fall-2017/f17f09.xml", + "comp2804/exam-fall-2017/f17f10.xml", + "comp2804/exam-fall-2017/f17f11.xml", + "comp2804/exam-fall-2017/f17f12.xml", + "comp2804/exam-fall-2017/f17f13.xml", + "comp2804/exam-fall-2017/f17f14.xml", + "comp2804/exam-fall-2017/f17f15.xml", + "comp2804/exam-fall-2017/f17f16.xml", + "comp2804/exam-fall-2017/f17f17.xml", + "comp2804/exam-fall-2017/f17f18.xml", + "comp2804/exam-fall-2017/f17f19.xml", + "comp2804/exam-fall-2017/f17f20.xml", + "comp2804/exam-fall-2017/f17f21.xml", + "comp2804/exam-fall-2017/f17f22.xml", + "comp2804/exam-fall-2017/f17f23.xml", + "comp2804/exam-fall-2017/f17f24.xml", + "comp2804/exam-fall-2017/f17f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f01.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f01.xml new file mode 100644 index 00000000..aab53076 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f01.xml @@ -0,0 +1,21 @@ + + + +
  • + the leftmost 10 bottles are all beer bottles, and +
    • +
  • + the rightmost 10 bottles are all cider bottles. +
    • + + How many such arrangements are there? (The order of the bottles matters.) + ]]> +     + ${20 \choose 10} \cdot {30 \choose 10} \cdot 30!$ + ${20 \choose 10} \cdot 10! \cdot {30 \choose 10} \cdot 10! \cdot 30!$ + ${20 \choose 10} \cdot 10! \cdot {30 \choose 10} \cdot 10!$ + $50!$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f02.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f02.xml new file mode 100644 index 00000000..e17f6121 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f02.xml @@ -0,0 +1,17 @@ + + + awesome if $a_2 = 2 \cdot a_1$.
    + + For example, if $n = 6$, then the permutation $3,6,4,1,5,2$ is awesome, whereas the permutation + $3,5,4,1,6,2$ is not awesome.
    + + How many awesome permutations of the set $\{1,2,\dots,n\}$ are there? + ]]> +     + $n \cdot (n-1)!$ + $n \cdot (n-2)!$ + ${\frac{n}{2}} \cdot (n-1)!$ + ${\frac{n}{2}} \cdot (n-2)!$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f03.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f03.xml new file mode 100644 index 00000000..a82386c5 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f03.xml @@ -0,0 +1,24 @@ + + + cool if + + or + + How many cool strings are there? + ]]> + + ${{40 \choose 8} \cdot 2^{32} + {40 \choose 7} \cdot 2^{33}}$ + ${{40 \choose 8} \cdot 2^{32} + {40 \choose 7} \cdot 2^{33} - {40 \choose 15} \cdot {15 \choose 8} \cdot 2^{25}}$ + ${{40 \choose 8} \cdot 2^{32} + {40 \choose 7} \cdot 2^{33} - {40 \choose 15} \cdot {15 \choose 8}}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f04.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f04.xml new file mode 100644 index 00000000..5a6d5bb7 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f04.xml @@ -0,0 +1,12 @@ + + + + + ${40 \choose 3}$ + $3^{40} - 1 - 40 \cdot 3^{39} - {40 \choose 2} \cdot 3^{38}$ + $3^{40} - 2^{40} - 40 \cdot 2^{39} - {40 \choose 2} \cdot 2^{38}$ + $3^{40} - 2^{40} - 40 \cdot 2^{39} - {40 \choose 2} \cdot 2^{38}\ -$ $ {40 \choose 3} \cdot 2^{37}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f05.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f05.xml new file mode 100644 index 00000000..a4abc7c2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f05.xml @@ -0,0 +1,15 @@ + + + + + The number of subsets having size 3 of a set consisting of $n$ positive numbers and $m$ negative numbers. + The number of subsets having size at most 3 of a set consisting of $n$ positive numbers and $m$ negative numbers. + The number of subsets having size 2 or 3 of a set consisting of $n$ positive numbers and $m$ negative numbers. + The number of subsets having size 2 and 3 of a set consisting of $n$ positive numbers and $m$ negative numbers. + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f06.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f06.xml new file mode 100644 index 00000000..fa305668 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f06.xml @@ -0,0 +1,23 @@ + + + +
  • + 13 students like Donald Trump, +
    • +
  • + 25 students like Justin Bieber, and +
    • +
  • + 8 students like Donald Trump and like Justin Bieber. +
    • + + How many students do not like Donald Trump and do not like Justin Bieber? + ]]> +     + 69 + 70 + 71 + 72 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f07.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f07.xml new file mode 100644 index 00000000..55b719c9 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f07.xml @@ -0,0 +1,14 @@ + + + + BOOKKEEPER +

    + ]]> +     + $2! \cdot 2! \cdot 3!$ + ${10 \choose 2} \cdot {8 \choose 2} \cdot {6 \choose 3} \cdot 3 \cdot 2$ + ${10 \choose 2} \cdot {8 \choose 2} \cdot {6 \choose 3} \cdot {3 \choose 2}$ + ${10 \choose 2} \cdot {8 \choose 2} \cdot {5 \choose 3} \cdot 3 \cdot 2$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f08.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f08.xml new file mode 100644 index 00000000..0830cff5 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f08.xml @@ -0,0 +1,18 @@ + + + + + For any integer $n \geq 1$, let $E_n$ be the number of such strings of length $n$ that contain an even + number of $c$'s, and let $O_n$ be the number of such strings of length $n$ that contain an odd + number of $c$'s. (Remember that 0 is an even number.)
    + + Which of the following is true for any integer $n \geq 2$? + ]]> +     + $E_n = 2 \cdot E_{n-1} + O_{n-1}$ + $E_n = 2 \cdot E_{n-1} + E_{n-1}$ + $E_n = 2 \cdot O_{n-1} + O_{n-1}$ + $E_n = 2 \cdot O_{n-1} + E_{n-1}$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f09.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f09.xml new file mode 100644 index 00000000..cf9bd12b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f09.xml @@ -0,0 +1,12 @@ + + + + + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-2}$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-3}$ + $S_n = S_{n-1} + S_{n-2} + 1$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f10.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f10.xml new file mode 100644 index 00000000..9eb38703 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f10.xml @@ -0,0 +1,30 @@ + + + + $\newcommand{\IFeelLikeSinging}{{\rm IF {\scriptsize EEL} L {\scriptsize IKE} S {\scriptsize INGING}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the recursive algorithm $\IFeelLikeSinging$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \IFeelLikeSinging(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \mathrm{sing}\ {\it O\ Canada}$
    + $\mathbf{else}\ \mathbf{if}\ n\ \text{is odd}$
    + $\elsesp \mathbf{then}\ \IFeelLikeSinging(n + 1)$
    + $\elsesp \mathbf{else}\ \IFeelLikeSinging(\frac{n}{2});$
    + $\elsesp \elsesp \IFeelLikeSinging(\frac{n}{2} - 1)$
    + $\elsesp \mathbf{endif};$
    + $\mathbf{endif}$ +

    + + If you run algorithm $\IFeelLikeSinging(9)$, how many times do you sing O Canada? + ]]> +     + 6 + 7 + 8 + 9 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f11.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f11.xml new file mode 100644 index 00000000..9b185bd6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f11.xml @@ -0,0 +1,21 @@ + + + + + Assume that you answer each question uniformly at random, where the choices for different questions are + independent of each other.
    + + What is the probability that you have exactly 17 correct answers? + ]]> +     + ${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^{17}$ + ${25 \choose 17} \cdot \left( 3 \middle/ 4 \right)^{17}$ + + ${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^{17} \cdot \left( 3 \middle/ 4 \right)^8$ + + + ${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^8 \cdot \left( 3 \middle/ 4 \right)^{17}$ + +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f12.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f12.xml new file mode 100644 index 00000000..bfb8a064 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f12.xml @@ -0,0 +1,19 @@ + + + palindrome if + $$ + b_1b_2{\dots}b_{n-1}b_n = b_nb_{n-1}{\dots}b_2b_1, + $$ + i.e., reading the string from left to right gives the same result as reading the string from right + to left.
    + + Let $n \geq 2$ be an even integer. You are given a uniformly random bitstring of length $n$. What is + the probability that this bitstring is a palindrome? + ]]> +     + $(1/2)^{n}$ + $(1/2)^{n/2}$ + $(1/2)^{2n}$ + $1/2$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f13.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f13.xml new file mode 100644 index 00000000..626aa388 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f13.xml @@ -0,0 +1,16 @@ + + + + + 3/16 + 4/16 + 5/16 + 6/16 + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f14.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f14.xml new file mode 100644 index 00000000..e1e5c7d6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f14.xml @@ -0,0 +1,19 @@ + + + + + Consider the string + $$ + a_1 + b_1,a_2 + b_2,\dots,a_n + b_n. + $$ + What is the probability that each element in this string is non-zero? + ]]> + + $1 - (1/4)^{n}$ + $1 - (3/4)^{n}$ + $(1/4)^{n}$ + $(3/4)^{n}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f15.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f15.xml new file mode 100644 index 00000000..a07ab805 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f15.xml @@ -0,0 +1,15 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f16.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f16.xml new file mode 100644 index 00000000..b4456ce3 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f16.xml @@ -0,0 +1,17 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f17.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f17.xml new file mode 100644 index 00000000..30a47175 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f17.xml @@ -0,0 +1,15 @@ + + + + + $1 - (1/2)^{7}$ + $(1/2)^{7}$ + $1 - \frac{{50 \choose 7}}{{100 \choose 7}}$ + $\frac{{50 \choose 7}}{{100 \choose 7}}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f18.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f18.xml new file mode 100644 index 00000000..38f195ef --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f18.xml @@ -0,0 +1,42 @@ + + + + + + + + + + + + + + + + + + + + + + + + +
    $X = \bigg\{$$1\ $if the red coin flip resulted in heads$,$
    $0\ $if the red coin flip resulted in tails$,$
    $Y = \bigg\{$$1\ $if the blue coin flip resulted in heads$,$
    $0\ $if the blue coin flip resulted in tails$,$
    + + and + [defs] + Z = $\max(X,Y).$ + [/defs] + What is the expected value $\mathbb{E}(Z)$ of the random variable $Z$? + ]]> +     + 1/4 + 1/2 + 3/4 + 1 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f19.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f19.xml new file mode 100644 index 00000000..3e27f13a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f19.xml @@ -0,0 +1,17 @@ + + + + + Define the random variable $X$ to be the number of indices $i$ with $1 \leq i < n$ for which $b_i \neq b_{i+1}$.
    + + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]> +     + $\frac{n-1}{2}$ + $\frac{n}{2}$ + $\frac{n+1}{2}$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f20.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f20.xml new file mode 100644 index 00000000..25b953af --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f20.xml @@ -0,0 +1,17 @@ + + + + + ${\sum_{k=1}^{100}} k \cdot \frac{k^{2}}{100^{2}}$ + ${\sum_{k=1}^{100}} k \cdot \frac{k(k-1)}{100^{2}}$ + ${\sum_{k=1}^{100}} k \cdot \frac{2k}{100^{2}}$ + ${\sum_{k=1}^{100}} k \cdot \left( \frac{1+2(k-1)}{100^{2}} \right)$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f21.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f21.xml new file mode 100644 index 00000000..edfab514 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f21.xml @@ -0,0 +1,20 @@ + + + + + The random variables $X$ and $Y$ are independent. + The random variables $X$ and $Y$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f22.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f22.xml new file mode 100644 index 00000000..da7942e1 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f22.xml @@ -0,0 +1,48 @@ + + + + $\newcommand{\WhoGoesFirst}{{\rm W {\scriptsize HO} G {\scriptsize OES} F {\scriptsize IRST}}} + \newcommand{\thensp}{\phantom{\mathbf{then}\ }}$ + + + Alexa and May want to play the game of Monopoly. They use the following recursive algorithm + to decide who goes first: + +
    + $\mathbf{Algorithm}\ \WhoGoesFirst(k)\mathrm{:}$
    + + + + + + + +
    $//$ + $k \geq 1,$ $\text{the die is fair},$ $\text{and all rolls are independent}$ +
    + $\text{Alexa rolls the die, let \(a\) be the result};$
    + $\text{May rolls the die, let \(m\) be the result};$
    + $\mathbf{if}\ a > m$
    + $\mathbf{then}\ \mathrm{print}\ {\tt Alexa\ goes\ first};$
    + $\thensp \mathrm{return}\ k$
    + $\mathbf{endif};$
    + $\mathbf{if}\ a < m$
    + $\mathbf{then}\ \mathrm{print}\ {\tt May\ goes\ first};$
    + $\thensp \mathrm{return}\ k$
    + $\mathbf{endif};$
    + $\mathbf{if}\ a = m$
    + $\mathbf{then}\ \WhoGoesFirst(k + 1)$
    + $\mathbf{endif}$ +
    + + The ladies run algorithm $\WhoGoesFirst(1)$, i.e., with $k = 1$. Define the random variable $X$ to + be the value of the output of this algorithm.
    + What is the expected value $\mathbb{E}(X)$ of the random variable $X$? + ]]> +     + 3/2 + 5/4 + 5/6 + 6/5 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f23.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f23.xml new file mode 100644 index 00000000..0d776fc0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f23.xml @@ -0,0 +1,14 @@ + + + + For any random variable $X$, $\mathbb{E}\left(1 \middle/ X \right) = 1 / \mathbb{E}(X)$. +

    + ]]> +     + The statement is true. + The statement is false. + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f24.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f24.xml new file mode 100644 index 00000000..0faf30e2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f24.xml @@ -0,0 +1,44 @@ + + + + The Carleton Computer Science Society (CCSS) is having their annual Christmas Holiday Season Party, which + is attended by $n$ students. + + + + + + + + + + + + + +
    (a) + $k$ of these $n$ students are politically correct and, thus, refuse to say Merry Christmas. + Instead, they say Happy Holidays. +
    (b) + $n - k$ of these $n$ students do not care about political correctness and, thus, they say + Merry Christmas. +
    + + Consider a uniformly random permutation of these $n$ students. The positions in this permutation are numbered + as $1,2,...,n$.
    + + Define the following random variable $X$: + [defs] + X = the number of positions $i$ with $1 \leq i \leq \left. n \middle/ 2 \right.$ such that both students at positions $i$ and $2i$ are politically correct. + [/defs] + + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]> +     + $n \cdot \frac{k(k - 1)}{n(n - 1)}$ + $n \cdot \frac{(k - 1)(k - 2)}{n(n - 1)}$ + $\frac{n}{2} \cdot \frac{k(k - 1)}{n(n - 1)}$ + $\frac{n}{2} \cdot \frac{(k - 1)(k - 2)}{n(n - 1)}$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2017/f17f25.xml b/app/src/data/questions/comp2804/exam-fall-2017/f17f25.xml new file mode 100644 index 00000000..d8bd1037 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2017/f17f25.xml @@ -0,0 +1,6 @@ + + + How do you feel about writing an exam on Friday evening? + + I would rather sit in the pub and have a beer. + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/exam-fall-2018.json b/app/src/data/questions/comp2804/exam-fall-2018/exam-fall-2018.json new file mode 100644 index 00000000..4a91b0f6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/exam-fall-2018.json @@ -0,0 +1,32 @@ +{ + "id": "exam-fall-2018", + "title": "Fall 2018 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-fall-2018/f18f01.xml", + "comp2804/exam-fall-2018/f18f02.xml", + "comp2804/exam-fall-2018/f18f03.xml", + "comp2804/exam-fall-2018/f18f04.xml", + "comp2804/exam-fall-2018/f18f05.xml", + "comp2804/exam-fall-2018/f18f06.xml", + "comp2804/exam-fall-2018/f18f07.xml", + "comp2804/exam-fall-2018/f18f08.xml", + "comp2804/exam-fall-2018/f18f09.xml", + "comp2804/exam-fall-2018/f18f10.xml", + "comp2804/exam-fall-2018/f18f11.xml", + "comp2804/exam-fall-2018/f18f12.xml", + "comp2804/exam-fall-2018/f18f13.xml", + "comp2804/exam-fall-2018/f18f14.xml", + "comp2804/exam-fall-2018/f18f15.xml", + "comp2804/exam-fall-2018/f18f16.xml", + "comp2804/exam-fall-2018/f18f17.xml", + "comp2804/exam-fall-2018/f18f18.xml", + "comp2804/exam-fall-2018/f18f19.xml", + "comp2804/exam-fall-2018/f18f20.xml", + "comp2804/exam-fall-2018/f18f21.xml", + "comp2804/exam-fall-2018/f18f22.xml", + "comp2804/exam-fall-2018/f18f23.xml", + "comp2804/exam-fall-2018/f18f24.xml", + "comp2804/exam-fall-2018/f18f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f01.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f01.xml new file mode 100644 index 00000000..72a0fcca --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f01.xml @@ -0,0 +1,12 @@ + + + + + $70 \cdot 2^{69}$ + $70 \cdot 3^{69}$ + $70 \cdot 2^{70}$ + $70 \cdot 3^{70}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f02.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f02.xml new file mode 100644 index 00000000..643cb34a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f02.xml @@ -0,0 +1,12 @@ + + + and exactly 30 letters $b$? + ]]> + + ${70 \choose 12} + {58 \choose 30}$ + ${70 \choose 12} \cdot {58 \choose 30}$ + ${70 \choose 12} \cdot {58 \choose 30} \cdot 2^{28}$ + ${70 \choose 12} \cdot {58 \choose 30} \cdot 3^{28}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f03.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f03.xml new file mode 100644 index 00000000..c7625bb6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f03.xml @@ -0,0 +1,12 @@ + + + or exactly 30 letters $b$? + ]]> + + ${70 \choose 12} + {70 \choose 30} - {58 \choose 30}$ + ${70 \choose 12} \cdot 2^{58} + {70 \choose 30} \cdot 2^{40}$ + ${70 \choose 12} \cdot 2^{58} + {70 \choose 30} \cdot 2^{40} - {70 \choose 12} \cdot {58 \choose 30}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f04.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f04.xml new file mode 100644 index 00000000..a2ea5fa3 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f04.xml @@ -0,0 +1,12 @@ + + + + + $3^{70} - 2^{70} - 70 \cdot 2^{69}$ + $3^{70} - 2^{70} - 70 \cdot 2^{69} - {70 \choose 2} \cdot 2^{68}$ + $\sum_{k=4}^{70} {70 \choose k} \cdot 2^{k}$ + $\sum_{k=4}^{70} {70 \choose k} \cdot 2^{70-k}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f05.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f05.xml new file mode 100644 index 00000000..85f2b9a9 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f05.xml @@ -0,0 +1,14 @@ + + + + + The number of subsets of $S$ that are non-empty. + The number of subsets of $S$ that contain at least two elements. + The number of bitstrings of length $n$ that contain at least one 0. + The number of bitstrings of length $n$ that contain at least three 0's. + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f06.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f06.xml new file mode 100644 index 00000000..5d4a231d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f06.xml @@ -0,0 +1,11 @@ + + + + + ${55 \choose 20} \cdot 5^{35} \cdot 3^{20}$ + $- {55 \choose 20} \cdot 5^{35} \cdot 3^{20}$ + ${55 \choose 35} \cdot 5^{20} \cdot 3^{35}$ + $- {55 \choose 35} \cdot 5^{20} \cdot 3^{35}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f07.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f07.xml new file mode 100644 index 00000000..336f0ad7 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f07.xml @@ -0,0 +1,17 @@ + + + + SUCCESS +

    + is called awesome, if the string contains the substring UE or EU. + Thus, both SCUESCS and SCSEUCS are awesome, whereas SUCCESS is not awesome. + What is the number of awesome strings? + ]]> +     + 30 + 60 + 90 + 120 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f08.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f08.xml new file mode 100644 index 00000000..0a0959d0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f08.xml @@ -0,0 +1,17 @@ + + + + + $S_n = S_{n-2} + S_{n-4}$ + $S_n = 2 \cdot S_{n-2} + S_{n-4}$ + $S_n = S_{n-2} + 4 \cdot S_{n-4}$ + $S_n = 2 \cdot S_{n-2} + 4 \cdot S_{n-4}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f09.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f09.xml new file mode 100644 index 00000000..e146ba52 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f09.xml @@ -0,0 +1,12 @@ + + + + + $S_n = S_{n-1} + S_{n-2} + 1$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-2}$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-3}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f10.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f10.xml new file mode 100644 index 00000000..2638ff06 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f10.xml @@ -0,0 +1,18 @@ + + + + + $T(n) = \frac{5}{2} \cdot 3^{n} - 1$ + $T(n) = \frac{3}{2} \cdot 3^{n} - \frac{1}{2}$ + $T(n) = \frac{5}{2} \cdot 3^{n} - \frac{1}{2}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f11.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f11.xml new file mode 100644 index 00000000..cf01ce24 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f11.xml @@ -0,0 +1,12 @@ + + + + + 2/3 + 1/2 + 1/3 + 1/4 + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f12.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f12.xml new file mode 100644 index 00000000..207dfc4f --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f12.xml @@ -0,0 +1,15 @@ + + + suit and a rank. + There are four suits (spades ♠, hearts ♡, clubs ♣, and diamonds ♢), + and 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King).
    + Assume you get a uniformly random hand consisting of 5 cards. What is the probability that the 5 + cards in this hand are all of the same suit? + ]]> +     + $\left. {13 \choose 5} \middle/ {52 \choose 5} \right.$ + $\left. {52 \choose 5} \middle/ {13 \choose 5} \right.$ + $4 \cdot \left. {13 \choose 5} \middle/ {52 \choose 5} \right.$ + $4 \cdot \left. {52 \choose 5} \middle/ {13 \choose 5} \right.$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f13.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f13.xml new file mode 100644 index 00000000..39c7ce64 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f13.xml @@ -0,0 +1,18 @@ + + + + + 2/7 + 1/7 + 2/11 + 1/11 + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f14.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f14.xml new file mode 100644 index 00000000..4591cc23 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f14.xml @@ -0,0 +1,20 @@ + + + + You are given two bitstrings $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ of length $n$. In both + bitstrings, each bit is 0 with probability 1/2, and 1 with probability 1/2 (independent of all + other bits).
    + Consider the bitstring + $$ + a_1 \cdot b_1,a_2 \cdot b_2,\dots,a_n \cdot b_n + $$ + where $a_i \cdot b_i$ is the result of multiplying the two bits $a_i$ and $b_i$. What is the probability that this + bitstring contains exactly $k$ many 1's? + ]]> +     + ${n \choose k} \cdot \left. 3^{n - k} \middle/ 4^n \right.$ + ${n \choose k} \cdot \left. 4^{n - k} \middle/ 3^n \right.$ + ${n \choose k} \cdot \left. 4^{n - k} \middle/ 3^k \right.$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f15.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f15.xml new file mode 100644 index 00000000..01e60019 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f15.xml @@ -0,0 +1,17 @@ + + + a_2$", + B = "$a_4 > a_3$". + [/defs] + Which of the following is correct? + ]]> + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f16.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f16.xml new file mode 100644 index 00000000..9d4dd721 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f16.xml @@ -0,0 +1,16 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f17.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f17.xml new file mode 100644 index 00000000..38f4fb99 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f17.xml @@ -0,0 +1,12 @@ + + + + + 5/6 + 2/3 + 3/4 + 1/3 + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f18.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f18.xml new file mode 100644 index 00000000..e24001c9 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f18.xml @@ -0,0 +1,19 @@ + + + + + 9/13 + 13/9 + 8/9 + 9/8 + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f19.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f19.xml new file mode 100644 index 00000000..a0ed5c5c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f19.xml @@ -0,0 +1,17 @@ + + + + Define the random variable $X$ to be the number of indices $i$ with $1 \leq i < n$ + for which $c_i = c_{i+1}$.
    + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]> +     + $\left. n \middle/ 25 \right.$ + $\left. (n - 1) \middle/ 25 \right.$ + $\left. n \middle/ 5 \right.$ + $\left. (n - 1) \middle/ 5 \right.$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f20.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f20.xml new file mode 100644 index 00000000..f1f3f0d4 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f20.xml @@ -0,0 +1,17 @@ + + + + + $\frac{2 {{n}\choose{2}} + n}{{{n+2}\choose{3}}}$ + $\frac{2 {{n}\choose{2}} + n + 1}{{{n+2}\choose{3}}}$ + $\frac{2 {{n}\choose{2}} + 2n}{{{n+2}\choose{3}}}$ + $\frac{2 {{n}\choose{2}} + n - 1}{{{n+2}\choose{3}}}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f21.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f21.xml new file mode 100644 index 00000000..ae7aba26 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f21.xml @@ -0,0 +1,29 @@ + + + + + + $\Pr(X = 0)$ + + $= \Pr(X = 1) = \Pr(Y = 0)$ + $= \Pr(Y = 1) = 1/2.$ + + + + + + Define the random variable + $$ + Z = (X + Y)\ \mathrm{mod}\ 2. + $$ + Which of the following is correct? + ]]> + + The random variables $X$ and $Z$ are independent. + The random variables $X$ and $Z$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f22.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f22.xml new file mode 100644 index 00000000..9769789e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f22.xml @@ -0,0 +1,55 @@ + + + + $\newcommand{\TriIsHungry}{{\rm T {\scriptsize RI} I {\scriptsize S} H {\scriptsize UNGRY}}}$ + + + Zoltan's Noodle House is a popular restaurant in downtown Ottawa. When you order the surprise dish, + you get Mi Quang with probability 1/4, Bun Cha Ca with probability 1/3, and + Banh Xeo with probability 5/12.
    + Tri enjoys going to this restaurant, because the food reminds him of his mommy's food back home in Da Nang.
    + Tri runs the following recursive algorithm: + + +
    + $\mathbf{Algorithm}\ \TriIsHungry\mathrm{:}$
    + + + + + + + +
    $//$ + $\text{the results of all}$ $\text{orders are independent}$ +
    + $\text{Tri orders the surprise dish;}$
    + $\mathbf{if}\ \mathrm{Tri}\ \mathrm{gets}\ \mathit{Mi}\ \mathit{Quang}$
    + $\mathbf{then}\ \text{Tri eats the dish;}$
    + $\qquad\ \ \TriIsHungry$
    + $\mathbf{else}\ \mathbf{if}\ \mathrm{Tri}\ \mathrm{gets}\ \mathit{Bun}\ \mathit{Cha}\ \mathit{Ca}$
    + $\qquad\, \mathbf{then}\ \text{Tri eats the dish;}$
    + $\qquad \qquad\ \ \, \TriIsHungry$
    + $\qquad\, \mathbf{else}\ \text{Tri eats the dish;}$
    + + + + + + + +
    $\hspace{4.05em}$$\text{Tri pays the bill}$ $\text{and goes home}$
    + $\qquad\, \mathbf{endif}$
    + $\mathbf{endif}$
    +
    + + Define the random variable $X$ to be the number of dishes that Tri eats when running algorithm $\TriIsHungry$.
    + What is the expected value $\mathbb{E}(X)$ of the random variable $X$? + ]]> +     + 5/12 + 12/5 + 5/7 + 7/5 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f23.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f23.xml new file mode 100644 index 00000000..9166ede4 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f23.xml @@ -0,0 +1,25 @@ + + + + + Jim is one of the students and answers the 25 questions, by choosing a uniformly random answer for + each question; the choices are independent of each other.
    + + Define the random variable + [defs] + X = the number of marks that Jim recevies. + [/defs] + For what value of $z$ is the expected value $\mathbb{E}(X)$ equal to 0?
    + + Hint: Use the Linearity of Expectation. + ]]> +     + $z = 1/4$ + $z = 1/3$ + $z = 1/2$ + $z = 3/4$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f24.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f24.xml new file mode 100644 index 00000000..5f48fcb9 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f24.xml @@ -0,0 +1,23 @@ + + + + + At the end of the exam, the proctor places the backpacks in a uniformly random order $b_1,b_2,\dots,b_n$, + and, for each $i = 1, 2, ..., n$, gives backpack $b_i$ to student $S_i$.
    + + Define the following random variable $X$: + [defs] + X = the number of students who get their own backpack. + [/defs] + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]> +     + 1 + 2 + $\left. (n + 1) \middle/ n \right.$ + $\left. (n - 1) \middle/ n \right.$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2018/f18f25.xml b/app/src/data/questions/comp2804/exam-fall-2018/f18f25.xml new file mode 100644 index 00000000..ff04a7f8 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2018/f18f25.xml @@ -0,0 +1,11 @@ + + + + + Professor Identity + Donald Trump + Alexandre-Théophile Vandermonde + Justin Bieber + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/exam-fall-2019.json b/app/src/data/questions/comp2804/exam-fall-2019/exam-fall-2019.json new file mode 100644 index 00000000..ef574092 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/exam-fall-2019.json @@ -0,0 +1,32 @@ +{ + "id": "exam-fall-2019", + "title": "Fall 2019 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-fall-2019/f19f01.xml", + "comp2804/exam-fall-2019/f19f02.xml", + "comp2804/exam-fall-2019/f19f03.xml", + "comp2804/exam-fall-2019/f19f04.xml", + "comp2804/exam-fall-2019/f19f05.xml", + "comp2804/exam-fall-2019/f19f06.xml", + "comp2804/exam-fall-2019/f19f07.xml", + "comp2804/exam-fall-2019/f19f08.xml", + "comp2804/exam-fall-2019/f19f09.xml", + "comp2804/exam-fall-2019/f19f10.xml", + "comp2804/exam-fall-2019/f19f11.xml", + "comp2804/exam-fall-2019/f19f12.xml", + "comp2804/exam-fall-2019/f19f13.xml", + "comp2804/exam-fall-2019/f19f14.xml", + "comp2804/exam-fall-2019/f19f15.xml", + "comp2804/exam-fall-2019/f19f16.xml", + "comp2804/exam-fall-2019/f19f17.xml", + "comp2804/exam-fall-2019/f19f18.xml", + "comp2804/exam-fall-2019/f19f19.xml", + "comp2804/exam-fall-2019/f19f20.xml", + "comp2804/exam-fall-2019/f19f21.xml", + "comp2804/exam-fall-2019/f19f22.xml", + "comp2804/exam-fall-2019/f19f23.xml", + "comp2804/exam-fall-2019/f19f24.xml", + "comp2804/exam-fall-2019/f19f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f01.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f01.xml new file mode 100644 index 00000000..98333659 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f01.xml @@ -0,0 +1,10 @@ + + + How many such strings have exactly 5 letters $c$? + ]]> + ${85 \choose 5} \cdot 3^{80}$ + ${85 \choose 5} \cdot 3^{85}$ + ${85 \choose 5} \cdot 4^{80}$ + ${85 \choose 5} \cdot 4^{85}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f02.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f02.xml new file mode 100644 index 00000000..00465258 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f02.xml @@ -0,0 +1,10 @@ + + + How many such strings have exactly 15 letters $a$ and exactly 30 letters $d$? + ]]> + ${85 \choose 15} \cdot {70 \choose 30} \cdot 2^{40}$ + ${85 \choose 15} \cdot {70 \choose 30} \cdot 3^{40}$ + ${85 \choose 15} \cdot {70 \choose 30} \cdot 4^{40}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f03.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f03.xml new file mode 100644 index 00000000..84e3db5d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f03.xml @@ -0,0 +1,12 @@ + + + How many such strings have exactly 15 letters $a$ or exactly 30 letters $d$? + ]]> + ${85 \choose 15} \cdot 3^{70} + {85 \choose 30} \cdot 3^{55}$ + + ${85 \choose 15} \cdot 3^{70} + {85 \choose 30} \cdot 3^{55} - {85 \choose 15} \cdot {70 \choose 30} \cdot 2^{40}$ + + ${85 \choose 15} \cdot 4^{70} + {85 \choose 30} \cdot 4^{55}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f04.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f04.xml new file mode 100644 index 00000000..e723fd58 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f04.xml @@ -0,0 +1,13 @@ + + + Consider subsets of these 70 bottles, that contain exactly 12 beer bottles (and any number of cider bottles) + or exactly 12 cider bottles (and any number of beer bottles). How many such subsets are there? + ]]> + ${20 \choose 12} + {50 \choose 12}$ + ${20 \choose 12} + {50 \choose 12} - {20 \choose 12} \cdot {50 \choose 12}$ + ${20 \choose 12} \cdot 2^{50} + {50 \choose 12} \cdot 2^{20}$ + + ${20 \choose 12} \cdot 2^{50} + {50 \choose 12} \cdot 2^{20} - {20 \choose 12} \cdot {50 \choose 12}$ + + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f05.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f05.xml new file mode 100644 index 00000000..9bede1e5 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f05.xml @@ -0,0 +1,13 @@ + + + Consider subsets of these 70 bottles, that contain at least 3 beer bottles (and any number of cider bottles). + How many such subsets are there? + ]]> + $2^{70} - 2^{50} - 20 - {20 \choose 2}$ + + $2^{70} - 2^{50} - 20 \cdot 2^{50} - {20 \choose 2} \cdot 2^{50}$ + + $2^{70} - 2^{50} - 20 \cdot 2^{50}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f06.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f06.xml new file mode 100644 index 00000000..670f6f9c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f06.xml @@ -0,0 +1,15 @@ + + + Let $m \geq 2$ and $n \geq 2$ be integers. What does + $$ + {m \choose 2} + {n \choose 2} + m \cdot n + $$ + count? + + The number of ways to choose a subset from a set consisting of $m + n$ elements. + The number of ways to choose an ordered pair of 2 elements from a set consisting of $m + n$ elements. + + The number of ways to choose a 2-element subset from a set consisting of $m + n$ elements. + + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f07.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f07.xml new file mode 100644 index 00000000..3ce098d8 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f07.xml @@ -0,0 +1,12 @@ + + + Nick eats lots of bananas. During a period of 7 days, Nick eats a total of 25 bananas. A banana schedule + is a sequence of 7 numbers, whose sum is equal to 25, and whose numbers indicate the number of bananas + that Nick eats on each day. Three examples of such schedules are (3,2,7,4,1,3,5), (2,3,7,4,1,3,5), and + (3,0,9,4,1,0,8). How many banana schedules are there? + + ${31 \choose 6}$ + ${31 \choose 7}$ + ${32 \choose 6}$ + ${32 \choose 7}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f08.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f08.xml new file mode 100644 index 00000000..9fea4814 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f08.xml @@ -0,0 +1,11 @@ + + awesome, if + the string does not contain the substring OO. Thus, GEOROB is awesome, whereas GREOOB is not awesome. + What is the number of awesome strings? + ]]> + $6 \cdot {5 \choose 2} \cdot 3 \cdot 2$ + $(6 \cdot {5 \choose 2} \cdot 3 \cdot 2) - 5!$ + $6! - 5!$ + $(6 \cdot {5 \choose 2} \cdot 3) - 5!$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f09.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f09.xml new file mode 100644 index 00000000..37d66145 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f09.xml @@ -0,0 +1,11 @@ + + + $S_n = S_{n - 1} + S_{n - 2} + S_{n - 3}$ + $S_n = S_{n - 1} + S_{n - 2} + 2^{n - 2}$ + $S_n = S_{n - 1} + S_{n - 2} + 2^{n - 3}$ + $S_n = S_{n - 1} + S_{n - 2} + 1$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f10.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f10.xml new file mode 100644 index 00000000..1fdc799e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f10.xml @@ -0,0 +1,26 @@ + + + $\newcommand{\IFeelLikeSinging}{{\rm IF {\scriptsize EEL} L {\scriptsize IKE} S {\scriptsize INGING}}}$ + + + Consider the recursive algorithm $\IFeelLikeSinging$, which takes as input an integer $n \geq 0$: + +

    + $\IFeelLikeSinging(n):$
    + $\quad \mathbf{if}\ n = 0\ \mathrm{or}\ n = 1\ \mathbf{then}$
    + $\quad \quad \text{sing O Canada}$
    + $\quad \mathbf{else}\ \mathbf{if}\ n\ \text{is odd}\ \mathbf{then}$
    + $\quad \quad \IFeelLikeSinging(n + 1)$
    + $\quad \mathbf{else}$
    + $\quad \quad \IFeelLikeSinging(\frac{n}{2})$
    + $\quad \quad \IFeelLikeSinging(\frac{n}{2} - 1)$
    +

    + + If you run algorithm $\IFeelLikeSinging(9)$, how many times do you sing O Canada? + ]]>     + 6 + 7 + 8 + 9 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f11.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f11.xml new file mode 100644 index 00000000..3c48d30f --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f11.xml @@ -0,0 +1,16 @@ + + + Assume you write a multiple-choice exam that has 25 questions. For each question, four options are given + to you, and exactly one of these options is the correct answer. Assume that you answer each question + uniformly at random, where the choices for different questions are independent of each other. What is the + probability that you have exactly 17 correct answers? + + ${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^{17}$ + ${25 \choose 17} \cdot \left( 3 \middle/ 4 \right)^{17}$ + + ${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^{17} \cdot \left( 3 \middle/ 4 \right)^8$ + + + ${25 \choose 17} \cdot \left( 1 \middle/ 4 \right)^8 \cdot \left( 3 \middle/ 4 \right)^{17}$ + + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f12.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f12.xml new file mode 100644 index 00000000..ba36ef52 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f12.xml @@ -0,0 +1,15 @@ + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f13.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f13.xml new file mode 100644 index 00000000..d2081ed7 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f13.xml @@ -0,0 +1,19 @@ + + + When Nika writes the exam, what is the expected number of questions that appear on the exam and that + Nika has practiced on? +

    + + (n.b., depending on your background, it may be helpful to observe that this can be modelled using the hypergeometric + distribution) + ]]>     + 2 + 5 + 10 + 20 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f14.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f14.xml new file mode 100644 index 00000000..1e4d0f5d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f14.xml @@ -0,0 +1,10 @@ + + + Continuing from the previous question, what is the probability that Nika has not practiced on any of the + actual exam questions? + + 0 + $\left. {225 \choose 50} \middle/ {250 \choose 50} \right.$ + $\left. {225 \choose 25} \middle/ {250 \choose 50} \right.$ + $\left. {225 \choose 25} \middle/ {225 \choose 50} \right.$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f15.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f15.xml new file mode 100644 index 00000000..3c8e0980 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f15.xml @@ -0,0 +1,16 @@ + + + $\frac{{4 \choose 2}}{{5 \choose 2} + {4 \choose 2}}$ + $\frac{{5 \choose 2} + {4 \choose 2}}{{4 \choose 2}}$ + $\frac{{4 \choose 2}}{{9 \choose 2}}$ + $\frac{{4 \choose 2}}{{5 \choose 2} \cdot {4 \choose 2}}$ + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f16.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f16.xml new file mode 100644 index 00000000..67e50890 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f16.xml @@ -0,0 +1,16 @@ + + + 3/10 + 1/3 + 1/2 + 2/3 + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f17.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f17.xml new file mode 100644 index 00000000..7165fd5a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f17.xml @@ -0,0 +1,15 @@ + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f18.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f18.xml new file mode 100644 index 00000000..d2aad426 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f18.xml @@ -0,0 +1,15 @@ + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f19.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f19.xml new file mode 100644 index 00000000..b851a637 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f19.xml @@ -0,0 +1,17 @@ + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f20.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f20.xml new file mode 100644 index 00000000..39052993 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f20.xml @@ -0,0 +1,12 @@ + + + 2 + 1 + 1/4 + 1/2 + diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f21.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f21.xml new file mode 100644 index 00000000..cff9248e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f21.xml @@ -0,0 +1,18 @@ + + + What is the expected value $\mathbb{E}(X)$ of the random variable $X$? +

    + + Hint: Use indicator random variables. + ]]>     + $\frac{2}{3} \cdot n$ + $\frac{2}{3} \cdot (n - 1)$ + $\frac{4}{9} \cdot n$ + $\frac{4}{9} \cdot (n - 1)$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f22.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f22.xml new file mode 100644 index 00000000..14584456 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f22.xml @@ -0,0 +1,21 @@ + + + Experiment: Roll each die once and take the sum of the two rolls. You repeat this experiment until + the sum of the two rolls is equal to 7. Consider the random variable +

    + + [defs] + X = the number of times you do the experiment. + [/defs] + + (This value $X$ includes the experiment in which the sum is 7 for the first time.)
    + What is the expected value $\mathbb{E}(X)$ of the random variable $X$? + ]]>     + 6 + 5 + 4 + 3 +     diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f23.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f23.xml new file mode 100644 index 00000000..a1ac8a97 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f23.xml @@ -0,0 +1,12 @@ + + + For any random variable $X$, $\mathbb{E}\left(1 \middle/ X \right) = 1 / \mathbb{E}(X)$. +

    + ]]>     + The statement is true. + The statement is false. + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f24.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f24.xml new file mode 100644 index 00000000..34b6e471 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f24.xml @@ -0,0 +1,46 @@ + + Christmas Holiday Season Party, which + is attended by $n$ students. + + + + + + + + + + + + + +
    (i) + $k$ of these $n$ students are politically correct and, thus, refuse to say Merry Christmas. + Instead, they say Happy Holidays. +
    (ii) + $n - k$ of these $n$ students do not care about political correctness and, thus, they say + Merry Christmas. +
    + + Consider a uniformly random permutation of these $n$ students. The positions in this permutation are numbered + as $1,2,...,n$. + +

    + Define the random variable $X$ as the number of positions $i$ with $1 \leq i \leq \left. n \middle/2 \right.$ + such that both students at positions $i$ and $2i$ are politically correct. +

    + + What is the expected value $\mathbb{E}(X)$ of the random variable $X$? + +

    + Hint: Use indicator random variables. +

    + ]]>     + $n \cdot \frac{k(k - 1)}{n(n - 1)}$ + $n \cdot \frac{(k - 1)(k - 2)}{n(n - 1)}$ + $\frac{n}{2} \cdot \frac{k(k - 1)}{n(n - 1)}$ + $\frac{n}{2} \cdot \frac{(k - 1)(k - 2)}{n(n - 1)}$ +     diff --git a/app/src/data/questions/comp2804/exam-fall-2019/f19f25.xml b/app/src/data/questions/comp2804/exam-fall-2019/f19f25.xml new file mode 100644 index 00000000..66df427a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-fall-2019/f19f25.xml @@ -0,0 +1,10 @@ + + + How do you feel about writing an exam on Friday evening? + + I would rather sit in the pub and have a beer. + I would rather sit in the pub and have a beer. + I would rather sit in the pub and have a beer. + I would rather sit in the pub and have a beer. + I would rather sit in the pub and have a beer. + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/exam-winter-2014.json b/app/src/data/questions/comp2804/exam-winter-2014/exam-winter-2014.json new file mode 100644 index 00000000..9c2ed18d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/exam-winter-2014.json @@ -0,0 +1,32 @@ +{ + "id": "exam-winter-2014", + "title": "Winter 2014 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-winter-2014/w14f01.xml", + "comp2804/exam-winter-2014/w14f02.xml", + "comp2804/exam-winter-2014/w14f03.xml", + "comp2804/exam-winter-2014/w14f04.xml", + "comp2804/exam-winter-2014/w14f05.xml", + "comp2804/exam-winter-2014/w14f06.xml", + "comp2804/exam-winter-2014/w14f07.xml", + "comp2804/exam-winter-2014/w14f08.xml", + "comp2804/exam-winter-2014/w14f09.xml", + "comp2804/exam-winter-2014/w14f10.xml", + "comp2804/exam-winter-2014/w14f11.xml", + "comp2804/exam-winter-2014/w14f12.xml", + "comp2804/exam-winter-2014/w14f13.xml", + "comp2804/exam-winter-2014/w14f14.xml", + "comp2804/exam-winter-2014/w14f15.xml", + "comp2804/exam-winter-2014/w14f16.xml", + "comp2804/exam-winter-2014/w14f17.xml", + "comp2804/exam-winter-2014/w14f18.xml", + "comp2804/exam-winter-2014/w14f19.xml", + "comp2804/exam-winter-2014/w14f20.xml", + "comp2804/exam-winter-2014/w14f21.xml", + "comp2804/exam-winter-2014/w14f22.xml", + "comp2804/exam-winter-2014/w14f23.xml", + "comp2804/exam-winter-2014/w14f24.xml", + "comp2804/exam-winter-2014/w14f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f01.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f01.xml new file mode 100644 index 00000000..237c5365 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f01.xml @@ -0,0 +1,13 @@ + + + a, e, i, o, or u). + How many passwords are there? + ]]> + + $5 \cdot 21^{27}$ + $28 \cdot 5 \cdot 21^{27}$ + $28 \cdot 5 \cdot 27^{21}$ + $28 \cdot 5 \cdot 26^{27}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f02.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f02.xml new file mode 100644 index 00000000..305b3e0a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f02.xml @@ -0,0 +1,14 @@ + + + + + ${10 \choose 1}26^9 + {10 \choose 2}26^8 + {10 \choose 3}26^7$ + $10 \cdot 26^9 + 10^2 \cdot 26^8 + 10^3 \cdot 26^7$ + + ${10 \choose 1} \cdot 10 \cdot 26^9 + {10 \choose 2} \cdot 10^2 \cdot 26^8\ +$ $ {10 \choose 3} \cdot 10^3 \cdot 26^7$ + + none of the above + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f03.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f03.xml new file mode 100644 index 00000000..97b8842d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f03.xml @@ -0,0 +1,14 @@ + + + + + $(-2)^{45}$ + $4^{45}$ + $(-8)^{45}$ + $10^{45}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f04.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f04.xml new file mode 100644 index 00000000..4248d008 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f04.xml @@ -0,0 +1,13 @@ + + + + + $7 \cdot 6!$ + $2 \cdot 7 \cdot 6!$ + $8 \cdot 6!$ + $2 \cdot 8 \cdot 6!$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f05.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f05.xml new file mode 100644 index 00000000..7567dc28 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f05.xml @@ -0,0 +1,13 @@ + + + + + $S_{n+1} = S_n + S_{n-1}\ \mathrm{for}\ n \geq 2.$ + $S_{n+1} = 2 \cdot S_n + S_{n-1}\ \mathrm{for}\ n \geq 2.$ + $S_{n+1} = S_n + 2 \cdot S_{n-1}\ \mathrm{for}\ n \geq 2.$ + $S_{n+1} = 2 \cdot S_n + 2 \cdot S_{n-1}\ \mathrm{for}\ n \geq 2.$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f06.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f06.xml new file mode 100644 index 00000000..1da6fd40 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f06.xml @@ -0,0 +1,23 @@ + + + +
  • + 30 weigh at least 150 pounds, +
    • +
  • + 60 are at least 6 feet tall, +
    • +
  • + 70 weigh less than 150 pounds and are less than 6 feet tall. +
    • + + How many people weigh at least 150 pounds and are at least 6 feet tall? + ]]> +     + 10 + 20 + 30 + 40 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f07.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f07.xml new file mode 100644 index 00000000..a0bc0ee8 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f07.xml @@ -0,0 +1,13 @@ + + + + + 64 + 74 + 84 + 94 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f08.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f08.xml new file mode 100644 index 00000000..c30601f2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f08.xml @@ -0,0 +1,15 @@ + + + + + $\frac{n!}{k(n-k)!}$ + $\frac{n!}{k!(n-k)!}$ + $\frac{n!}{(n-k)!}$ + $\frac{2 \cdot n!}{(n-k)!}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f09.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f09.xml new file mode 100644 index 00000000..88271ee2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f09.xml @@ -0,0 +1,12 @@ + + + + + ${780 \choose 3}$ + ${780 \choose 4}$ + ${781 \choose 3}$ + ${781 \choose 4}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f10.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f10.xml new file mode 100644 index 00000000..7b0f0a5d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f10.xml @@ -0,0 +1,26 @@ + + + + + + $f(0)$ + $\;=\;$ + $3,$ + + + $f(n + 1)$ + $\;=\;$ + $f(n) + 10n + 2\; \ \text{for all}$ $\text{integers}\ n \geq 0.$ + + + + Which of the following is true? + ]]> + + for all $n \geq 0$: $f(n) = 5n^{2} - 3n + 2$ + for all $n \geq 0$: $f(n) = 5n^{2} - 3n + 3$ + for all $n \geq 0$: $f(n) = 5n^{2} + 3n + 3$ + for all $n \geq 0$: $f(n) = 5n^{2} - 2n + 3$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f11.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f11.xml new file mode 100644 index 00000000..18421c17 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f11.xml @@ -0,0 +1,30 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + The Fibonacci numbers are defined as follows: $f_0 = 0$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for + $n \geq 2$.
    + + Consider the recursive algorithm $\Fib$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + For $n \geq 5$, run algorithm $\Fib(n)$ and let $a_n$ be the number of times that $\Fib(4)$ is called.
    + Which of the following is true? + ]]> +     + for all $n \geq 5$, $a_n = f_n$ + for all $n \geq 5$, $a_n = f_{n-1}$ + for all $n \geq 5$, $a_n = f_{n-2}$ + for all $n \geq 5$, $a_n = f_{n-3}$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f12.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f12.xml new file mode 100644 index 00000000..3d95ae0a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f12.xml @@ -0,0 +1,12 @@ + + + + + 4/16 + 4/32 + 5/16 + 5/32 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f13.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f13.xml new file mode 100644 index 00000000..008ee8f9 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f13.xml @@ -0,0 +1,12 @@ + + + + + $\frac{1}{100}$ + $\frac{1}{100 \cdot 99}$ + $\frac{1}{100 \cdot 100}$ + $\frac{1}{{100 \choose 2}}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f14.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f14.xml new file mode 100644 index 00000000..ecd3322e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f14.xml @@ -0,0 +1,13 @@ + + + + + 4/8 + 5/8 + 6/8 + 7/8 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f15.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f15.xml new file mode 100644 index 00000000..d8f8c3e0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f15.xml @@ -0,0 +1,16 @@ + + + + + 1/2 + 1/4 + 1/8 + 1/9 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f16.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f16.xml new file mode 100644 index 00000000..a186337c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f16.xml @@ -0,0 +1,18 @@ + + + + + 1/3 + 1/2 + 2/3 + none of the above + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f17.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f17.xml new file mode 100644 index 00000000..ce14074f --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f17.xml @@ -0,0 +1,12 @@ + + + + + 1/5 + 2/5 + 3/5 + 4/5 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f18.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f18.xml new file mode 100644 index 00000000..59c42744 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f18.xml @@ -0,0 +1,13 @@ + + + + + $1 - \frac{12!}{12^{4} \cdot 7!}$ + $1 - \frac{11!}{12^{4} \cdot 7!}$ + $1 - \frac{11!}{12^{4} \cdot 6!}$ + $1 - \frac{11!}{12^{4} \cdot 8!}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f19.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f19.xml new file mode 100644 index 00000000..6764d249 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f19.xml @@ -0,0 +1,13 @@ + + + + + $\frac{1}{3}$ + $\frac{1}{4}$ + $\frac{1}{4} \cdot \frac{1}{3} + \frac{3}{4} \cdot \frac{1}{4}$ + none of the above + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f20.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f20.xml new file mode 100644 index 00000000..28865f75 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f20.xml @@ -0,0 +1,14 @@ + + + + + 3 + 2 + 5/2 + 3/2 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f21.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f21.xml new file mode 100644 index 00000000..4158cb5c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f21.xml @@ -0,0 +1,19 @@ + + + + + Yes + No + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f22.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f22.xml new file mode 100644 index 00000000..af3ac037 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f22.xml @@ -0,0 +1,18 @@ + + + + + 27/4 + 28/4 + 29/4 + 30/4 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f23.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f23.xml new file mode 100644 index 00000000..e6f5cf95 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f23.xml @@ -0,0 +1,13 @@ + + + + + 4 + 5 + 6 + 7 + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f24.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f24.xml new file mode 100644 index 00000000..8365f4cf --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f24.xml @@ -0,0 +1,15 @@ + + + + + $n/2$ + $n/4$ + $n/8$ + $0$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2014/w14f25.xml b/app/src/data/questions/comp2804/exam-winter-2014/w14f25.xml new file mode 100644 index 00000000..b9c4d8cd --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2014/w14f25.xml @@ -0,0 +1,15 @@ + + + + + Justin Bieber + Fibonacci + Carl Friedrich Gauss + Simon Pratt + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/exam-winter-2015.json b/app/src/data/questions/comp2804/exam-winter-2015/exam-winter-2015.json new file mode 100644 index 00000000..e6a0de60 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/exam-winter-2015.json @@ -0,0 +1,32 @@ +{ + "id": "exam-winter-2015", + "title": "Winter 2015 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-winter-2015/w15f01.xml", + "comp2804/exam-winter-2015/w15f02.xml", + "comp2804/exam-winter-2015/w15f03.xml", + "comp2804/exam-winter-2015/w15f04.xml", + "comp2804/exam-winter-2015/w15f05.xml", + "comp2804/exam-winter-2015/w15f06.xml", + "comp2804/exam-winter-2015/w15f07.xml", + "comp2804/exam-winter-2015/w15f08.xml", + "comp2804/exam-winter-2015/w15f09.xml", + "comp2804/exam-winter-2015/w15f10.xml", + "comp2804/exam-winter-2015/w15f11.xml", + "comp2804/exam-winter-2015/w15f12.xml", + "comp2804/exam-winter-2015/w15f13.xml", + "comp2804/exam-winter-2015/w15f14.xml", + "comp2804/exam-winter-2015/w15f15.xml", + "comp2804/exam-winter-2015/w15f16.xml", + "comp2804/exam-winter-2015/w15f17.xml", + "comp2804/exam-winter-2015/w15f18.xml", + "comp2804/exam-winter-2015/w15f19.xml", + "comp2804/exam-winter-2015/w15f20.xml", + "comp2804/exam-winter-2015/w15f21.xml", + "comp2804/exam-winter-2015/w15f22.xml", + "comp2804/exam-winter-2015/w15f23.xml", + "comp2804/exam-winter-2015/w15f24.xml", + "comp2804/exam-winter-2015/w15f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f01.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f01.xml new file mode 100644 index 00000000..103c29d5 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f01.xml @@ -0,0 +1,12 @@ + + + + + ${20 \choose 7}$ + ${5\choose 3} + {15 \choose 4}$ + ${5 \choose 3}{15 \choose 4}$ + ${5 \choose 4}{15 \choose 3}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f02.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f02.xml new file mode 100644 index 00000000..c5d6b998 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f02.xml @@ -0,0 +1,15 @@ + + + + + + ${5 \choose 5}{15 \choose 2} + {5 \choose 2}{15 \choose 5} + {5 \choose 1}{15 \choose 6} + {5 \choose 0}{15 \choose 7}$ + + ${5 \choose 5}{15 \choose 2} + {5 \choose 2}{15 \choose 5} - {5 \choose 5}{15 \choose 5}$ + ${20 \choose 7} - {5 \choose 4}{15 \choose 4}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f03.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f03.xml new file mode 100644 index 00000000..f3835277 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f03.xml @@ -0,0 +1,11 @@ + + + + + $2^{73} + 2^{73}$ + $2^{74} - 2^{69}$ + $2^{77} - 2^{69}$ + $2^{77} - 2^{73} - 2^{73}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f04.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f04.xml new file mode 100644 index 00000000..35a47dd1 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f04.xml @@ -0,0 +1,11 @@ + + + + + ${15 \choose 11} \cdot 2^4 \cdot 7^{11}$ + $- {15 \choose 11} \cdot 2^4 \cdot 7^{11}$ + ${15 \choose 11} \cdot 2^{11} \cdot 7^4$ + $- {15 \choose 11} \cdot 2^{11} \cdot 7^4$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f05.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f05.xml new file mode 100644 index 00000000..6ef089f9 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f05.xml @@ -0,0 +1,23 @@ + + + +
  • + 6 are blond, +
    • +
  • + 7 have green eyes, +
    • +
  • + 11 are not blond and do not have green eyes. +
    • + + How many people in this group are blond and have green eyes? + ]]> +     + 3 + 4 + 5 + 6 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f06.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f06.xml new file mode 100644 index 00000000..dcfccace --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f06.xml @@ -0,0 +1,18 @@ + + + + + Because both sides count the number of pairs $(A,B)$ of subsets of the $n$ people, such that $|A| = 2$, $|B| = 5$, and $A \subseteq B$. + Because both sides count the number of pairs $(A,B)$ of subsets of the $n$ people, such that $|A| = 5$, $|B| = 2$, and $A \subseteq B$. + Because both sides count the number of ways to choose 2 committees in a group of $n$ people, one committee has 5 members, the other committee has 2 members, and a person can be on both committees. + + no person can be on both committees. + ]]> + + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f07.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f07.xml new file mode 100644 index 00000000..44d1da8c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f07.xml @@ -0,0 +1,12 @@ + + + + + $S_n = S_{n-1} + S_{n-2}$ + $S_n = 2 \cdot S_{n-1} + S_{n-2}$ + $S_n = S_{n-1} + (3^{n-1} - S_{n-1})$ + $S_n = 2 \cdot S_{n-1} + (3^{n-1} - S_{n-1})$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f08.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f08.xml new file mode 100644 index 00000000..24289148 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f08.xml @@ -0,0 +1,16 @@ + + + + + For all $n \geq 0$: $f(n) = 2n^2 + 7$ + For all $n \geq 0$: $f(n) = 3n^2 + 7$ + For all $n \geq 0$: $f(n) = 4n^2 + 7$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f09.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f09.xml new file mode 100644 index 00000000..c411bf35 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f09.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + If we run algorithm $\Fib(77)$, how many calls are there to $\Fib(73)$? + ]]> +     + 3 + 4 + 5 + 6 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f10.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f10.xml new file mode 100644 index 00000000..fd8da1ea --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f10.xml @@ -0,0 +1,23 @@ + + + +
  • + $(1,6) \in S$. +
    • +
  • + If $(a,b) \in S$ then $(a+3, b+4) \in S$. +
    • +
  • + If $(a,b) \in S$ then $(a+5, b+2) \in S$. +
    • + + Which of the following is true? + ]]> +     + For every element $(a,b)$ in $S$, $a + b$ is divisible by 7. + For every element $(a,b)$ in $S$, $a + b$ is not divisible by 7. + $S = \{(a,b) : a > 0$ and $b > 0$ are integers and $a + b$ is divisible by 7$\}$. + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f11.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f11.xml new file mode 100644 index 00000000..9c9a9c75 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f11.xml @@ -0,0 +1,12 @@ + + + + + ${102 \choose 3}$ + ${102 \choose 4}$ + ${103 \choose 3}$ + ${103 \choose 4}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f12.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f12.xml new file mode 100644 index 00000000..237810d0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f12.xml @@ -0,0 +1,12 @@ + + + + + 5/15 + 6/15 + 7/15 + 8/15 + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f13.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f13.xml new file mode 100644 index 00000000..8f70765a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f13.xml @@ -0,0 +1,18 @@ + + + + + + $\Pr\bigl(A \cup (\overline{B} \cap \overline{C})\bigr) = $ + $\Pr(A) + \Pr(B)\ - $ $\Pr(C).$ + + + + ]]> + + True + False + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f14.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f14.xml new file mode 100644 index 00000000..a6f514c6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f14.xml @@ -0,0 +1,13 @@ + + + + + $\left. {30 \choose 1} \middle/ {100 \choose 20} \right.$ + $\left. {30 \choose 1}{70 \choose 19} \middle/ {100 \choose 20} \right.$ + $1 - \left. {70 \choose 20} \middle/ {100 \choose 20} \right.$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f15.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f15.xml new file mode 100644 index 00000000..f0ce5637 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f15.xml @@ -0,0 +1,13 @@ + + + + + $\frac{1}{{100 \choose 20}} \sum_{k=0}^{20} \left({70 \choose k} + {30 \choose 20 - k}\right)$ + $\frac{1}{{100 \choose 20}} \sum_{k=0}^{20} {70 \choose k}{30 \choose 20 - k}$ + $\frac{1}{{100 \choose 20}} \sum_{k=0}^{10} \left({70 \choose 2k} + {30 \choose 20 - 2k}\right)$ + $\frac{1}{{100 \choose 20}} \sum_{k=0}^{10} {70 \choose 2k}{30 \choose 20 - 2k}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f16.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f16.xml new file mode 100644 index 00000000..68cd9145 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f16.xml @@ -0,0 +1,13 @@ + + + + + ${77 \choose 15}(1/3)^{15}(2/3)^{62}$ + ${77 \choose 15}((1/3)^{15} + (2/3)^{62})$ + ${77 \choose 15}(1/3)^{62}(2/3)^{15}$ + ${77 \choose 15}((1/3)^{62} + (2/3)^{15})$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f17.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f17.xml new file mode 100644 index 00000000..1af5f9bc --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f17.xml @@ -0,0 +1,16 @@ + + + + + $\left. {5 \choose 2} \middle/ 2^3 \right.$ + $\left. {5 \choose 2} \middle/ 2^4 \right.$ + $\left. {5 \choose 2} \middle/ 2^5 \right.$ + $\left. 2^5 \middle/ {5 \choose 2} \right.$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f18.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f18.xml new file mode 100644 index 00000000..94466448 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f18.xml @@ -0,0 +1,18 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f19.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f19.xml new file mode 100644 index 00000000..f1cae8cf --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f19.xml @@ -0,0 +1,13 @@ + + + + + $n/2$ + $n$ + $3n/2$ + $2n$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f20.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f20.xml new file mode 100644 index 00000000..ca434366 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f20.xml @@ -0,0 +1,15 @@ + + + + + $m/2$ + $m/4$ + $m/8$ + $m/16$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f21.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f21.xml new file mode 100644 index 00000000..3a46b32f --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f21.xml @@ -0,0 +1,20 @@ + + + + + The random variables $X$ and $Y$ are independent. + The random variables $X$ and $Y$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f22.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f22.xml new file mode 100644 index 00000000..cb3acef4 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f22.xml @@ -0,0 +1,20 @@ + + + + + $\mathbb{E}(L) = 50$ and $\mathbb{E}(\max(L,S)) = 50$. + $\mathbb{E}(L) = 100$ and $\mathbb{E}(\max(L,S)) = 100$. + $\mathbb{E}(L) = 50$ and $\mathbb{E}(\max(L,S)) = 100$. + $\mathbb{E}(L) = 100$ and $\mathbb{E}(\max(L,S)) = 50$. + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f23.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f23.xml new file mode 100644 index 00000000..31a70880 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f23.xml @@ -0,0 +1,38 @@ + + + + $\newcommand{\TwoTails}{{\rm T {\scriptsize WO} T {\scriptsize AILS}}}$ + + + Consider the following recursive algorithm $\TwoTails$, which takes as input a positive integer $k$: + +
    + $\mathbf{Algorithm}\ \TwoTails(k)\mathrm{:}$
    + + + + + + + +
    $//$ + $\text{all coin flips made are mutually}$ $\text{independent}$ +
    + $\text{flip a fair coin twice};$
    + $\mathbf{if}\ \text{the coin came up heads exactly twice}$
    + $\mathbf{then}\ \mathrm{return}\ 2^k$
    + $\mathbf{else}\ \TwoTails(k + 1)$
    + $\mathbf{endif}$
    +
    + + You run algorithm $\TwoTails(1)$, i.e., with $k = 1$. Define the random variable $X$ to be the value of the + output of this algorithm. + Let $k \geq 1$ be an integer. What is $\Pr(X = 2^k)$? + ]]> +     + $(1/4)^{k} \cdot 3/4$ + $(1/4)^{k-1} \cdot 3/4$ + $(3/4)^{k} \cdot 1/4$ + $(3/4)^{k-1} \cdot 1/4$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f24.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f24.xml new file mode 100644 index 00000000..e0ccfe5b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f24.xml @@ -0,0 +1,13 @@ + + + + + 36 + 30 + 20 + 12 + diff --git a/app/src/data/questions/comp2804/exam-winter-2015/w15f25.xml b/app/src/data/questions/comp2804/exam-winter-2015/w15f25.xml new file mode 100644 index 00000000..d32cabb0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2015/w15f25.xml @@ -0,0 +1,15 @@ + + Who is Lindsay Bangs? + A world-renowned rapper. + + The Hamburglar. + ]]> + + President of the Carleton Computer Science Society. + + Leo's Early Breakfast IPA. + ]]> + + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/exam-winter-2017.json b/app/src/data/questions/comp2804/exam-winter-2017/exam-winter-2017.json new file mode 100644 index 00000000..90fe6f61 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/exam-winter-2017.json @@ -0,0 +1,32 @@ +{ + "id": "exam-winter-2017", + "title": "Winter 2017 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-winter-2017/w17f01.xml", + "comp2804/exam-winter-2017/w17f02.xml", + "comp2804/exam-winter-2017/w17f03.xml", + "comp2804/exam-winter-2017/w17f04.xml", + "comp2804/exam-winter-2017/w17f05.xml", + "comp2804/exam-winter-2017/w17f06.xml", + "comp2804/exam-winter-2017/w17f07.xml", + "comp2804/exam-winter-2017/w17f08.xml", + "comp2804/exam-winter-2017/w17f09.xml", + "comp2804/exam-winter-2017/w17f10.xml", + "comp2804/exam-winter-2017/w17f11.xml", + "comp2804/exam-winter-2017/w17f12.xml", + "comp2804/exam-winter-2017/w17f13.xml", + "comp2804/exam-winter-2017/w17f14.xml", + "comp2804/exam-winter-2017/w17f15.xml", + "comp2804/exam-winter-2017/w17f16.xml", + "comp2804/exam-winter-2017/w17f17.xml", + "comp2804/exam-winter-2017/w17f18.xml", + "comp2804/exam-winter-2017/w17f19.xml", + "comp2804/exam-winter-2017/w17f20.xml", + "comp2804/exam-winter-2017/w17f21.xml", + "comp2804/exam-winter-2017/w17f22.xml", + "comp2804/exam-winter-2017/w17f23.xml", + "comp2804/exam-winter-2017/w17f24.xml", + "comp2804/exam-winter-2017/w17f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f01.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f01.xml new file mode 100644 index 00000000..1b224fdc --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f01.xml @@ -0,0 +1,20 @@ + + + +
  • + $a_1,a_3,a_5,a_7,a_9$ are all odd and +
    • +
  • + $a_2,a_4,a_6,a_8,a_{10}$ are all even. +
    • + + How many such permutations are there? + ]]> +     + $10!$ + $5^5 \cdot 5^5$ + $(5!)^2$ + $2 \cdot (5!)^2$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f02.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f02.xml new file mode 100644 index 00000000..a1d9992c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f02.xml @@ -0,0 +1,12 @@ + + + + + $\frac{n!}{2}$ + $n!$ + $2{n \choose 2} \cdot (n-2)!$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f03.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f03.xml new file mode 100644 index 00000000..efd10353 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f03.xml @@ -0,0 +1,24 @@ + + + +
  • + exactly 5 beer bottles and zero or more cider bottles, +
    • + + or + + How many such subsets are there? + ]]> +     + ${17 \choose 5} \cdot 2^{28} + 2^{17} \cdot {28 \choose 5}$ + ${17 \choose 5} \cdot 2^{28} + 2^{17} \cdot {28 \choose 5} - {17 \choose 5} \cdot {28 \choose 5}$ + $2^{45} - {17 \choose 5} - {28 \choose 5}$ + $2^{45} - {17 \choose 5} \cdot {28 \choose 5}$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f04.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f04.xml new file mode 100644 index 00000000..f1797acc --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f04.xml @@ -0,0 +1,13 @@ + + + + + $2^{n-1}$ + $2^{n-2}$ + $2^{(n-1)/2}$ + $2^{(n+1)/2}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f05.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f05.xml new file mode 100644 index 00000000..f058dd4c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f05.xml @@ -0,0 +1,11 @@ + + + + + The number of bitstrings of length $n$ in which the first bit is 0 or the last bit is 1. + The number of bitstrings of length $n$ in which the first bit is 0 and the last bit is 1. + The number of bitstrings of length $n$ in which the first bit is equal to the last bit. + The number of bitstrings of length $n$ in which the first bit is not equal to the last bit. + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f06.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f06.xml new file mode 100644 index 00000000..f0dc99e1 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f06.xml @@ -0,0 +1,23 @@ + + + +
  • + 63 students like beer, +
    • +
  • + 71 students like cider, and +
    • +
  • + 25 students do not like beer and do not like cider. +
    • + + How many students like beer and cider? + ]]> +     + 57 + 58 + 59 + 60 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f07.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f07.xml new file mode 100644 index 00000000..cfd0109d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f07.xml @@ -0,0 +1,24 @@ + + + + + For any even integer $n$, let $S_n$ be the number of bitstrings of length $n$ that have both of the + following two properties: + + Which of the following is true for all even integers $n \geq 6$? + ]]> + + $S_n = S_{n/2} + S_{(n/2)-3}$ + $S_n = S_{n-1} + S_{n-3}$ + $S_n = S_{n-2} + S_{n-3}$ + $S_n = S_{n-2} + S_{n-4}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f08.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f08.xml new file mode 100644 index 00000000..edc3ebbc --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f08.xml @@ -0,0 +1,12 @@ + + + + + $S_n = S_{n-1} + S_{n-2} + 2^{n-2}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3} + 2^{n-3}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3} + 2^{n-4}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f09.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f09.xml new file mode 100644 index 00000000..00a5430a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f09.xml @@ -0,0 +1,32 @@ + + + + $\newcommand{\Hello}{{\rm H {\scriptsize ELLO}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }} + \newcommand{\thensp}{\phantom{\mathbf{then}\ }}$ + + + Consider the recursive algorithm $\Hello$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Hello(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \text{print ``hello''}$
    + $\mathbf{else}\ \mathbf{if}\ n\ \mathrm{is}\ \mathrm{even}$
    + $\elsesp \mathbf{then}\ \Hello(\frac{n}{2});$
    + $\elsesp \thensp \Hello(\frac{n}{2} - 1)$
    + $\elsesp \mathbf{else}\ \Hello(n - 1);$
    + $\elsesp \elsesp \Hello(n - 2)$
    + $\elsesp \mathbf{endif};$
    + $\mathbf{endif}$ +

    + + If we run algorithm $\Hello(7)$, how many times is the word "hello" printed? + ]]> +     + 9 + 10 + 11 + 12 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f10.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f10.xml new file mode 100644 index 00000000..f35240ce --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f10.xml @@ -0,0 +1,16 @@ + + + + + $1 - (21/26)^{14}$ + $1 - (26/21)^{14}$ + $5 \cdot (5/26) \cdot (21/26)^{13}$ + $14 \cdot (5/26) \cdot (21/26)^{13}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f11.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f11.xml new file mode 100644 index 00000000..d5353f48 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f11.xml @@ -0,0 +1,13 @@ + + + + + $11 \cdot {6 \choose 2} \cdot 10!$ + $11 \cdot 6 \cdot 5 \cdot 10!$ + $12 \cdot 6 \cdot 5 \cdot 10!$ + $13 \cdot 6 \cdot 5 \cdot 10!$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f12.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f12.xml new file mode 100644 index 00000000..7da2d186 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f12.xml @@ -0,0 +1,17 @@ + + + + + 4/8 + 5/8 + 4/9 + 5/9 + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f13.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f13.xml new file mode 100644 index 00000000..06599cf0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f13.xml @@ -0,0 +1,16 @@ + + + + + 1/3 + 2/3 + 2/5 + 1/4 + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f14.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f14.xml new file mode 100644 index 00000000..62201d34 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f14.xml @@ -0,0 +1,16 @@ + + + a_1$". + [/defs] + Which of the following is true? + ]]> + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f15.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f15.xml new file mode 100644 index 00000000..07e9c81e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f15.xml @@ -0,0 +1,17 @@ + + + + + ${\frac{1}{n}} - {\frac{1}{n(n-1)}}$ + ${\frac{2}{n}} - {\frac{1}{n(n-1)}}$ + ${\frac{2}{n}} - {\frac{1}{n^{2}}}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f16.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f16.xml new file mode 100644 index 00000000..21d4f2e3 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f16.xml @@ -0,0 +1,19 @@ + + + 0. + \end{align} + $$ + Which of the following is true? + ]]> + + $\Pr(A) < 1/2$ + $\Pr(A) > 1/2$ + $\Pr(A) < 1$ + $\Pr(A) > 0$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f17.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f17.xml new file mode 100644 index 00000000..ec6c7703 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f17.xml @@ -0,0 +1,15 @@ + + + + + 1/3 + 2/3 + 1/5 + 1/6 + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f18.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f18.xml new file mode 100644 index 00000000..67c68335 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f18.xml @@ -0,0 +1,18 @@ + + + + + 5/18 + 1/6 + 1/4 + 1/3 + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f19.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f19.xml new file mode 100644 index 00000000..755beacd --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f19.xml @@ -0,0 +1,19 @@ + + + + + Define the random variable $X$ to be the number of unordered triples $\{P_i,P_j,P_k\}$ of people + (i.e., subsets consisting of three people) that have the same birthday.
    + + What is the expected value $\mathbb{E}(X)$ of $X$?
    + Hint: Use indicator random variables. + ]]> +     + ${\frac{1}{365^2}} \cdot {n \choose 3}$ + ${\frac{1}{365^3}} \cdot {n \choose 3}$ + ${\frac{1}{365^2}} \cdot n^3$ + ${\frac{1}{365^3}} \cdot n^3$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f20.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f20.xml new file mode 100644 index 00000000..a474e7da --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f20.xml @@ -0,0 +1,14 @@ + + + + + What is the expected value $\mathbb{E}(X)$ of $X$? + ]]> + + 80 + 100 + 120 + 140 + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f21.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f21.xml new file mode 100644 index 00000000..e4e844aa --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f21.xml @@ -0,0 +1,20 @@ + + + + + The random variables $X$ and $Y$ are independent. + The random variables $X$ and $Y$ are not independent. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f22.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f22.xml new file mode 100644 index 00000000..30f6675c --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f22.xml @@ -0,0 +1,38 @@ + + + + $\newcommand{\TwoTails}{{\rm T {\scriptsize WO} T {\scriptsize AILS}}}$ + + + Consider the following recursive algorithm $\TwoTails$, which takes as input a positive integer $k$: + +
    + $\mathbf{Algorithm}\ \TwoTails(k)\mathrm{:}$
    + + + + + + + +
    $//$ + $\text{all coin flips made are mutually}$ $\text{independent}$ +
    + $\text{flip a fair coin twice};$
    + $\mathbf{if}\ \text{the coin came up heads exactly twice}$
    + $\mathbf{then}\ \mathrm{return}\ 2^k$
    + $\mathbf{else}\ \TwoTails(k + 1)$
    + $\mathbf{endif}$
    +
    + + You run algorithm $\TwoTails(1)$, i.e., with $k = 1$. Define the random variable $X$ to be the value of the + output of this algorithm.
    + Let $m \geq 1$ be an integer. What is $\Pr(X = 2^m)$? + ]]> +     + $(1/4)^{m} \cdot 3/4$ + $(1/4)^{m-1} \cdot 3/4$ + $(3/4)^{m} \cdot 1/4$ + $(3/4)^{m-1} \cdot 1/4$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f23.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f23.xml new file mode 100644 index 00000000..2c88d0c6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f23.xml @@ -0,0 +1,14 @@ + + + + For any two random variables $X$ and $Y$, + $\mathbb{E}(X \cdot Y) = \mathbb{E}(X) \cdot \mathbb{E}(Y)$. +

    + ]]> +     + The statement is true. + The statement is false. + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f24.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f24.xml new file mode 100644 index 00000000..3d882b35 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f24.xml @@ -0,0 +1,22 @@ + + + + + Elisa orders 7 ciders, one cider at a time. Let $(D_1,D_2,\dots,D_7)$ be the sequence of drinks + that Tan serves. Define the following random variable $X$: + [defs] + X = the number of indices $i$ such that $D_i$ is a cider and $D_{i+1}$ is not a cider. + [/defs] + What is the expected value $\mathbb{E}(X)$ of $X$? + ]]> + + 44/144 + 55/144 + 66/144 + 77/144 + diff --git a/app/src/data/questions/comp2804/exam-winter-2017/w17f25.xml b/app/src/data/questions/comp2804/exam-winter-2017/w17f25.xml new file mode 100644 index 00000000..480fc2d6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2017/w17f25.xml @@ -0,0 +1,4 @@ + + How do you feel about writing an exam on Sunday afternoon? + I hate it! + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/exam-winter-2018.json b/app/src/data/questions/comp2804/exam-winter-2018/exam-winter-2018.json new file mode 100644 index 00000000..dea1f659 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/exam-winter-2018.json @@ -0,0 +1,32 @@ +{ + "id": "exam-winter-2018", + "title": "Winter 2018 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-winter-2018/w18f01.xml", + "comp2804/exam-winter-2018/w18f02.xml", + "comp2804/exam-winter-2018/w18f03.xml", + "comp2804/exam-winter-2018/w18f04.xml", + "comp2804/exam-winter-2018/w18f05.xml", + "comp2804/exam-winter-2018/w18f06.xml", + "comp2804/exam-winter-2018/w18f07.xml", + "comp2804/exam-winter-2018/w18f08.xml", + "comp2804/exam-winter-2018/w18f09.xml", + "comp2804/exam-winter-2018/w18f10.xml", + "comp2804/exam-winter-2018/w18f11.xml", + "comp2804/exam-winter-2018/w18f12.xml", + "comp2804/exam-winter-2018/w18f13.xml", + "comp2804/exam-winter-2018/w18f14.xml", + "comp2804/exam-winter-2018/w18f15.xml", + "comp2804/exam-winter-2018/w18f16.xml", + "comp2804/exam-winter-2018/w18f17.xml", + "comp2804/exam-winter-2018/w18f18.xml", + "comp2804/exam-winter-2018/w18f19.xml", + "comp2804/exam-winter-2018/w18f20.xml", + "comp2804/exam-winter-2018/w18f21.xml", + "comp2804/exam-winter-2018/w18f22.xml", + "comp2804/exam-winter-2018/w18f23.xml", + "comp2804/exam-winter-2018/w18f24.xml", + "comp2804/exam-winter-2018/w18f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f01.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f01.xml new file mode 100644 index 00000000..8434375f --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f01.xml @@ -0,0 +1,13 @@ + + + + + ${20 \choose 12} \cdot {38 \choose 18}$ + ${20 \choose 12} \cdot {50 \choose 18}$ + ${50 \choose 12} \cdot {20 \choose 18}$ + ${70 \choose 30} \cdot {20 \choose 12}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f02.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f02.xml new file mode 100644 index 00000000..5ddbbc89 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f02.xml @@ -0,0 +1,14 @@ + + + + + ${20 \choose 12} + {50 \choose 12}$ + ${20 \choose 12} + {50 \choose 12} - {20 \choose 12} \cdot {50 \choose 12}$ + ${20 \choose 12} \cdot 2^{50} + {50 \choose 12} \cdot 2^{20}$ + ${20 \choose 12} \cdot 2^{50} + {50 \choose 12} \cdot 2^{20} - {20 \choose 12} \cdot {50 \choose 12}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f03.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f03.xml new file mode 100644 index 00000000..6a5e4d6d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f03.xml @@ -0,0 +1,13 @@ + + + + + $2^{70} - 2^{50} - 20 - {20 \choose 2}$ + $2^{70} - 2^{50} - 20 \cdot 2^{50} - {20 \choose 2} \cdot 2^{50}$ + $2^{70} - 2^{50} - 20 \cdot 2^{50}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f04.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f04.xml new file mode 100644 index 00000000..61b4e4af --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f04.xml @@ -0,0 +1,12 @@ + + + + + $2 \cdot {40 \choose 5} \cdot 3^{35} - {40 \choose 5} \cdot {35 \choose 5} \cdot 2^{30}$ + $2 \cdot {40 \choose 5} \cdot 3^{35}$ + ${40 \choose 5} + {35 \choose 5} - {40 \choose 5} \cdot {35 \choose 5}$ + $2 \cdot {40 \choose 5} - {40 \choose 5} \cdot {35 \choose 5}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f05.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f05.xml new file mode 100644 index 00000000..5d138168 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f05.xml @@ -0,0 +1,15 @@ + + + + + The number of ways to choose a subset from a set consisting of $m + n$ elements. + The number of ways to choose an ordered pair of 2 elements from a set consisting of $m + n$ elements. + The number of ways to choose a 2-element subset from a set consisting of $m + n$ elements. + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f06.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f06.xml new file mode 100644 index 00000000..ecf6995b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f06.xml @@ -0,0 +1,14 @@ + + + + + ${31 \choose 6}$ + ${31 \choose 7}$ + ${32 \choose 6}$ + ${32 \choose 7}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f07.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f07.xml new file mode 100644 index 00000000..1a64422a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f07.xml @@ -0,0 +1,17 @@ + + + + BOOGER +

    + is called awesome, if the string does not contain the substring OO. + Thus, GEOROB is awesome, whereas GREOOB is not awesome. + What is the number of awesome strings? + ]]> +     + $6 \cdot {5 \choose 2} \cdot 3 \cdot 2$ + $(6 \cdot {5 \choose 2} \cdot 3 \cdot 2) - 5!$ + $6! - 5!$ + $(6 \cdot {5 \choose 2} \cdot 3) - 5!$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f08.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f08.xml new file mode 100644 index 00000000..56d9e86b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f08.xml @@ -0,0 +1,15 @@ + + + valid, if it does not contain $aa$, it does not contain $bb$, + it does not contain $cc$, and it does not contain $dd$.
    + + For any integer $n \geq 2$, what is the number of valid strings of length $n$? + ]]> +     + $4^{n} - 4(n-1)$ + $4^{n} - 4n$ + $4 \cdot 3^{n}$ + $4 \cdot 3^{n-1}$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f09.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f09.xml new file mode 100644 index 00000000..e146ba52 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f09.xml @@ -0,0 +1,12 @@ + + + + + $S_n = S_{n-1} + S_{n-2} + 1$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-2}$ + $S_n = S_{n-1} + S_{n-2} + 2^{n-3}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f10.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f10.xml new file mode 100644 index 00000000..b22563bd --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f10.xml @@ -0,0 +1,48 @@ + + + + $\newcommand{\SundayEveningExam}{{\rm S {\scriptsize UNDAY} E {\scriptsize VENING} E {\scriptsize XAM}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the recursive algorithm $\SundayEveningExam$, which takes as input an integer $n \geq 1$: + +
    + $\mathbf{Algorithm}\ \SundayEveningExam(n)\mathrm{:}$
    + $\mathbf{if}\ n = 1$
    + + + + + + + +
    + $\mathbf{then}\ \mathrm{print}\ \text{``}$ + $\text{I don't like Sunday}$ $\text{evening exams''}$
    + $\mathbf{else}\ \mathbf{for}\ i = 1\ \mathbf{to}\ n$
    + + + + + + + +
    + $\elsesp \mathbf{do}\ \mathrm{print}\ \text{``}$ + $\text{I don't like Sunday}$ $\text{evening exams''}$
    + $\elsesp \mathbf{endfor};$
    + $\elsesp \SundayEveningExam(n - 1)$
    + $\mathbf{endif}$ +
    + + Let $P(n)$ be the number of times the line "I don't like Sunday evening exams" is printed when running + algorithm $\SundayEveningExam(n)$. Which of the following is true for all $n \geq 1$? + ]]> +     + $P(n) = 1 + \frac{n(n-1)}{2}$ + $P(n) = \frac{n(n-1)}{2}$ + $P(n) = \frac{n(n+1)}{2}$ + $P(n) = 1 + \frac{n(n+1)}{2}$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f11.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f11.xml new file mode 100644 index 00000000..f3cc2f1d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f11.xml @@ -0,0 +1,12 @@ + + + + + ${18 \choose 3} \cdot \left(5 \middle/ 6 \right)^{18}$ + $18 \cdot \left. 5^{15} \middle/ 6^{18} \right.$ + ${18 \choose 3} \cdot \left. 5^{15} \middle/ 6^{18} \right.$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f12.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f12.xml new file mode 100644 index 00000000..ee2cec45 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f12.xml @@ -0,0 +1,20 @@ + + + palindrome, if + $$ + s_1s_2{\dots}s_{n-1}s_n = s_ns_{n-1}{\dots}s_2s_1, + $$ + i.e., reading the string from left to right gives the + same result as reading the string from right to left.
    + + Let $n \geq 3$ be an odd integer. You are given a string of length $n$, in which each character is a + uniformly random element of $\{a,b,c\}$. The characters are independent of each other. What is the + probability that this string is a palindrome? + ]]> +     + $(1/3)^{(n+1)/2}$ + $(1/3)^{(n-1)/2}$ + $(1/2)^{(n+1)/2}$ + $(1/2)^{(n-1)/2}$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f13.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f13.xml new file mode 100644 index 00000000..0950b6aa --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f13.xml @@ -0,0 +1,16 @@ + + + + + 12/16 + 13/16 + 14/16 + 15/16 + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f14.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f14.xml new file mode 100644 index 00000000..c8527378 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f14.xml @@ -0,0 +1,19 @@ + + + + + Consider the string + $$ + a_1-b_1,a_2-b_2,\dots,a_{77}-b_{77}. + $$ + What is the probability that each element in this string is non-zero? + ]]> + + $(3/8)^{77}$ + $(4/8)^{77}$ + $(5/8)^{77}$ + $(6/8)^{77}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f15.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f15.xml new file mode 100644 index 00000000..d1a127ee --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f15.xml @@ -0,0 +1,16 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f16.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f16.xml new file mode 100644 index 00000000..8ca755b2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f16.xml @@ -0,0 +1,16 @@ + + + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f17.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f17.xml new file mode 100644 index 00000000..9cb1f17a --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f17.xml @@ -0,0 +1,16 @@ + + + + + $\frac{2(2n-1)}{n-1}$ + $\frac{n-1}{4n}$ + $\frac{n-1}{2(2n-1)}$ + $\frac{n}{2(2n-1)}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f18.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f18.xml new file mode 100644 index 00000000..5b3bf191 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f18.xml @@ -0,0 +1,42 @@ + + + + + + + + + + + + + + + + + + + + + + + + +
    $X = \bigg\{$$1\ $if the red coin flip resulted in heads$,$
    $0\ $if the red coin flip resulted in tails$,$
    $Y = \bigg\{$$1\ $if the blue coin flip resulted in heads$,$
    $0\ $if the blue coin flip resulted in tails$,$
    + + and + [defs] + Z = $\min(X,Y).$ + [/defs] + What is the expected value $\mathbb{E}(Z)$ of the random variable $Z$? + ]]> +     + 1/4 + 1/2 + 3/4 + 1 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f19.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f19.xml new file mode 100644 index 00000000..a9826872 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f19.xml @@ -0,0 +1,18 @@ + + + + + Define the random variable $X$ to be the number of indices $i$ with $1 \leq i < n$ for which $b_i + \cdot b_{i+1} = 0$.
    + + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]> +     + $n/4$ + $(n-1)/4$ + $3n/4$ + $3(n-1)/4$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f20.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f20.xml new file mode 100644 index 00000000..a8e26cee --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f20.xml @@ -0,0 +1,17 @@ + + + + + 1 + $\frac{n+2}{n+1}$ + $\frac{n+1}{n}$ + $\frac{n+3}{n+1}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f21.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f21.xml new file mode 100644 index 00000000..c69def89 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f21.xml @@ -0,0 +1,25 @@ + + + + + + $\Pr(X = 0)$ + + $= \Pr(X = 1) = \Pr(Y = 0)$ + $= \Pr(Y = 1) = 1/2.$ + + + + + + Define the random variable $Z = X \cdot Y$. Which of the following is correct? + ]]> + + The random variables $X$ and $Z$ are independent. + The random variables $X$ and $Z$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f22.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f22.xml new file mode 100644 index 00000000..6bac896d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f22.xml @@ -0,0 +1,48 @@ + + + + $\newcommand{\WhoGoesFirst}{{\rm W {\scriptsize HO} G {\scriptsize OES} F {\scriptsize IRST}}} + \newcommand{\thensp}{\phantom{\mathbf{then}\ }}$ + + + Alexa and Shelly want to play the game of Monopoly. They use the following recursive algorithm + to decide who goes first: + +
    + $\mathbf{Algorithm}\ \WhoGoesFirst(k)\mathrm{:}$
    + + + + + + + +
    $//$ + $k \geq 1,$ $\text{the die is fair},$ $\text{and all rolls are independent}$ +
    + $\text{Alexa rolls the die, let \(a\) be the result};$
    + $\text{Shelly rolls the die, let \(s\) be the result};$
    + $\mathbf{if}\ a > s$
    + $\mathbf{then}\ \mathrm{print}\ {\tt Alexa\ goes\ first};$
    + $\thensp \mathrm{return}\ k$
    + $\mathbf{endif};$
    + $\mathbf{if}\ a < s$
    + $\mathbf{then}\ \mathrm{print}\ {\tt Shelly\ goes\ first};$
    + $\thensp \mathrm{return}\ k$
    + $\mathbf{endif};$
    + $\mathbf{if}\ a = s$
    + $\mathbf{then}\ \WhoGoesFirst(k + 1)$
    + $\mathbf{endif}$ +
    + + The ladies run algorithm $\WhoGoesFirst(1)$, i.e., with $k = 1$. Define the random variable $X$ to + be the value of the output of this algorithm.
    + What is the expected value $\mathbb{E}(X)$ of the random variable $X$? + ]]> +     + 3/2 + 5/4 + 5/6 + 6/5 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f23.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f23.xml new file mode 100644 index 00000000..3804449b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f23.xml @@ -0,0 +1,17 @@ + + + + + The statement is true. + The statement is false. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f24.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f24.xml new file mode 100644 index 00000000..b64b8b37 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f24.xml @@ -0,0 +1,21 @@ + + + + Define the following random variable $X$: + [defs] + X = the number of students who get exactly 2 ciders. + [/defs] + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]> +     + $2^{4} \cdot 3^{2} \cdot \left. 5^{2} \middle/ 3^{8} \right.$ + $2^{4} \cdot 3^{2} \cdot \left. 2^{5} \middle/ 8^{3} \right.$ + $2^{4} \cdot 3^{2} \left. \cdot 5^{2} \middle/ 8^{3} \right.$ + $3^{2} \cdot \left. 5^{2} \middle/ 8^{3} \right.$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2018/w18f25.xml b/app/src/data/questions/comp2804/exam-winter-2018/w18f25.xml new file mode 100644 index 00000000..d8134dc0 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2018/w18f25.xml @@ -0,0 +1,4 @@ + + Are you happy that this is the last question? + Yes. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/exam-winter-2019.json b/app/src/data/questions/comp2804/exam-winter-2019/exam-winter-2019.json new file mode 100644 index 00000000..b9323073 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/exam-winter-2019.json @@ -0,0 +1,32 @@ +{ + "id": "exam-winter-2019", + "title": "Winter 2019 Exam", + "Author": "Michiel Smid", + "questions": [ + "comp2804/exam-winter-2019/w19f01.xml", + "comp2804/exam-winter-2019/w19f02.xml", + "comp2804/exam-winter-2019/w19f03.xml", + "comp2804/exam-winter-2019/w19f04.xml", + "comp2804/exam-winter-2019/w19f05.xml", + "comp2804/exam-winter-2019/w19f06.xml", + "comp2804/exam-winter-2019/w19f07.xml", + "comp2804/exam-winter-2019/w19f08.xml", + "comp2804/exam-winter-2019/w19f09.xml", + "comp2804/exam-winter-2019/w19f10.xml", + "comp2804/exam-winter-2019/w19f11.xml", + "comp2804/exam-winter-2019/w19f12.xml", + "comp2804/exam-winter-2019/w19f13.xml", + "comp2804/exam-winter-2019/w19f14.xml", + "comp2804/exam-winter-2019/w19f15.xml", + "comp2804/exam-winter-2019/w19f16.xml", + "comp2804/exam-winter-2019/w19f17.xml", + "comp2804/exam-winter-2019/w19f18.xml", + "comp2804/exam-winter-2019/w19f19.xml", + "comp2804/exam-winter-2019/w19f20.xml", + "comp2804/exam-winter-2019/w19f21.xml", + "comp2804/exam-winter-2019/w19f22.xml", + "comp2804/exam-winter-2019/w19f23.xml", + "comp2804/exam-winter-2019/w19f24.xml", + "comp2804/exam-winter-2019/w19f25.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f01.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f01.xml new file mode 100644 index 00000000..3aeb8662 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f01.xml @@ -0,0 +1,10 @@ + + + ${85 \choose 5} \cdot 3^{80}$ + ${85 \choose 5} \cdot 3^{85}$ + ${85 \choose 5} \cdot 4^{80}$ + ${85 \choose 5} \cdot 4^{85}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f02.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f02.xml new file mode 100644 index 00000000..55e1c157 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f02.xml @@ -0,0 +1,10 @@ + + + ${85 \choose 15} \cdot {70 \choose 30} \cdot 2^{40}$ + ${85 \choose 15} \cdot {70 \choose 30} \cdot 3^{40}$ + ${85 \choose 15} \cdot {70 \choose 30} \cdot 4^{40}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f03.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f03.xml new file mode 100644 index 00000000..7ab01009 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f03.xml @@ -0,0 +1,10 @@ + + + ${55 \choose 10} - {30 \choose 10} - {30 \choose 9}$ + ${55 \choose 10} - {30 \choose 10} - 25 \cdot {29 \choose 9}$ + ${55 \choose 10} - {30 \choose 10} - 25 \cdot {30 \choose 9}$ + ${55 \choose 10} - {30 \choose 10} - 25 \cdot {30 \choose 10}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f04.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f04.xml new file mode 100644 index 00000000..75c7aec8 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f04.xml @@ -0,0 +1,12 @@ + + awesome, if $x$ is even or $y$ is even. + What is the number of awesome elements in $S$? + ]]> + 104 + 105 + 106 + 107 + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f05.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f05.xml new file mode 100644 index 00000000..623bd7ec --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f05.xml @@ -0,0 +1,18 @@ + + + Let $m \geq 2$ and $n \geq 2$ be integers. Why does the identity + $$ + {m + n \choose 2} = {m \choose 2} + {n \choose 2} + mn + $$ + hold? + + + Because both sides count the number of ways $m$ men and $n$ women can be arranged on a line, + such that no two men are standing next to each other. + + Because both sides count the number of ordered pairs in a set of size $m + n$. + + Because both sides count the number of 2-element subsets of a set of size $m + n$. + + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f06.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f06.xml new file mode 100644 index 00000000..06ba83be --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f06.xml @@ -0,0 +1,9 @@ + + + ${55 \choose 20} \cdot 5^{35} \cdot 3^{20}$ + $-{55 \choose 20} \cdot 5^{35} \cdot 3^{20}$ + ${55 \choose 35} \cdot 5^{20} \cdot 3^{35}$ + $-{55 \choose 35} \cdot 5^{20} \cdot 3^{35}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f07.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f07.xml new file mode 100644 index 00000000..287b7041 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f07.xml @@ -0,0 +1,20 @@ + + POOPERSCOOPER

    + is called cool, if each occurrence of E has the letter R to its left or + right, and each occurrence of R has the letter E to its left or right. Thus, both +

    POOPERSCOOPER

    + and +

    OPRECSOOOERPP

    + are cool, whereas +

    EPOOPRSCOOPER

    + is not cool. What is the number of cool strings? + ]]>     + ${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1}$ + ${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1} \cdot 2$ + ${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1} \cdot 3$ + + ${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1} \cdot 4$ + +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f08.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f08.xml new file mode 100644 index 00000000..1059ea69 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f08.xml @@ -0,0 +1,13 @@ + + free, if it does not contain two 0's next to each other. + In class, we have seen that for any $m \geq 1$, the number of 00-free bitstrings of length $m$ + is equal to the $(m + 2)$-th Fibonacci number $f_{m+2}$.
    + What is the number of 00-free bitstrings of length 20 that have 1 at position 11 and 0 at + position 20? (The positions are numbered $1,2,...,20$.) + ]]>     + $f_{10} \cdot f_{12}$ + $f_9 \cdot f_{12}$ + $f_8 \cdot f_{11}$ + $f_7 \cdot f_{10}$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f09.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f09.xml new file mode 100644 index 00000000..2be5c9b6 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f09.xml @@ -0,0 +1,11 @@ + + + $S_n = 1 + S_{n - 1}$ + $S_n = 1 + S_{n - 2}$ + $S_n = S_{n - 1} + S_{n - 3}$ + $S_n = S_{n - 2} + S_{n - 3}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f10.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f10.xml new file mode 100644 index 00000000..6f8a645b --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f10.xml @@ -0,0 +1,16 @@ + + + $f(n) = \frac{5}{2} \cdot 3^n - 1$ + $f(n) = \frac{3}{2} \cdot 3^n - \frac{1}{2}$ + $f(n) = \frac{5}{2} \cdot 3^n - \frac{1}{2}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f11.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f11.xml new file mode 100644 index 00000000..f91fa4bb --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f11.xml @@ -0,0 +1,11 @@ + + + You flip a fair coin 6 times; the flips are independent of each other. + What is the probability that in these 6 coin flips, the coin comes up heads exactly + 3 times? + + 3/16 + 4/16 + 5/16 + 6/16 + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f12.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f12.xml new file mode 100644 index 00000000..8836d561 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f12.xml @@ -0,0 +1,13 @@ + + suit and a rank. + There are four suits (spades ♠, hearts ♡, clubs ♣, and diamonds ♢), and 13 ranks + (Ace, 2,3,4,5,6,7,8,9,10, Jack, Queen, and King).
    + Assume you get a uniformly random hand consisting of 5 cards. What is the probability + that the 5 cards in this hand consist of 3 Aces and 2 Queens? + ]]>     + $\left. 24 \middle/ {52 \choose 5} \right.$ + $\left. 25 \middle/ {52 \choose 5} \right.$ + $\left. 26 \middle/ {52 \choose 5} \right.$ + $\left. 27 \middle/ {52 \choose 5} \right.$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f13.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f13.xml new file mode 100644 index 00000000..6c5d773d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f13.xml @@ -0,0 +1,16 @@ + + + 17/57 + 18/57 + 19/57 + 20/57 + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f14.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f14.xml new file mode 100644 index 00000000..470799ef --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f14.xml @@ -0,0 +1,17 @@ + + + You are given two bitstrings $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$, both of length $n$. + In both bitstrings, each bit is 0 with probability 1/2, and 1 with probability 1/2 + (independent of all other bits).
    + For each $i$ with $1 \leq i \leq n$, let $c_i = a_i - b_i$. Consider the set + $$ + P = \{i : 1 \leq i \leq n \text{ and } c_i \geq 0\}. + $$ + What is the probability that the size of the set $P$ is equal to $k$? + ]]>     + ${n \choose k} \cdot (3/4)^{n - k} \cdot (1/4)^k$ + ${n \choose k} \cdot (3/4)^k \cdot (1/4)^{n - k}$ + ${n \choose k} \cdot (3/4)^k \cdot (1/2)^k$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f15.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f15.xml new file mode 100644 index 00000000..bec46e75 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f15.xml @@ -0,0 +1,17 @@ + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f16.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f16.xml new file mode 100644 index 00000000..98a1cd8e --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f16.xml @@ -0,0 +1,17 @@ + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f17.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f17.xml new file mode 100644 index 00000000..a9414c08 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f17.xml @@ -0,0 +1,17 @@ + + + The events $A$ and $B$ are independent. + The events $A$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f18.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f18.xml new file mode 100644 index 00000000..0ce3ba83 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f18.xml @@ -0,0 +1,11 @@ + + + 3/4 + 2/3 + 1/2 + 1/3 + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f19.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f19.xml new file mode 100644 index 00000000..cec56320 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f19.xml @@ -0,0 +1,33 @@ + + + Step 1: Choose a uniformly random number from + the red box, and denote it by $x$.
    + Step 2: + + Consider the random variables + + [defs] + X = the number $x$ you choose in Step 1, + Y = the number $y$ you choose in Step 2, + Z = $\max(X,Y).$ + [/defs] + + What is the expected value $\mathbb{E}(Z)$ of the random variable $Z$? + ]]>     + 1 + 3/2 + 2 + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f20.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f20.xml new file mode 100644 index 00000000..850e83a2 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f20.xml @@ -0,0 +1,14 @@ + + + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]>     + $\frac{2}{3} \cdot n$ + $\frac{2}{3} \cdot (n - 1)$ + $\frac{4}{9} \cdot n$ + $\frac{4}{9} \cdot (n - 1)$ +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f21.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f21.xml new file mode 100644 index 00000000..11ff4f36 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f21.xml @@ -0,0 +1,17 @@ + + + $\frac{n}{n + 2} + \frac{2n + 6}{(n + 1)(n + 2)}$ + $\frac{n}{n + 2} + \frac{2n + 2}{(n + 1)(n + 2)}$ + $\frac{n}{n + 2} + \frac{4n + 6}{(n + 1)(n + 2)}$ + $\frac{n}{n + 2} + \frac{4n + 2}{(n + 1)(n + 2)}$ + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f22.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f22.xml new file mode 100644 index 00000000..1f055969 --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f22.xml @@ -0,0 +1,21 @@ + + + The random variables $G$ and $B$ are independent. + The random variables $G$ and $B$ are not independent. + None of the above. + All of the above. + diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f23.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f23.xml new file mode 100644 index 00000000..ae3f1f3d --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f23.xml @@ -0,0 +1,19 @@ + + Experiment: Roll each die once and take the sum of the two rolls.

    + You repeat this experiment until the sum of the two rolls is equal to 7.
    + Consider the random variable + + [defs] + X = the number of times you do the experiment. + [/defs] + + (This value $X$ includes the experiment in which the sum is 7 for the first time.)
    + What is the expected value $\mathbb{E}(X)$ of the random variable $X$? + ]]>     + 6 + 5 + 4 + 3 +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f24.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f24.xml new file mode 100644 index 00000000..791391ad --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f24.xml @@ -0,0 +1,27 @@ + + + Consider $n$ students $S_1,S_2,...,S_n$ who hand in an assignment for COMP 2804. This assignment has 2 questions. + For $i = 1,2,...,n$ let $A_i$ be the assignment handed in by student $S_i$.
    + Alexa and Zoltan are TAs for the course. Alexa is supposed to mark the first question, whereas Zoltan + is supposed to mark the second question. Since both TAs are lazy, they use the following scheme + (all random choices are mutually independent):
    + For $i = 1,2,...,n$, + + Consider the random variable + + [defs] + X = the number of assignments that are marked by both Alexa and Zoltan. + [/defs] + + What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
    + Hint: Use indicator random variables. + ]]>     + $(p + q) \cdot n$ + $\frac{n}{p \cdot q}$ + $p \cdot q \cdot n$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/exam-winter-2019/w19f25.xml b/app/src/data/questions/comp2804/exam-winter-2019/w19f25.xml new file mode 100644 index 00000000..a71893fb --- /dev/null +++ b/app/src/data/questions/comp2804/exam-winter-2019/w19f25.xml @@ -0,0 +1,11 @@ + + + Who discovered the Fibonacci numbers? + + Donald Trump + + The Italian mathematician Fibonacci, also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano + + Alexandre-Théophile Vandermonde + Keith Richards (the guitarist of the greatest rock band in the universe) + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m01.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m01.xml new file mode 100644 index 00000000..cc74191a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m01.xml @@ -0,0 +1,12 @@ + + + + + ${n \choose 2} \cdot 2^{n-2}$ + $n \cdot 2^{n-1}$ + $2^{n} - 1 - n$ + $2^{n} - n$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m02.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m02.xml new file mode 100644 index 00000000..50db1f9c --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m02.xml @@ -0,0 +1,11 @@ + + + + + $2^{91}$ + $2^{95} + 2^{95}$ + $2^{96} - 2^{91}$ + $2^{99} - 2^{96}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m03.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m03.xml new file mode 100644 index 00000000..8ff10b58 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m03.xml @@ -0,0 +1,12 @@ + + + + + $17! + 17!$ + $2(17! + 17!)$ + $(17!)^2$ + $2(17!)^2$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m04.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m04.xml new file mode 100644 index 00000000..9b757822 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m04.xml @@ -0,0 +1,25 @@ + + + +
  • + all boys stand next to each other, +
    • +
  • + all girls stand next to each other, +
    • +
  • + all dogs stand next to each other, +
    • +
  • + none of the boys are standing next to any of the girls. +
    • + + ]]> +     + $12! + 17! + 25!$ + $2(12! + 17! + 25!)$ + $(12!)(17!)(25!)$ + $2(12!)(17!)(25!)$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m05.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m05.xml new file mode 100644 index 00000000..de11e9e0 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m05.xml @@ -0,0 +1,10 @@ + + + + + True + False + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m06.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m06.xml new file mode 100644 index 00000000..516fbf4b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m06.xml @@ -0,0 +1,12 @@ + + + + + At least two of these $n$ points are at distance at most $1$. + At least two of these $n$ points are at distance at most $\left. 1 \middle/ \sqrt{2} \right.$. + At least two of these $n$ points are at distance at most $\sqrt{2}$. + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m07.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m07.xml new file mode 100644 index 00000000..b2af2669 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m07.xml @@ -0,0 +1,11 @@ + + + + + $-{100 \choose 80} 5^{20}3^{80}$ + ${100 \choose 80} 5^{20}3^{80}$ + ${100 \choose 80} 5^{80}3^{20}$ + ${80 \choose 100} 5^{20}3^{80}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m08.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m08.xml new file mode 100644 index 00000000..a0f1bfdc --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m08.xml @@ -0,0 +1,12 @@ + + + + + ${30 \choose 3}$ + ${30 \choose 4}$ + ${31 \choose 3}$ + ${31 \choose 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m09.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m09.xml new file mode 100644 index 00000000..c2f6e282 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m09.xml @@ -0,0 +1,27 @@ + + + valid, if in each step, it either goes one unit to the right or one unit upwards.
    + In the example below, you see a valid path for the case when $m = 5$ and $n = 3$. + + + + How many valid paths are there? + ]]> +     + ${m + n \choose n - 1}$ + ${m + n \choose n}$ + $2^{m + n}$ + $2^m + 2^n$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m10.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m10.xml new file mode 100644 index 00000000..116cc109 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m10.xml @@ -0,0 +1,18 @@ + + + + + $f(n) = 2n^{2} - 5n + 7$ + $f(n) = 2n^{2} - 5n + 8$ + $f(n) = 3n^{2} - 5n + 8$ + $f(n) = 4n^{2} - 5n + 8$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m11.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m11.xml new file mode 100644 index 00000000..33581606 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m11.xml @@ -0,0 +1,12 @@ + + + + + $B_n = B_{n-1} + B_{n-2} + B_{n-3}$ + $B_n = B_{n-1} + B_{n-2} + B_{n-3} + B_{n-4}$ + $B_n = 2^{n} - 2^{n-4}$ + $B_n = 2^{n} - (n-3) \cdot 2^{n-4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m12.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m12.xml new file mode 100644 index 00000000..236fef2f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m12.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the following recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + When running $\Fib(9)$, how many calls are there to $\Fib(4)$? + ]]> +     + 6 + 7 + 8 + 9 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m13.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m13.xml new file mode 100644 index 00000000..00a48f13 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m13.xml @@ -0,0 +1,33 @@ + + + + $\newcommand{\Silly}{{\rm S \scriptsize ILLY}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the following recursive algorithm $\Silly$, which takes as input an integer $n \geq 1$ which is a power of 2: + +

    + $\mathbf{Algorithm}\ \Silly(n)\mathrm{:}$
    + $\mathbf{if}\ n = 1$
    + $\mathbf{then}\ \text{drink one pint of beer}$
    + $\mathbf{else}\ \mathbf{if}\ n = 2$
    + $\elsesp \mathbf{then}\ \text{fart once}$
    + $\elsesp \mathbf{else}\ \text{fart once};$
    + $\elsesp \elsesp \Silly(n / 2);$
    + $\elsesp \elsesp \text{fart once}$
    + $\elsesp \mathbf{endif}$
    + $\mathbf{endif}$ +

    + + For $n$ a power of 2, let $F(n)$ be the number of times you fart when running algorithm $\Silly(n)$. + Which of the following is true?
    + (n.b., $\log$ denotes the base-2 logarithm) + ]]> +     + $F(n) = \log n\ \mathrm{for}\ n \geq 1.$ + $F(n) = 2 \log n\ \mathrm{for}\ n \geq 1.$ + $F(n) = (2 \log n) - 1\ \mathrm{for}\ n \geq 1.$ + $F(n) = (2 \log n) - 1\ \mathrm{for}\ n \geq 2.$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m14.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m14.xml new file mode 100644 index 00000000..ec2c0b23 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m14.xml @@ -0,0 +1,12 @@ + + + + + 1/4 + 1/2 + 3/4 + 1 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m15.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m15.xml new file mode 100644 index 00000000..d4658b4e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m15.xml @@ -0,0 +1,18 @@ + + + + + $\Pr(A) = \Pr(B)$ + $\Pr(A) < \Pr(B)$ + $\Pr(A) > \Pr(B)$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m16.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m16.xml new file mode 100644 index 00000000..38c39218 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m16.xml @@ -0,0 +1,12 @@ + + + + + $\frac{10!}{10^{10}}$ + $\frac{10^{10}}{10!}$ + $1 - \frac{10 \cdot (9/10)^{10}}{10^{10}}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/f14m17.xml b/app/src/data/questions/comp2804/midterm-fall-2014/f14m17.xml new file mode 100644 index 00000000..65ea8110 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/f14m17.xml @@ -0,0 +1,13 @@ + + + + + $\frac{100}{4^{100}}$ + $\frac{3^{99}}{4^{100}}$ + $\frac{100 + 3^{99}}{4^{100}}$ + $\frac{100 \cdot 3^{99}}{4^{100}}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2014/midterm-fall-2014.json b/app/src/data/questions/comp2804/midterm-fall-2014/midterm-fall-2014.json new file mode 100644 index 00000000..354c7d04 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2014/midterm-fall-2014.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-fall-2014", + "title": "Fall 2014 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-fall-2014/f14m01.xml", + "comp2804/midterm-fall-2014/f14m02.xml", + "comp2804/midterm-fall-2014/f14m03.xml", + "comp2804/midterm-fall-2014/f14m04.xml", + "comp2804/midterm-fall-2014/f14m05.xml", + "comp2804/midterm-fall-2014/f14m06.xml", + "comp2804/midterm-fall-2014/f14m07.xml", + "comp2804/midterm-fall-2014/f14m08.xml", + "comp2804/midterm-fall-2014/f14m09.xml", + "comp2804/midterm-fall-2014/f14m10.xml", + "comp2804/midterm-fall-2014/f14m11.xml", + "comp2804/midterm-fall-2014/f14m12.xml", + "comp2804/midterm-fall-2014/f14m13.xml", + "comp2804/midterm-fall-2014/f14m14.xml", + "comp2804/midterm-fall-2014/f14m15.xml", + "comp2804/midterm-fall-2014/f14m16.xml", + "comp2804/midterm-fall-2014/f14m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m01.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m01.xml new file mode 100644 index 00000000..c7c13a7d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m01.xml @@ -0,0 +1,15 @@ + + + + + $n{n \choose 2}{n \choose 5}$ + $(n-2){n\choose 2}{n-2 \choose 5}$ + $(n-5){n \choose 2}{n-1 \choose 5}$ + $(n-7){n \choose 2}{n-2 \choose 5}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m02.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m02.xml new file mode 100644 index 00000000..fbf44223 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m02.xml @@ -0,0 +1,12 @@ + + + + + $2^{25} - 2^{22}$ + $2^{25} - 2^{24} + 2^{23}$ + $2^{22}$ + $2^{23}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m03.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m03.xml new file mode 100644 index 00000000..52f9cf9c --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m03.xml @@ -0,0 +1,12 @@ + + + + + $5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$ + $6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$ + $7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$ + $8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m04.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m04.xml new file mode 100644 index 00000000..d69f490a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m04.xml @@ -0,0 +1,12 @@ + + + + + $S_n = 2^{n-2}$ + $S_n = 2^{n-1}$ + $S_n = 2^{n} - 2^{n-2}$ + $S_n = 2^{n} - 2^{n-1} + 2^{n-2}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m05.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m05.xml new file mode 100644 index 00000000..b8b653d1 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m05.xml @@ -0,0 +1,12 @@ + + + + + $3^{96} + 3^{96}$ + $3^{99} - 2 \cdot 3^{96}$ + $2 \cdot 3^{96} - 3^{93}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m06.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m06.xml new file mode 100644 index 00000000..ff59120b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m06.xml @@ -0,0 +1,15 @@ + + + + + The number of non-empty subsets of a set of size $m$. + The number of subsets of a set of size $m$. + The number of bitstrings of length $m$ having exactly $k$ many 1s. + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m07.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m07.xml new file mode 100644 index 00000000..78674a6e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m07.xml @@ -0,0 +1,17 @@ + + + + $\newcommand{\ShortLastName}{{\rm S {\scriptsize HORT} L {\scriptsize AST} N {\scriptsize AME}}}$ + + + In the city of $\ShortLastName$, every person has a last name consisting of two characters, the first one being + an uppercase letter and the second one being a lowercase letter. What is the minimum number of people + needed so that we can guarantee that at least 4 of them have the same last name? + ]]> + + $3 \cdot 26^{2}$ + $4 \cdot 26^{2}$ + $3 \cdot 26^{2} + 1$ + $4 \cdot 26^{2} + 1$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m08.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m08.xml new file mode 100644 index 00000000..9dacc0c0 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m08.xml @@ -0,0 +1,11 @@ + + + + + ${88 \choose 7} \cdot 3^{81} \cdot 17^7$ + $- {88 \choose 7} \cdot 3^{81} \cdot 17^7$ + ${88 \choose 7} \cdot 3^7 \cdot 17^{81}$ + $- {88 \choose 7} \cdot 3^7 \cdot 17^{81}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m09.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m09.xml new file mode 100644 index 00000000..69ca408c --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m09.xml @@ -0,0 +1,12 @@ + + + + + ${58 \choose 3}$ + ${58 \choose 4}$ + ${59 \choose 3}$ + ${59 \choose 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m10.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m10.xml new file mode 100644 index 00000000..21051dc9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m10.xml @@ -0,0 +1,18 @@ + + + + + $f(n) = 4n^2 - 2n + 7$ + $f(n) = 4n^2 - n + 7$ + $f(n) = 5n^2 - 2n + 7$ + $f(n) = 5n^2 - n + 7$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m11.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m11.xml new file mode 100644 index 00000000..26e812c2 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m11.xml @@ -0,0 +1,12 @@ + + + + + $S_n = S_{n-1} + S_{n-2} + S_{n-3} + S_{n-4}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3} + S_{n-4}\ +$ $ 2^{n-4}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3} + 2^{n-3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m12.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m12.xml new file mode 100644 index 00000000..36013787 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m12.xml @@ -0,0 +1,35 @@ + + + +
  • + the order in which the 1s and 2s occur in the sum matters, and +
    • +
  • + it is not allowed to have two consecutive 2s. +
    • + + For example, if $n = 7$, then both + $$ + 7 = 1 + 2 + 1 + 2 + 1 + $$ + and + $$ + 7 = 2 + 1 + 1 + 2 + 1 + $$ + are allowed, whereas + $$ + 7 = 1 + 2 + 2 + 1 + 1 + $$ + is not allowed.
    + + Which of the following is true? + ]]> +     + $S_n = S_{n-1} + S_{n-2}$ + $S_n = S_{n-1} + S_{n-3}$ + $S_n = S_{n-2} + S_{n-3}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m13.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m13.xml new file mode 100644 index 00000000..56caa3ef --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m13.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the following recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + When running $\Fib(55)$, how many calls are there to $\Fib(50)$? + ]]> +     + 6 + 7 + 8 + 9 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m14.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m14.xml new file mode 100644 index 00000000..db4a18b9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m14.xml @@ -0,0 +1,53 @@ + + + + $\newcommand{\JustinBieber}{{\rm J {\scriptsize USTIN} B {\scriptsize IEBER}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the following recursive algorithm $\JustinBieber$, which takes as input an integer $n \geq 1$, + which is a power of 2: + +
    + $\mathbf{Algorithm}\ \JustinBieber(n)\mathrm{:}$
    + $\mathbf{if}\ n = 1$
    + $\mathbf{then}\ \text{order chicken wings}$
    + $\mathbf{else}\ \mathbf{if}\ n = 2$ + + + + + + + +
    $\elsesp \mathbf{then}$ + $\text{drink one pint of}$ $\text{India Pale Ale}$ +
    + + + + + + + +
    + $\elsesp \mathbf{else}\ \mathrm{print}\ \text{``}$ + + $\text{I don't like}$ $\text{Justin Bieber''};$ +
    + $\elsesp \elsesp \JustinBieber(n / 2)$
    + $\elsesp \mathbf{endif}$
    + $\mathbf{endif}$ +
    + + For $n$ a power of 2, let $B(n)$ be the number of times you print "I don't like Justin Bieber" when + running algorithm $\JustinBieber(n)$. Which of the following is true?
    + (n.b., $\log$ denotes the base-2 logarithm) + ]]> +     + $B(n) = \log n - 1\ \text{for all}\ n \geq 2.$ + $B(n) = \log n - 1\ \text{for all}\ n \geq 1.$ + $B(n) = \log n\ \text{for all}\ n \geq 2.$ + $B(n) = n - 2\ \text{for all}\ n \geq 2.$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m15.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m15.xml new file mode 100644 index 00000000..37d9376e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m15.xml @@ -0,0 +1,15 @@ + + + + + $\frac{2^5 + 2}{2^7}$ + $1/2$ + $1/3$ + $1/4$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m16.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m16.xml new file mode 100644 index 00000000..dce84e8f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m16.xml @@ -0,0 +1,19 @@ + + + + + $\Pr(A) = \Pr(B)$ + $\Pr(A) < \Pr(B)$ + $\Pr(A) > \Pr(B)$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/f15m17.xml b/app/src/data/questions/comp2804/midterm-fall-2015/f15m17.xml new file mode 100644 index 00000000..f1f4dec3 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/f15m17.xml @@ -0,0 +1,17 @@ + + + + + 7/15 + 15/7 + 3/7 + 7/3 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2015/midterm-fall-2015.json b/app/src/data/questions/comp2804/midterm-fall-2015/midterm-fall-2015.json new file mode 100644 index 00000000..05663a3d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2015/midterm-fall-2015.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-fall-2015", + "title": "Fall 2015 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-fall-2015/f15m01.xml", + "comp2804/midterm-fall-2015/f15m02.xml", + "comp2804/midterm-fall-2015/f15m03.xml", + "comp2804/midterm-fall-2015/f15m04.xml", + "comp2804/midterm-fall-2015/f15m05.xml", + "comp2804/midterm-fall-2015/f15m06.xml", + "comp2804/midterm-fall-2015/f15m07.xml", + "comp2804/midterm-fall-2015/f15m08.xml", + "comp2804/midterm-fall-2015/f15m09.xml", + "comp2804/midterm-fall-2015/f15m10.xml", + "comp2804/midterm-fall-2015/f15m11.xml", + "comp2804/midterm-fall-2015/f15m12.xml", + "comp2804/midterm-fall-2015/f15m13.xml", + "comp2804/midterm-fall-2015/f15m14.xml", + "comp2804/midterm-fall-2015/f15m15.xml", + "comp2804/midterm-fall-2015/f15m16.xml", + "comp2804/midterm-fall-2015/f15m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m01.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m01.xml new file mode 100644 index 00000000..4e6b5654 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m01.xml @@ -0,0 +1,15 @@ + + + + + ${f + m \choose 4}$ + $f \cdot {f + m - 1 \choose 3}$ + $f \cdot {f + m \choose 3}$ + $(f - 1) \cdot {f + m \choose 3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m02.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m02.xml new file mode 100644 index 00000000..d9879574 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m02.xml @@ -0,0 +1,12 @@ + + + and do not contain 2? + ]]> + + ${n - 1 \choose k - 1}$ + ${n - 1 \choose k}$ + ${n - 2 \choose k - 2}$ + ${n - 2 \choose k}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m03.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m03.xml new file mode 100644 index 00000000..05571924 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m03.xml @@ -0,0 +1,12 @@ + + + or do not contain 2? + ]]> + + ${n - 1 \choose k} + {n - 1 \choose k}$ + ${n - 2 \choose k}$ + ${n \choose k} - {n - 2 \choose k - 2}$ + ${n \choose k} - {n - 1 \choose k - 1} - {n - 1 \choose k - 1}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m04.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m04.xml new file mode 100644 index 00000000..f9f20b95 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m04.xml @@ -0,0 +1,12 @@ + + + + + $B_n = 2 \cdot 2^{n - 3}$ + $B_n = 6 \cdot 2^{n - 3}$ + $B_n = 2^n - 2$ + $B_n = 2^n - 6$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m05.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m05.xml new file mode 100644 index 00000000..0c855c14 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m05.xml @@ -0,0 +1,12 @@ + + + + + 17 + 18 + 19 + 20 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m06.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m06.xml new file mode 100644 index 00000000..866d1b9f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m06.xml @@ -0,0 +1,12 @@ + + + + + $4^{n - 2} + 4^{n - 3} - 4^{n - 5}$ + $4^{n - 2} + 4^{n - 3}$ + $4^n - 4^{n - 5}$ + $4^n - 4^{n - 2} - 4^{n - 3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m07.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m07.xml new file mode 100644 index 00000000..18a72f34 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m07.xml @@ -0,0 +1,25 @@ + + + What does + $$ + {w \choose 3} \cdot {m \choose 2} + {w \choose 4} \cdot m + {w \choose 5} + $$ + count? + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at most 3 women can be chosen. + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at most 3 men can be chosen. + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at least 3 women must be chosen. + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at least 3 men must be chosen. + + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m08.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m08.xml new file mode 100644 index 00000000..3f04f2a5 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m08.xml @@ -0,0 +1,13 @@ + + + + + $n - 1$ + $n$ + $n + 1$ + $n + 2$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m09.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m09.xml new file mode 100644 index 00000000..f623ae77 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m09.xml @@ -0,0 +1,11 @@ + + + + + $- {50 \choose 24} \cdot 5^{26} \cdot 7^{24}$ + $- {50 \choose 26} \cdot 5^{24} \cdot 7^{26}$ + ${50 \choose 24} \cdot 5^{26} \cdot 7^{24}$ + ${50 \choose 26} \cdot 5^{24} \cdot 7^{26}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m10.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m10.xml new file mode 100644 index 00000000..93a4fed7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m10.xml @@ -0,0 +1,18 @@ + + + + + $f(n) = 7 \cdot \frac{n!}{3^n}$ + $f(n) = 7^n \cdot \frac{n!}{3^n}$ + $f(n) = 7 \cdot \frac{(n + 1)!}{3^n}$ + $f(n) = 7^n \cdot \frac{(n + 1)!}{3^n}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m11.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m11.xml new file mode 100644 index 00000000..82fa1abb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m11.xml @@ -0,0 +1,13 @@ + + + + + $B_n = B_{n - 1} + B_{n - 2}$ + $B_n = B_{n - 1} + B_{n - 3}$ + $B_n = B_{n - 2} + B_{n - 3}$ + $B_n = B_{n - 2} + B_{n - 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m12.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m12.xml new file mode 100644 index 00000000..08974e67 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m12.xml @@ -0,0 +1,28 @@ + + + + + + $11$ + + $= 3 + 4 + 4 = 4 + 3 + 4$ + $= 4 + 4 + 3.$ + + + + + + Which of the following is true for any $n \geq 5$? + ]]> + + $S_n = 2 \cdot S_{n - 1}$ + $S_n = S_{n - 1} + S_{n - 2}$ + $S_n = S_{n - 2} + S_{n - 3}$ + $S_n = S_{n - 3} + S_{n - 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m13.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m13.xml new file mode 100644 index 00000000..38e9221a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m13.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the following recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + When running $\Fib(12)$, how many calls are there to $\Fib(8)$? + ]]> +     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m14.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m14.xml new file mode 100644 index 00000000..d5ea4d54 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m14.xml @@ -0,0 +1,31 @@ + + + + $\newcommand{\ElisaDrinksCider}{{\rm E {\scriptsize LISA} D {\scriptsize RINKS} C {\scriptsize IDER}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the following recursive algorithm $\ElisaDrinksCider$, which takes as input an integer + $n \geq 1$, which is a power of 2: + +

    + $\mathbf{Algorithm}\ \ElisaDrinksCider(n)\mathrm{:}$
    + $\mathbf{if}\ n = 1$
    + $\mathbf{then}\ \text{order Fibonachos}$
    + $\mathbf{else}\ \ElisaDrinksCider(n / 2);$
    + $\elsesp \text{drink \(n\) bottles of cider};$
    + $\elsesp \ElisaDrinksCider(n / 2)$
    + $\mathbf{endif}$ +

    + + For $n$ a power of 2, let $C(n)$ be the total number of bottles of cider that you drink when running algorithm + $\ElisaDrinksCider(n)$. Which of the following is true for any $n \geq 1$ that is a power of 2?
    + (n.b., $\log$ denotes the base-2 logarithm) + ]]> +     + $C(n) = n \log n - 1$ + $C(n) = n \log n + 1$ + $C(n) = n \log n$ + $C(n) = 2n \log n$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m15.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m15.xml new file mode 100644 index 00000000..2fa3e89d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m15.xml @@ -0,0 +1,16 @@ + + + + + 1/4 + 1/3 + 1/2 + 1 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m16.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m16.xml new file mode 100644 index 00000000..b7ca9ccb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m16.xml @@ -0,0 +1,20 @@ + + + + + ${4 \choose 2} \cdot \frac{1}{365}$ + + ${4 \choose 2} \cdot \frac{1}{365} + {4 \choose 3} \cdot \frac{1}{365^2} + {4 \choose 4} \cdot \frac{1}{365^3}$ + + $1 - \frac{361 \cdot 362 \cdot 363 \cdot 364}{365^4}$ + $1 - \frac{362 \cdot 363 \cdot 364}{365^3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/f16m17.xml b/app/src/data/questions/comp2804/midterm-fall-2016/f16m17.xml new file mode 100644 index 00000000..94774750 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/f16m17.xml @@ -0,0 +1,18 @@ + + + Hearthstone, you are given a deck of 18 distinct cards. One of the cards is the Raven Idol. + You choose a uniformly random hand of 3 cards. Define the event + + [defs] + A = $\text{``}$the hand of 3 cards contains the Raven Idol$\,\text{''}.$ + [/defs] + + What is $\Pr(A)$? + ]]> + + 3/17 + 3/18 + 3/19 + 4/19 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2016/midterm-fall-2016.json b/app/src/data/questions/comp2804/midterm-fall-2016/midterm-fall-2016.json new file mode 100644 index 00000000..4a595c72 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2016/midterm-fall-2016.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-fall-2016", + "title": "Fall 2016 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-fall-2016/f16m01.xml", + "comp2804/midterm-fall-2016/f16m02.xml", + "comp2804/midterm-fall-2016/f16m03.xml", + "comp2804/midterm-fall-2016/f16m04.xml", + "comp2804/midterm-fall-2016/f16m05.xml", + "comp2804/midterm-fall-2016/f16m06.xml", + "comp2804/midterm-fall-2016/f16m07.xml", + "comp2804/midterm-fall-2016/f16m08.xml", + "comp2804/midterm-fall-2016/f16m09.xml", + "comp2804/midterm-fall-2016/f16m10.xml", + "comp2804/midterm-fall-2016/f16m11.xml", + "comp2804/midterm-fall-2016/f16m12.xml", + "comp2804/midterm-fall-2016/f16m13.xml", + "comp2804/midterm-fall-2016/f16m14.xml", + "comp2804/midterm-fall-2016/f16m15.xml", + "comp2804/midterm-fall-2016/f16m16.xml", + "comp2804/midterm-fall-2016/f16m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m01.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m01.xml new file mode 100644 index 00000000..d8074714 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m01.xml @@ -0,0 +1,12 @@ + + + + + ${n/2 \choose 4} \cdot {n/2 \choose 3}$ + ${n \choose 4} \cdot {n \choose 3}$ + ${n/2 \choose 4} + {n/2 \choose 3}$ + ${n \choose 4} + {n \choose 3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m02.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m02.xml new file mode 100644 index 00000000..9879809d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m02.xml @@ -0,0 +1,26 @@ + + + +
  • + $s$ students who are dressed up as Superman, +
    • +
  • + $t$ students who are dressed up as Donald Trump, +
    • +
  • + $k$ students who are dressed up as Kim Jong Un. +
    • + + These $s+t+k$ students are arranged on a line, such that all Supermen are standing next to each + other, all Trumps are standing next to each other, all Kims are standing next to each other, and no + Trump is standing next to any Kim. How many such arrangements are there? + ]]> +     + $\left. (s+t+k)! \middle/ (s! \cdot t! \cdot k!) \right.$ + $\left. (s+t+k)! \middle/ (2 \cdot s! \cdot t! \cdot k!) \right.$ + $s! \cdot t! \cdot k!$ + $2 \cdot s! \cdot t! \cdot k!$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m03.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m03.xml new file mode 100644 index 00000000..186d74a0 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m03.xml @@ -0,0 +1,16 @@ + + + + + $(7n)^{n}$ + $n^{7n}$ + $n^{n}$ + $7^{n}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m04.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m04.xml new file mode 100644 index 00000000..167a9c88 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m04.xml @@ -0,0 +1,14 @@ + + + + + ${n \choose 5} + {n \choose 3} \cdot {m \choose 2} + n \cdot {m \choose 4}$ + ${m \choose 5} + {m \choose 3} \cdot {n \choose 2} + m \cdot {n \choose 4}$ + ${m+n \choose 5} - {n \choose 5}$ + ${m \choose 5} \cdot {n \choose 5}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m05.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m05.xml new file mode 100644 index 00000000..269d43a7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m05.xml @@ -0,0 +1,12 @@ + + + + + $(n/2) \cdot 2^{n/2}$ + $2^{n/2} + 2^{n/2}$ + $2^{n} - 2^{n/2}$ + $2^{n} + 2^{n/2}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m06.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m06.xml new file mode 100644 index 00000000..91c7aee6 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m06.xml @@ -0,0 +1,12 @@ + + + + + $4^{n-3} + 4^{n-4}$ + $4^{n-3} + 4^{n-4} - 4^{n-7}$ + $2 \cdot 4^{n-3} - 4^{n-7}$ + $2 \cdot 4^{n-4} - 4^{n-7}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m07.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m07.xml new file mode 100644 index 00000000..e5396e04 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m07.xml @@ -0,0 +1,15 @@ + + + + + The number of strings of length $n$, where each character is $a$ or $b$, that contain at least one $a$. + The number of strings of length $n$, where each character is $a$ or $b$, that contain at least 2 many $a$'s. + The number of strings of length $n$, where each character is $a$, $b$, or $c$, that contain at least one $a$. + The number of strings of length $n$, where each character is $a$, $b$, or $c$, that contain at least 2 many $a$'s. + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m08.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m08.xml new file mode 100644 index 00000000..82306d7b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m08.xml @@ -0,0 +1,13 @@ + + + + + 4 + 5 + 6 + 7 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m09.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m09.xml new file mode 100644 index 00000000..24a4d334 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m09.xml @@ -0,0 +1,11 @@ + + + + + ${100 \choose 80} \cdot 7^{20} \cdot 13^{80}$ + $- {100 \choose 80} \cdot 7^{20} \cdot 13^{80}$ + ${100 \choose 20} \cdot 7^{80} \cdot 13^{20}$ + $- {100 \choose 20} \cdot 7^{80} \cdot 13^{20}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m10.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m10.xml new file mode 100644 index 00000000..9f2df3d2 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m10.xml @@ -0,0 +1,12 @@ + + + free if it does not contain 01. + Let $n \geq 2$ be an integer. How many 01-free bitstrings of length $n$ are there? + ]]> + + $n-1$ + $n$ + $n+1$ + $n+2$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m11.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m11.xml new file mode 100644 index 00000000..291e9da1 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m11.xml @@ -0,0 +1,16 @@ + + + free if it does not contain two 0's next to each other. + In class, we have seen that for any $m \geq 1$, the number of 00-free + bitstrings of length $m$ is equal to the $(m + 2)$-th Fibonacci number $f_{m+2}$.
    + + What is the number of 00-free bitstrings of length 20 that have 0 at position 7? + (The positions are numbered $1,2,\dots,20$.) + ]]> +     + $f_{7} \cdot f_{15}$ + $f_{8} \cdot f_{14}$ + $f_{8} \cdot f_{15}$ + $f_{7} \cdot f_{14}$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m12.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m12.xml new file mode 100644 index 00000000..ae58c3c0 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m12.xml @@ -0,0 +1,38 @@ + + + + + + $f(m, 0)$ + $= m\ \;$ + $\mathrm{if}\ m \geq 0,$ + + + $f(m, n)$ + $= 1 + f(m, n - 1)\ \;$ + $\mathrm{if}\ m \geq 0$ $\mathrm{and}\ n \geq 1,$ + + + $g(0)$ + $= 1,$ + + + $g(n)$ + $= 5 \cdot g(n - 1)\ \;$ + $\mathrm{if}\ n \geq 1.$ + + + + + For any integer $n \geq 0$, what is $f(g(n), g(n))$? + ]]> + + $2 \cdot 5^{n}$ + $2 \cdot 5^{n-1}$ + $(5^{n})^{2}$ + $(5^{n-1})^{2}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m13.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m13.xml new file mode 100644 index 00000000..14fb77bb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m13.xml @@ -0,0 +1,14 @@ + + + ccc-free + if it does not contain ccc.
    + + For any integer $n \geq 4$, let $B_n$ be the number of ccc-free strings of length $n$. Which of the following is true? + ]]> +     + $B_n = B_{n-1} + B_{n-2} + B_{n-3}$ + $B_n = 2 \cdot B_{n-1} + 2 \cdot B_{n-2} + 2 \cdot B_{n-3}$ + $B_n = 2 \cdot B_{n-1} + 2 \cdot B_{n-2} + B_{n-3}$ + $B_n = B_{n-1} + B_{n-2} + 2 \cdot B_{n-3}$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m14.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m14.xml new file mode 100644 index 00000000..c6cd52d4 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m14.xml @@ -0,0 +1,35 @@ + + + + + A brick is a horizontal or vertical board consisting of 2 square cells; see the bottom part of the figure + above. A tiling of the board $B_n$ is a placement of bricks on the board such that + + The figure below shows a tiling of $B_{13}$. + + + + Let $T_n$ be the number of different tilings of the board $B_n$. Which of the following is true for any $n \geq 3$? + ]]> + + $T_n = T_{n + 1} + T_{n + 2}$ + $T_n = T_{n - 1} + T_{n - 2}$ + $T_n = 2 \cdot T_{n - 1} + T_{n - 2}$ + $T_n = T_{n - 1} + 2 \cdot T_{n - 2}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m15.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m15.xml new file mode 100644 index 00000000..3968cc1a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m15.xml @@ -0,0 +1,28 @@ + + + + $\newcommand{\Hello}{{\rm H {\scriptsize ELLO}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the recursive algorithm $\Hello$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Hello(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \text{print ``hello''}$
    + $\mathbf{else}\ \Hello(n - 1);$
    + $\elsesp \text{print ``hello''};$
    + $\elsesp \Hello(n - 2)$
    + $\mathbf{endif}$ +

    + + If you run algorithm $\Hello(5)$, how many times is the word "hello" printed? + ]]> +     + 13 + 14 + 15 + 16 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m16.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m16.xml new file mode 100644 index 00000000..4b755133 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m16.xml @@ -0,0 +1,16 @@ + + + + + $\frac{{19 \choose 6} + {19 \choose 6}}{{20 \choose 7}}$ + $\frac{2 \cdot {20 \choose 6} - {20 \choose 5}}{{20 \choose 7}}$ + $\frac{2 \cdot {19 \choose 7} - {18 \choose 7}}{{20 \choose 7}}$ + $\frac{2 \cdot {19 \choose 6} - {18 \choose 5}}{{20 \choose 7}}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/f17m17.xml b/app/src/data/questions/comp2804/midterm-fall-2017/f17m17.xml new file mode 100644 index 00000000..5f653ad9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/f17m17.xml @@ -0,0 +1,22 @@ + + + + + Let $m \geq 7$ and $n \geq 7$ be integers. There are $m$ houses whose owners hand out candy and + there are $n$ houses whose owners do not hand out candy.
    + + The ladies choose a uniformly random subset of 7 houses and visit these 7 houses. Define the event + [defs] + A = "the ladies throw rotten eggs at exactly 2 of the 7 chosen houses". + [/defs] + What is $\Pr(A)$? + ]]> +     + $\frac{{m \choose 5} + {n \choose 2}}{{m+n \choose 7}}$ + $\frac{{m \choose 5} \cdot {n \choose 2}}{{m+n \choose 7}}$ + $\frac{{7 \choose 2}}{{m \choose 5} \cdot {n \choose 2}}$ + $\frac{{7 \choose 2}}{{m+n \choose 7}}$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2017/midterm-fall-2017.json b/app/src/data/questions/comp2804/midterm-fall-2017/midterm-fall-2017.json new file mode 100644 index 00000000..b94578dd --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2017/midterm-fall-2017.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-fall-2017", + "title": "Fall 2017 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-fall-2017/f17m01.xml", + "comp2804/midterm-fall-2017/f17m02.xml", + "comp2804/midterm-fall-2017/f17m03.xml", + "comp2804/midterm-fall-2017/f17m04.xml", + "comp2804/midterm-fall-2017/f17m05.xml", + "comp2804/midterm-fall-2017/f17m06.xml", + "comp2804/midterm-fall-2017/f17m07.xml", + "comp2804/midterm-fall-2017/f17m08.xml", + "comp2804/midterm-fall-2017/f17m09.xml", + "comp2804/midterm-fall-2017/f17m10.xml", + "comp2804/midterm-fall-2017/f17m11.xml", + "comp2804/midterm-fall-2017/f17m12.xml", + "comp2804/midterm-fall-2017/f17m13.xml", + "comp2804/midterm-fall-2017/f17m14.xml", + "comp2804/midterm-fall-2017/f17m15.xml", + "comp2804/midterm-fall-2017/f17m16.xml", + "comp2804/midterm-fall-2017/f17m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m01.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m01.xml new file mode 100644 index 00000000..677cfce9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m01.xml @@ -0,0 +1,13 @@ + + + + + $2^n$ + $3^n$ + $4^n$ + $5^n$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m02.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m02.xml new file mode 100644 index 00000000..48d78487 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m02.xml @@ -0,0 +1,20 @@ + + + +
  • + the bit at each even position is 0, or +
    • +
  • + the bitstring starts with 1010? +
    • + + ]]> +     + 54 + 56 + 58 + 60 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m03.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m03.xml new file mode 100644 index 00000000..eb895664 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m03.xml @@ -0,0 +1,12 @@ + + + + + $52^{15} - 26^{15} - 15 \cdot 26^{14}$ + $52^{15} - 26^{15} - 15 \cdot 26^{15}$ + $52^{15} - 15 \cdot 26^{15}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m04.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m04.xml new file mode 100644 index 00000000..5392dc35 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m04.xml @@ -0,0 +1,16 @@ + + + + In how many ways can Elisa place these orders of cider, such that exactly 4 of her friends get a + cider of type $C_3$? + ]]> + + ${7 \choose 4} \cdot 7^3$ + ${7 \choose 4} \cdot 8^3$ + ${7 \choose 4} \cdot 7^4$ + ${7 \choose 4} \cdot 8^4$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m05.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m05.xml new file mode 100644 index 00000000..5b1f19de --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m05.xml @@ -0,0 +1,16 @@ + + + + + ${22 \choose 5}$ + ${22 \choose 4}$ + ${21 \choose 5}$ + ${21 \choose 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m06.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m06.xml new file mode 100644 index 00000000..84952642 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m06.xml @@ -0,0 +1,12 @@ + + + + What is the minimum size of this subset $S$, such that there must be at least two elements in $S$ whose sum is equal to 51? + ]]> + + 25 + 26 + 27 + 28 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m07.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m07.xml new file mode 100644 index 00000000..373819b9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m07.xml @@ -0,0 +1,13 @@ + + + + How many such subsets have the property that $x_3 = 7$? + ]]> + + ${6 \choose 2} \cdot {10 \choose 2}$ + ${6 \choose 2} \cdot {9 \choose 2}$ + ${7 \choose 2} \cdot {10 \choose 2}$ + ${7 \choose 2} \cdot {9 \choose 2}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m08.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m08.xml new file mode 100644 index 00000000..c234af29 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m08.xml @@ -0,0 +1,17 @@ + + + + + Consider subsets $X$ of $\mathcal{B} \cup \mathcal{C}$, such that $X$ consists of exactly 5 beer + bottles and all cider bottles in $X$ have an even index.
    + + How many such subsets $X$ are there? + ]]> +     + ${12 \choose 5} \cdot 2^6$ + ${13 \choose 5} \cdot 2^5$ + ${13 \choose 5} \cdot 2^6$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m09.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m09.xml new file mode 100644 index 00000000..4c349692 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m09.xml @@ -0,0 +1,16 @@ + + + palindrome, if $$s_1s_2...s_{n-1}s_n = s_ns_{n-1}...s_2s_1,$$ + i.e., reading the string from left to right gives the same string as when reading from right to left.
    + + For any integer $n \geq 1$, let $P_n$ be the number of bitstrings of length $n$ that are palindromes.
    + + Which of the following is true for any integer $n \geq 3$? + ]]> +     + $P_n = 2 + P_{n-2}$ + $P_n = 2 \cdot P_{n/2}$ + $P_n = 2 \cdot P_{n-1}$ + $P_n = 2 \cdot P_{n-2}$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m10.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m10.xml new file mode 100644 index 00000000..f17e8d26 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m10.xml @@ -0,0 +1,19 @@ + + + + + For any integer $k$ with $0 \leq k \leq n$, consider subsets $X$ of $\mathcal{B} \cup \mathcal{C}$, + such that $X$ consists of exactly $k$ bottles and no two bottles in $X$ have the same index. (For + example, if $B_n \in X$, then $C_n \notin X$.)
    + + Let $F(n,k)$ be the number of such subsets X.
    + Which of the following is true for all integers $n \geq 2$ and $k$ with $1 \leq k \leq n - 1$? + ]]> +     + $F(n,k) = F(n-1,k)\ +$ $ F(n-1,k-1)$ + $F(n,k) = F(n-1,k)\ +$ $ 2 \cdot F(n-1,k-1)$ + $F(n,k) = F(n,k-1)\ +$ $ F(n-1,k-1)$ + $F(n,k) = F(n,k-1)\ +$ $ 2 \cdot F(n-1,k-1)$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m11.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m11.xml new file mode 100644 index 00000000..8872d37f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m11.xml @@ -0,0 +1,16 @@ + + + free, if it does not contain two 0's next to each other. + In class, we have seen that for any $m \geq 1$, + the number of 00-free bitstrings of length $m$ is equal to the $(m+2)$-th Fibonacci number $f_{m+2}$.
    + + What is the number of 00-free bitstrings of length 55 that have 0 at position 9, and 1 at position 40? + (The positions are numbered $1,2,\dots,55$.) + ]]> +     + $f_7 \cdot f_{29} \cdot f_{15}$ + $f_8 \cdot f_{30} \cdot f_{16}$ + $f_9 \cdot f_{31} \cdot f_{17}$ + $f_{10} \cdot f_{32} \cdot f_{18}$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m12.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m12.xml new file mode 100644 index 00000000..422e38e9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m12.xml @@ -0,0 +1,21 @@ + + + + + $3 + 5 \cdot 2^{n}$ + $5 + 3 \cdot 2^{n}$ + $2^{3 + 5n}$ + $2^{5 + 3n}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m13.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m13.xml new file mode 100644 index 00000000..82ff6b78 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m13.xml @@ -0,0 +1,15 @@ + + + -free, if it does not contain two $q$'s next to each other. + For any integer $n \geq 1$, let $Q_n$ be the number of $qq$-free strings of length $n$.
    + + Which of the following is true for any integer $n \geq 3$? + ]]> +     + $Q(n) = 26 \cdot Q(n-1) + 26 \cdot Q(n-2)$ + $Q(n) = 25 \cdot Q(n-1) + 25 \cdot Q(n-2)$ + $Q(n) = 26 \cdot Q(n-1) + 25 \cdot Q(n-2)$ + $Q(n) = 25 \cdot Q(n-1) + 26 \cdot Q(n-2)$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m14.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m14.xml new file mode 100644 index 00000000..aaf25f2b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m14.xml @@ -0,0 +1,33 @@ + + + + $\newcommand{\Fart}{{\rm F {\scriptsize ART}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }} + \newcommand{\thensp}{\phantom{\mathbf{then}\ }}$ + + + Consider the recursive algorithm $\Fart$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fart(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \text{eat one can of beans}$
    + $\mathbf{else}\ \mathbf{if}\ n\ \mathrm{is}\ \mathrm{even}$
    + $\elsesp \mathbf{then}\ \text{fart once};$
    + $\elsesp \thensp \Fart \left( n \middle/ 2 \right)$
    + $\elsesp \mathbf{else}\ \Fart(n + 1);$
    + $\elsesp \elsesp \text{fart once};$
    + $\elsesp \elsesp \Fart(n - 1)$
    + $\elsesp \mathbf{endif};$
    + $\mathbf{endif}$ +

    + + If you run algorithm $\Fart(9)$, how many times do you fart? + ]]> +     + 11 + 12 + 13 + 14 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m15.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m15.xml new file mode 100644 index 00000000..e47b28ab --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m15.xml @@ -0,0 +1,29 @@ + + + +
  • + one student is dressed up as Donald Trump, +
    • +
  • + one student is dressed up as Kim Jong Un, +
    • +
  • + the remaining 57 students are dressed up as Kim Kardashian. +
    • + + These students are arranged, uniformly at random, on a line. +
    + Define the event, + [defs] + A = "Donald Trump is standing next to Kim Jong Un". + [/defs] + What is $\Pr(A)$? + ]]> +     + 1/58 + 2/58 + 1/59 + 2/59 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m16.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m16.xml new file mode 100644 index 00000000..33bbf22b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m16.xml @@ -0,0 +1,18 @@ + + + + + Define the event + [defs] + A = "Alexa, Tri, and Zoltan have different birthdays". + [/defs] + What is $\text{Pr}(A)$? + ]]> + + $\frac{365^2}{363 \cdot 364}$ + $\frac{363^3}{362 \cdot 363 \cdot 364}$ + $\frac{363 \cdot 364}{365^2}$ + $\frac{362 \cdot 363 \cdot 364}{365^3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/f18m17.xml b/app/src/data/questions/comp2804/midterm-fall-2018/f18m17.xml new file mode 100644 index 00000000..444d39eb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/f18m17.xml @@ -0,0 +1,18 @@ + + + + + Define the event + [defs] + A = "you answer exactly 7 questions correctly". + [/defs] + What $\Pr(A)$? + ]]> + + $\frac{{17 \choose 7} \cdot 3^{10}}{4^{17}}$ + $\frac{{17 \choose 7} \cdot 2^{10}}{4^{17}}$ + $\frac{{17 \choose 7}}{4^{17}}$ + $\frac{4^{17}}{{17 \choose 7}}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2018/midterm-fall-2018.json b/app/src/data/questions/comp2804/midterm-fall-2018/midterm-fall-2018.json new file mode 100644 index 00000000..b78c5e38 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2018/midterm-fall-2018.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-fall-2018", + "title": "Fall 2018 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-fall-2018/f18m01.xml", + "comp2804/midterm-fall-2018/f18m02.xml", + "comp2804/midterm-fall-2018/f18m03.xml", + "comp2804/midterm-fall-2018/f18m04.xml", + "comp2804/midterm-fall-2018/f18m05.xml", + "comp2804/midterm-fall-2018/f18m06.xml", + "comp2804/midterm-fall-2018/f18m07.xml", + "comp2804/midterm-fall-2018/f18m08.xml", + "comp2804/midterm-fall-2018/f18m09.xml", + "comp2804/midterm-fall-2018/f18m10.xml", + "comp2804/midterm-fall-2018/f18m11.xml", + "comp2804/midterm-fall-2018/f18m12.xml", + "comp2804/midterm-fall-2018/f18m13.xml", + "comp2804/midterm-fall-2018/f18m14.xml", + "comp2804/midterm-fall-2018/f18m15.xml", + "comp2804/midterm-fall-2018/f18m16.xml", + "comp2804/midterm-fall-2018/f18m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m01.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m01.xml new file mode 100644 index 00000000..7031cbd4 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m01.xml @@ -0,0 +1,13 @@ + + + ${f + m \choose 3}$ + $f \cdot {f + m - 1 \choose 3}$ + $f \cdot {f + m \choose 3}$ + $(f - 1) \cdot {f + m \choose 3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m02.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m02.xml new file mode 100644 index 00000000..f9addcde --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m02.xml @@ -0,0 +1,10 @@ + + and do not contain 2? + ]]> + ${n - 1 \choose k - 1}$ + ${n - 1 \choose k}$ + ${n - 2 \choose k - 2}$ + ${n - 2 \choose k}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m03.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m03.xml new file mode 100644 index 00000000..873a746d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m03.xml @@ -0,0 +1,10 @@ + + or do not contain 2? + ]]> + ${n - 1 \choose k} + {n - 1 \choose k}$ + ${n - 2 \choose k}$ + ${n \choose k} - {n - 2 \choose k - 2}$ + ${n \choose k} - {n - 1 \choose k - 1} - {n - 1 \choose k - 1}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m04.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m04.xml new file mode 100644 index 00000000..ecb78afb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m04.xml @@ -0,0 +1,10 @@ + + + $B_n = 2 \cdot 2^{n - 3}$ + $B_n = 6 \cdot 2^{n - 3}$ + $B_n = 2^n - 2$ + $B_n = 2^n - 6$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m05.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m05.xml new file mode 100644 index 00000000..18377f1a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m05.xml @@ -0,0 +1,10 @@ + + + 17 + 18 + 19 + 20 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m06.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m06.xml new file mode 100644 index 00000000..801080e7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m06.xml @@ -0,0 +1,10 @@ + + + $4^{n - 2} + 4^{n - 3} - 4^{n - 5}$ + $4^{n - 2} + 4^{n - 3}$ + $4^n - 4^{n - 5}$ + $4^n - 4^{n - 2} - 4^{n - 3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m07.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m07.xml new file mode 100644 index 00000000..18a72f34 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m07.xml @@ -0,0 +1,25 @@ + + + What does + $$ + {w \choose 3} \cdot {m \choose 2} + {w \choose 4} \cdot m + {w \choose 5} + $$ + count? + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at most 3 women can be chosen. + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at most 3 men can be chosen. + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at least 3 women must be chosen. + + + The number of ways to choose 5 people out of a group consisting of $w$ women and $m$ men, + where at least 3 men must be chosen. + + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m08.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m08.xml new file mode 100644 index 00000000..e8634d05 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m08.xml @@ -0,0 +1,11 @@ + + + $n - 1$ + $n$ + $n + 1$ + $n + 2$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m09.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m09.xml new file mode 100644 index 00000000..855e005b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m09.xml @@ -0,0 +1,9 @@ + + + $- {50 \choose 24} \cdot 5^{26} \cdot 7^{24}$ + $- {50 \choose 26} \cdot 5^{24} \cdot 7^{26}$ + ${50 \choose 24} \cdot 5^{26} \cdot 7^{24}$ + ${50 \choose 26} \cdot 5^{24} \cdot 7^{26}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m10.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m10.xml new file mode 100644 index 00000000..3d09cc24 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m10.xml @@ -0,0 +1,16 @@ + + + $f(n) = 7 \cdot \frac{n!}{3^n}$ + $f(n) = 7^n \cdot \frac{n!}{3^n}$ + $f(n) = 7 \cdot \frac{(n + 1)!}{3^n}$ + $f(n) = 7^n \cdot \frac{(n + 1)!}{3^n}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m11.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m11.xml new file mode 100644 index 00000000..e3ea3e87 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m11.xml @@ -0,0 +1,11 @@ + + + $B_n = B_{n - 1} + B_{n - 2}$ + $B_n = B_{n - 1} + B_{n - 3}$ + $B_n = B_{n - 2} + B_{n - 3}$ + $B_n = B_{n - 2} + B_{n - 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m12.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m12.xml new file mode 100644 index 00000000..d8ce881a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m12.xml @@ -0,0 +1,26 @@ + + + + + $11$ + + $= 3 + 4 + 4 = 4 + 3 + 4$ + $= 4 + 4 + 3.$ + + + + + + Which of the following is true for any $n \geq 5$? + ]]> + $S_n = 2 \cdot S_{n - 1}$ + $S_n = S_{n - 1} + S_{n - 2}$ + $S_n = S_{n - 2} + S_{n - 3}$ + $S_n = S_{n - 3} + S_{n - 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m13.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m13.xml new file mode 100644 index 00000000..cb6e3504 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m13.xml @@ -0,0 +1,23 @@ + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + Consider the following recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\Fib(n):$
    + $\quad \mathbf{if}\ n = 0\ \mathrm{or}\ n = 1\ \mathbf{then}$
    + $\quad \quad f = n$
    + $\quad \mathbf{else}$
    + $\quad \quad f = \Fib(n - 1) + \Fib(n - 2)$
    + $\quad \mathbf{return}\ f$ +

    + + When running $\Fib(12)$, how many calls are there to $\Fib(8)$? + ]]>     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m14.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m14.xml new file mode 100644 index 00000000..8413f79b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m14.xml @@ -0,0 +1,28 @@ + + + $\newcommand{\ElisaDrinksCider}{{\rm E {\scriptsize LISA} D {\scriptsize RINKS} C {\scriptsize IDER}}}$ + + + Consider the following recursive algorithm $\ElisaDrinksCider$, which takes as input an integer + $n \geq 1$, which is a power of 2: + +

    + $\ElisaDrinksCider(n):$
    + $\quad \mathbf{if}\ n = 1\ \mathbf{then}$
    + $\quad \quad \text{then order Fibonachos}$
    + $\quad \mathbf{else}$
    + $\quad \quad \ElisaDrinksCider\left(n \middle/ 2 \right)$
    + $\quad \quad \mathrm{drink}\ n\ \text{bottles of cider}$
    + $\quad \quad \ElisaDrinksCider\left(n \middle/ 2 \right)$ +

    + + For $n$ a power of 2, let $C(n)$ be the total number of bottles of cider that you drink when running algorithm + $\ElisaDrinksCider(n)$. Which of the following is true for any $n \geq 1$ that is a power of 2?
    + (n.b., $\log$ denotes the base-2 logarithm) + ]]>     + $C(n) = n \log n - 1$ + $C(n) = n \log n + 1$ + $C(n) = n \log n$ + $C(n) = 2n \log n$ +     diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m15.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m15.xml new file mode 100644 index 00000000..4734ec9d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m15.xml @@ -0,0 +1,14 @@ + + + 1/4 + 1/3 + 1/2 + 1 + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m16.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m16.xml new file mode 100644 index 00000000..eb177623 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m16.xml @@ -0,0 +1,18 @@ + + + ${4 \choose 2} \cdot \frac{1}{365}$ + + ${4 \choose 2} \cdot \frac{1}{365} + {4 \choose 3} \cdot \frac{1}{365^2} + {4 \choose 4} \cdot \frac{1}{365^3}$ + + $1 - \frac{361 \cdot 362 \cdot 363 \cdot 364}{365^4}$ + $1 - \frac{362 \cdot 363 \cdot 364}{365^3}$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/f19m17.xml b/app/src/data/questions/comp2804/midterm-fall-2019/f19m17.xml new file mode 100644 index 00000000..06c198a7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/f19m17.xml @@ -0,0 +1,16 @@ + + Raven Idol. + You choose a uniformly random hand of 3 cards. Define the event + + [defs] + A = "the hand of 3 cards contains the Raven Idol". + [/defs] + + What is $\Pr(A)$? + ]]> + 3/17 + 3/18 + 4/19 + $1 - (17/18)^3$ + diff --git a/app/src/data/questions/comp2804/midterm-fall-2019/midterm-fall-2019.json b/app/src/data/questions/comp2804/midterm-fall-2019/midterm-fall-2019.json new file mode 100644 index 00000000..d794868b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-fall-2019/midterm-fall-2019.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-fall-2019", + "title": "Fall 2019 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-fall-2019/f19m01.xml", + "comp2804/midterm-fall-2019/f19m02.xml", + "comp2804/midterm-fall-2019/f19m03.xml", + "comp2804/midterm-fall-2019/f19m04.xml", + "comp2804/midterm-fall-2019/f19m05.xml", + "comp2804/midterm-fall-2019/f19m06.xml", + "comp2804/midterm-fall-2019/f19m07.xml", + "comp2804/midterm-fall-2019/f19m08.xml", + "comp2804/midterm-fall-2019/f19m09.xml", + "comp2804/midterm-fall-2019/f19m10.xml", + "comp2804/midterm-fall-2019/f19m11.xml", + "comp2804/midterm-fall-2019/f19m12.xml", + "comp2804/midterm-fall-2019/f19m13.xml", + "comp2804/midterm-fall-2019/f19m14.xml", + "comp2804/midterm-fall-2019/f19m15.xml", + "comp2804/midterm-fall-2019/f19m16.xml", + "comp2804/midterm-fall-2019/f19m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/midterm-winter-2014.json b/app/src/data/questions/comp2804/midterm-winter-2014/midterm-winter-2014.json new file mode 100644 index 00000000..e49eddee --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/midterm-winter-2014.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-winter-2014", + "title": "Winter 2014 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-winter-2014/w14m01.xml", + "comp2804/midterm-winter-2014/w14m02.xml", + "comp2804/midterm-winter-2014/w14m03.xml", + "comp2804/midterm-winter-2014/w14m04.xml", + "comp2804/midterm-winter-2014/w14m05.xml", + "comp2804/midterm-winter-2014/w14m06.xml", + "comp2804/midterm-winter-2014/w14m07.xml", + "comp2804/midterm-winter-2014/w14m08.xml", + "comp2804/midterm-winter-2014/w14m09.xml", + "comp2804/midterm-winter-2014/w14m10.xml", + "comp2804/midterm-winter-2014/w14m11.xml", + "comp2804/midterm-winter-2014/w14m12.xml", + "comp2804/midterm-winter-2014/w14m13.xml", + "comp2804/midterm-winter-2014/w14m14.xml", + "comp2804/midterm-winter-2014/w14m15.xml", + "comp2804/midterm-winter-2014/w14m16.xml", + "comp2804/midterm-winter-2014/w14m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m01.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m01.xml new file mode 100644 index 00000000..0b766122 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m01.xml @@ -0,0 +1,15 @@ + + + + + ${m + n + k \choose 7} - (m + n + k)$ + ${m + n + k \choose 7} - {m \choose 2} - {n \choose 2} - {k \choose 2}$ + + ${m \choose 3}{n \choose 2}{k \choose 2} + {m \choose 2}{n \choose 3}{k \choose 2} + {m \choose 2}{n \choose 2}{k \choose 3}$ + + ${m \choose 2}{n \choose 2}{k \choose 2}(m + n + k)$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m02.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m02.xml new file mode 100644 index 00000000..5c787ee9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m02.xml @@ -0,0 +1,12 @@ + + + + + $\frac{24!}{9!15!}$ + ${24 \choose 9}(9!)(15!)$ + $(9!)(15!)$ + $2(9!)(15!)$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m03.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m03.xml new file mode 100644 index 00000000..55f1294c --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m03.xml @@ -0,0 +1,12 @@ + + + and $y$, but do not contain $z$? + ]]> + + $2^{33}$ + $2^{34}$ + $2^{35}$ + $2^{37} - 2^{35} - 2^{36}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m04.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m04.xml new file mode 100644 index 00000000..e6baff36 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m04.xml @@ -0,0 +1,12 @@ + + + or $y$, but do not contain $z$? + ]]> + + $2^{36} - 2^{34}$ + $2^{36} - 2^{35}$ + $2^{37} - 2^{34}$ + $2^{37} - 2^{35}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m05.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m05.xml new file mode 100644 index 00000000..923a9dec --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m05.xml @@ -0,0 +1,12 @@ + + + + + $10^{12} + 10^{13} - 9^{12} - 9^{13}$ + $12^{10} + 13^{10} - 12^{9} - 13^{9}$ + $10^{12} + 10^{13} - 7^{12} - 7^{13}$ + $12^{10} + 13^{10} - 12^{7} - 13^{7}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m06.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m06.xml new file mode 100644 index 00000000..1c5d4362 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m06.xml @@ -0,0 +1,14 @@ + + + + + $2^k - 1 < {n \choose 7}$ + $2^k - 1 \geq {n \choose 7}$ + $2^k - 1 < \left. 2^n \middle/ {n \choose n-7} \right.$ + $2^k - 1 \geq \left. 2^n \middle/ {n \choose n-7} \right.$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m07.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m07.xml new file mode 100644 index 00000000..fb8824fb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m07.xml @@ -0,0 +1,11 @@ + + + + + ${25 \choose 16} 7^{16}21^{9}$ + ${16 \choose 25} 7^{9}21^{16}$ + ${25 \choose 16} 7^{25}3^{16}$ + none of the above + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m08.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m08.xml new file mode 100644 index 00000000..7dd154c5 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m08.xml @@ -0,0 +1,12 @@ + + + + + ${19 \choose 16}$ + ${19 \choose 17}$ + ${20 \choose 16}$ + ${20 \choose 17}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m09.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m09.xml new file mode 100644 index 00000000..cab38630 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m09.xml @@ -0,0 +1,14 @@ + + + + POOPERSCOOPER +

    + ]]> +     + $13!$ + ${13 \choose 4}{9 \choose 2}{7 \choose 2}{5 \choose 3}$ + ${13 \choose 4}{9 \choose 3}{6 \choose 2}{4 \choose 2}$ + ${13 \choose 1}{12 \choose 4}{8 \choose 2}{6 \choose 1}{5 \choose 2}{3 \choose 3}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m10.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m10.xml new file mode 100644 index 00000000..664f1085 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m10.xml @@ -0,0 +1,18 @@ + + + + + $f(n) = 3n^{2} - 5n + 2$ + $f(n) = 3n^{2} + 5n + 2$ + $f(n) = 2n^{2} - 5n + 2$ + $f(n) = 2n^{2} + 5n + 2$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m11.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m11.xml new file mode 100644 index 00000000..d0d16ba0 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m11.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the following recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + When running $\Fib(7)$, how many calls are there to $\Fib(3)$? + ]]> +     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m12.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m12.xml new file mode 100644 index 00000000..e9e765d7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m12.xml @@ -0,0 +1,26 @@ + + + + + Consider again the recursive algorithm $\Fib$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + For $n \geq 3$, run algorithm $\Fib(n)$ and let $a_n$ be the number of times that $\Fib(2)$ is called.
    + Which of the following is true? + ]]> +     + For $n \geq 3$, $a_n = f_{n - 1}$ + For $n \geq 3$, $a_n = f_n$ + For $n \geq 3$, $a_n = f_{n + 1}$ + For $n \geq 3$, $a_n = -1 + f_n$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m13.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m13.xml new file mode 100644 index 00000000..c7825785 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m13.xml @@ -0,0 +1,12 @@ + + + + + $B_n = B_{n-1} + B_{n-2} + 2^{n-3}$ + $B_n = B_{n-1} + B_{n-2} + 2^{n-3} - B_{n-3}$ + $B_n = B_{n-1} + B_{n-2} + B_{n-3}$ + $B_n = B_{n-1} + B_{n-2} + B_{n-3} + 2^{n-4}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m14.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m14.xml new file mode 100644 index 00000000..9d2c95c2 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m14.xml @@ -0,0 +1,12 @@ + + + + + $1 - \left. {48 \choose 7} \middle/ {52 \choose 9} \right.$ + $\left. \bigl\{ {4 \choose 2} + {48 \choose 7} \bigr\} \middle/ {52 \choose 9} \right.$ + $\left. {52 \choose 9} \middle/ \bigl\{ {4 \choose 2}{48 \choose 7} \bigr\} \right.$ + $\left. {4 \choose 2}{48 \choose 7} \middle/ {52 \choose 9} \right.$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m15.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m15.xml new file mode 100644 index 00000000..5f617a87 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m15.xml @@ -0,0 +1,12 @@ + + + + + $1 - (1/2)^{25} - 25(1/2)^{25}$ + $1 + (1/2)^{25} - 25(1/2)^{25}$ + $\sum_{k=2}^{25} {{25}\choose{k}}(1/2)^{k}$ + none of the above + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m16.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m16.xml new file mode 100644 index 00000000..bcded6f7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m16.xml @@ -0,0 +1,12 @@ + + + + + $1 - \frac{365^2}{364 \cdot 363}$ + $1 - \frac{364 \cdot 363}{365^2}$ + $1 - \left. {3 \choose 2} \middle/ 365^3 \right.$ + $1 - \left. \bigl\{ {3\choose 2} + {3 \choose 3} \bigr\} \middle/ 365^3 \right.$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2014/w14m17.xml b/app/src/data/questions/comp2804/midterm-winter-2014/w14m17.xml new file mode 100644 index 00000000..1dffa4b4 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2014/w14m17.xml @@ -0,0 +1,11 @@ + + + + + Herbal tea + India Pale Ale + Poutine + None of the above, because Simon doesn't like beer + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/midterm-winter-2015.json b/app/src/data/questions/comp2804/midterm-winter-2015/midterm-winter-2015.json new file mode 100644 index 00000000..5ff68ce3 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/midterm-winter-2015.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-winter-2015", + "title": "Winter 2015 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-winter-2015/w15m01.xml", + "comp2804/midterm-winter-2015/w15m02.xml", + "comp2804/midterm-winter-2015/w15m03.xml", + "comp2804/midterm-winter-2015/w15m04.xml", + "comp2804/midterm-winter-2015/w15m05.xml", + "comp2804/midterm-winter-2015/w15m06.xml", + "comp2804/midterm-winter-2015/w15m07.xml", + "comp2804/midterm-winter-2015/w15m08.xml", + "comp2804/midterm-winter-2015/w15m09.xml", + "comp2804/midterm-winter-2015/w15m10.xml", + "comp2804/midterm-winter-2015/w15m11.xml", + "comp2804/midterm-winter-2015/w15m12.xml", + "comp2804/midterm-winter-2015/w15m13.xml", + "comp2804/midterm-winter-2015/w15m14.xml", + "comp2804/midterm-winter-2015/w15m15.xml", + "comp2804/midterm-winter-2015/w15m16.xml", + "comp2804/midterm-winter-2015/w15m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m01.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m01.xml new file mode 100644 index 00000000..d58688c4 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m01.xml @@ -0,0 +1,14 @@ + + + + + $(n - 7) + {n \choose 7}$ + $n + {n \choose 7}$ + $(n - 7) \cdot {n \choose 7}$ + $n \cdot {n \choose 7}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m02.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m02.xml new file mode 100644 index 00000000..2852335f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m02.xml @@ -0,0 +1,12 @@ + + + + + $7^{15}$ + $15^7$ + $15!/7!$ + ${15 \choose 7}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m03.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m03.xml new file mode 100644 index 00000000..f2d46aa5 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m03.xml @@ -0,0 +1,12 @@ + + + + + $8 \cdot 9 \cdot 10 \cdot 11 \cdot 12 \cdot 13 \cdot 14 \cdot 15$ + $9 \cdot 10 \cdot 11 \cdot 12 \cdot 13 \cdot 14 \cdot 15$ + $10 \cdot 11 \cdot 12 \cdot 13 \cdot 14 \cdot 15$ + $9 \cdot 10 \cdot 11 \cdot 12 \cdot 13 \cdot 14$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m04.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m04.xml new file mode 100644 index 00000000..1ccc17f6 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m04.xml @@ -0,0 +1,16 @@ + + + valid, + if no two consecutive characters are equal. Thus, $abacbac$ is valid, whereas $abaccac$ is not + valid.
    + + Let $n \geq 1$ be an integer and let $V_n$ be the number of valid strings of length $n$. Which of + the following is true? + ]]> +     + $V_n = 3 \cdot 2^{n-1}$ + $V_n = 3^n - (n - 1) \cdot 3$ + $V_n = 3^n - (n - 1) \cdot 3 \cdot 3^{n-2}$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m05.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m05.xml new file mode 100644 index 00000000..b8b653d1 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m05.xml @@ -0,0 +1,12 @@ + + + + + $3^{96} + 3^{96}$ + $3^{99} - 2 \cdot 3^{96}$ + $2 \cdot 3^{96} - 3^{93}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m06.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m06.xml new file mode 100644 index 00000000..89d25dea --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m06.xml @@ -0,0 +1,19 @@ + + + + $\newcommand{\Silly}{{\rm S \scriptsize ILLY}}$ + + + What does + $$ + \sum_{k=2}^{n-1} (k-1)(n-k) + $$ + count? + ]]> + + The number of 3-element subsets of an $(n-1)$-element set. + The number of 3-element subsets of an $n$-element set. + The number of 3-element subsets of an $(n+1)$-element set. + The number of times you fart when running algorithm $\Silly(n)$. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m07.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m07.xml new file mode 100644 index 00000000..7e66d076 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m07.xml @@ -0,0 +1,12 @@ + + + + + $365^2$ + $365^2 + 1$ + $2 \cdot 365$ + $2 \cdot 365 + 1$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m08.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m08.xml new file mode 100644 index 00000000..61e5b18f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m08.xml @@ -0,0 +1,11 @@ + + + + + $2^{16} \cdot 17^{82} \cdot {99 \choose 16}$ + $2^{17} \cdot 17^{82} \cdot {99 \choose 17}$ + $2^{82} \cdot 17^{17} \cdot {99 \choose 17}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m09.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m09.xml new file mode 100644 index 00000000..88dbcd57 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m09.xml @@ -0,0 +1,12 @@ + + + + + ${101 \choose 2}$ + ${101 \choose 3}$ + ${102 \choose 2}$ + ${102 \choose 3}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m10.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m10.xml new file mode 100644 index 00000000..ffaec5bf --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m10.xml @@ -0,0 +1,18 @@ + + + + + $f(n) = 3n^{2} + 2n + 15$ + $f(n) = 3n^{2} - 2n + 15$ + $f(n) = 3n^{2} + n + 15$ + $f(n) = 3n^{2} - n + 15$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m11.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m11.xml new file mode 100644 index 00000000..42479d5f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m11.xml @@ -0,0 +1,13 @@ + + + valid, + if it does not contain the substring $aaa$. Let $S_n$ be the number of valid strings of length + $n$. Which of the following is true? + ]]> + + $S_n = 2 \cdot S_{n-1} + 2 \cdot S_{n-2}$ + $S_n = 2 \cdot S_{n-1} + 2 \cdot S_{n-2} + S_{n-3}$ + $S_n = 2 \cdot S_{n-1} + S_{n-2} + 2 \cdot S_{n-3}$ + $S_n = 2 \cdot S_{n-1} + 2 \cdot S_{n-2} + 2 \cdot S_{n-3}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m12.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m12.xml new file mode 100644 index 00000000..026bac49 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m12.xml @@ -0,0 +1,30 @@ + + + +
  • + the order in which the 1s and 2s occur in the sum matters, and +
    • +
  • + it is not allowed to have two consecutive 1s. +
    • + + For example, if $n = 7$, then + $$ + 7 = 1 + 2 + 1 + 2 + 1 + $$ is allowed, whereas + $$ + 7 = 1 + 2 + 1 + 1 + 2 + $$ + is not allowed.
    + + Which of the following is true? + ]]> +     + $S_n = S_{n-1} + S_{n-2}$ + $S_n = S_{n-1} + S_{n-3}$ + $S_n = S_{n-2} + S_{n-3}$ + $S_n = S_{n-1} + S_{n-2} + S_{n-3}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m13.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m13.xml new file mode 100644 index 00000000..bb0c3c29 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m13.xml @@ -0,0 +1,27 @@ + + + + $\newcommand{\Fib}{{\rm F \scriptsize IB}}$ + + + Consider the following recursive algorithm $\Fib$, which takes as input an + integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ f = n$
    + $\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$
    + $\mathbf{endif};$
    + $\mathbf{return}\ f$
    +

    + + When running $\Fib(99)$, how many calls are there to $\Fib(95)$? + ]]> +     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m14.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m14.xml new file mode 100644 index 00000000..2719e04e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m14.xml @@ -0,0 +1,31 @@ + + + + $\newcommand{\NationalAnthem}{{\rm N {\scriptsize ATIONAL} A {\scriptsize NTHEM}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the following recursive algorithm $\NationalAnthem$, which takes as input an integer $n \geq 1$, + which is a power of 2: + +

    + $\mathbf{Algorithm}\ \NationalAnthem(n)\mathrm{:}$
    + $\mathbf{if}\ n = 1$
    + $\mathbf{then}\ \mathrm{sing}\ {\it O\ Canada}\ \mathrm{once}$
    + $\mathbf{else}\ \NationalAnthem(n / 2);$
    + $\elsesp \mathrm{sing}\ {\it O\ Canada}\ \mathrm{once};$
    + $\elsesp \NationalAnthem(n / 2)$
    + $\mathbf{endif}$ +

    + + For $n$ a power of 2, let $S(n)$ be the number of times you sing O Canada when running algorithm + $\NationalAnthem(n)$. Which of the following is true?
    + (n.b., $\log$ denotes the base-2 logarithm) + ]]> +     + $S(n) = 1 + \log n$ + $S(n) = 1 + n \log n$ + $S(n) = 2n + 1$ + $S(n) = 2n - 1$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m15.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m15.xml new file mode 100644 index 00000000..4e1c1e67 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m15.xml @@ -0,0 +1,15 @@ + + + + + 3/8 + 4/8 + 5/8 + 6/8 + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m16.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m16.xml new file mode 100644 index 00000000..9f9f1285 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m16.xml @@ -0,0 +1,19 @@ + + + + + $\Pr(A) = \Pr(B)$ + $\Pr(A) < \Pr(B)$ + $\Pr(A) > \Pr(B)$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2015/w15m17.xml b/app/src/data/questions/comp2804/midterm-winter-2015/w15m17.xml new file mode 100644 index 00000000..8d5fd101 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2015/w15m17.xml @@ -0,0 +1,25 @@ + + + +
  • + There are 7 beers of the pilsner type. +
    • +
  • + There are 5 beers of the India pale ale type. +
    • +
  • + There are 4 beers of the German wheatbeer type. +
    • + + You order 4 different beers with at least one beer of each type. What is the number of ways in which + you can do this? (The order in which you order the beers does not matter.) + ]]> +     + ${16 \choose 4}$ + ${7 \choose 2} \cdot 5 \cdot 4 + 7 \cdot {5 \choose 2} \cdot 4 + 7 \cdot 5 \cdot {4 \choose 2}$ + ${16 \choose 4} - {7 \choose 3} - {5 \choose 3} - {4 \choose 3}$ + None of the above. +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/midterm-winter-2017.json b/app/src/data/questions/comp2804/midterm-winter-2017/midterm-winter-2017.json new file mode 100644 index 00000000..fb9549bb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/midterm-winter-2017.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-winter-2017", + "title": "Winter 2017 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-winter-2017/w17m01.xml", + "comp2804/midterm-winter-2017/w17m02.xml", + "comp2804/midterm-winter-2017/w17m03.xml", + "comp2804/midterm-winter-2017/w17m04.xml", + "comp2804/midterm-winter-2017/w17m05.xml", + "comp2804/midterm-winter-2017/w17m06.xml", + "comp2804/midterm-winter-2017/w17m07.xml", + "comp2804/midterm-winter-2017/w17m08.xml", + "comp2804/midterm-winter-2017/w17m09.xml", + "comp2804/midterm-winter-2017/w17m10.xml", + "comp2804/midterm-winter-2017/w17m11.xml", + "comp2804/midterm-winter-2017/w17m12.xml", + "comp2804/midterm-winter-2017/w17m13.xml", + "comp2804/midterm-winter-2017/w17m14.xml", + "comp2804/midterm-winter-2017/w17m15.xml", + "comp2804/midterm-winter-2017/w17m16.xml", + "comp2804/midterm-winter-2017/w17m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m01.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m01.xml new file mode 100644 index 00000000..b3fb363e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m01.xml @@ -0,0 +1,16 @@ + + + + + $f \cdot {f+m-1 \choose 7}$ + $f + {f+m-1 \choose 5} + {f+m-6 \choose 2}$ + $f \cdot {f+m-1 \choose 5} \cdot {f+m-6 \choose 2}$ + $f \cdot {f+m \choose 5} \cdot {f+m \choose 2}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m02.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m02.xml new file mode 100644 index 00000000..9aafaf45 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m02.xml @@ -0,0 +1,12 @@ + + + + + $(b + g)!$ + $g \cdot (b + g - 1)!$ + $(b + g)! / b$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m03.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m03.xml new file mode 100644 index 00000000..2b6f2a1b --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m03.xml @@ -0,0 +1,12 @@ + + + + + $(b+g)!$ + $(b+g)! - b \cdot (b+g-1)! -\ $ $ g \cdot (b+g-1)!$ + $g \cdot (b+g-1)! + b \cdot (b+g-1)!\ -$ $ b \cdot g \cdot (b+g-1)!$ + $g \cdot (b+g-1)! + b \cdot (b+g-1)!\ -$ $ b \cdot g \cdot (b+g-2)!$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m04.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m04.xml new file mode 100644 index 00000000..53ea2ebe --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m04.xml @@ -0,0 +1,12 @@ + + + + + ${n \choose 12} \cdot {12 \choose 7}$ + ${n \choose 7} + {n \choose 5}$ + ${n \choose 7} + {n \choose 5} - {n \choose 5} \cdot {n - 5 \choose 7}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m05.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m05.xml new file mode 100644 index 00000000..12e49b2d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m05.xml @@ -0,0 +1,12 @@ + + + + + 79 + 80 + 81 + 82 + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m06.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m06.xml new file mode 100644 index 00000000..a6cf4752 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m06.xml @@ -0,0 +1,12 @@ + + + + + $4^{n}$ + $4^{n-1}$ + $4^{n-2}$ + $4^{n-3}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m07.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m07.xml new file mode 100644 index 00000000..067b46bf --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m07.xml @@ -0,0 +1,15 @@ + + + + + The number of ways to choose an ordered pair $(x,y)$ from a group consisting of $m$ men and $n$ women, where $x$ must be a man and $y$ must be a woman. + The number of ways to choose an ordered pair $(x,y)$ from a group consisting of $m$ men and $n$ women, where $x$ and $y$ cannot both be men. + The number of ways to choose an unordered pair of people from a group consisting of $m$ men and $n$ women, where at least one man must be chosen. + The number of ways to choose an unordered pair of people from a group consisting of $m$ men and $n$ women. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m08.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m08.xml new file mode 100644 index 00000000..60dff348 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m08.xml @@ -0,0 +1,15 @@ + + + + + What is the minimum number of students such that we can guarantee that at least two of them have the + same student number? + ]]> + + $1 + 10^{5}$ + $1 + 5^{10}$ + $1 + \frac{10!}{5!}$ + $1 + \frac{5!}{10!}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m09.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m09.xml new file mode 100644 index 00000000..24a4d334 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m09.xml @@ -0,0 +1,11 @@ + + + + + ${100 \choose 80} \cdot 7^{20} \cdot 13^{80}$ + $- {100 \choose 80} \cdot 7^{20} \cdot 13^{80}$ + ${100 \choose 20} \cdot 7^{80} \cdot 13^{20}$ + $- {100 \choose 20} \cdot 7^{80} \cdot 13^{20}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m10.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m10.xml new file mode 100644 index 00000000..4e416946 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m10.xml @@ -0,0 +1,14 @@ + + + + POOPERSCOOPER +

    + ]]> +     + $13!$ + ${13 \choose 4}{13 \choose 3}{13 \choose 2}{13 \choose 2}{13 \choose 1}$ + ${13 \choose 4}{9 \choose 3}{6 \choose 2}{4 \choose 2}$ + ${13 \choose 4}{9 \choose 3}{6 \choose 2}{4 \choose 2}{2 \choose 1}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m11.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m11.xml new file mode 100644 index 00000000..e7f8dfd4 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m11.xml @@ -0,0 +1,17 @@ + + + free if it does not contain two 0's next to each other. + In class, we have seen that for any $m \geq 1$, + the number of 00-free bitstrings of length $m$ is equal to the $(m+2)$-th Fibonacci number $f_{m+2}$.
    + + Let $n \geq 3$ be an integer. What is the number of 00-free bitstrings of length $3n-1$ that have 0 at position $n$ + and 1 at position $2n$? (The positions are numbered $1,2,\dots,3n-1)$.
    + (n.b., $f^2_x = f_x \cdot f_x$) + ]]> +     + ${f^2_{n+1}} \cdot f_n$ + ${f^2_{n}} \cdot f_{n+1}$ + ${f^2_{n+2}} \cdot f_{n+1}$ + ${f^2_{n+1}} \cdot f_{n+2}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m12.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m12.xml new file mode 100644 index 00000000..571c114d --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m12.xml @@ -0,0 +1,18 @@ + + + + + $f(n) = \frac{2^{n}}{(n-1)^{2}}$ + $f(n) = \frac{2^{n-1}}{n^{2}}$ + $f(n) = \frac{2^{n}}{n^{2}}$ + $f(n) = \frac{n^{2}}{2^{n}}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m13.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m13.xml new file mode 100644 index 00000000..5062d61e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m13.xml @@ -0,0 +1,12 @@ + + + + + $B_n = n$ + $B_n = n+1$ + $B_n = f_{n+1}$, the $(n+1)$-st Fibonacci number. + $B_n = f_{n+2}$, the $(n+2)$-nd Fibonacci number. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m14.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m14.xml new file mode 100644 index 00000000..e5b634f5 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m14.xml @@ -0,0 +1,28 @@ + + + + + + $11$ + + $= 3 + 4 + 4 = 4 + 3 + 4$ + $= 4 + 4 + 3.$ + + + + + + Which of the following is true for any $n \geq 5$? + ]]> + + $S_n = 2 \cdot S_{n - 1}$ + $S_n = S_{n - 1} + S_{n - 2}$ + $S_n = S_{n - 2} + S_{n - 3}$ + $S_n = S_{n - 3} + S_{n - 4}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m15.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m15.xml new file mode 100644 index 00000000..0c05a838 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m15.xml @@ -0,0 +1,30 @@ + + + + $\newcommand{\Hello}{{\rm H {\scriptsize ELLO}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the recursive algorithm $\Hello$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \Hello(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \text{print ``hello''}$
    + $\mathbf{else}\ \mathbf{if}\ n\ \mathrm{is}\ \mathrm{even}$
    + $\elsesp \mathbf{then}\ \Hello(n / 2)$
    + $\elsesp \mathbf{else}\ \Hello(n - 1);$
    + $\elsesp \elsesp \Hello(n - 2)$
    + $\elsesp \mathbf{endif};$
    + $\mathbf{endif}$ +

    + + If we run algorithm $\Hello(7)$, how many times is the word "hello" printed? + ]]> +     + 4 + 5 + 6 + 7 +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m16.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m16.xml new file mode 100644 index 00000000..0c5b237c --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m16.xml @@ -0,0 +1,15 @@ + + + + + 1/2 + 1/4 + 1/6 + 1/8 + diff --git a/app/src/data/questions/comp2804/midterm-winter-2017/w17m17.xml b/app/src/data/questions/comp2804/midterm-winter-2017/w17m17.xml new file mode 100644 index 00000000..b64a296a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2017/w17m17.xml @@ -0,0 +1,16 @@ + + + + + $\left. n^2 \middle/ {2n \choose 2} \right.$ + $\left. {2n \choose 2} \middle/ n^2 \right.$ + $\left. n^2 \middle/ {n \choose 2} \right.$ + $\left. {n \choose 2} \middle/ n^2 \right.$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/midterm-winter-2018.json b/app/src/data/questions/comp2804/midterm-winter-2018/midterm-winter-2018.json new file mode 100644 index 00000000..017b4060 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/midterm-winter-2018.json @@ -0,0 +1,23 @@ +{ + "id": "midterm-winter-2018", + "title": "Winter 2018 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-winter-2018/w18m01.xml", + "comp2804/midterm-winter-2018/w18m02.xml", + "comp2804/midterm-winter-2018/w18m03.xml", + "comp2804/midterm-winter-2018/w18m04.xml", + "comp2804/midterm-winter-2018/w18m05.xml", + "comp2804/midterm-winter-2018/w18m06.xml", + "comp2804/midterm-winter-2018/w18m07.xml", + "comp2804/midterm-winter-2018/w18m08.xml", + "comp2804/midterm-winter-2018/w18m09.xml", + "comp2804/midterm-winter-2018/w18m10.xml", + "comp2804/midterm-winter-2018/w18m11.xml", + "comp2804/midterm-winter-2018/w18m12.xml", + "comp2804/midterm-winter-2018/w18m13.xml", + "comp2804/midterm-winter-2018/w18m14.xml", + "comp2804/midterm-winter-2018/w18m15.xml", + "comp2804/midterm-winter-2018/w18m16.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m01.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m01.xml new file mode 100644 index 00000000..105ebdfb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m01.xml @@ -0,0 +1,22 @@ + + + + + How many such strings have the property that + + ]]> + + $6^6$ + $5^{12}$ + $6^3 \cdot 6^2$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m02.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m02.xml new file mode 100644 index 00000000..57519780 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m02.xml @@ -0,0 +1,22 @@ + + + + + How many such strings have the property that + + ]]> + + $6^{3} \cdot 6^{5} + 6^{2} \cdot 6^{5}$ + $3^{6} \cdot 5^{6} + 2^{6} \cdot 5^{6}$ + $3^{6} \cdot 5^{6} + 2^{5} \cdot 5^{6} - 2^{6} \cdot 3^{6}$ + $6^{3} \cdot 6^{5} + 6^{2} \cdot 6^{5} - 6^{2} \cdot 6^{3}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m03.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m03.xml new file mode 100644 index 00000000..ed926745 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m03.xml @@ -0,0 +1,14 @@ + + + + + How many such strings contain at least two $a$'s? + ]]> + + $5^{12} - 4^{12} - 12 \cdot 4^{11}$ + $5^{12} - 4^{12} - 12 \cdot 4^{12}$ + $12^{5} - 12^{4} - 12 \cdot 11^{4}$ + $12^{5} - 12^{4} - 12 \cdot 12^{4}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m04.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m04.xml new file mode 100644 index 00000000..6a0977ea --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m04.xml @@ -0,0 +1,16 @@ + + + + + In how many ways can Elisa place these orders, such that exactly 4 people get a beer? + ]]> + + ${6 \choose 4} \cdot b^{4} \cdot c^{2}$ + ${6 \choose 4} \cdot b^{4} \cdot c^{3}$ + ${7 \choose 4} \cdot b^{4} \cdot c^{3}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m05.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m05.xml new file mode 100644 index 00000000..e833c9ab --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m05.xml @@ -0,0 +1,20 @@ + + + +
  • + 60 do not like Donald Trump and do not like Kim Jong Un, and +
    • +
  • + 25 like Kim Jong Un. +
    • + + How many people in this group like Donald Trump and do not like Kim Jong Un? + ]]> +     + 10 + 15 + 20 + 25 +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m06.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m06.xml new file mode 100644 index 00000000..398cbda7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m06.xml @@ -0,0 +1,16 @@ + + + ShortLastName, every person has a last name consisting of one uppercase letter, + followed by two lowercase letters. No two letters in a last name can be equal. Thus, Lin + is a valid last name, whereas Xax is not a valid last name.
    + + What is the minimum size of the population of ShortLastName, such that there must be at least two people + who have the same last name? + ]]> +     + $1 + 26!$ + $1 + 26^{3}$ + $1 + \frac{24 \cdot 25 \cdot 26}{3!}$ + $1 + 24 \cdot 25 \cdot 26$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m07.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m07.xml new file mode 100644 index 00000000..d7ff1fc8 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m07.xml @@ -0,0 +1,17 @@ + + + + + The number of bitstrings of length $n$. + The number of bitstrings of length $\left. (n-2) \middle/ 2 \right.$. + The number of bitstrings of length $n$ having an odd number of 0's. + + The number of bitstrings of length $n$ in which the number of 0's plus the number of 1's is at most $\left. (n-1) \middle/ 2 \right.$. + + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m08.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m08.xml new file mode 100644 index 00000000..0a53bf73 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m08.xml @@ -0,0 +1,23 @@ + + + + + Consider subsets $X$ of $A \cup B$, such that $|X| = k, |X \cap A| = 17$, and all elements of $X \cap A$ are even.
    + + How many such subsets $X$ are there? + ]]> +     + ${m/2 \choose 17} \cdot {n \choose k-17}$ + ${m \choose 17} \cdot {n \choose k-17}$ + ${m+n \choose 17} \cdot {n \choose k-17}$ + ${m/2+n \choose 17} \cdot {n \choose k-17}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m09.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m09.xml new file mode 100644 index 00000000..85d89aea --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m09.xml @@ -0,0 +1,16 @@ + + + + + ${36 \choose 4}$ + ${36 \choose 3}$ + ${37 \choose 4}$ + ${37 \choose 3}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m10.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m10.xml new file mode 100644 index 00000000..31dce6eb --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m10.xml @@ -0,0 +1,21 @@ + + + + + Since having finished marking assignment 2, Nick has been working hard to improve his Spiderman-skills: + In one step, Nick can move up either two floors or three floors. (Nick lost his ability to move up one floor in one step.)
    + + Let $n \geq 2$ be an integer and consider a building with $n$ floors, numbered $1,2,\dots,n$. (The first floor has number 1; + this is not the ground floor.) Nick is standing in front of this building, at the ground level.
    + + Let $N_n$ be the number of different ways in which Nick can climb to the $n$-th floor. + Which of the following is true for any $n \geq 5$? + ]]> +     + $N_n = N_{n-1} + N_{n-2} + N_{n-3}$ + $N_n = N_{n-1} + N_{n-2}$ + $N_n = N_{n-1} + N_{n-3}$ + $N_n = N_{n-2} + N_{n-3}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m11.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m11.xml new file mode 100644 index 00000000..4518855a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m11.xml @@ -0,0 +1,16 @@ + + + free, if it does not contain two 0's next to each other. + In class, we have seen that for any $m \geq 1$, + the number of 00-free bitstrings of length $m$ is equal to the $(m+2)$-th Fibonacci number $f_{m+2}$.
    + + What is the number of 00-free bitstrings of length 55 that have 0 at position 9, and 1 at position 40? + (The positions are numbered $1,2,\dots,55$.) + ]]> +     + $f_7 \cdot f_{29} \cdot f_{15}$ + $f_8 \cdot f_{30} \cdot f_{16}$ + $f_9 \cdot f_{31} \cdot f_{17}$ + $f_{10} \cdot f_{32} \cdot f_{18}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m12.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m12.xml new file mode 100644 index 00000000..6d9d760e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m12.xml @@ -0,0 +1,21 @@ + + + + + $7^{2n}$ + $7^{n}$ + $(2n)^{7}$ + $n^{7}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m13.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m13.xml new file mode 100644 index 00000000..d15ec03a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m13.xml @@ -0,0 +1,15 @@ + + + ccc-free, if it does not contain ccc.
    + + For any integer $n \geq 4$, let $B_n$ be the number of ccc-free bitstrings of length $n$. + Which of the following is true? + ]]> +     + $B_n = B_{n-1} + B_{n-2} + B_{n-3}$ + $B_n = 2 \cdot B_{n-1} + 2 \cdot B_{n-2} + 2 \cdot B_{n-3}$ + $B_n = 2 \cdot B_{n-1} + 2 \cdot B_{n-2} + B_{n-3}$ + $B_n = B_{n-1} + B_{n-2} + 2 \cdot B_{n-3}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m14.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m14.xml new file mode 100644 index 00000000..9eb38703 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m14.xml @@ -0,0 +1,30 @@ + + + + $\newcommand{\IFeelLikeSinging}{{\rm IF {\scriptsize EEL} L {\scriptsize IKE} S {\scriptsize INGING}}} + \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$ + + + Consider the recursive algorithm $\IFeelLikeSinging$, which takes as input an integer $n \geq 0$: + +

    + $\mathbf{Algorithm}\ \IFeelLikeSinging(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \mathrm{sing}\ {\it O\ Canada}$
    + $\mathbf{else}\ \mathbf{if}\ n\ \text{is odd}$
    + $\elsesp \mathbf{then}\ \IFeelLikeSinging(n + 1)$
    + $\elsesp \mathbf{else}\ \IFeelLikeSinging(\frac{n}{2});$
    + $\elsesp \elsesp \IFeelLikeSinging(\frac{n}{2} - 1)$
    + $\elsesp \mathbf{endif};$
    + $\mathbf{endif}$ +

    + + If you run algorithm $\IFeelLikeSinging(9)$, how many times do you sing O Canada? + ]]> +     + 6 + 7 + 8 + 9 +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m15.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m15.xml new file mode 100644 index 00000000..36fbd3c7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m15.xml @@ -0,0 +1,15 @@ + + + + + ${12 \choose 2} \cdot \left. 5^{10} \middle/ 6^{12} \right.$ + $12^{2} \cdot \left. 5^{10} \middle/ 6^{12} \right.$ + ${12 \choose 2} \cdot \left. 5^{12} \middle/ 6^{12} \right.$ + $12^{2} \cdot \left. 5^{12} \middle/ 6^{12} \right.$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2018/w18m16.xml b/app/src/data/questions/comp2804/midterm-winter-2018/w18m16.xml new file mode 100644 index 00000000..e896e1a4 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2018/w18m16.xml @@ -0,0 +1,19 @@ + + + + + Nick buys 25 bananas at Alexa's Banana Emporium (ABE) and 30 bananas at Shelly's Fruit Market (SFM). + Nick chooses, uniformly at random, a 15-element subset of these bananas. Define the event + [defs] + A = "the subset chosen by Nick contains exactly 7 bananas from ABE". + [/defs] + What is $\Pr(A)$? + ]]> + + $\frac{{25 \choose 8} + {30 \choose 7}}{55 \choose 15}$ + $\frac{{25 \choose 8} \cdot {30 \choose 7}}{55 \choose 15}$ + $\frac{{25 \choose 7} + {30 \choose 8}}{55 \choose 15}$ + $\frac{{25 \choose 7} \cdot {30 \choose 8}}{55 \choose 15}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/midterm-winter-2019.json b/app/src/data/questions/comp2804/midterm-winter-2019/midterm-winter-2019.json new file mode 100644 index 00000000..02db5f3a --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/midterm-winter-2019.json @@ -0,0 +1,24 @@ +{ + "id": "midterm-winter-2019", + "title": "Winter 2019 Midterm", + "Author": "Michiel Smid", + "questions": [ + "comp2804/midterm-winter-2019/w19m01.xml", + "comp2804/midterm-winter-2019/w19m02.xml", + "comp2804/midterm-winter-2019/w19m03.xml", + "comp2804/midterm-winter-2019/w19m04.xml", + "comp2804/midterm-winter-2019/w19m05.xml", + "comp2804/midterm-winter-2019/w19m06.xml", + "comp2804/midterm-winter-2019/w19m07.xml", + "comp2804/midterm-winter-2019/w19m08.xml", + "comp2804/midterm-winter-2019/w19m09.xml", + "comp2804/midterm-winter-2019/w19m10.xml", + "comp2804/midterm-winter-2019/w19m11.xml", + "comp2804/midterm-winter-2019/w19m12.xml", + "comp2804/midterm-winter-2019/w19m13.xml", + "comp2804/midterm-winter-2019/w19m14.xml", + "comp2804/midterm-winter-2019/w19m15.xml", + "comp2804/midterm-winter-2019/w19m16.xml", + "comp2804/midterm-winter-2019/w19m17.xml" + ] +} \ No newline at end of file diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m01.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m01.xml new file mode 100644 index 00000000..6f301d16 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m01.xml @@ -0,0 +1,12 @@ + + + + + 32 + 64 + 128 + 256 + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m02.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m02.xml new file mode 100644 index 00000000..e9957c2c --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m02.xml @@ -0,0 +1,12 @@ + + + + How many such permutations are there? + ]]> + + $7! - 6!$ + $7! - 5!$ + $7! - 4!$ + $7! - 3!$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m03.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m03.xml new file mode 100644 index 00000000..304b400f --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m03.xml @@ -0,0 +1,12 @@ + + + + + $52^{15} - 26^{15}$ + $52^{15} - 2 \cdot 26^{15}$ + $52^{15} - 3 \cdot 26^{15}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m04.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m04.xml new file mode 100644 index 00000000..a6de04dd --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m04.xml @@ -0,0 +1,13 @@ + + + + + ${n \choose k} - {n/2 \choose k - 1} - \frac{n}{2} \cdot {n/2 \choose k}$ + ${n \choose k} - {n/2 \choose k - 1} - \frac{n}{2} \cdot {n/2 \choose k - 1}$ + ${n \choose k} - {n/2 \choose k} - \frac{n}{2} \cdot {n/2 \choose k}$ + ${n \choose k} - {n/2 \choose k} - \frac{n}{2} \cdot {n/2 \choose k - 1}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m05.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m05.xml new file mode 100644 index 00000000..9a6891b9 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m05.xml @@ -0,0 +1,23 @@ + + + +
  • + 37 students like beer, +
    • +
  • + 18 students like cider, +
    • +
  • + 55 students do not like beer and do not like cider. +
    • + + How many students like beer and cider? + ]]> +     + 8 + 9 + 10 + 11 +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m06.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m06.xml new file mode 100644 index 00000000..fb8d956e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m06.xml @@ -0,0 +1,13 @@ + + + + What is the minimum value for $n$, such that there must be at least four students who receive the same grade? + ]]> + + 14 + 15 + 16 + 17 + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m07.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m07.xml new file mode 100644 index 00000000..05bfcbc7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m07.xml @@ -0,0 +1,12 @@ + + + + How many such subsets have the property that the largest element in the subset is equal to 30? + ]]> + + ${{29}\choose{16}}$ + ${29 \choose 17}$ + ${30 \choose 16}$ + ${30 \choose 17}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m08.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m08.xml new file mode 100644 index 00000000..5adedbec --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m08.xml @@ -0,0 +1,27 @@ + + + + + + + (beer bottle, cider bottle, wine bottle) +

    + in a set consisting of $n$ beer bottles, $n$ cider bottles, and $n$ wine bottles. + ]]> +     + + The number of ways to choose a 3-element subset of a set consisting of $n$ beer bottles, $n$ cider bottles, and $n$ wine bottles. + + + The number of ways to choose 3 elements (with repetitions allowed) in a set consisting of $n$ beer bottles, $n$ cider bottles, and $n$ wine bottles. + + None of the above. +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m09.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m09.xml new file mode 100644 index 00000000..fd8b472e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m09.xml @@ -0,0 +1,24 @@ + + + +
  • + each character is an element of $\{a, b, c\}$, +
    • +
  • + the number of $a$'s is equal to $k$, and +
    • +
  • + each $a$ is at an even position. +
    • + + How many such strings are there? + ]]> +     + ${n/2 \choose k} \cdot 2^{n-k}$ + ${n/2 \choose k} \cdot 2^{n/2}$ + ${n \choose k} \cdot 2^{n-k}$ + ${n \choose k} \cdot 2^{n/2}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m10.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m10.xml new file mode 100644 index 00000000..7a5ea8e1 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m10.xml @@ -0,0 +1,15 @@ + + + free, if it does not contain two 0's next to each other. + In class, we have seen that for any $m \geq 1$, + the number of 00-free bitstrings of length $m$ is equal to the $(m+2)$-th Fibonacci number + $f_{m+2}$.
    + What is the number of 00-free bitstrings of length 77 that have 0 at position 59? (The positions are numbered $1,2,\dots,77$.) + ]]> +     + $f_{17} \cdot f_{57}$ + $f_{18} \cdot f_{58}$ + $f_{19} \cdot f_{59}$ + $f_{20} \cdot f_{60}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m11.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m11.xml new file mode 100644 index 00000000..8d0161f7 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m11.xml @@ -0,0 +1,18 @@ + + + + + $f(n) = 6 \cdot 4^{n} - 2^{n}$ + $f(n) = 7 \cdot 4^{n} - 2^{n}$ + $f(n) = 8 \cdot 4^{n} - 2^{n+1}$ + None of the above. + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m12.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m12.xml new file mode 100644 index 00000000..e9c68100 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m12.xml @@ -0,0 +1,14 @@ + + + awesome, if it does not contain $aa$, and does not contain $aba$, and does not contain $abb$.
    + For any integer $n \geq 1$, let $A_n$ be the number of awesome strings of length $n$.
    + Which of the following is true for any integer $n \geq 4$? + ]]> +     + $A_n = A_{n-1} + 2 \cdot A_{n-2} + 2 \cdot A_{n-3}$ + $A_n = A_{n-1} + 2 \cdot A_{n-2} + A_{n-3}$ + $A_n = 2 \cdot A_{n-1} + A_{n-2} + 2 \cdot A_{n-3}$ + $A_n = 2 \cdot A_{n-1} + A_{n-2} + A_{n-3}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m13.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m13.xml new file mode 100644 index 00000000..c19e2fbc --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m13.xml @@ -0,0 +1,23 @@ + + + $1 \atop 1$-column, + if both bits in the column are 1.
    + For any integer $n \geq 1$ and integer $k$ with $0 \leq k \leq n$, let $M(n,k)$ be the number of binary $2 \times n$ matrices + + Which of the following is true for all integers $n \geq 2$ and $k$ with $1 \leq k \leq n-1$? + ]]> +     + $M(n,k) = M(n-1,k)\ +$$\ M(n-1,k-1)$ + $M(n,k) = M(n-1,k)\ +$$\ 2 \cdot M(n-1,k-1)$ + $M(n,k) = M(n,k-1)\ +$$\ M(n-1,k-1)$ + $M(n,k) = M(n,k-1)\ +$$\ 2 \cdot M(n-1,k-1)$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m14.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m14.xml new file mode 100644 index 00000000..27f07e22 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m14.xml @@ -0,0 +1,26 @@ + + + $\newcommand{\HelloWorld}{{\rm H {\scriptsize ELLO} W {\scriptsize ORLD}}}$ + + Consider the recursive algorithm $\HelloWorld$, which takes as input an integer $n \geq 0$: +

    + $\mathbf{Algorithm}\ \HelloWorld(n)\mathrm{:}$
    + $\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$
    + $\mathbf{then}\ \mathrm{print}\ \mathit{Hello}\ \mathit{World}$
    + $\mathbf{else}\ \mathbf{if}\ n \text{ is a multiple of } 3$
    + $\qquad \mathbf{then}\ \HelloWorld\left( n \middle/ 3 \right);$
    + $\qquad \qquad \ \ \mathrm{print}\ \mathit{Hello}\ \mathit{World};$
    + $\qquad \qquad \ \ \HelloWorld\left( 2n \middle/ 3 \right)$
    + $\qquad \mathbf{else}\ \HelloWorld(n + 1)$
    + $\qquad \mathbf{endif};$
    + $\mathbf{endif}$ +

    + Which of the following is correct? + ]]> +     + For any integer $n \geq 0$, algorithm $\HelloWorld(n)$ terminates. + There exists an integer $n \geq 0$, for which algorithm $\HelloWorld(n)$ does not terminate. + All of the above. + None of the above. +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m15.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m15.xml new file mode 100644 index 00000000..102fa33e --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m15.xml @@ -0,0 +1,30 @@ + + + +
  • + Porterhouse Brewing Co. Oyster Stout, +
    • +
  • + Guinness Extra Stout, +
    • +
  • + Magners Original Irish Cider. +
    • + + There is an unlimited supply for each of these types.
    + There are 75 students at the party, and each of them gets one drink, which is chosen uniformly at random from these three types.
    + Let $A$ be the event + [defs] + A = "exactly 50 students get Magners Original Irish Cider". + [/defs] + What is $\Pr(A)$? + ]]> +     + $\frac{{ 75 \choose 50 } \cdot 2^{25}}{3^{75}}$ + $\frac{3^{75}}{{75 \choose 50} \cdot 2^{25}}$ + $\frac{75 \choose 50}{3^{75}}$ + $\frac{{75 \choose 50} \cdot 3^{25}}{3^{75}}$ +     diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m16.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m16.xml new file mode 100644 index 00000000..7c7245cf --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m16.xml @@ -0,0 +1,15 @@ + + + + + $1 - {\frac{8 \cdot 5^{7}}{6^{8}}}$ + $1 - {\frac{6^{8}}{5^{8} + 8 \cdot 5^{7}}}$ + $1 - {\frac{5^{8} + 8 \cdot 5^{7}}{6^{8}}}$ + $1 - {\frac{8^{5} + 8 \cdot 7^{5}}{8^{6}}}$ + diff --git a/app/src/data/questions/comp2804/midterm-winter-2019/w19m17.xml b/app/src/data/questions/comp2804/midterm-winter-2019/w19m17.xml new file mode 100644 index 00000000..b284aaa0 --- /dev/null +++ b/app/src/data/questions/comp2804/midterm-winter-2019/w19m17.xml @@ -0,0 +1,17 @@ + + + + Let A be the event + [defs] + A = "you answer at least 16 questions correctly". + [/defs] + What is $\Pr(A)$? + ]]> + + $\frac{4^{17}}{52}$ + $\frac{51}{4^{17}}$ + $\frac{49}{4^{17}}$ + $\frac{52}{4^{17}}$ +