# Recent questions tagged cmi2018-datascience

1 vote
1
If $P$ is an invertible matrix and $A=PBP^{-1},$ then which of the following statements are necessarily true? $B=P^{-1}AP$ $|A|=|B|$ $A$ is invertible if and only if $B$ is invertible $B^T=A^T$
2
Let $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}, C=\begin{bmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix}$ and $D=\begin{bmatrix} -1 & 2 & 3 \\ 4 & -5 & 6 \\ 7 & 8 & -9 \end{bmatrix}$. Which of the following statements are true? $|A|=|B|$ $|C|=|D|$ $|B|=-|C|$ $|A|=-|D|$
3
Let $x=\begin{bmatrix} 3& 1 & 2 \end{bmatrix}$. Which of the following statements are true? $x^Tx$ is a $3\times 3$ matrix $xx^T$ is a $3\times 3$ matrix $xx^T$ is a $1\times 1$ matrix $xx^T=x^Tx$
4
A $n\times n$ matrix $A$ is said to be $symmetric$ if $A^T=A$. Suppose $A$ is an arbitrary $2\times 2$ matrix. Then which of the following matrices are symmetric (here $0$ denotes the $2\times 2$ matrix consisting of zeros): $A^TA$ $\begin{bmatrix} 0&A^T \\ A & 0 \end{bmatrix}$ $AA^T$ $\begin{bmatrix} A & 0 \\ 0 & A^T \end{bmatrix}$
5
$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a),$ where $\Phi(a=-2)=0.02,$ ... less than 0.02 $\int^{18} _{-\infty} \frac{1}{\sqrt{2\pi .2^2}}\exp\{-\frac{1}{2}(\frac{x-24}{2})^2\}dx$
6
$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a),$ where $\Phi(a=-2)=0.02,$ ... more than 0.4 less than 0.5
7
$\text{Description for the following question:}$ If $Z$ is a continuous random variable which follows normal distribution with mean=$0$ and standard deviation=$1$, then $\mathbb{P}(Z\leq a)=\int^a _{-\infty} \frac{\exp\{\frac{-z^2}{2}\}}{\sqrt{2\pi}}dz=\Phi(a),$ ... will last more than $26$ months approximately equals $16\%$ is more than $15\%$ is less than $14\%$ is between $10\%$ and $15\%$
8
$\text{Description for the following question:}$ Given below is the time table for a transcontinental train cutting across several time zones. All times given below are local time in respective cities. It is given that the average speed between any two cities is the same for both ... Which of the following pairs of cities are in the same time zone? Zut and Yag Xum and Wip Vaq and Uap Tix and Sab
9
$\text{Description for the following question:}$ Given below is the time table for a transcontinental train cutting across several time zones. All times given below are local time in respective cities. It is given that the average speed between any two cities is the same for both way ... Write down the total time taken in minutes by the train to go from Zut to Raz.
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$\text{Description for the following question:}$ ... $12:00$ noon at Zut.
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$\text{Description for the following question:}$ ... $12:00$ noon at Raz.
12
In an entrance examination with multiple choice questions, with each question having four options and a single correct answer, suppose that only $20\%$ candidates think they know the answer to one difficult question and only half of them know it correctly and the ... tick the same. If a candidate has correctly answered the question, what is the (conditional) probability that she knew the answer?
13
There are $n$ songs segregated into $3$ play lists. Assume that each play list has at least one song.The number of ways of choosing three songs consisting of one song from each play list is: $>\frac{n^3}{27}$ for all $n$ $\leq \frac{n^3}{27}$ for all $n$ $\left( \begin{array}{c} n \\ 3 \end{array} \right)$ for all $n$ $n^3$ for all $n$
14
Consider the following functions defined from the interval $(0,1)$ to real numbers. Which of these functions attain their maximum value in the interval $(0,1)?$ $f(x)=\frac{1}{x(1-x)}$ $g(x)=-(x-0.75)^2$ $u(x)=\sin(\frac{\pi x}{2})$ $v(x)=x^2+2x$
15
A farmer owns $50$ papaya trees. Each tree produces $600$ papayas in a year. For each additional tree planted in the orchard, the output of each tree (including the pre-existing ones) drops by $5$ papayas. How many trees should be added to the existing orchard in order to maximize the total production of papayas?
16
Let $a,b$ be numbers between $1$ and $2$ and let $c,d$ be numbers between $3$ and $4$. Let $u=a^{-1},v=b^{-1},w=c^{-1}$ and $x=d^{-1}$. Say which of the following inequalities are true: $(a+b+c+d)(u+v+w+x)>16$ $(a^4+b^4+c^4+d^4)\leq 4abcd$ $(a^2+b^2)wx\leq (c^2+d^2)uv$ $d(a^3+b^3+c^3)<3abc$
17
In the code fragment below, $\text{start}$ and $\text{end}$ are integer values and $\text{square(x)}$ is a function that returns $\text{Ture}$ if $\text{x}$ is a perfect square and $\text{False}$ otherwise. i := 0; j := 0; k :=0; for m in [start,start+1,...,end] { if(square(m)=True) ... $\text{j = k-i if (end - start)}$ is even $\text{i = k-j if (end - start)}$ is odd
18
Given the following definition of the function $foo$, what does $foo(1037,2)$ return? Note that $a//b$ denotes the quotient (integer part) of $a\div b$, for integers $a$ and $b$. For instance $7//3$ is $2$. function foo(n,d) { x:=0; while(n>=1) { x:=x+1; n:=n//d; } return(x); }
19
A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet three inhabitants: Ramesh, Bharath and Menaka. Ramesh claims, Bharat is a knave. Bharat says, Menaka and I are both knights or both ... following are correct? Bharat is a knave Ramesh is a knight Exactly two of the three are knaves Exactly two of the three are knights
20
There are $7$ switches on a switchboard, some of which are $on$ and some of which are $off$. In one move, you pick any $2$ switches and toggle each of them-if the switch you pick is currently $off$, you turn it $on$, if it is $on$, you turn it $off$. Your aim is to execute a sequence ... $\text{(off,on,on,on,off,on,off)}$ $\text{(off,on,off,off,on,off,on)}$ $\text{(off,on,off,off,off,on,off)}$
21
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Let $N=\{1,2,3,...\}$ be the set of natural integers and let $f:N\times N \mapsto N$ be defined by $f(m,n)=(2m-1)*2^n.$Is $f$ injective? Is $f$ surjective? Give reasons.
22
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Suppose $A,B$ and $C$ are $m\times m$ matrices. What does the following algorithm compute? (Here $A(i,j)$ denotes the $(i.j)^{th}$ entry of matrix $A$.) for i=1 to m for j=1 to m for k=1 to m C(i,j)=A(i,k)*B(k,j)+C(i,j) end end end
23
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Find $A^{10}$ where $A$ is the matrix $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$. Justify your answer.
24
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. In computing, a floating point operation (flop) is any one of the following operations performed by a computer ... $c_{ij}=\displaystyle\sum^5 _{k=1} a_{ik} b_{kj}$. How does this number change if both the matrices are upper triangular?
25
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A function $f$ from the set $A$ to itself is said to have a fixed point if $f(i)=i$ for some $i$ in $A$. Suppose $A$ is the set $\{a,b,c,d\}$. Find the number of bijective functions from the set $A$ to itself having no fixed point.
26
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A $4$-digit number is represented as $abcd$ i.e. $a\times 10^3 +b\times 10^2 +c\times10+d,$ where $a\neq0.$ Suppose the number $dcba$, obtained by reversing the digits of $abcd$, is $9$ times $abcd$. Find the number $abcd$.
27
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A computer password requires you to use exactly $1$ uppercase letter, $3$ lowercase letters, $3$ digits and $2$ special characters (there are $33$ special characters that can be used). In how many ways can you create such a password$?$
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of successes ... one credit default in a year. You can assume that whether a given debtor will default or not is independent of the behavior of other debtors.
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of successes out of ... $\mathbb{E}(X)=np$. For the situation in the previous problem, what is the expected number defaults?
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of successes out of $n$ ... on his/her credit than the bank loses the entire loan amount. What is the expected revenue of the bank from a loan of $Rs. 100,000?$