# Recent questions tagged cmi2019-datascience 1
Let $X=\{ x_1,x_2,\dots,x_n\}$ and $Y=\{y_1,y_2\}$. The number of surjective functions from $X$ to $Y$ equals $2^n$ $2^n-1$ $2^n-2$ $2^{n/2}$
2
If $P(A\cup B)=0.7$ and $P(A\cup B^c)=0.9$ then find $P(A).$
3
Which of the following statements are true for all $n\times n$ matrices $A,B:$ $(A^T)^T=A$ $|A^T|=|A|$ $(AB)^T=A^TB^T$ $(A+B)^T=A^T+B^T$
4
Let $A=\begin{bmatrix} 1& 1& 1\\0&2&2\\0&0&3 \end{bmatrix}, B=\begin{bmatrix} 5&5&5\\0&10&10\\0&0&15\end{bmatrix}, C=\begin{bmatrix} 3&0&0\\3&6&0\\3&6&9 \end{bmatrix}$. Which of the following statements are true? $|A|=|B|$ $|B|=125|A|$ $|C|=27|A|$ $|C|=\frac{|A|}{3}$
5
Consider the polynomials $p(x)=(5x^2+6x+1)(x+1)(2x+3)$ and $q(x)=(5x^2-9x-2)(2x^2+5x+3)$. The set of common divisors of $p(x)$ and $q(x)$ is: $\{2x+3,\;x+1,\;5x+1\}$ $\{2x+3,\;x-1,\;5x+1\}$ $\{x+3,\;2x+1,\;x-2\}$ $\{2x-3,\;x+1,\;5x+1\}$
6
Let $R$ denote the set of real numbers and let $A=\{x\in R:x\neq 3\}$. For $x\in A$, let $f(x)=\frac{2x+1}{x-3}.$ Let $B$ denote the range of $f$. Then $B=\{x\in R:x \neq -2\} \;and \;f^{-1}(x)=\frac{3x-1}{x+2};$ $B=\{x\in R:x \neq 2\}\; and\; f^{-1}(x)=\frac{3x+1}{x-2};$ $B=\{x\in R:x \neq 2\}\; and\; f^{-1}(x)=\frac{3x-1}{x-2};$ $f^{-1}(x)$ does not exist because $f$ is not injective.
7
We need to choose a team of $11$ from a pool of $15$ players and also select a captain. The number of different ways this can be done is: $\begin{pmatrix}15\\11 \end{pmatrix}$ $11\cdot \begin{pmatrix}15\\11 \end{pmatrix}$ ... $(15\cdot 14\cdot 13 \cdot 12 \cdot 11\cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5)\cdot 11$
8
Consider the following Venn diagram. The universal set $U$ is the set of all natural numbers from $1$ to $1000.$ The sets $A,B,C$ contain integers in $U$ that are multiples of $6,7,8$ respectively. The number of elements in the shaded region is: $12$ $15$ $16$ $17$
9
In the code fragment below, $\text{start}$ and $\text{end}$ are integer values and $\text{prime(x)}$ is a function that returns $\text{True}$ if $\text{x}$ is a prime number and $\text{False}$ otherwise. i = 0; j = 0; k = 0; for m=start to end { if prime(m)==True { i = i + 1; # ... $\text{k = k + 1}$ Statement 1: $\text{k = k - 1}$ and Statement 2: $\text{k = k - 1}$
10
$\text{Description for the following question:}$ The following table gives the budget allocation (in Rupees Crores) to $5$ ... As a percentage of total allocation, the maximum quarterly expenditure ( in any quarter) was shown by $D2$ $D5$ $D1$ $D3$
11
$\text{Description for the following question:}$ The following table gives the budget allocation (in Rupees Crores) to $5$ ... as expenditure in each quarter is concerned? $Q4<Q1<Q3<Q2$ $Q1<Q4<Q2<Q3$ $Q4<Q1<Q2<Q3$ $Q1<Q4<Q3<Q2$
12
Three boxes are presented to you. At most one of them contains some gold. Each box has printed on it a clue about its contents. The clues are: $\textbf{(Box 1)}\; The\; gold\; is\; not\; here.$ $\textbf{(Box 2)} \; The\; gold\; is\; not\; here.$ ... Only one clue is true; the other two are false. Which box has the gold? Box 1 Box 2 Box 3 None of them has the gold
13
Abha and Vibha both have white and yellow handkerchieves. To distinguish them, their mother has marked Abha's handkerchieves with the letter $A$ and Vibha's handkerchieves with letter $V$. There are $8$ white handkerchieves of which $3$ belong to Abha, and $11$ yellow handkerchieves of ... it was marked $V$. What is the probability that the handkerchief was yellow? $5/12$ $7/19$ $7/12$ $11/19$
14
We need to choose a team of $11$ from a pool of $15$ players and also select a captain. The number of different ways this can be done is: $\begin{pmatrix}15\\11 \end{pmatrix}$ $11\cdot \begin{pmatrix}15\\11 \end{pmatrix}$ $15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8 \cdot 7\cdot 6 \cdot 5$ $(15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8 \cdot 7\cdot 6 \cdot 5)\cdot 11$
15
The sum of the diagonal elements of a matrix $A$ is called the trace of $A$ and is denoted by $tr(A)$. Which of the following statements about the trace are true? $\text{tr(A+B)=tr(A)+tr(B)}$ $\text{tr(2A)=2tr(A)}$ $\text{tr($A^T$)=tr(A)}$ $\text{tr($A^{-1}$)=tr(A)}$
16
An upper triangular matrix is a square matrix with all entries below the diagonal being zero. Suppose $A$ and $B$ are upper triangular matrices. Which of the following statements are true? The matrix $A+B$ is upper triangular. The matrix $A^T$ is upper triangular. The matrix $A^{-1}$ is upper triangular. The matrix $AB$ is upper triangular.
17
If $Z$ is a continuous random variable which follows a Gaussian distribution with mean=$0$ and standard deviation=1, then $\mathbb{P}(Z \leq a)= \int^a_{-\infty}\frac{\exp \{ -z^2 / 2 \}} {\sqrt{2 \pi} }dz=\Phi (a)$ ... $\mu-2\sigma=16\;\text{and}\;\mu+\sigma=82$; $\mu=53.33\;\text{and}\;\sigma=13.33$; $\mu=50\;\text{and}\;\sigma=15$;
18
If $Z$ is a continuous random variable which follows a Gaussian distribution with mean=$0$ and standard deviation=1, then $\mathbb{P}(Z \leq a)= \int^a_{-\infty}\frac{\exp \{ -z^2 / 2 \}} {\sqrt{2 \pi} }dz=\Phi (a)$ ... than $49$ is more than $50\%$ The probability that the average score of the group of $225$ students is greater than $57.5$ is more than $16\%$
19
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. For positive numbers $a,b,c,$ show that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3$
20
$\text{Description of the following question:}$ Suppose $X$ is the number of successes out of $n$ trials, where the trails are independent of each other. The probability of success at every trial is $p$. The probability that there will be exactly $k$ successes out of $n$ ... probability of the event that the gambler will lose all one million times that she/he will try. Note: $1$ million =$10^6$.
21
Suppose $X$ is a continuous distribution with probability density function $f(x)=k(x-x^2),\;0\leq x\leq 1,$ where $k$ is the normalizing constant. Find the value of $k$ and the expected value of the distribution.
22
Four friends attending an opera leave their coats at the checkroom. When they return each one is handed a coat that does not belong to her. In how many ways can this happen?
23
If the lines $2x+y=2,-5x+3y=4$ and $ax+by=1$ are concurrent then prove that the line $9x-10y=1$ passes through $(a,b).$
24
Let $\theta=\log_e(2).$ For $-\theta\leq x\leq\theta,$ let $f(x)=\frac{\exp(x)}{1+\exp(x)}$. Compute $\alpha$ and $\beta$ where $\alpha=\displaystyle \max_{-\theta\leq x\leq\theta}f(x)\;\text{and}\; \beta=\displaystyle \min_{-\theta\leq x\leq\theta}f(x).$ Justify your answers.
25
Ani is training for the olympics with Usain Bolt. After a few days of training Usain challenges Ani to catch him. Usain sets off running very slowly with a view to encourage Ani. He covers $\text{70m}$ the first minute, $\text{100m}$ the next minute, ... Usain at an integral multiple of a minute. How many minutes did Ani run before catching up with Usain. What were their respective speeds?
26
A small circular fire is spreading with its radius increasing at the rate of $1.5$ meters per minute. When the radius of the fire is $5$ metres, how fast is the burned area growing?
27
Show that among any set of $7$ distinct integers there must exist $2$ integers whose sum or difference is divisible by $10$.
Let $p(x)$ be a polynomial with integer coefficients. Let $n$ be a positive integer and suppse $a$ and $b$ are two integers such that $a \equiv b(\text{mod}\;n)$. Is it true that $p(a)\equiv p(b)(\text{mod}\;n)$? Justify your answer.
A thin piece of metal of length $20$ cm and width $16$ cm is to be used to construct an open-topped box. A square will be cut from each corner and the sides will be folded up. What size corner should be cut so that the volume of the box is maximized?
Let $n,k$ be positive integers. The expansion of $(x_1+\dots+x_k)^n$ is given by $(x_1+\dots+x_k)^n=\sum\frac{n!}{n_1!n_2!\dots n_k!}x_1^{n_1}x_2^{n_2}\dots x_k^{n_k},$ where the sum is taken over all sequences $n_1,n_2,\dots,n_k$ of non-negative integers such that $n_1,n_2+\dots+n_k=n$. What is the coefficient of $x^5$ in the expansion of $(1+3x+2x^2)^4$?