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1501
CMI-2019-DataScience-A: 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: $\{2x+3,\;x+1,\;5x+1\}$ $\{2x+3,\;x-1,\;5x+1\}$ $\{x+3,\;2x+1,\;x-2\}$ $\{2x-3,\;x+1,\;5x+1\}$
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1502
CMI-2019-DataScience-A: 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 $B=\{x\in R:x \neq -2\} \;and \;f^{-1}(x)=\frac{3x-1}{x+2};$ ... $B=\{x\in R:x \neq 2\}\; and\; f^{-1}(x)=\frac{3x-1}{x-2};$ $f^{-1}(x)$ does not exist because $f$ is not injective.
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1503
CMI-2019-DataScience-A: 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: $\begin{pmatrix}15\\11 \end{pmatrix}$ $11\cdot \begin{pmatrix}15\\11 \end{pmatrix}$ ... $(15\cdot 14\cdot 13 \cdot 12 \cdot 11\cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5)\cdot 11$
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1504
CMI-2019-DataScience-A: 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: $12$ $15$ $16$ $17$
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1505
CMI-2019-DataScience-A: 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 ... $\text{k = k + 1}$ Statement 1: $\text{k = k - 1}$ and Statement 2: $\text{k = k - 1}$
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1506
CMI-2019-DataScience-A: 10
$\text{Description for the following question:}$ The following table gives the budget allocation (in Rupees Crores) to $5$ ... percentage of total allocation, the maximum quarterly expenditure ( in any quarter) was shown by $D2$ $D5$ $D1$ $D3$
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1507
CMI-2019-DataScience-A: 11
$\text{Description for the following question:}$ The following table gives the budget allocation (in Rupees Crores) to $5$ ... quarter is concerned? $Q4<Q1<Q3<Q2$ $Q1<Q4<Q2<Q3$ $Q4<Q1<Q2<Q3$ $Q1<Q4<Q3<Q2$
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1508
CMI-2019-DataScience-A: 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
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1509
CMI-2019-DataScience-A: 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 ... marked $V$. What is the probability that the handkerchief was yellow? $5/12$ $7/19$ $7/12$ $11/19$
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1510
CMI-2019-DataScience-A: 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: $\begin{pmatrix}15\\11 \end{pmatrix}$ $11\cdot \begin{pmatrix}15\\11 \end{pmatrix}$ ... $(15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8 \cdot 7\cdot 6 \cdot 5)\cdot 11$
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1511
CMI-2019-DataScience-A: 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}$)=tr(A)}$
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1512
CMI-2019-DataScience-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.
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1513
CMI-2019-DataScience-A: 17
If $Z$ is a continuous random variable which follows a Gaussian distribution with mean=$0$ and standard deviation=1, then $\mathbb{P}(Z \leq a)= \int^a_{-\infty}\frac{\exp \{ -z^2 / 2 \}} {\sqrt{2 \pi} }dz=\Phi (a)$ ... $\mu=53.33\;\text{and}\;\sigma=13.33$; $\mu=50\;\text{and}\;\sigma=15$;
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1514
CMI-2019-DataScience-A: 18
If $Z$ is a continuous random variable which follows a Gaussian distribution with mean=$0$ and standard deviation=1, then $\mathbb{P}(Z \leq a)= \int^a_{-\infty}\frac{\exp \{ -z^2 / 2 \}} {\sqrt{2 \pi} }dz=\Phi (a)$ ... $50\%$ The probability that the average score of the group of $225$ students is greater than $57.5$ is more than $16\%$
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1515
CMI-2019-DataScience-B: 1
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 $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3$
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1516
CMI-2019-DataScience-B: 2
$\text{Description of the following question:}$ 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 ... of the event that the gambler will lose all one million times that she/he will try. Note: $1$ million =$10^6$.
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1517
CMI-2019-DataScience-B: 3
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.
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1518
CMI-2019-DataScience-B: 4
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?
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1519
CMI-2019-DataScience-B: 5
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 $(a,b).$
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1520
CMI-2019-DataScience-B: 6
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.
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