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1

CMI-2019-DataScience-A: 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 $2^n$ $2^n-1$ $2^n-2$ $2^{n/2}$

soujanyareddy13 asked in Others Jan 29, 2021

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2

CMI-2019-DataScience-A: 2
If $P(A\cup B)=0.7$ and $P(A\cup B^c)=0.9$ then find $P(A).$

soujanyareddy13 asked in Others Jan 29, 2021

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3

CMI-2019-DataScience-A: 3
Which of the following statements are true for all $n\times n$ matrices $A,B:$ $(A^T)^T=A$ $|A^T|=|A|$ $(AB)^T=A^TB^T$ $(A+B)^T=A^T+B^T$

soujanyareddy13 asked in Others Jan 29, 2021

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4

CMI-2019-DataScience-A: 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? $|A|=|B|$ $|B|=125|A|$ $|C|=27|A|$ $|C|=\frac{|A|}{3}$

soujanyareddy13 asked in Others Jan 29, 2021

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5

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: $\{2x+3,\;x+1,\;5x+1\}$ $\{2x+3,\;x-1,\;5x+1\}$ $\{x+3,\;2x+1,\;x-2\}$ $\{2x-3,\;x+1,\;5x+1\}$

soujanyareddy13 asked in Others Jan 29, 2021

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6

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.

soujanyareddy13 asked in Others Jan 29, 2021

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7

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: $\begin{pmatrix}15\\11 \end{pmatrix}$ $11\cdot \begin{pmatrix}15\\11 \end{pmatrix}$ ... $(15\cdot 14\cdot 13 \cdot 12 \cdot 11\cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5)\cdot 11$

soujanyareddy13 asked in Others Jan 29, 2021

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8

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: $12$ $15$ $16$ $17$

soujanyareddy13 asked in Others Jan 29, 2021

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9

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: $\text{k = k - 1}$

soujanyareddy13 asked in Others Jan 29, 2021

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10

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 $D2$ $D5$ $D1$ $D3$

soujanyareddy13 asked in Others Jan 29, 2021

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11

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? $Q4<Q1<Q3<Q2$ $Q1<Q4<Q2<Q3$ $Q4<Q1<Q2<Q3$ $Q1<Q4<Q3<Q2$

soujanyareddy13 asked in Others Jan 29, 2021

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12

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

soujanyareddy13 asked in Others Jan 29, 2021

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13

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? $5/12$ $7/19$ $7/12$ $11/19$

soujanyareddy13 asked in Others Jan 29, 2021

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14

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: $\begin{pmatrix}15\\11 \end{pmatrix}$ $11\cdot \begin{pmatrix}15\\11 \end{pmatrix}$ ... $(15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8 \cdot 7\cdot 6 \cdot 5)\cdot 11$

soujanyareddy13 asked in Others Jan 29, 2021

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15

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}$)=tr(A)}$

soujanyareddy13 asked in Others Jan 29, 2021

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16

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.

soujanyareddy13 asked in Others Jan 29, 2021

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17

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 $\mathbb{P}(Z \leq a)= \int^a_{-\infty}\frac{\exp \{ -z^2 / 2 \}} {\sqrt{2 \pi} }dz=\Phi (a)$ ... $\mu=53.33\;\text{and}\;\sigma=13.33$; $\mu=50\;\text{and}\;\sigma=15$;

soujanyareddy13 asked in Others Jan 29, 2021

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18

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 $16\%$

soujanyareddy13 asked in Others Jan 29, 2021

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19

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 $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3$

soujanyareddy13 asked in Others Jan 29, 2021

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20

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 =$10^6$.

soujanyareddy13 asked in Others Jan 29, 2021

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21

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.

soujanyareddy13 asked in Others Jan 29, 2021

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22

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?

soujanyareddy13 asked in Others Jan 29, 2021

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23

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 $(a,b).$

soujanyareddy13 asked in Others Jan 29, 2021

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24

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.

soujanyareddy13 asked in Others Jan 29, 2021

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25

CMI-2019-DataScience-B: 7
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 ... at an integral multiple of a minute. How many minutes did Ani run before catching up with Usain. What were their respective speeds?

soujanyareddy13 asked in Others Jan 29, 2021

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26

CMI-2019-DataScience-B: 8
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?

soujanyareddy13 asked in Others Jan 29, 2021

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27

CMI-2019-DataScience-B: 9
Show that among any set of $7$ distinct integers there must exist $2$ integers whose sum or difference is divisible by $10$.

soujanyareddy13 asked in Others Jan 29, 2021

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28

CMI-2019-DataScience-B: 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.

soujanyareddy13 asked in Others Jan 29, 2021

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29

CMI-2019-DataScience-B: 11
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?

soujanyareddy13 asked in Others Jan 29, 2021

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30

CMI-2019-DataScience-B: 12
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 $(1+3x+2x^2)^4$?

soujanyareddy13 asked in Others Jan 29, 2021

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