Recent questions tagged cmi2019-datascience

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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 $2^n$ $2^n-1$ $2^n-2$ $2^{n/2}$

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 $2^n$ $2^n-1$ $2^n-2$ $2^{n/2}$

asked Jan 29 in Others soujanyareddy13 19 views

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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 $P(A).$

If $P(A\cup B)=0.7$ and $P(A\cup B^c)=0.9$ then find $P(A).$

asked Jan 29 in Others soujanyareddy13 22 views

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3

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

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

asked Jan 29 in Others soujanyareddy13 14 views

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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? $|A|=|B|$ $|B|=125|A|$ $|C|=27|A|$ $|C|=\frac{|A|}{3}$

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? $|A|=|B|$ $|B|=125|A|$ $|C|=27|A|$ $|C|=\frac{|A|}{3}$

asked Jan 29 in Others soujanyareddy13 25 views

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

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

asked Jan 29 in Others soujanyareddy13 18 views

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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.

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.

asked Jan 29 in Others soujanyareddy13 10 views

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

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

asked Jan 29 in Others soujanyareddy13 15 views

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

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

asked Jan 29 in Others soujanyareddy13 15 views

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

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

asked Jan 29 in Others soujanyareddy13 20 views

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

$\text{Description for the following question:}$ 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 $D2$ $D5$ $D1$ $D3$

asked Jan 29 in Others soujanyareddy13 15 views

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

$\text{Description for the following question:}$ The following table gives the budget allocation (in Rupees Crores) to $5$ ... as expenditure in each quarter is concerned? $Q4<Q1<Q3<Q2$ $Q1<Q4<Q2<Q3$ $Q4<Q1<Q2<Q3$ $Q1<Q4<Q3<Q2$

asked Jan 29 in Others soujanyareddy13 10 views

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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

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

asked Jan 29 in Others soujanyareddy13 10 views

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

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

asked Jan 29 in Others soujanyareddy13 19 views

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

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

asked Jan 29 in Others soujanyareddy13 14 views

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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}$)=tr(A)}$

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}$)=tr(A)}$

asked Jan 29 in Others soujanyareddy13 19 views

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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.

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.

asked Jan 29 in Others soujanyareddy13 10 views

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

If $Z$ is a continuous random variable which follows a Gaussian distribution with mean=$0$ and standard deviation=1, then $\mathbb{P}(Z \leq a)= \int^a_{-\infty}\frac{\exp \{ -z^2 / 2 \}} {\sqrt{2 \pi} }dz=\Phi (a)$ ... $\mu-2\sigma=16\;\text{and}\;\mu+\sigma=82$; $\mu=53.33\;\text{and}\;\sigma=13.33$; $\mu=50\;\text{and}\;\sigma=15$;

asked Jan 29 in Others soujanyareddy13 24 views

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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 $16\%$

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 $16\%$

asked Jan 29 in Others soujanyareddy13 22 views

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

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

asked Jan 29 in Others soujanyareddy13 12 views

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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 =$10^6$.

$\text{Description of the following question:}$ 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 =$10^6$.

asked Jan 29 in Others soujanyareddy13 16 views

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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.

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.

asked Jan 29 in Others soujanyareddy13 12 views

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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?

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?

asked Jan 29 in Others soujanyareddy13 15 views

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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 $(a,b).$

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 $(a,b).$

asked Jan 29 in Others soujanyareddy13 9 views

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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.

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.

asked Jan 29 in Others soujanyareddy13 14 views

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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?

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?

asked Jan 29 in Others soujanyareddy13 17 views

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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?

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?

asked Jan 29 in Others soujanyareddy13 12 views

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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 $10$.

Show that among any set of $7$ distinct integers there must exist $2$ integers whose sum or difference is divisible by $10$.

asked Jan 29 in Others soujanyareddy13 12 views

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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.

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.

asked Jan 29 in Others soujanyareddy13 13 views

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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?

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?

asked Jan 29 in Others soujanyareddy13 14 views

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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 $(1+3x+2x^2)^4$?

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 $(1+3x+2x^2)^4$?

asked Jan 29 in Others soujanyareddy13 12 views

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