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1

CMI-2021-DataScience-B: 1
Consider the following code. Function foo(n) { answer = 0; x = n; while (x > 0) { y = x % 10; answer = (answer * 10) + y; x = x // 10; } x = answer; answer = (answer * 10) + 1; while (x > 0) { y = x % 10; answer = (answer * 10 ... For example, $23 \% 10=3$. $\text{a // b}$ represents integer division. For example, $23 / / 10=2$. What will $\textsf{foo(2021)}$ return?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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2

CMI-2021-DataScience-B: 2
Food delivery agents Aman, Boni, Chan, Dong and Eman were assessed on five parameters $\text{P1}$ to $\text{P5.}$ They received an integer rating between $1$ and $5$ for each parameter. None of them received the same rating in ... could have achieved across all the five parameters? What is the minimum average rating Chan could have received across all the five parameters?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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3

CMI-2021-DataScience-B: 3
Consider the following code where $\text{A}$ is an array indexed from $0.$ function foo(A, year, n) { 1 = 0, r = n - 1, c = 0; while (1 <= r) { c = c + 1; m = 1 + (r - 1) // 2; if (A[m] == year) { return(c * m); } ... print (result); } Here, $\text{a // b}$ represents integer division. For example, $23 / / 10=2$. What will be printed when $\textsf{bar()}$ is executed?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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4

CMI-2021-DataScience-B: 4
The $\text{CMI MSc DS 2021}$ batch of $60$ students is holding an online event to celebrate their joining $\text{CMI.}$ Student Aruni is in charge of organizing the musical section, and she sends out an online form where each student has ... of whom twenty students agreed to do both. How many students agreed to do exactly one activity, and how many declined to participate?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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5

CMI-2021-DataScience-B: 5
Show that every selection of $503$ numbers from $\{1,2,3, \dots,987\}$ has two numbers with g.c.d. $1$. Recall that the g.c.d. of two positive integers $x, y$ is the largest positive integer smaller than $x, y$ which divides both $x$ and $y$.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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6

CMI-2021-DataScience-B: 6
Let $S$ be the sphere given by the equation $x^{2} + y^{2} + z^{2} – 6x + 4y – 4z – 272=0$. What are the centre and radius of $S?$ What is the minimum distance between $S$ and the plane $2x + 2y + z = 214?$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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7

CMI-2021-DataScience-B: 7
We draw two rows of $n$ points with labels $\{ 1,2, \dots,n \}$, as shown in the figure below. We connect each point on the top with a point on the bottom at random, making sure that no two points on the top are connected to the same point on the bottom. ... to point $j$ below. Let $X$ be the number of special points. What is $\mathbb{E}(X)$, the expected value of $X?$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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8

CMI-2021-DataScience-B: 8
Let $T_{n}$ denote the set of all sequences over $\{ 0, 1, 2 \}$ of length $n$. If we pick an element $x$ in $T_{n}$ uniformly at random what is the expected number of $\text{1's}$ in $x?$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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9

CMI-2021-DataScience-B: 9
What is the smallest possible value of $\displaystyle{}\sum_{i=1}^{n} \frac{1}{d_{i}+1}$, if the $d_{i} \text{'s}$ are constrained to be non-negative real numbers and satisfy $\displaystyle{}\sum_{i=1}^{n} d_{i}=N.$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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10

CMI-2021-DataScience-B: 10
Let $A$ be the $3 \times 3$ real matrix $\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & I \end{array}\right)$. Suppose $x^{T} Ax \geq 0$ for every $x \in \mathbb{R}^{3}$. Then show that all of $a, e$ and $i$ are non-negative.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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11

CMI-2021-DataScience-B: 11
In the finale of the Indian Idol programme, Ladies Special, there are four women contestants - Arunima, Priyamani, Razi and Shriya. The organisers send all participants a list of $7$ songs, asking each one to pick one song for the finale. All the ... allowed to pick that song. This is agreed to. What is the probability that Shriya gets to sing the song Pyar Hua Chupkese?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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12

CMI-2021-DataScience-B: 12
A Non-Banking Finance Corporation $\text{(NBFC)}$ declares fixed annual rates of simple interest on their auto and housing loans each year. The rates of interest offered by the company differ from year to year depending on the variation in macro economic indicators like ... by the customer after one year was $ ₹ \; 7.36 \text{ lakhs.}$ What was the housing loan amount?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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13

CMI-2021-DataScience-B: 13
A Non-Banking Finance Corporation $\text{(NBFC)}$ declares fixed annual rates of simple interest on their auto and housing loans each year. The rates of interest offered by the company differ from year to year depending on the variation in macro economic ... on the prevailing interest rate for that year. Which of them paid more interest over $4$ years? How much more?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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14

CMI-2021-DataScience-B: 14
Let $Y$ be a continuous random variable that takes values in $[0,1]$. $g(y)$ is a probability density function that can be used to model $Y$, $ g(y)=c \cdot y^{2}(1-y), \quad 0 \leq y \leq 1 $ Determine $c$. Using $g(y)$, find the probability that $Y$ is greater than $0.8$.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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15

CMI-2021-DataScience-B: 15
The state government has decided to regulate the sale and distribution of solar power. There are multiple suppliers and consumers, as described in the tables below. The supplier table lists the amount of electricity generated by each supplier, in megawatts ... Find the value of $p$ at which the maximum amount of solar power can be bought and sold on the market.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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16

CMI-2021-DataScience-B: 16
If a polynomial $f(x)$ has roots $r_{1}, r_{2}, \dots r_{n}$, what are the roots of the polynomial $f(x+1)?$ Let $f$ be a polynomial of degree three. Set $g(x)=f(x-1)$. Assume that $-1$ is not a root of $f$. What are the roots of the polynomial $x^{3} g(1 / x)?$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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17

CMI-2021-DataScience-B: 17
Is the following statement true? If $f(x) \geq 0$ for all $x$, and $\displaystyle{}\int_{-\infty}^{\infty} f(x) d x<\infty$ then $\displaystyle{}\int_{-\infty}^{\infty} x^{2} f(x) d x \geq \epsilon^{2} \int_{\epsilon}^{\infty} f(x) d x$, for all $\epsilon>0$.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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18

CMI-2021-DataScience-B: 18
Consider the function $f(x)=\frac{\ln x}{x}$. Find the critical point(s) of $f$ and say whether $f$ has a maximum or minimum at that point. Prove that for integers $a>b \geq 3$, we must have $a^{b}<b^{a}$.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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19

CMI-2021-DataScience-B: 19
Recall that if $h$ is a function from $X$ to $Y$ and $g$ is a function from $Y$ to $Z$ then, $g \circ h$ is the function from $X$ to $Z$ such that $(g \circ h)(x)=g(h(x))$, for all $x \in X$. Let $S$ be the set of all functions ... $f \circ f=f$. Compute the number of functions $f \in S$ whose range has three elements. What is the cardinality of $S$?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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20

CMI-2021-DataScience-B: 20
A pinhole camera is placed between an object and a sensor. Recall that images from a pinhole camera are inverted (see figure below). Assume that the object lies on a plane. Also assume that the plane containing the object, the plane containing the ... the sensor plane is $30 \text{ cm}$. What are the heights of their images after projection through the pinhole camera?

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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21

CMI-2021-DataScience-A: 1
There are two longest subsequences, not necessarily contiguous, common to the strings $\text{ ARTIFICIAL"}$ and $\text{ INTELLIGENCE."}$ They are $\text{ IIC"}$ and $\text{ TIC"}$ which are of length three. Consider two strings $\text{S1 = CORONAVIRUS"}$ ... $x$ between $\text{S} 1$ and $\text{S} 2$. What is $x+5y?$ $13$ $15$ $14$ $16$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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22

CMI-2021-DataScience-A: 2
The roots of the polynomial $p(x)=x^{4}-2x^{3}-2x^{2}+8x-8$ are: $1, -1, 2, 2+3 i$ $1+i, 1-i, 2, -2$ $1, -1+i, 2, 2+3 i$ $1+i, -1+i, 2, -2$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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23

CMI-2021-DataScience-A: 3
Consider the following code, in which $\text{A}$ is an array indexed from $0.$ function foo (A , n) { m = A [0] ; x = 0 ; for i = 0 to n – 1 { x = x + A [i] ; if (m < x) { m = x ; } if (x < 0) { x = 0 ; } } return (m) ; } If $\text{A}=[-12, -3, 5, 10, 8, -16, -23, 12, -5, 7]$, what will $\textsf{foo(A, 10)}$ return? $23$ $17$ $-17$ $35$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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24

CMI-2021-DataScience-A: 4
A student has an average score of $80$ from her first four Mathematics tests, and $88$ from her first five Physics tests. How much must she score in her upcoming tests to raise her average score in Mathematics and Physics to $82$ and $89,$ ... Mathematics and $90$ in Physics. $88$ in Mathematics and $92$ in Physics. $90$ in Mathematics and $94$ in Physics.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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25

CMI-2021-DataScience-A: 5
A binary tree starts with a single root node at the top of the tree. Each node can have either a left child or a right child, or both, or neither. The children of a node are drawn below it, connected by edges. Here are the five possible binary trees with ... a right child, and so on. How many different binary trees can be constructed with four nodes? $13$ $15$ $14$ $30$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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26

CMI-2021-DataScience-A: 6
Consider the following code, in which $a$ is an array indexed from $0.$ function foo (A, n) { x=0; for i=0 to n-1 { x=A[i]^x; } return (x); } Here, $a^{\wedge} b$ represents the bitwise Exclusive $\text{OR}$ function over variables $a$ and $b$. Given integers $a$ ... $a=[2, 3, 7, 2, 3]$, what will $\textsf{foo (A}, 5)$ return? $0$ $2$ $3$ $7$

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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27

CMI-2021-DataScience-A: 7
Fifteen telephones are received at a service center. Of these, $5$ are mobile, $6$ are cordless, and $4$ are wired. These $15$ phones are randomly numbered from $1$ to $15$ to establish the order in which they are serviced. Which of the following statement(s) is/are ... .

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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28

CMI-2021-DataScience-A: 8
Which of the following statement(s) is/are true? For $k=69597$ and $x_{1}, x_{2}, \dots, x_{k} \in (0, \infty), \dfrac{x_{1}+ x_{2}+ \dots + x_{2^{k}}} {2^{k}} \leq (x_{1} x_{2} \dots x_{2^k})^{2^\frac{1}{k}} .$ ... $(n^{3} - n)$ is divisible by $3$ for all positive integers $n$.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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29

CMI-2021-DataScience-A: 9
Which of the following statement(s) is/are true for an arbitrary $n \times n$ matrix $A?$ Exchanging two rows of $A$ does not change its determinant. Exchanging two rows of $A$ does not change its trace. Replacing each diagonal element of $A$ with a $1$ does not change its determinant. Exchanging two columns of $A$ negates its determinant.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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30

CMI-2021-DataScience-A: 10
Let $A, B$ be $n \times n$ matrices. Which of the following properties of $A$ and $B$ is/are preserved under matrix multiplication? Being upper triangular. All diagonal elements being zero. Being diagonal. Being symmetric.

Lakshman Patel RJIT asked in Others Jul 23, 2022

by Lakshman Patel RJIT

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