CMI Entrance Exam 2019 Question Paper for M.Sc in Data Science

CHENNAI MATHEMATICAL INSTITUTE
M.Sc. Data Science Entrance Examination
15th May 2019

Enter your Admit Card Number : D − −

IMPORTANT INSTRUCTIONS
• This booklet has 9 printed sheets, including this cover page. Sheet 2 (page 1) contains space to answer
part A, sheets 3 to 5 contain eighteen questions in part A and sheets 6 to 9 contain eighteen questions in
part B.
– For questions in part (A), you only have to write your answer on the designated page on the appro-
priate line. For example, if the answer is parts (a) and (c), write only (a) and (c) on the line and
if the answer is 2781, write this number on the line.
– For questions in part (B), you have to write your answer with short explanation in the space provided
below the question.
– For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.: nr ,


n P , n! etc.
r

For rough work use the blank pages at the end.

• Time allowed is 3 hours. Total points: 90 = 36 for part A + 54 for part B.

• Part A will be used for screening. Part B will be graded only if you score a certain minimum in part
A. However your scores in both parts will be used while making the final decision.

For office use only

Points Remarks

Part A

Part B

Total

B1 B2 B3 B4 B5 B6 B7 B8 B9 Total

B10 B11 B12 B13 B14 B15 B16 B17 B18 Total

Part A - Answers

This is the only place that will be seen for grading part A. So carefully and clearly write the
answers to each question on the designated line below. Write only the final answers, do not show any
intermediate work. Illegible/unclear answers will not be considered.

A1. A10.

A2. A11.

A3. A12.

A4. A13.

A5. A14.

A6. A15.

A7. A16.

A8. A17.

A9. A18.

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Notation
• A function f from a set A to a set B is said to be injective (or one-to-one) if f (x) = f (y)
implies x = y for all x, y ∈ A;

• f is said to be surjective (or onto) if for every y ∈ B there exists x ∈ A such that f (x) = y;

• f is said to be bijective if it is both injective and surjective;

• f is said to be invertible if there exists a function g from B to A such that f (g(y)) = y for all
y ∈ B and g(f (x)) = x for all x ∈ A and then g is said to be inverse of f and is denoted by
f −1 .

• For a matrix A, |A| denotes the determinant of A and AT denotes the transpose of A.

Part A - Questions

This section consists of some questions requiring a single answer and some multiple choice
questions. For questions that ask you to calculate a value, you will be assessed based on the
answer you provide. No explanation is required. In multiple choice questions, there may be
multiple correct choices. You have to select all the correct options and no incorrect option
to get full marks. There is no partial credit. Write the correct options / answer in the space
provided on page 1. Only page 1 will be seen for grading part A.

1. Let X = {x1 , x2 , . . . , xn } and Y = {y1 , y2 }. The number of surjective functions from X to Y
equals

(a) 2n
(b) 2n − 1
(c) 2n − 2
(d) 2n/2

2. If P (A ∪ B) = 0.7 and P (A ∪ B c ) = 0.9 then find P (A).

3. Which of the following statements are true for all n × n matrices A, B:

(a) (AT )T = A;
(b) |AT | = |A|;
(c) (AB)T = AT B T ;
(d) (A + B)T = AT + B T .

3

     
1 1 1 5 5 5 3 0 0
4. Let A =  0 2 2 , B =  0 10 10 , C =  3 6 0 . Which of the following state-
0 0 3 0 0 15 3 6 9
ments are true?

(a) |A| = |B|;
(b) |B| = 125|A|;
(c) |C| = 27|A|;
|A|
(d) |C| = .
3

5. Consider the polynomials p(x) = (5x2 + 6x + 1)(x + 1)(2x + 3) and q(x) = (5x2 − 9x − 2)(2x2 +
5x + 3). The set of common divisors of p(x) and q(x) is:

(a) {2x + 3, x + 1, 5x + 1}
(b) {2x + 3, x − 1, 5x + 1}
(c) {x + 3, 2x + 1, x − 2}
(d) {2x − 3, x + 1, 5x + 1}

6. Let R denote the set of real numbers and let A = {x ∈ R : x 6= 3}. For x ∈ A, let f (x) = 2x+1
x−3 .
Let B denote the range of f . Then

(a) B = {x ∈ R : x 6= −2} and f −1 (x) = 3x−1
x+2 ;
(b) B = {x ∈ R : x 6= 2} and f −1 (x) = 3x+1
x−2 ;
(c) B = {x ∈ R : x 6= 2} and f −1 (x) = 3x−1
x−2 ;
(d) 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:

(a) 15


11
(b) 11 · 15


11
(c) 15 · 14 · 13 · 12 · 11 · 10 · 9 · 8 · 7 · 6 · 5
(d) (15 · 14 · 13 · 12 · 11 · 10 · 9 · 8 · 7 · 6 · 5) · 11

8. Consider the following Venn diagram. The universal set U is the set of all natural numbers
from 1 to 1000.

4

A B

C

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:

(a) 12
(b) 15
(c) 16
(d) 17

9. In the code fragment below, start and end are integer values and prime(x) is a function that
returns True if x is a prime number and False otherwise.

i = 0;
j = 0;
k = 0;
for m = start to end {
if prime(m) == True {
i = i + 1;
# Statement 1
}else{
j = j - 1 ;
# Statement 2
}
}

We wish to maintain the invariant k == i - j after each iteration of the for loop. What
should we insert at Statement 1 and Statement 2?

(a) Statement 1: k = k + 1 and Statement 2: k = k + 1
(b) Statement 1: k = k + 1 and Statement 2: k = k - 1
(c) Statement 1: k = k - 1 and Statement 2: k = k + 1
(d) Statement 1: k = k - 1 and Statement 2: k = k - 1

5

Description for the following two questions:
The following table gives the budget allocation (in Rupees Crores) to 5 different departments
and their quarterly expenditure for a particular year.

Department Allocation Q1 Q2 Q3 Q4
D1 240 25 60 70 45
D2 120 10 40 30 20
D3 200 30 40 60 20
D4 125 20 35 20 25
D5 180 40 20 60 30

10. As a percentage of total allocation, the maximum quarterly expenditure (in any quarter) was
shown by

(a) D2
(b) D5
(c) D1
(d) D3

11. Looking at the combined expenditure of all the five departments, which of the following gives
the correct order as far as expenditure in each quarter is concerned?

(a) Q4 < Q1 < Q3 < Q2
(b) Q1 < Q4 < Q2 < Q3
(c) Q4 < Q1 < Q2 < Q3
(d) 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:

(Box 1) The gold is not here.
(Box 2) The gold is not here.
(Box 3) The gold is in Box 1.

Only one clue is true; the other two are false. Which box has the gold?

(a) Box 1
(b) Box 2
(c) Box 3
(d) 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 the letter
V. There are 8 white handkerchieves of which 3 belong to Abha, and 11 yellow handkerchieves

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of which 4 belong to Abha. All 19 handkerchieves were packed together in a bag for a trip.
Their mother pulled out a handkerchief from the bag without looking and found it was marked
V. What is the probability that the handkerchief was yellow?

(a) 5/12
(b) 7/19
(c) 7/12
(d) 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:

(a) 15


11
(b) 11 · 15


11
(c) 15 · 14 · 13 · 12 · 11 · 10 · 9 · 8 · 7 · 6 · 5
(d) (15 · 14 · 13 · 12 · 11 · 10 · 9 · 8 · 7 · 6 · 5) · 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?

(a) tr(A + B) = tr(A) + tr(B);
(b) tr(2A) = 2tr(A);
(c) tr(AT ) = tr(A);
(d) 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?

(a) The matrix A + B is upper triangular.
(b) The matrix AT is upper triangular.
(c) The matrix A−1 is upper triangular.
(d) The matrix AB is upper triangular.

Description for the following 2 questions: If Z is a continuous random variable which
follows a Gaussian distribution with mean = 0 and standard deviation = 1, then
Z a
exp{−z 2 /2}
P(Z ≤ a) = √ dz = Φ(a),
−∞ 2π
where Φ(a = −2) = 0.02, Φ(a = −1.5) = 0.067, Φ(a = −1) = 0.16, Φ(a = −0.5) = 0.31,
Φ(a = 0) = 0.50, Φ(a = 0.5) = 0.69, Φ(a = 1) = 0.84, Φ(a = 1.5) = 0.933, and Φ(a = 2) = 0.98.
Note that X = µ + σZ follows Gaussian distribution with mean µ and standard deviation σ.
Suppose the score distribution of an exam has a Gaussian distribution with mean µ and standard
deviation σ. A candidate fails if she/he obtains less than 35% marks. However she/he must

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Result Percentage of Students
Passed with distinction 2
Passed without distinction 82

Table 1: Result of a particular exam taken by 225 students.

obtain more that 80% marks in order to pass with distinction. The exam is taken by a group
of 225 students and the results are given in Table 1..
You may use the fact that the sample mean X̄ follows a Gaussian distribution with mean µ and
√
standard deviation σ/ n.

17. Which of the following statements are true?

(a) µ − σ = 35 and µ + 2σ = 82;
(b) µ − 2σ = 16 and µ + σ = 82;
(c) µ = 53.33 and σ = 13.33;
(d) µ = 50 and σ = 15.

18. Which of the following statements are true?

(a) The probability that the average score of the group of 225 students is less than 48 is 2%.
(b) The probability that the average score of the group of 225 students is less than 42.5 is
6.7%.
(c) The probability that the average score of the group of 225 students is greater than 49 is
more than 50%.
(d) The probability that the average score of the group of 225 students is greater than 57.5 is
more than 16%.

8

Part B

For questions in part (B), you have to write your answer with a short explanation in the space provided
below the question.

n For numerical answers, the following forms are acceptable: fractions, decimals,
n
symbolic e.g.: r , Pr , n! etc.

1. For positive numbers a, b, c, show that
a b c
+ + ≥ 3.
b c a

Description for the following question:
Suppose X is the number of successes out of n trials, where the trials 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 trials is
 
n k
P(X = k) = p (1 − p)n−k , k = 0, 1, . . . , n.
k
The expected number of successes is E(X) = np. If n → ∞ and p → 0
λk
 
n k
p (1 − p)n−k → e−λ , k = 0, 1, . . . ,
k k!
where λ = np.
2. Each time a gambler plays a game, there is a one-in-million chance of winning. The gambler
plays the game one million times. Find the probability of winning the game zero times, i.e. find
the probability of the event that the gambler will lose all one million times that she/he will try.
Note : 1 million = 106

3. Suppose X is a continuous distribution with probability density function

f (x) = k(x − x2 ), 0 ≤ x ≤ 1,

where k is the normalizing constant. Find the value of k and the expected value of the distri-
bution.

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?

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).

exp(x)
6. Let θ = loge (2). For −θ ≤ x ≤ θ, let f (x) = 1+exp(x) . Compute α and β where

α = max f (x) and β = min f (x).
−θ≤x≤θ −θ≤x≤θ

Justify your answers.

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
70m the first minute, 100m the next minute, then 130m the minute afterwards and so on. Ani
is told to start 3 minutes later. Having trained hard he runs 100m the first minute, 150 m the
second Minute, then 200m and so on. Ani catches Usain at an integral multiple of a minute.
How many minutes did Ani run before catching up with Usain. What were their respective
speeds?

8. A small circular fire is spreading with its radius increasing at the rate of 1.5 metres per minute.
When the radius of the fire is 5 metres, how fast is the burned area growing?

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

10. Let p(x) be a polynomial with integer coefficients. Let n be a positive integer and suppose a
and b are two integers such that a ≡ b(mod n). Is it true that p(a) ≡ p(b)(mod n)? Justify
your answer.

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?

12. Let n, k be positive integers. The expansion of (x1 + · · · + xk )n is given by
X n!
(x1 + · · · + xk )n = xn1 xn2 . . . xnk k ,
n1 !n2 ! . . . nk ! 1 2
where the sum is taken over all sequences n1 , n2 , . . . , nk of non-negative integers such that
n1 + n2 + · · · + nk = n. What is the coefficient of x5 in the expansion of (1 + 3x + 2x2 )4 ?

13. How many positive integers are factors of 2310?

14. A thief picks a wallet and starts running at a speed of 5 m/s with zero acceleration. In 5
seconds, a policewoman notices the alarm and starts following the thief with a speed of 3 m/s
with a uniform acceleration of 1 m/s2 . How many seconds will it take the policewoman to catch
the thief, assuming she follows the same path as that of the thief? (A body moving with initial
velocity v0 m/s and uniform acceleration a m/s2 will cover a distance of (at2 /2 + v0 t) meters
in t seconds.)

15. Consider the following pseudocode for a function that operates on an N element array
A[1], A[2], . . . , A[N ] of integers.

function mystery (A[1..N]) {
int i,j,position,tmp;
for i = 1 to N {
position = i;
for j = i+1 to N {
if (A[j] < A[position]) {
position = j;
}
}
tmp = A[i];
A[i] = A[position];
A[position] = tmp;
}
}

(a) Explain what effect the function has on the input array A.

(b) If N = 100, how many times is the comparison A[i] < A[position] checked?

Description for next three questions:
The break-up of the colour of cars sold by an Indian company in a given year is provided in
the pie-chart, see the Figure (1). The bar chart shows the number of grey coloured cars sold in
different cities, see the Figure (2).

Figure 1: The break-up of the colour of cars sold in a given year.

Figure 2: The number of grey cars sold in cities.

16. How many blue cars were sold in that year?

17. If the proportion of grey cars sold in Delhi is the same as the proportion of red cars sold in
Delhi, how many red cars were sold in Delhi?

18. 12% of all cars were sold in Chennai. What is the largest possible percentage of brown cars sold
in Chennai, rounded off to the nearest integer?