CMI Entrance Exam 2026 Question Paper for M.Sc Data Science

FOR CMI EXAM PREPARATION

CMI 2026
Question Paper · M.Sc
Data Science
EXAM YEAR TYPE SUBJECT

CMI 2026 Question Paper M.Sc Data Science

Notes · Sample Papers · Previous Year Papers · Mock Tests

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m .co CHENNAI MATHEMATICAL INSTITUTE
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M.Sc. Data Science Entrance Examination
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IMPORTANT INSTRUCTIONS

• Part (A) consists of 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.

• For questions in part (A), you have to provide the answers on the computer. You only have to choose
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the appropriate answer(s) from the choices provided. If the answer is parts (a) and (c), choose only (a)
and (c).
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• For questions in part (B), you have to write your answer with a short explanation in the space provided

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

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

Notation and terminology
• 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
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x = y for all x, y ∈ A;

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

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

• For a matrix A, AT denotes the transpose of A. For a square matrix A, |A| denotes the determinant of
A and trace(A) denotes the trace of A — namely the sum of the diagonal elements of A.

• A diagonal matrix is a square matrix D with all off-diagonal entries equal to zero i.e. dij = 0 for all
i ̸= j.

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• An upper triangular matrix is a square matrix A for which all entries below the diagonal are zero, i.e.

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aij = 0 for i > j.
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• .A symmetric matrix is a square matrix S for which s = s for all i ̸= j.
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a 1 Notes & Syllabus visit
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Part (A) - Multiple-choice questions

1. An insurance company offers a special policy covering two independent risk events in a policyholder’s life
over one year: a medical emergency and a vehicle accident. Based on historical actuarial data:
1 1
P (medical emergency) = , P (vehicle accident) =
10 20
A claim is considered valid if at least one of the two events occurs during the year. Each policyholder
pays an annual premium of Rs. 4,800, and if a valid claim is filed, the company pays out Rs. 6,000 (no
payout otherwise). Which of the following statements are correct?
29
(a) The probability of valid claim is 200 .
(b) A single policyholder renews the same policy for four consecutive years. The probability that she
files a valid claim in exactly one of the four years equals

29 3 171 1
    
4
≈ 0.0095
1 200 200

(c) A family of four members each independently hold this policy for a year. The probability that
exactly one of the four members files a valid claim equals

29 1 171 3
    
4
≈ 0.3257
1 200 200

(d) If 800 policyholders are enrolled under this policy in a given year, the expected profit of the insurance
company for the year equals Rs. 19,44,000.

2. Consider the single variable, real-valued function f (x) = x3 − 3x. Which of the following statements
about f are true?

(a) f is increasing for all x.
(b) f has exactly 2 critical points.
(c) f has exactly one inflection point.
(d) f is bounded below.

3. There are 10 switches labeled 1 to 10, all initially turned OFF. You perform the following operation: at
step k, you toggle (an ON switch is turned OFF and vice versa) every switch whose label is divisible by
k. After performing 10 such steps, which switches are in ON position?

(a) 1, 4, 9
(b) 2, 4, 6, 8
(c) 1, 4, 9, 10
(d) 1, 3, 6, 9

4. Which of the following inequalities involving real valued functions are true?

(a) | sin x| ≤ |x|, ∀x ∈ R
√ √
(b) e x ≤ x, ∀x ≥ 0
(c) ln(1 + x) ≤ x, ∀x > −1
(d) tan x ≥ x, for all x where defined

5. Let S be the set of all functions from {1, 2, 3, 4} → {1, 2, 3}. Which of the following are true?

(a) The number of functions in S whose image has exactly 2 elements is 3 · (24 − 2)
(b) The number of functions in S whose image has exactly 3 elements is 34 − 3 · 24 + 3
(c) The number of injective functions in S is nonzero
(d) The number of surjective functions in S with f (1) = 1 and f (2) = 2 is 5.

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c. o6. A box contains 3 red balls and 2 blue balls. Two balls are drawn one safter
e manother without replacement.
m Let A be the event that both balls are red, and B the event thatlathe first ball is red. Which of the
se following are true?
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(a) P (A) = 10
(b) P (B) = 53
(c) P (B c | Ac ) = 32 (here X c is the event that X does not take place).
(d) P (A | B) = 35

7. Five people, A, B, C, D, E sit in a row. Two particular people, A and B, refuse to sit next to each other.
Which of the following are true?
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(a) The number of valid arrangements is 5! − 4! · 2

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(b) The probability that A and B are not adjacent is 35
(c) The number e ofm a
arrangements (where A and B are not adjacent) in which A and D sit nextlto
seach
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g of arrangements (where A and B are not adjacent) in which D and E sit next to each
other is 36
(d) The a
number
other is 48.

8. Let A be an n × n real matrix. Which of the following statements are always true?

(a) If A is invertible, then the equation Ax = b has a unique solution for every b ∈ Rn .
(b) If Ax = 0 has only the trivial solution, then A is invertible.
(c) If A2 = 0, then A = 0.

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(d) If AB = 0 for some n × n matrix B ̸= 0, then A is not invertible.

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9. How many 3-element subsets of {1, 2, . . . , 8} contain exactly one pair of consecutive integers?

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(a) 20
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(b) 36
(c) 30
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(d) 56

10. Let a and b be the outcomes of two independent rolls of a fair die. Define X = max{a, b}. Which of the
following are true?
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(a) P (X = 6) = 36
1
(b) P (X = 1) = 36
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(c) E[X] > 4
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c. (d)o P (X ≤ 3) = m
8
36

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m Starting with the number n = 1, we generate a sequence of numbers. In lthea second step we replace 1 by
s e11. either 2 = 2 × 1 or 3 = 2 × 1 + 1. In general, replace n by either 2n org2n + 1 to get a new value for n .
l a a fashion?
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Which of the following numbers can be obtained as values of n in this

(a) 7
(b) 10
(c) 15
(d) 2026

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12. Arrange the numbers 1, 2, 3, 4, 5 in a row. We say a number i, 1 ≤ i ≤ 5 is well-placed in the given
arrangement if it appears in position i, starting from the left. For example, in the arrangement 4, 1, 3, 2, 5,
the numbers 3 and 5 are well-placed. We are interested in arrangements where no number is well-placed.
How many such arrangements exist?

(a) 20
(b) 44
(c) 60
(d) 45

13. Three statements are made:

• S1: “Exactly one of S2 and S3 is true.”
• S2: “S1 is false.”
• S3: “S2 is true.”

Which of the following are possible?

(a) Only S1 is true.
(b) Only S2 is true.
(c) Only S3 is true.
(d) Exactly two statements are true.

14. A subset (possibly empty) of {1, 2, 3, 4, 5, 6} is chosen uniformly at random. What is the probability that
the chosen subset contains at least one even number and at least one odd number?
1
(a)
2
3
(b)
4
7
(c)
8
49
(d)
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15. A student writes the following code to compute the average of numbers A[1], . . . , A[n]:
sum = 0
for i = 1 to n :
sum = sum + A [ i ]
avg = sum / n
return avg

Which of the following statements about this piece of code are correct?

(a) The code always returns the correct average.
(b) The output depends on the order of the input values
(c) In general, the code doesn’t calculate the correct average.
(d) The variable avg is unnecessarily updated.

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c. o16. In the following code, the operator % denotes the remainder after integer
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m function mystery ( n ) { la
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for i from 1 to n {
if ( i % 2 == 0) {
result = result + i ;
} else {
result = result - i ;
}
}
return ( result );
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What does mystery(10) return?
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(a) −5

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(b) 5
(c) 10
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(d) 0

Questions (17) to (19) are based on the following code.
The arguments to the function enigma(A, n) in the code below are: (i) an integer array A indexed from
0, and (ii) the number n of elements in A.
function enigma (A , n ) {
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for i from 0 to n - 2 {
for j from 0 to n - 2 - i {
if ( A [ j ] > A [ j +1]) {
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temp = A [ j ];
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A [ j ] = A [ j +1];
A [ j +1] = temp ;
}
}
}
return ( A [0]);
}

Answer the next three questions about this function.

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17. What does enigma([3, 1, 4, 2, 5], 5) return?

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(a)
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18. Which of the following statements about the function enigma(A,n) are true?

(a) After execution, the array A is sorted in non-decreasing order.
(b) The function returns the smallest element of the array.
(c) The function modifies the input array A.
(d) The function breaks down if the array contains repeated elements.

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19. Which of the following statements about the number of operations performed in enigma(A,n) are true?
Note that assigning a value to a variable is not considered as an operation.

(a) The number of comparisons performed is proportional to n2 .
(b) The total number of operations depends on the initial order of the array.
(c) The function always performs the same number of comparisons for a given n.
(d) The function may terminate early if the array is already sorted.

20. The argument to the function steps(n) is a positive integer n. The operator % denotes the remainder
after integer division.
function steps ( n ) {
count = 0;
while ( n != 1) {
if ( n % 2 == 0) {
n = n / 2;
} else {
n = 3 * n + 1;
}
count = count + 1;
}
return ( count );
}

What does steps(6) return?

(a) 6
(b) 8
(c) 3
(d) 4

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m .co Part (B) - Short-answer questions
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For questions in part (B), you have to write your answer withga short explanation in the space
a space, you may continue on the
provided for the question in your answer sheet. If you need more
pages provided for rough work. Any such overflows must be clearly labeled.

1. Let A be a 2 × 2 real matrix such that A2 = I. Determine all possible values of trace(A).

2. How many binary strings of length 8 contain exactly three 1s such that no two 1s are adjacent.
Questions (3) and (4) are based on the following information

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Let f : R → R be a continuous function that satisfies

c o m f (x + y) = f (x) · f (y), ∀x, y ∈ R,
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and that f (0) = 1. The. following two questions are based on this information. s e
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3. Show that for a
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any rational number r, we have f (r) = (f (1)) . (Hint: Consider the reduced g
r
a expression

ag p, q are integers).
r = p/q, where

4. Assuming that f (x) = (f (1))x holds for all real x, determine all values of f (1) for which f is strictly
increasing.

5. Determine all natural numbers n ≥ 1 such that
n
X n
X
k divides k2 .
k=1 k=1

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You may express your answer as a sequence of integers, giving the first few integers n for which the
statement is true.

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Question (6) and (7) are based on the following information

g l a numbers 1, 2, . . . , n in a random order, with all orderings
A list (a , . . . , a ) is an arrangement of natural
equally likely. We say that a position iais a new maximum if a > a for all j < i. For example, in the
1 n
i j
list
(3, 1, 4, 2, 5)
positions 1, 3, 5 are new maxima. Note that position 1 is always a new maximum. Now answer the two
questions below based on this information.

6. For a given i, determine the probability that position i is a new maximum.

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7. What is the expected number of new maxima in a list?

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Questions (8) and (9) are based on the following description.
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. For positive integers
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In m
edivided by b.
the pseudocode below, the function Steps(n) takes a positive integer n as input.
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e function Steps ( n ) { a
b, the expression a % b evaluates to the remainder obtained when a is
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while ( n > 1) {
if ( n % 2 == 0) {
n = n / 2;
} else {
n = n - 1;
}
count = count + 1;
}
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return ( count );
}
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8. Compute the value of Steps(13).

9. Let n be a positive integer with exactly k digits in its binary representation. What is the maximum
possible value of Steps(n)? For which value(s) of n (expressed in terms of k) is this maximum attained?
Questions (10) and (11) are based on the following information.
A finite sequence of n ≥ 1 integers a1 , a2 , . . . , an is processed one by one. Two variables M and c are
maintained as follows:

• Initially, M = a1 and c = 1.
• For each subsequent number ai , if ai > M , set M = ai and c = 1; if ai = M , set c = c + 1; otherwise,
do nothing.

10. For the input sequence 3, 1, 4, 1, 5, 9, 2, 6, 5, 9, write down the values of M and c after all elements have
been processed.

11. What do the variables M and c represent at the end of processing? Justify your answer in one or two
sentences.

12. Find the sum of all integers from 1 to 100 which are neither multiples of 3 nor multiples of 7.
Questions (13) and (14) are based on the following information.
Let p be a prime and n be a natural number such that

p divides n2 + n + 1.

13. Find all possible remainders of n3 upon division by p.

14. Suppose p > 3. Show that the numbers 1, n, n2 leave distinct remainders upon division by p.

15. Find all two-digit positive integers that are equal to three times the sum of their digits.

16. At a gathering of 5 people (named A, B, C, D, E), each person records how many other people they shook
hands with. Each number must be between 0 and 4. Find one possible set of 5 recorded numbers, with
at least 2 repeated values, but not all values equal, and describe a possible configuration of handshakes.

17. An LLM has generated 3 deep learning models M1 , M2 , M3 . Exactly one of these models is faulty. After
running internal diagnostic tests each model produced the following output:

(a) M1 : Model M2 is faulty.
(b) M2 : Model M3 is faulty.
(c) M3 : Model M1 is not faulty.

You know that only faulty models produce false statements and only non-faulty models produce true
statements. Which model is faulty? Explain.

18. Let A be an n × n real matrix such that
A2 = A.

(a) Show that if x is in the image of A, then Ax = x.
(b) Show that every vector v ∈ Rn can be uniquely written as

v = u + w,

where Au = u and Aw = 0.

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c. o19. Let X , X , X be independent and identically distributed Uniform(0,seθ)mrandom variables with proba-
m bility density function (pdf), la
se
1 2 3

1
f (x) = , 0 < x < θ, a g
θ
where unknown θ > 0. Let T = max(X1 , X2 , X3 ).

(a) Derive the cdf and pdf of T
(b) Find E(T ) and Var(T ).

20. Chebyshev’s Inequality states: For any random variable X with mean µ and variance σ 2 , and for
any k > 0:
1 1
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P (|X − µ| ≥ kσ) ≤ 2 equivalently, P (|X − µ| < kσ) ≥ 1 − 2
m k k

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Let X ∼ Exponential(λ) with mean and variance 2 . A student claims:
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λ λ
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1 2 3

ag P X−
λ
<
λ
≥
4

Is this claim true? Verify by computing the exact probability and comment on what your answer reveals
about Chebyshev’s inequality.

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