Back to Sections

Type:ALLMCQNATMSQ

Discrete Mathematics(19)Linear Algebra(11)Calculus(7)Probability and Statistics(8)

Discrete Mathematics

19 Questions

GATE→2026→Engineering Mathematics→Discrete Mathematics

Q1

MCQ[1 Mark]

Consider 4x4 matrices with their elements from {0,1}. The number of such matrices with even number of 1s in every row and every column is

(A)

512

(B)

1025

(C)

1023

(D)

255

Explanation

GATE→2026→Engineering Mathematics→Discrete Mathematics

Q2

NAT[2 Marks]

Let G be an undirected graph which is a path on 8 vertices. The number of matchings in G is

Explanation

GATE→2025→Engineering Mathematics→Discrete Mathematics

Q3

MCQ[1 Mark]

g: A->B, f: B->C, fog: A->C. If f and fog are onto, what is TRUE about g?

(A)

g(.) must be an onto (surjective) function.

(B)

g(.) must be a one-to-one (injective) function.

(C)

g(.) must be a bijective function, that is, both one-to-one and onto.

(D)

g(.) is not required to be a one-to-one or onto function.

Explanation

GATE→2025→Engineering Mathematics→Discrete Mathematics

Q4

NAT[1 Mark]

Number of ternary strings of length 5 containing at least one occurrence of two consecutive identical symbols ("aa", "bb" or "cc").

Explanation

GATE→2025→Engineering Mathematics→Discrete Mathematics

Q5

MSQ[2 Marks]

Predicate logic representations for "Everyone has exactly one mother".

(A)

$\forall x\exists y\exists z(mother(y,x)\wedge\neg mother(z,x))$

(B)

$\forall x\exists y[mother(y,x)\wedge\forall z(noteq(z,y)\rightarrow\neg mother(z,x))]$

(C)

$\forall x\forall y[mother(y,x)\rightarrow\exists z(mother(z,x)\wedge\neg noteq(z,y))]$

(D)

$\forall x\exists y[mother(y,x)\wedge\neg\exists z(noteq(z,y)\wedge mother(z,x))]$

Explanation

GATE→2025→Engineering Mathematics→Discrete Mathematics

Q6

MSQ[2 Marks]

A = {0, 1, 2, ...}, F = functions A->A. (f1 @ f2)(n) = f1(n) + f2(n). Structure (F, @)?

(A)

(F,O) is an Abelian group.

(B)

(F, ) is an Abelian monoid.

(C)

(F,O) is a non-Abelian group.

(D)

(F, ) is a non-Abelian monoid.

Explanation

GATE→2024→Engineering Mathematics→Discrete Mathematics

Q7

MSQ[1 Mark]

Which of the following statements is/are TRUE?

(A)

(B)

(C)

(D)

Explanation

GATE→2024→Engineering Mathematics→Discrete Mathematics

Q8

NAT[1 Mark]

Let A and B be non-empty finite sets such that there exist one-to-one and onto functions (i) from A to B and (ii) from A to A union B. Number of possible values of |A| is

Explanation

GATE→2024→Engineering Mathematics→Discrete Mathematics

Q9

NAT[1 Mark]

Number of spanning trees in a complete graph of 4 vertices labelled A, B, C, and D is

Explanation

GATE→2024→Engineering Mathematics→Discrete Mathematics

Q10

MSQ[2 Marks]

The chromatic number of a graph is the minimum number of colours used in a proper colouring. Let G be any graph with n vertices and chromatic number k. Which of the following statements is/are always TRUE?

(A)

G contains a complete subgraph with k vertices

(B)

G contains an independent set of size at least n/k

(C)

G contains at least k(k-1)/2 edges

(D)

G contains a vertex of degree at least k

Explanation

GATE→2024→Engineering Mathematics→Discrete Mathematics

Q11

MSQ[2 Marks]

Consider the operators o and [] defined by a o b = ab, a [] b = a + 2b, for positive integers. Which of the following properties hold?

(A)

(B)

(C)

(D)

Explanation

GATE→2023→Engineering Mathematics→Discrete Mathematics

Q12

MCQ[1 Mark]

The Lucas sequence L_n is defined by the recurrence relation: L_n = L_(n-1) + L_(n-2), for n >= 3, with L1 = 1 and L2 = 3. Which one of the options given is TRUE?

(A)

L_n = ((1+\sqrt{5})/2)^n + ((1-\sqrt{5})/2)^n

(B)

L_n = ((1+\sqrt{5})/2)^n - ((1-\sqrt{5})/3)^n

(C)

L_n = ((1+\sqrt{5})/2)^n + ((1-\sqrt{5})/3)^n

(D)

L_n = ((1+\sqrt{5})/2)^n - ((1-\sqrt{5})/2)^n

Explanation

GATE→2023→Engineering Mathematics→Discrete Mathematics

Q13

MSQ[1 Mark]

Geetha has a conjecture about integers, of the form \forall x (P(x) \Rightarrow \exists y Q(x,y)). Which of the following option(s) would imply Geetha's conjecture?

(A)

\exists x(P(x)\wedge\forall yQ(x,y))

(B)

\forall x\forall yQ(x,y)

(C)

\exists y\forall x(P(x)\Rightarrow Q(x,y))

(D)

\exists x(P(x)\wedge\exists yQ(x,y))

Explanation

GATE→2023→Engineering Mathematics→Discrete Mathematics

Q14

MCQ[2 Marks]

Let U = {1, 2, ..., n}. Let A, B be subsets with |A|=|B|=k and A \cap B = \emptyset. How many permutations of U separate A from B?

(A)

n!

(B)

C(n, 2k)(n-2k)!

(C)

C(n, 2k)(n-2k)!(k!)^2

(D)

2 C(n, 2k)(n-2k)!(k!)^2

Explanation

GATE→2023→Engineering Mathematics→Discrete Mathematics

Q15

MSQ[2 Marks]

Let f: A -> B be an onto function. Define equivalence relation ~ and mapping F: E -> B. Which of the following statements is/are TRUE?

(A)

F is NOT well-defined.

(B)

F is an onto (or surjective) function.

(C)

F is a one-to-one (or injective) function.

(D)

F is a bijective function.

Explanation

GATE→2023→Engineering Mathematics→Discrete Mathematics

Q16

MSQ[2 Marks]

Let X be a set and 2^X denote powerset. Define binary operation \Delta. H = (2^X, \Delta). Which statements are correct?

(A)

H is a group.

(B)

Every element in H has an inverse, but H is NOT a group.

(C)

For every A in 2^X the inverse of A is the complement of A.

(D)

For every A in 2^X the inverse of A is A.

Explanation

GATE→2022→Engineering Mathematics→Discrete Mathematics

Q17

MSQ[1 Mark]

Which of the following statements is/are TRUE for a group G?

(A)

If for all x, y \in G, (xy)^2 = x^2 y^2, then G is commutative.

(B)

If for all x \in G, x^2 = 1 then G is commutative. Here, 1 is the identity element.

(C)

If the order of G is 2, then G is commutative.

(D)

If G is commutative, then a subgroup of G need not be commutative.

Explanation

GATE→2022→Engineering Mathematics→Discrete Mathematics

Q18

NAT[1 Mark]

The number of arrangements of six identical balls in three identical bins is

Explanation

GATE→2022→Engineering Mathematics→Discrete Mathematics

Q19

MCQ[2 Marks]

Which one of the following is the closed form for the generating function of the sequence {a_n}_{n>=0} defined below?

(A)

\frac{x(1+x^2)}{(1-x^2)^2} + \frac{1}{1-x}

(B)

\frac{x(3-x^2)}{(1-x^2)^2} + \frac{1}{1-x}

(C)

\frac{2x}{(1-x^2)^2} + \frac{1}{1-x}

(D)

\frac{x}{(1-x^2)^2} + \frac{1}{1-x}

Explanation

Linear Algebra

11 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q20

MCQ[1 Mark]

For n > 1, the maximum multiplicity of any eigenvalue of an n x n matrix with elements from R is

(A)

n

(B)

n - 1

(C)

1

(D)

n + 1

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q21

MSQ[1 Mark]

Let n > 1. Consider an n x n matrix M with its elements from R. Let vector (0, 1, 0, ..., 0)^T be in the null space of M. Which of the following options is/are always correct?

(A)

Determinant of M is 1

(B)

Determinant of M is 0

(C)

Rank of M is 1

(D)

There are at least two non-zero vectors in the null space of M

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q22

MSQ[1 Mark]

System: x + ky = 1, kx + y = -1. Which are CORRECT?

(A)

There is exactly one value of k for which the above system of equations has no solution.

(B)

There exist an infinite number of values of k for which the system of equations has no solution.

(C)

There exists exactly one value of k for which the system of equations has exactly one solution.

(D)

There exists exactly one value of k for which the system of equations has an infinite number of solutions.

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q23

MCQ[2 Marks]

Matrix A = [[1, 1], [1, -1]]. Eigenvalues of A^13?

(A)

1, -1

(B)

$2\sqrt{2}, -2\sqrt{2}$

(C)

$4\sqrt{2}, -4\sqrt{2}$

(D)

$64\sqrt{2}, -64\sqrt{2}$

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q24

MCQ[1 Mark]

The product of all eigenvalues of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] is

(A)

0

(B)

0 [Wait, option B is 1, let's check determinant]

(C)

2

(D)

3

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q25

MSQ[2 Marks]

Let A be n x m matrix with m > n. Which of the following statements is/are TRUE for Ax = 0?

(A)

There exist at least m-n linearly independent solutions to this system

(B)

There exist m-n linearly independent vectors such that every solution is a linear combination...

(C)

There exists a non-zero solution in which at least m-n variables are 0

(D)

There exists a solution in which at least n variables are non-zero

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q26

MCQ[1 Mark]

Let A and B be 4x4 matrices given... Which one of the options given below is TRUE?

(A)

det(A) = det(B)

(B)

det(B) = -det(A)

(C)

det(A) = 0

(D)

det(AB) = det(A) + det(B)

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q27

NAT[1 Mark]

Let A be the adjacency matrix of the graph with vertices {1,2,3,4,5}. Let \lambda_1, ..., \lambda_5 be eigenvalues. The value of \lambda_1 + ... + \lambda_5 =

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q28

MCQ[1 Mark]

Consider the following two statements with respect to the matrices A_{m x n}, B_{n x m}, C_{n x n} and D_{n x n}... Which one of the following holds?

(A)

Statement 1 is correct and Statement 2 is wrong.

(B)

Statement 1 is wrong and Statement 2 is correct.

(C)

Both Statement 1 and Statement 2 are correct.

(D)

Both Statement 1 and Statement 2 are wrong.

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q29

MCQ[2 Marks]

Which one of the following is the correct combination of values for L32, U33, and x1?

(A)

L32 = 2, U33 = -1/2, x1 = -1

(B)

L32 = 2, U33 = 2, x1 = -1

(C)

L32 = -1/2, U33 = 2, x1 = 0

(D)

L32 = -1/2, U33 = -1/2, x1 = 0

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q30

MSQ[2 Marks]

Which of the following is/are the eigenvector(s) for the matrix given below?

(A)

(B)

(C)

(D)

Explanation

Calculus

7 Questions

GATE→2026→Engineering Mathematics→Calculus

Q31

NAT[1 Mark]

Consider the function f: R -> R defined as f(x) = c1 e^x - c2 log_e(1/x) if x > 0, 6 otherwise. If f is continuous at x = 0, then c1 + c2 =

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q32

MSQ[2 Marks]

Let f: R -> R be defined as f(x) = (|x|/2 - x)(x - |x|/2). Which of the following statements is/are true?

(A)

f has a local maximum

(B)

f has a local minimum

(C)

f' is continuous over R

(D)

f' is not differentiable over R

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q33

NAT[1 Mark]

If f(x) = ax + b for x < 1, x^2 + x + 1 for x >= 1 is differentiable everywhere, value of b?

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q34

MCQ[1 Mark]

Let f: R -> R be a function such that f(x) = max{x, x^3}, x in R. The set of all points where f(x) is NOT differentiable is

(A)

{-1, 1, 2}

(B)

{-2, -1, 1}

(C)

{0, 1}

(D)

{-1, 0, 1}

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q35

MSQ[1 Mark]

Let f(x) = x^3 + 15x^2 - 33x - 36 be a real-valued function. Which of the following statements is/are TRUE?

(A)

f(x) does not have a local maximum.

(B)

f(x) has a local maximum.

(C)

f(x) does not have a local minimum.

(D)

f(x) has a local minimum.

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q36

NAT[1 Mark]

The value of the definite integral \int_{-3}^{3}\int_{-2}^{2}\int_{-1}^{1}(4x^{2}y-z^{3})dz~dy~dx is

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q37

NAT[1 Mark]

The value of the following limit is lim_{x -> 0+} sqrt(x) / (1 - e^(2 sqrt(x)))

Explanation

Probability and Statistics

8 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q38

MCQ[1 Mark]

An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that there are equal number of red and blue balls after two steps is

(A)

1/4

(B)

1/3

(C)

1/2

(D)

2/3

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q39

NAT[2 Marks]

Let X be a random variable taking values in {1, 2, 3, 4, 5, 6, 7, 8} with Pr(X=1)=Pr(X=2)=Pr(X=5)=Pr(X=7)=1/6 and Pr(X=3)=Pr(X=4)=Pr(X=6)=Pr(X=8)=1/12. The expected value E[X] is

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q40

NAT[1 Mark]

Box with 4 regular coins and 1 fake coin. Toss twice, get two heads. Probability coin is fake?

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q41

NAT[2 Marks]

Probability density function P(x) = C x^2 for 1 <= x <= 4, 0 otherwise. Probability P(2 <= x <= 3)?

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q42

MCQ[1 Mark]

Consider a permutation sampled uniformly at random from {1, 2, ..., n} for some n >= 4. Let X be the event that 1 occurs before 2, and Y that 3 occurs before 4. Which one of the following is TRUE?

(A)

X must occur

(B)

Events X and Y are independent

(C)

Y must occur

(D)

Event X is more likely than Y

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q43

MSQ[1 Mark]

Let A and B be two events in a probability space with P(A)=0.3, P(B)=0.5, and P(A intersect B)=0.1. Which of the following statements is/are TRUE?

(A)

The two events A and B are independent

(B)

P(A \cup B) = 0.7

(C)

P(A \cap B^c) = 0.2

(D)

P(A^c \cap B^c) = 0.4

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q44

NAT[2 Marks]

A bag contains ... balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probability both drawn are red is

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q45

MSQ[2 Marks]

Consider a random experiment where two fair coins are tossed. A, B, C defined... Which of the following statements is/are TRUE?

(A)

A and B are independent.

(B)

A and C are independent.

(C)

B and C are independent.

(D)

Prob(B|C) = Prob(B)

Explanation