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Linear Algebra(16)Calculus(14)Differential Equations(5)Complex Variables(4)Probability and Statistics(8)

Linear Algebra

16 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MSQ[1 Mark]

Matrices A and B share eigenvalue 2 with the same eigenvector x. Which statements are correct?

(A)

Determinant(A - 2I) = 0

(B)

Determinant(B - 2I) = 0

(C)

Determinant(A + B - 2I) = 0

(D)

Determinant(A + B - 4I) = 0

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

NAT[1 Mark]

For a real skew-symmetric matrix A and unit vector x, evaluate x^T A x.

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q3

MCQ[2 Marks]

For a collection of p column vectors with n real entries, which statement is always correct?

(A)

If p > n then the column vectors must be linearly dependent

(B)

If p > n then the column vectors must be linearly independent

(C)

If p = n then the column vectors must be orthogonal

(D)

If p < n then the column vectors must be linearly independent

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q4

MSQ[2 Marks]

For a real n x n orthogonal matrix A with unit column norms, which statements are correct?

(A)

The value of the determinant of A is either +1 or -1

(B)

The eigenvalues of A have modulus 1

(C)

||Ax|| = ||x|| for all x, and (Ax)^T(Ay) != x^T y for distinct x, y

(D)

||Ax|| = ||x|| for all x, and (Ax)^T(Ay) = x^T y for all x, y

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q5

MSQ[2 Marks]

If Ax = b has a unique solution for square matrix A, which statements are correct?

(A)

A^{-1} exists

(B)

The system of equations A^m x = b also has a unique solution for m = 1, 2, 3...

(C)

rank(A) < rank([A|b])

(D)

rank(A) = rank(A^m)

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q6

MCQ[1 Mark]

Let v_1 and v_2 be eigenvectors corresponding to distinct eigenvalues of a 3x3 real symmetric matrix. Which statement is true?

(A)

v_1^T v_2 != 0

(B)

v_1^T v_2 = 0

(C)

v_1 + v_2 = 0

(D)

v_1 - v_2 = 0

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

Given matrix A and vector b, determine whether the system Ax = b has a unique, infinite, finite, or no solution.

(A)

a unique solution.

(B)

infinitely many solutions.

(C)

a finite number of solutions.

(D)

no solution.

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q8

MCQ[1 Mark]

Let P be the given 3x3 matrix and I be the identity matrix. P^2 is equal to

(A)

2P - I

(B)

P + I

(C)

P

(D)

I

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q9

MCQ[1 Mark]

Which of the given 3x3 matrices has an inverse (i.e., non-zero determinant)?

(A)

(A)

(B)

(B)

(C)

(C)

(D)

(D)

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q10

NAT[1 Mark]

Find the sum of eigenvalues of the matrix A = ([1 2; 3 4])^2.

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q11

MCQ[1 Mark]

For vector w = [1 2 3]^T, find a vector normal to the plane defined by w^T X = 1.

(A)

[-2 -2 2]^T

(B)

[3 0 -1]^T

(C)

[3 2 1]^T

(D)

[1 2 3]^T

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q12

MCQ[1 Mark]

Find the 2D rotation/transformation matrix A that relates vector u to vector v (Au = v).

(A)

[4/5 3/5; -3/5 4/5]

(B)

[3/5 4/5; -4/5 3/5]

(C)

[4/5 -3/5; 3/5 4/5]

(D)

[3/5 -4/5; 4/5 3/5]

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q13

MSQ[2 Marks]

For the curve |x_1|^p + |x_2|^p = 1 in 2-D space for p > 0, which statements are true?

(A)

When p = 2 the area enclosed by the curve is pi.

(B)

When p tends to infinity, the area enclosed by the curve tends to 4.

(C)

When p tends to 0, the area enclosed by the curve is 1.

(D)

When p = 1, the area enclosed by the curve is 2.

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q14

MCQ[1 Mark]

Consider a 3x3 matrix A with elements a_{i,j} = (i - j)^3. The matrix A is

(A)

symmetric.

(B)

skew-symmetric.

(C)

unitary.

(D)

null.

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q15

MCQ[2 Marks]

For square matrix A with eigenvalue lambda and eigenvector v, are 'e^lambda is eigenvalue of e^A' and 'v is eigenvector of e^A' correct?

(A)

Statement 1 is true and statement 2 is false.

(B)

Statement 1 is false and statement 2 is true.

(C)

Both the statements are correct.

(D)

Both the statements are false.

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q16

MCQ[2 Marks]

For matrix A = [1 0 0; 0 4 -2; 0 1 1], if 6A^{-1} = A^2 + cA + dI, find (c + d).

(A)

5

(B)

17

(C)

-6

(D)

11

Explanation

Calculus

14 Questions

GATE→2026→Engineering Mathematics→Calculus

Q17

NAT[1 Mark]

Evaluate the surface integral of F(x, y, z) = sin(y) x_hat + cos(x) y_hat + 5 z_hat over the unit sphere S centered at the origin.

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q18

MCQ[2 Marks]

For the periodic waveform f(x) constructed from g(x), if the sum of third harmonic Fourier coefficients is given, find k.

(A)

1

(B)

1/2

(C)

1/3

(D)

1/4

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q19

NAT[2 Marks]

Evaluate the improper integral (1/pi) * integral from 0 to infinity of (x^2026 / ((1 + x^2026)(1 + x^2))) dx.

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q20

MCQ[1 Mark]

Consider points in R^2 minimizing f(x, y) = (x + y - 1)^2 + (x + y)^2. What is true about set S?

(A)

The number of elements in the set S is finite and more than one.

(B)

The number of elements in the set S is infinite.

(C)

The set S is empty.

(D)

The number of elements in the set S is exactly one.

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q21

NAT[2 Marks]

Find the directional derivative of V(x, y) = x^2 + x + y^2 + 1 at the origin in the direction of point (1, 2).

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q22

MCQ[2 Marks]

Find the directional derivative of f(x,y,z) = 2 ln(xy) + ln(yz) + 3 ln(xz) at (1,1,1) in the direction of vector u = 2i + j + 2k.

(A)

0

(B)

7/(5*sqrt(2))

(C)

7

(D)

21

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q23

MSQ[2 Marks]

If f(t) has a positive second derivative everywhere (-inf < t < inf), which statements are always true?

(A)

f(t) has at least one local minimum.

(B)

f(t) cannot have two distinct local minima.

(C)

f(t) has at least one local maximum.

(D)

The minimum value of f(t) cannot be negative.

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q24

MSQ[2 Marks]

For f(t) = (max(0, t))^2, which of the following statements is/are true?

(A)

f(t) is not differentiable.

(B)

f(t) is differentiable and its derivative is continuous.

(C)

f(t) is differentiable but its derivative is not continuous.

(D)

f(t) and its derivative are differentiable.

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q25

NAT[2 Marks]

Find the magnitude of the line integral of F = -y i + x j around the cardioid r = 1 + cos(theta).

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q26

NAT[2 Marks]

Find the magnitude of the maximum rate of change of f(x_1, x_2) = x_1^2 + 2x_2^2 + 3x_1 + 3x_2 + x_1 x_2 + 1 at the point (1,1).

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q27

MCQ[1 Mark]

For H(s) = (As + B)/(s^2 + Cs + D) with positive constants, which filter type can this system NOT operate as?

(A)

low pass filter.

(B)

high pass filter.

(C)

band pass filter.

(D)

an integrator.

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q28

MCQ[2 Marks]

For f(x) = integral from 0 to x of e^t (t-1)(t-2) dt, in which interval does f(x) decrease?

(A)

x in (1, 2)

(B)

x in (2, 3)

(C)

x in (0, 1)

(D)

x in (0.5, 1)

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q29

MCQ[2 Marks]

Find the direction in which f(x,y,z) = 4x^2 + 7xy + 3xz^2 increases most rapidly at point P(1, 0, 2).

(A)

20i + 7j

(B)

20i + 7j + 12k

(C)

20i + 12k

(D)

20i

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q30

MCQ[2 Marks]

Let R be a region enclosed by closed curve C counter-clockwise. Which expression does NOT represent the area of R?

(A)

double_integral_R dxdy

(B)

oint_C x dy

(C)

oint_C y dx

(D)

0.5 * oint_C (x dy - y dx)

Explanation

Differential Equations

5 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q31

MCQ[1 Mark]

Consider the differential equation t^2 (d^2y/dt^2) + 7t (dy/dt) + 8ty = 10 sin(t). Which statement is correct?

(A)

It is a linear differential equation

(B)

It is a nonlinear differential equation

(C)

It is a time-invariant differential equation

(D)

It is a second-order partial differential equation

Explanation

GATE→2026→Engineering Mathematics→Differential Equations

Q32

MCQ[2 Marks]

Solve d^2y/dx^2 + dy/dx + y = 0 with y(0) = 1, y'(0) = 1.

(A)

y(x) = exp(-x/2)(cos((sqrt(3)x)/2) + sqrt(3)sin((sqrt(3)x)/2))

(B)

y(x) = exp(-x/2)(cos((sqrt(3)x)/2) + (1/sqrt(3))sin((sqrt(3)x)/2))

(C)

y(x) = exp(-x/2)(cos((sqrt(3)x)/2) - (1/sqrt(3))sin((sqrt(3)x)/2))

(D)

y(x) = exp(-x/2)(cos((sqrt(3)x)/2) - sqrt(3)sin((sqrt(3)x)/2))

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q33

NAT[2 Marks]

Solve the system of differential equations with given initial conditions at t = 1.

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q34

MSQ[2 Marks]

Which of the given differential equations are nonlinear?

(A)

t x(t) + dx(t)/dt = t^2 e^t, x(0) = 0

(B)

0.5 e^t + x(t) dx(t)/dt = 0, x(0) = 0

(C)

x(t) cos(t) - (dx(t)/dt) sin(t) = 1

(D)

x(t) + e^{(dx(t)/dt)} = 1

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q35

NAT[1 Mark]

Solve dy/dt = e^{-alpha t} / (2 + alpha t) with alpha = 0.01 and y(0) = 0 at t = 1.

Explanation

Complex Variables

4 Questions

GATE→2026→Engineering Mathematics→Complex Variables

Q36

NAT[2 Marks]

Evaluate the magnitude of the contour integral oint_C ((z+1)^2 / ((z-i)(z-2))) dz over |z - 2 - i| = 3/2.

Explanation

GATE→2025→Engineering Mathematics→Complex Variables

Q37

NAT[2 Marks]

Evaluate the contour integral of f(z) = jz over clockwise circle |z| = 1.

Explanation

GATE→2024→Engineering Mathematics→Complex Variables

Q38

MSQ[1 Mark]

Which of the given complex functions are analytic on the entire complex plane?

(A)

f(z) = j Re(z)

(B)

f(z) = Im(z)

(C)

f(z) = e^|z|

(D)

f(z) = z^2 - z

Explanation

GATE→2024→Engineering Mathematics→Complex Variables

Q39

NAT[1 Mark]

Find the coefficient of z^5 in the Taylor series expansion of f(z) = cos(z) + e^{z^2} about the origin.

Explanation

Probability and Statistics

8 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q40

NAT[2 Marks]

Find the value of alpha that minimizes E((X - alpha Y)^2) given expectations of X, Y, X^2, Y^2, and XY.

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q41

MCQ[1 Mark]

Given discrete random variables X and Y with specified joint probabilities, find E[Y | X = 1].

(A)

1/4

(B)

1/2

(C)

1/8

(D)

1/3

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q42

MCQ[2 Marks]

If Y = X^2 and X is uniform on (0, 1], find the probability density function P_Y(y).

(A)

P_Y(y) = 1/(2*sqrt(y)) for y in (0, 1], 0 otherwise

(B)

P_Y(y) = 1 for y in (0, 1], 0 otherwise

(C)

P_Y(y) = 1.5*sqrt(y) for y in (0, 1], 0 otherwise

(D)

P_Y(y) = 2y for y in (0, 1], 0 otherwise

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q43

MSQ[1 Mark]

For X uniformly distributed over integers from -10 to 10, which transformations are uniformly distributed?

(A)

X^2

(B)

X^3

(C)

(X-5)^2

(D)

(X+10)^2

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q44

MCQ[1 Mark]

For sample variance estimators hat{sigma}_1^2 (divided by N) and hat{sigma}_2^2 (divided by N-1), which statement is true?

(A)

E(hat{sigma}_2^2) = sigma_0^2

(B)

E(hat{sigma}_2) = sigma_0

(C)

E(hat{sigma}_1^2) = sigma_0^2

(D)

E(hat{sigma}_1) = E(hat{sigma}_2)

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q45

NAT[1 Mark]

Find the slope of the best-fit straight line in the least squares sense for points (-1, 0.8), (0, 2.2), and (1, 2.8).

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q46

NAT[2 Marks]

For a biased coin where the expected number of trials for the first head is 4, find the probability that the first head occurs on the second trial.

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q47

NAT[2 Marks]

If the probability density function of random variable x is f(x) = a * e^{-2|x|}, find the value of a.

Explanation