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Linear Algebra & Calculus(27)Differential Equations, Complex Analysis & Probability(19)

Linear Algebra & Calculus

27 Questions

GATE→2026→Engineering Mathematics→Linear Algebra & Calculus

Q1

MCQ[1 Mark]

The determinant of the matrix $$\begin{pmatrix} 1 & p & q+r \\ 1 & q & r+p \\ 1 & r & p+q \end{pmatrix}$$ is

(A)

pqr

(B)

$p+q+r$

(C)

$pq+qr+rp$

(D)

0

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra & Calculus

Q2

MCQ[1 Mark]

The gradient of a scalar field is

(A)

a vector

(B)

a scalar

(C)

a second order tensor

(D)

always zero

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra & Calculus

Q3

MCQ[1 Mark]

Which ONE of the following curves represents the equation $y = e^{-x}$? Figures not to scale

(A)

Curve A

(B)

Curve B

(C)

Curve C

(D)

Curve D

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra & Calculus

Q4

MSQ[2 Marks]

Which of the following statements is/are TRUE regarding matrices?

(A)

Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

(B)

The determinant of a square matrix is same as that of its transpose.

(C)

The product of a square matrix and its inverse results in the identity matrix.

(D)

Eigenvalues of a square matrix are always real and positive.

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra & Calculus

Q5

NAT[2 Marks]

A differentiable function $f(x)$ is such that $f(-0.1) = 2$, $f(0) = 1$, and $f(0.1) = 2$. Then, $\frac{d^2f}{dx^2}$ at $x = 0$ is (answer in integer).

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra & Calculus

Q6

NAT[2 Marks]

Consider the function $f(x,y) = x^3 + y^2$. If $\vec{F} = \nabla f$, then $\int \vec{F} \cdot d\mathbf{r}$ evaluated from $(0, 0)$ to $(1, 2)$ is (answer in integer).

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra & Calculus

Q7

MCQ[1 Mark]

The value of $\lim_{t\to 1}\frac{\ln t}{t^{2}-1}$ is

(A)

$\frac{1}{2}$

(B)

0

(C)

1

(D)

08

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra & Calculus

Q8

MSQ[1 Mark]

Consider the function $f(x)=|x|-1$. Which of the following statements is/are true in the interval $[-10, 10]$?

(A)

The function is differentiable in the domain.

(B)

The function is continuous in the domain.

(C)

The function is not differentiable in the domain.

(D)

The function is not continuous in the domain.

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra & Calculus

Q9

NAT[2 Marks]

The value of $\iint x^{2}y\,dx\,dy$ evaluated over the specified domain is (answer in integer).

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra & Calculus

Q10

NAT[2 Marks]

Let $M = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ and $K = \begin{bmatrix} 2 & -1 \\ -1 & 1 \end{bmatrix}$ satisfy the eigenvalue problem given by $[M - \alpha K]\varphi = 0$.\nThe lowest eigenvalue $\alpha$ is (rounded off to two decimal places).

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra & Calculus

Q11

NAT[2 Marks]

The gradient of $y = 3x^{2}\sin(2x)$ at $x = 0.2$ is (rounded off to three decimal places).

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra & Calculus

Q12

MCQ[1 Mark]

The tangent plane to the surface $x^{2}+y^{2}+z=9$ at the point $(1, 2, 4)$ is

(A)

$x+4y+z=14$

(B)

$4x+2y+z=12$

(C)

$x+4y+2z=17$

(D)

$4x+y+2z=14$

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra & Calculus

Q13

MCQ[1 Mark]

The value of the line integral $\oint x^{2}dx+2x\,dy$ along the ellipse $4x^{2}+y^{2}=4$ oriented in the counterclockwise sense is

(A)

$\pi$

(B)

$2\pi$

(C)

$4\pi$

(D)

$8\pi$

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra & Calculus

Q14

MCQ[1 Mark]

The system of linear equations: $x + 2y + 3z = 4$ $2x - y - 2z = a^2$ $-x - 7y - 11z = a$ is consistent for specific values of a.

(A)

Option A

(B)

Option B

(C)

Option C

(D)

-3 and 4

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra & Calculus

Q15

NAT[1 Mark]

The arc length of the one arch of the cycloid given by $x=t-\sin t$ and $y=1-\cos t$ is

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra & Calculus

Q16

MCQ[2 Marks]

Consider the matrices $M = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & 0 \end{pmatrix}$. Which one of the following is true?

(A)

M is not diagonalizable but N is diagonalizable

(B)

Both M and N are not diagonalizable

(C)

Both M and N are diagonalizable

(D)

M is diagonalizable but N is not diagonalizable

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra & Calculus

Q17

MSQ[2 Marks]

Consider the function $f(x,y) = x^{4} + y^{4} - 4xy + 1$. Which of the following is/are correct?

(A)

The minimum value of f occurs at (0,0)

(B)

The point (0,0) is a point of inflection

(C)

f has three critical points

(D)

The minimum value of f is -1

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra & Calculus

Q18

NAT[2 Marks]

The value of the triple integral $\iint (xy^{2}+yz^{3})\,dx\,dy\,dz$ over the region given by $-1 \le x \le 1$, $3 \le y \le 4$, $0 \le z \le 2$, is

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra & Calculus

Q19

MCQ[1 Mark]

Consider the function $z = \tan^{-1}\left(\frac{x}{y}\right)$ where $x = u\sin v$ and $y = u\cos v$.\nThe partial derivative $\frac{\partial z}{\partial v}$ is

(A)

0

(B)

1

(C)

2

(D)

3

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra & Calculus

Q20

MCQ[1 Mark]

Consider the function $z = x^{3}-2x^{2}y+xy^{2}+1$. The directional derivative of z at the point $(1, 2)$ along the direction $3\hat{i}+4\hat{j}$ is

(A)

0

(B)

-1

(C)

1

(D)

-2

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra & Calculus

Q21

MCQ[2 Marks]

The value of the surface integral $\iint (x^{2}\,dy\,dz+y^{2}\,dz\,dx+z^{2}\,dx\,dy)$ over the surface of the cube given by $0 \le x \le 2$, $0 \le y \le 2$, $0 \le z \le 2$ is

(A)

12

(B)

24

(C)

36

(D)

48

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra & Calculus

Q22

MCQ[2 Marks]

If the system of linear equations, $x-ay-z=0$, $ax-y-z=0$, $x+y-z=0$, has infinite number of solutions, then the possible values of a are

(A)

0, 1

(B)

-1, 2

(C)

-1, 1

(D)

0, -1

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra & Calculus

Q23

MSQ[2 Marks]

Let $L = \begin{pmatrix} 3 & -1 & -1 & -1 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 3 & 0 \\ -1 & 0 & -1 & 1 \end{pmatrix}$. Which of the following are TRUE?

(A)

The matrix L is row equivalent to the specified matrix option A

(B)

The linear system Lx=b has a solution for all b

(C)

For b != [1, 1, 1, 1]^T, the system has a solution

(D)

Rank of the matrix L is 3

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra & Calculus

Q24

MCQ[1 Mark]

Let A be a real non-zero square matrix of order n. If the homogeneous system of linear equations $Ax=0$ has only trivial solution, then

(A)

matrix A is singular

(B)

determinant of A is zero

(C)

$\lambda=0$ is an eigenvalue of A

(D)

for any n-vector b, the system of linear equations $Ax=b$ has a unique solution

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra & Calculus

Q25

NAT[1 Mark]

Let $f(x,y,z)=xy+yz+xz$. If a point $(0, 0, \alpha)$ lies on the tangent plane to the surface $f(x,y,z)=3$ at the point $(1, 1, 1)$, then the value of $\alpha$ is

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra & Calculus

Q26

MCQ[2 Marks]

The value of the line integral $\oint_{c}-3y\,dx+3x\,dy+z\,dz$\nalong the circle C: $x^{2}+y^{2}=1$, $z=1$ oriented in the clockwise sense as seen from the origin, is

(A)

$2\pi$

(B)

$4\pi$

(C)

$6\pi$

(D)

$8\pi$

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra & Calculus

Q27

MSQ[2 Marks]

Let $M = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{pmatrix}$.\nWhich of the following are TRUE?

(A)

M is singular

(B)

$M^{-1}=\frac{1}{4}M^{2}-\frac{3}{2}M+\frac{9}{4}I$ where I is the identity matrix of order 3

(C)

M has three distinct eigenvalues

(D)

M has three linearly independent eigenvectors

Explanation

Differential Equations, Complex Analysis & Probability

19 Questions

GATE→2026→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q28

MCQ[1 Mark]

The Fourier series expansion of the function $f(x) = |\cos(x)|$ in the interval $(-\pi, \pi)$ is $\alpha + \beta \left[\frac{1}{3}\cos(2x) - \frac{1}{15}\cos(4x) + \dots\right]$. The values of $\alpha$ and $\beta$ respectively are

(A)

$2/\pi$ and $4/\pi$

(B)

$4/\pi$ and $2/\pi$

(C)

$\pi/2$ and $\pi/4$

(D)

$\pi/4$ and $\pi/2$

Explanation

GATE→2026→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q29

MSQ[1 Mark]

Which of the following ordinary differential equations is/are linear?

(A)

$(x+1)\frac{dy}{dx} - y = e^x(x+1)^2$

(B)

$\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$

(C)

$\frac{d^2y}{dx^2} + n^2x = 0$, where n is a constant

(D)

$xy\frac{dy}{dx} = 1 + x + y + xy$

Explanation

GATE→2026→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q30

NAT[2 Marks]

Consider a case where two identical six-faced fair dice are rolled simultaneously. The probability of getting even numbers in both AND a sum equal to 5 and above is (rounded off to two decimal places).

Explanation

GATE→2025→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q31

MCQ[1 Mark]

The Fourier series expansion of $f(x)=\sin^{2}x$ in the interval $(-\pi,\pi)$ is

(A)

$f(x)=\frac{1}{2}-\frac{1}{2}\cos 2x$

(B)

$f(x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\cos nx}{n^{2}}$

(C)

$f(x)=2\sum_{n=1}^{\infty}\frac{\sin 2nx}{n}$

(D)

$f(x)=\frac{1}{2}-\frac{4}{\pi^{2}}\sum_{n=1}^{\infty}\frac{\cos(2n-1)\pi x}{(2n-1)^{2}}$

Explanation

GATE→2025→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q32

MCQ[1 Mark]

For all real values of x and y, the partial differential equation in terms of $\psi(x,y)$ given by $\frac{\partial^{2}\psi}{\partial x^{2}}+2\frac{\partial^{2}\psi}{\partial x\partial y}-3\frac{\partial^{2}\psi}{\partial y^{2}}=0$ is

(A)

hyperbolic

(B)

parabolic

(C)

elliptic

(D)

elliptic within the region, $x^{2}-y<0$

Explanation

GATE→2025→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q33

NAT[1 Mark]

A box contains 12 red and 8 blue balls. Two balls are drawn randomly from the box without replacement.\nThe probability of drawing a pair of balls having the same color is (rounded off to three decimal places).

Explanation

GATE→2025→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q34

MCQ[2 Marks]

Consider $f(t)=\cos(at)$, where a is a real constant. The Laplace transform of $f(t)$ is

(A)

$\frac{a}{s^{2}+a^{2}}$

(B)

$\frac{s}{s^{2}+a^{2}}$

(C)

$\frac{a}{s^{2}-a^{2}}$

(D)

$\frac{s}{s^{2}-a^{2}}$

Explanation

GATE→2024→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q35

MCQ[1 Mark]

The value of the contour integral $\oint \frac{dz}{2z - z^{2}}$ along the circle $|z|=1$, oriented in the counterclockwise sense is

(A)

$\pi i$

(B)

0

(C)

$2\pi i$

(D)

$4\pi i$

Explanation

GATE→2024→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q36

MSQ[2 Marks]

Consider the $2\pi$-periodic function defined by $f(x) = \begin{cases} -1 & \text{if } -\pi < x \le 0 \\ 1 & \text{if } 0 < x < \pi \end{cases}$ Which of the following is/are correct about its Fourier series expansion $\frac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx)$?

(A)

$a_n = \frac{1}{n} \forall n = 1, 2, \dots$

(B)

$a_0 = 0$

(C)

$b_n = \frac{4}{n\pi}$ if n is odd

(D)

$b_n = -\frac{4}{n\pi}$ if n is even

Explanation

GATE→2024→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q37

NAT[2 Marks]

If X is a Poisson random variable with mean $\mu = 1$ then the conditional probability of the event $\{X \ge 2\}$ given that the event $\{X \ge 4\}$ has occurred, is (rounded off to two decimal places).

Explanation

GATE→2023→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q38

MSQ[1 Mark]

Let X be any random variable and $Y = -2X + 3$.\nIf $E[Y] = 1$ and $E[Y^2] = 9$, then which of the following are TRUE?

(A)

$E[X] = 1$

(B)

$E[X] = -2$

(C)

$\text{Var}(X) = 1$

(D)

$\text{Var}(X) = 2$

Explanation

GATE→2023→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q39

MSQ[1 Mark]

Consider the contour integral $\oint \frac{dz}{z^{4}+z^{3}-2z^{2}}$ along the curve $|z|=3$ oriented in the counterclockwise direction. If $\text{Res}[f,z_{0}]$ denotes the residue of $f(z)$ at the point $z_{0}$, then which of the following are TRUE?

(A)

$\text{Res}[f,0] = -1/4$

(B)

$\text{Res}[f,1] = 1/3$

(C)

$\text{Res}[f,-2] = -1/12$

(D)

$\text{Res}[f,2] = -1$

Explanation

GATE→2023→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q40

NAT[1 Mark]

If a random variable X has the probability density function $f(x) = \begin{cases} \frac{5}{32}x^{4} & \text{if } 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases}$\nand if $Y = X^{2}$ then the expected value of Y is (round off to one decimal place)

Explanation

GATE→2023→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q41

NAT[2 Marks]

If $y(x)$ is the solution of the differential equation $(1+x^{2})y''-2xy'=0$\nsatisfying $y(0)=0$ and $y'(0)=3$, then $y(1)$ equals

Explanation

GATE→2022→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q42

MCQ[1 Mark]

Let $z = x+iy$, where x and y are real numbers. Consider the complex functions: $f(z) = (x^{2}-y^{2})+i~2xy$\nand $g(z) = 2xy+i(x^{2}-y^{2})$.\nThen on the complex plane,

(A)

$f(z)$ is analytic and $g(z)$ is not analytic

(B)

both $f(z)$ and $g(z)$ are analytic

(C)

both $f(z)$ and $g(z)$ are not analytic

(D)

$f(z)$ is not analytic and $g(z)$ is analytic

Explanation

GATE→2022→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q43

MCQ[1 Mark]

If a population has exponential distribution with mean 1, then its median is

(A)

e

(B)

1

(C)

$\log_{e}2$

(D)

$\log_{e}3$

Explanation

GATE→2022→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q44

MSQ[1 Mark]

If $\frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos nx$ is the Fourier cosine series of the function $f(x)=\sin x$, $0<x<\pi$, then which of the following are TRUE?

(A)

$a_{0}+a_{1}=4/\pi$

(B)

$a_{0}=4/\pi$

(C)

$a_{0}+a_{1}=2/\pi$

(D)

$a_{1}=2/\pi$

Explanation

GATE→2022→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q45

NAT[2 Marks]

Let $y(x)$ be the solution of the differential equation $y^{\prime\prime}-4y^{\prime}-12y=3e^{5x}$\nsatisfying $y(0)=\frac{18}{7}$ and $y^{\prime}(0)=-\frac{1}{7}$.\nThen $y(1)$ is (rounded off to nearest integer).

Explanation

GATE→2022→Engineering Mathematics→Differential Equations, Complex Analysis & Probability

Q46

NAT[2 Marks]

We have two coins. One is biased with the probability for head being 1.0 and the other is a fair coin. One coin is chosen at random and is tossed twice. If we obtain head both times, then the probability of the chosen coin being a fair coin is (correct to one decimal place).

Explanation