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Linear Algebra(9)Calculus(6)Differential Equations(5)Vector Analysis(6)Complex Analysis(8)Probability and Statistics(7)

Linear Algebra

9 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MSQ[1 Mark]

Consider the matrix $M=[\begin{matrix}2&1&1\\ 1&3&0\\ -1&a&b\end{matrix}].$ Which of the following options is/are TRUE if $det(M)\ne0$?

(A)

$b=-\frac{1}{2}$ $a=-\frac{1}{2}$

(B)

$a=\frac{1}{2}$ and $b=\frac{1}{2}$

(C)

$a=-3$ and $b=0$

(D)

$a=\frac{1}{2}$ and $b=-3$

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

Consider the matrix A below: $A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix}$ For which of the following combinations of $\alpha$, $\beta$, and $\gamma$, is the rank of A at least three?

(A)

Only (i), (iii), and (iv)

(B)

Only (iv)

(C)

Only (ii)

(D)

Only (i) and (iii)

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q3

NAT[2 Marks]

Consider the vectors $a = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, $b = \begin{bmatrix} 0 \\ 3\sqrt{2} \end{bmatrix}$. For real-valued scalar variable X, the value of $\min_x \|ax - b\|_2$ is ______ (rounded off to two decimal places). ($\|\cdot\|_2$ denotes the Euclidean norm)

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q4

NAT[1 Mark]

Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors $\{[2 \quad -3 \quad \alpha], [3 \quad -1 \quad 3], [1 \quad -5 \quad 7]\}$ does not form a basis of $\mathbb{R}^3$ is

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q5

MSQ[2 Marks]

Consider the matrix $\begin{bmatrix} 1 & k \\ 2 & 1 \end{bmatrix}$, where k is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

(A)

$\begin{bmatrix} 1 \\ -\sqrt{2/k} \end{bmatrix}$

(B)

$\begin{bmatrix} 1 \\ \sqrt{2/k} \end{bmatrix}$

(C)

$\begin{bmatrix} \sqrt{2k} \\ 1 \end{bmatrix}$

(D)

$\begin{bmatrix} \sqrt{2k} \\ -1 \end{bmatrix}$

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q6

MCQ[1 Mark]

Let $v_1 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}$ be two vectors. The value of the coefficient $\alpha$ in the expression $v_1 = \alpha v_2 + e$ which minimizes the length of the error vector $e$, is

(A)

$\frac{7}{2}$

(B)

$\frac{-2}{7}$

(C)

$\frac{2}{7}$

(D)

$\frac{-7}{2}$

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

Let the sets of eigenvalues and eigenvectors of a matrix B be $\{\lambda_k | 1 \le k \le n\}$ and $\{v_k | 1 \le k \le n\}$, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where $B = P^{-1}AP$, respectively, are

(A)

$\{\lambda_k \det(A) | 1 \le k \le n\}$ and $\{Pv_k | 1 \le k \le n\}$

(B)

$\{\lambda_k | 1 \le k \le n\}$ and $\{v_k | 1 \le k \le n\}$

(C)

$\{\lambda_k | 1 \le k \le n\}$ and $\{Pv_k | 1 \le k \le n\}$

(D)

$\{\lambda_k | 1 \le k \le n\}$ and $\{P^{-1}v_k | 1 \le k \le n\}$

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q8

MCQ[1 Mark]

Consider a system of linear equations $Ax = b$, where $A = \begin{bmatrix} 1 & -\sqrt{2} & 3 \\ -1 & \sqrt{2} & -3 \end{bmatrix}$, $b = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$. This system of equations admits

(A)

a unique solution for x

(B)

infinitely many solutions for x

(C)

no solutions for x

(D)

exactly two solutions for x

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q9

MCQ[2 Marks]

Let $\alpha, \beta$ be two non-zero real numbers and $v_1, v_2$ be two non-zero real vectors of size $3 \times 1$. Suppose that $v_1^T v_2 = 0$, $v_1^T v_1 = 1$, and $v_2^T v_2 = 1$. Let A be the $3 \times 3$ matrix given by: $A = \alpha v_1 v_1^T + \beta v_2 v_2^T$. The eigenvalues of A are

(A)

$0, \alpha, \beta$

(B)

$0, \alpha+\beta, \alpha-\beta$

(C)

$\alpha, \frac{\alpha+\beta}{2}, \sqrt{\alpha\beta}$

(D)

$0, 0, \sqrt{\alpha^2+\beta^2}$

Explanation

Calculus

6 Questions

GATE→2026→Engineering Mathematics→Calculus

Q10

NAT[2 Marks]

Consider the square region R in the X-Y plane as shown with the dark shading in the Figure. The value of $\iint_{R}(x^{2}+y^{2}-1)dxdy$ is ______ (rounded off to two decimal places)

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q11

MSQ[1 Mark]

Consider the function $f:\mathbb{R} \to \mathbb{R}$ defined as $f(x) = 2x^3 - 3x^2 - 12x + 1$. Which of the following statements is/are correct? (Here, $\mathbb{R}$ is the set of real numbers.)

(A)

f has no global maximizer

(B)

f has no global minimizer

(C)

$x = -1$ is a local minimizer of f

(D)

$x = 2$ is a local maximizer of f

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q12

MSQ[2 Marks]

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval [2, 8]. Let M and m denote, respectively, the maximum and the minimum values of $f(x)$ over the interval. Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality $\beta \le \int_2^8 f(x) dx \le \alpha$ is guaranteed to hold.

(A)

$\beta = 5m$, $\alpha = 7M$

(B)

$\beta = 6m$, $\alpha = 5M$

(C)

$\beta = 7m$, $\alpha = 6M$

(D)

$\beta = 7m$, $\alpha = 5M$

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q13

NAT[2 Marks]

The value of the integral $\iint_R xy \, dx \, dy$ over the region R, given in the figure, is _______ (rounded off to the nearest integer).

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q14

MCQ[2 Marks]

The function $f(x) = 8 \log_e x - x^2 + 3$ attains its minimum over the interval $[1, e]$ at $x =

(A)

2

(B)

1

(C)

e

(D)

$\frac{1+e}{2}$

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q15

NAT[2 Marks]

The value of the integral $\iint_D 3(x^2 + y^2) dx dy$, where D is the shaded triangular region shown in the diagram, is _______ (rounded off to the nearest integer).

Explanation

Differential Equations

5 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q16

MCQ[1 Mark]

Consider the differential equation $\dot{\vec{w}}=A\vec{w}$ , with $\vec{w}(t=0)=[\begin{matrix}1\\ 1\end{matrix}]$ If $\vec{w}(t)=e^{t}\vec{u}_{x}+e^{-2t}\vec{u}_{y}$ be the solution to the equation where $\vec{u}_{x}$ and $\vec{u}_{y}$ are unit vectors along the positive x and y axes respectively, then which of the following options is the correct matrix representing A?

(A)

$[\begin{matrix}1&0\\ 0&-2\end{matrix}]$

(B)

$[\begin{matrix}-1&0\\ 0&2\end{matrix}]$

(C)

$[\begin{matrix}0&-2\\ 1&0\end{matrix}]$

(D)

$[\begin{matrix}0&2\\ -1&0\end{matrix}]$

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q17

NAT[1 Mark]

The function $y(t)$ satisfies $t^2 y''(t) - 2ty'(t) + 2y(t) = 0$, where $y'(t)$ and $y''(t)$ denote the first and second derivatives of $y(t)$, respectively. Given $y'(0) = 1$ and $y'(1) = -1$, the maximum value of $y(t)$ over [0,1] is ______ (rounded off to two decimal places).

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q18

MCQ[1 Mark]

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where A and B are real constants determined by the initial condition. The corresponding differential equation is

(A)

$\frac{d^2 y}{dt^2} + 4\frac{dy}{dt} + 4y = f(t)$

(B)

$\frac{d^2 y}{dt^2} + 4y = f(t)$

(C)

$\frac{d^2 y}{dt^2} + 3\frac{dy}{dt} + 2y = f(t)$

(D)

$\frac{d^2 y}{dt^2} + 5\frac{dy}{dt} + 6y = f(t)$

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q19

NAT[1 Mark]

$x^2 - 12x + 37 = 0$. In a number system of base r, the equation has $x = 8$ as one of its solutions. The value of r is

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q20

MSQ[1 Mark]

Consider the following partial differential equation (PDE) $a \frac{\partial^2 f(x,y)}{\partial x^2} + b \frac{\partial^2 f(x,y)}{\partial y^2} = f(x,y)$, where a and b are distinct positive real numbers. Select the combination(s) of values of the real parameters $\xi$ and $\eta$ such that $f(x,y) = e^{\xi x + \eta y}$ is a solution of the given PDE.

(A)

$\xi = \frac{1}{\sqrt{2a}}, \eta = \frac{1}{\sqrt{2b}}$

(B)

$\xi = \frac{1}{\sqrt{a}}, \eta = 0$

(C)

$\xi = 0, \eta = 0$

(D)

$\xi = \frac{1}{\sqrt{a}}, \eta = \frac{1}{\sqrt{b}}$

Explanation

Vector Analysis

6 Questions

GATE→2026→Engineering Mathematics→Vector Analysis

Q21

MCQ[1 Mark]

A surface is given by $z^{2}=2x^{2}-y^{2}$ and $\vec{n}$ and $-\vec{n}$ are unit normal vectors to the surface at the point $\vec{P}=\hat{i}+\sqrt{2}\hat{k}.$ Which of the following vectors can be $\vec{n}$ where î, and $\hat{k}$ are the unit vectors along x, y and z axes, respectively?

(A)

$\hat{i}-\sqrt{2}\hat{k}$

(B)

$\frac{2}{3}\hat{i}-\frac{1}{3}\hat{k}$

(C)

$\sqrt{2}\hat{i}-\sqrt{3}\hat{k}$

(D)

$\frac{\sqrt{2}\hat{i}-\hat{k}}{\sqrt{3}}$

Explanation

GATE→2024→Engineering Mathematics→Vector Analysis

Q22

MSQ[1 Mark]

Let $\rho(x,y,z,t)$ and $u(x,y,z,t)$ represent density and velocity. Which of the following equations is/are equivalent to $\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho u) = 0$?

(A)

$\int_V \frac{\partial\rho}{\partial t} dv = - \iint_S \rho u \cdot \hat{n} ds$

(B)

$\int_V \frac{\partial\rho}{\partial t} dv = \iint_S \rho u \cdot \hat{n} ds$

(C)

$\int_V \frac{\partial\rho}{\partial t} dv = - \int_V \nabla \cdot (\rho u) dv$

(D)

$\int_V \frac{\partial\rho}{\partial t} dv = \int_V \nabla \cdot (\rho u) dv$

Explanation

GATE→2024→Engineering Mathematics→Vector Analysis

Q23

MCQ[2 Marks]

Consider the Earth to be a perfect sphere of radius R. Then the surface area of the region, enclosed by the $60^{\circ}\text{N}$ latitude circle, that contains the north pole in its interior is

(A)

$(2 - \sqrt{3})\pi R^2$

(B)

$\frac{(\sqrt{2}-1)\pi R^2}{2}$

(C)

$\frac{2\pi R^2}{3}$

(D)

$\frac{(2+\sqrt{3})\pi R^2}{8\sqrt{2}}$

Explanation

GATE→2024→Engineering Mathematics→Vector Analysis

Q24

MSQ[2 Marks]

Let $F_1, F_2,$ and $F_3$ be functions of $(x,y,z)$. Suppose that for every given pair of points A and B in space, the line integral is path-independent. Then which of the following is/are true?

(A)

For every closed path $\Gamma$, we have $\oint_\Gamma (F_1 dx + F_2 dy + F_3 dz) = 0$.

(B)

There exists a differentiable scalar function $f(x,y,z)$ such that $F_1 = \frac{\partial f}{\partial x}, F_2 = \frac{\partial f}{\partial y}, F_3 = \frac{\partial f}{\partial z}$.

(C)

$\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} = 0$.

(D)

$\frac{\partial F_3}{\partial y} = \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} = \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} = \frac{\partial F_1}{\partial y}$.

Explanation

GATE→2023→Engineering Mathematics→Vector Analysis

Q25

MCQ[1 Mark]

The rate of increase of a scalar field $f(x, y, z) = xyz$ in the direction $v = (2, 1, 2)$ at a point $(0, 2, 1)$ is

(A)

$\frac{2}{3}$

(B)

$\frac{4}{3}$

(C)

2

(D)

4

Explanation

GATE→2022→Engineering Mathematics→Vector Analysis

Q26

MCQ[1 Mark]

Consider the two-dimensional vector field $\vec{F}(x,y) = x\vec{i} + y\vec{j}$. A contour C in the x-y plane... The value of the closed path integral $\oint_C \vec{F}(x,y) \cdot (dx\vec{i} + dy\vec{j})$ is

(A)

0

(B)

1

(C)

$8 + 2\pi$

(D)

-1

Explanation

Complex Analysis

8 Questions

GATE→2026→Engineering Mathematics→Complex Analysis

Q27

MCQ[2 Marks]

Consider the two series, $S_{A}$ and $S_{B}$ where $S_{A}=\Sigma_{n=1}^{\infty}\frac{n^{2}}{2^{n}}$ $S_{B}=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\cdot\cdot\cdot$ Which of the following statements is correct for the two given series?

(A)

Both $S_{A}$ and $S_{B}$ converge.

(B)

Neither $S_{A}$ nor SB converges.

(C)

$S_{A}$ converges but $S_{B}$ does not converge.

(D)

$S_{B}$ converges but $S_{A}$ does not converge.

Explanation

GATE→2025→Engineering Mathematics→Complex Analysis

Q28

MCQ[1 Mark]

Consider the following series: (i) $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ (ii) $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ (iii) $\sum_{n=1}^{\infty} \frac{1}{n!}$ Choose the correct option.

(A)

Only (ii) converges

(B)

Only (ii) and (iii) converge

(C)

Only (iii) converges

(D)

All three converge

Explanation

GATE→2025→Engineering Mathematics→Complex Analysis

Q29

MSQ[2 Marks]

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle C in the complex plane is/are TRUE?

(A)

$\oint_C e^z dz = 0$

(B)

$\oint_C z^n dz = 0$, where n is an even integer

(C)

$\oint_C \cos z dz \neq 0$

(D)

$\oint_C \sec z dz \neq 0$

Explanation

GATE→2024→Engineering Mathematics→Complex Analysis

Q30

MCQ[2 Marks]

Let z be a complex variable. If $f(z) = \frac{\sin(\pi z)}{z^2(z-2)}$ and C is the circle in the complex plane with $|z| = 3$, then $\oint_C f(z)dz$ is

(A)

$\pi^2 j$

(B)

$j\pi(\frac{1}{2} - \pi)$

(C)

$j\pi(\frac{1}{2} + \pi)$

(D)

$-\pi^2 j$

Explanation

GATE→2023→Engineering Mathematics→Complex Analysis

Q31

MCQ[1 Mark]

Let $w^4 = 16j$. Which of the following cannot be a value of w?

(A)

$2e^{\frac{j2\pi}{8}}$

(B)

$2e^{\frac{j\pi}{8}}$

(C)

$2e^{\frac{j5\pi}{8}}$

(D)

$2e^{\frac{j9\pi}{8}}$

Explanation

GATE→2023→Engineering Mathematics→Complex Analysis

Q32

MCQ[1 Mark]

The value of the contour integral, $\oint_C \left(\frac{z+2}{z^2+2z+2}\right)dz$, where the contour C is $\{z : |z + 1 - \frac{3}{2}j| = 1\}$ taken in the counter clockwise direction, is

(A)

$-\pi(1+j)$

(B)

$\pi(1+j)$

(C)

$\pi(1-j)$

(D)

$-\pi(1-j)$

Explanation

GATE→2022→Engineering Mathematics→Complex Analysis

Q33

NAT[1 Mark]

A simple closed path C in the complex plane is shown in the figure. If $\oint_C \frac{2^z}{z^2-1} dz = -i\pi A$, where $i = \sqrt{-1}$, then the value of A is _______ (rounded off to two decimal places).

Explanation

GATE→2022→Engineering Mathematics→Complex Analysis

Q34

MSQ[2 Marks]

Consider the following series: $\sum_{n=1}^{\infty} \frac{n^d}{c^n}$. For which of the following combinations of c, d values does this series converge?

(A)

$c = 1, d = -1$

(B)

$c = 2, d = 1$

(C)

$c = 0.5, d = -10$

(D)

$c = 1, d = -2$

Explanation

Probability and Statistics

7 Questions

GATE→2025→Engineering Mathematics→Probability and Statistics

Q35

MCQ[1 Mark]

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

(A)

$\frac{2}{3}$

(B)

$\frac{1}{3}$

(C)

$\frac{1}{2}$

(D)

1

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q36

MSQ[2 Marks]

The random variable X takes values in $\{-1, 0, 1\}$ with probabilities $P(X=-1) = P(X=1) = \alpha$ and $P(X=0) = 1-2\alpha$, where $0 < \alpha < \frac{1}{2}$: Let $g(\alpha)$ denote the entropy function of X (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

(A)

$g(0.4) > g(0.3)$

(B)

$g(0.3) > g(0.4)$

(C)

$g(0.3) > g(0.25)$

(D)

$g(0.25) > g(0.3)$

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q37

NAT[2 Marks]

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable X denote the sum of the outcomes obtained. The expectation of X is ______ (rounded off to two decimal places).

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q38

NAT[2 Marks]

X and Y are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by: $P(X=0, Y=0) = 0.06$, $P(X=0, Y=1) = 0.14$, $P(X=1, Y=0) = 0.24$, $P(X=1, Y=1) = 0.56$. The mutual information $I(X;Y)$ is ______ (rounded off to two decimal places).

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q39

NAT[1 Mark]

Suppose X and Y are independent and identically distributed random variables that are distributed uniformly in the interval [0,1]. The probability that $X \ge Y$ is

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q40

NAT[1 Mark]

The bar graph shows the frequency of the number of wickets taken in a match by a bowler in her career. The median number of wickets taken by the bowler in a match is _______ (rounded off to one decimal place).

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q41

NAT[2 Marks]

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, $f(x)$ as shown in the figure... If $\alpha^*$ and $\beta^*$ are the respective choices for $\alpha$ and $\beta$ that minimize the mean square quantization error, then $(\alpha^* - \beta^*) =$ _______ (rounded off to two decimal places).

Explanation