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Linear Algebra(8)Calculus(14)Differential Equations(6)Complex Variables(4)Probability and Statistics(6)Numerical Methods(5)

Linear Algebra

8 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

Which of the following options is CORRECT for the eigenvalues ($\lambda$) of the given matrix? $$\begin{bmatrix} 8 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$$

(A)

$\lambda = 1, 1, 1$

(B)

$\lambda = 8, 5, 7$

(C)

$\lambda = 0, 0, 0$

(D)

$\lambda = 40, 1, 35$

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

For a given matrix $A = \begin{bmatrix} 4 & 4 \\ 0 & 4 \\ -4 & 4 \end{bmatrix}$, and $A^T$ representing its transpose, which of the following options is the CORRECT representation of $A^T A$?

(A)

$\begin{bmatrix} 4 & 0 \\ 0 & -4 \end{bmatrix}$

(B)

$\begin{bmatrix} 32 & 0 \\ 0 & 48 \end{bmatrix}$

(C)

$\begin{bmatrix} 32 & 0 \\ 0 & 0 \\ 0 & 48 \end{bmatrix}$

(D)

$\begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & -4 \end{bmatrix}$

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q3

MCQ[1 Mark]

Let $\vec{A} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{B} = \hat{i} + \hat{j}$ where $\hat{i}$, $\hat{j}$, and $\hat{k}$ are unit vectors. The projection of $\vec{B}$ on $\vec{A}$ is

(A)

$\frac{1}{\sqrt{12}}$

(B)

$\frac{1}{\sqrt{6}}$

(C)

$\sqrt{6}$

(D)

$\frac{1}{\sqrt{2}}$

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q4

MCQ[2 Marks]

The eigenvalues of the matrix $\begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix}$ are $\lambda_1$, $\lambda_2$, and $\lambda_3$. The value of $\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)$ is

(A)

11

(B)

45

(C)

495

(D)

135

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q5

MSQ[1 Mark]

For the given matrix $Q = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{bmatrix}$, which of the following statements is/are true?

(A)

Q is an orthogonal matrix

(B)

$Q^T = Q^{-1}$

(C)

Q is a singular matrix

(D)

Q is a symmetric matrix

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q6

NAT[1 Mark]

If $P = \begin{bmatrix} 2 & -1 \\ 2 & 2 \end{bmatrix}$, the product of the eigenvalues of $P$ is

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

NAT[1 Mark]

Let $X$ be a $3 \times 3$ matrix with $\det(X) = 5$. The determinant of matrix $Y$ obtained by multiplying row 2 by 2 and row 3 by 3 is

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q8

MSQ[1 Mark]

Which of the following properties of Matrix $A = \begin{bmatrix} 1 & 0.5 & 0 \\ 0.5 & 1 & 0.5 \\ 0 & 0.5 & 1 \end{bmatrix}$ are CORRECT?

(A)

Singular

(B)

Positive definite

(C)

Symmetric

(D)

Diagonal

Explanation

Calculus

14 Questions

GATE→2026→Engineering Mathematics→Calculus

Q9

MCQ[1 Mark]

Given two vectors $\vec{U}$ and $\vec{V}$, which of the following options is CORRECT for the expansion of divergence, $\nabla \cdot (\vec{U} + \vec{V})$?

(A)

$\nabla \times \vec{U} + \nabla \times \vec{V}$

(B)

$\nabla \times \vec{U} - \nabla \times \vec{V}$

(C)

$\nabla \cdot \vec{U} + \nabla \cdot \vec{V}$

(D)

$\nabla \cdot \vec{U} - \nabla \cdot \vec{V}$

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q10

NAT[1 Mark]

The solution of the given expression is: $$\lim_{x \to \pi/2} \frac{\sin(\cos(x)) - \cos(x)}{\left(\frac{\pi}{2} - x\right)}$$

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q11

NAT[2 Marks]

Consider a vector function $\vec{F} = (y-z+2)\hat{i} + (yz+8)\hat{j} - xz\hat{k}$, on the surface (S) in the XY plane at $z=0$ (as shown in the figure). Considering Stokes' theorem for space, the ABSOLUTE value of surface integral $\iint_{S} (\nabla \times \vec{F}) \cdot \hat{n} ds$ is

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q12

MCQ[1 Mark]

The value of $\lim_{x \to \pi/2} \left( \frac{\cos x}{x - \frac{\pi}{2}} \right)$ is

(A)

1

(B)

-1

(C)

0

(D)

$\pi$

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q13

MSQ[2 Marks]

If a function $f(x)$ is continuous in the closed interval $[a, b]$ and the first derivative $f'(x)$ exists in the open interval $(a, b)$, then according to Lagrange's mean value theorem $\frac{f(b) - f(a)}{b-a} = f'(c)$. If $a=0$, $b=1.5$, and $f(x) = x(x-1)(x-2)$, then the value(s) of $c$ in $[a, b]$ is/are

(A)

0.50

(B)

0.75

(C)

1.00

(D)

1.50

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q14

MCQ[2 Marks]

Let $r(t) = \frac{\sin 3t}{t} \hat{i} + (t+2)^4 \hat{j} + (t+1)\frac{\sin t}{t} \hat{k}$. The value of $\lim_{t \to 0} r(t)$ is

(A)

0

(B)

$\hat{i} + 32\hat{j} - \hat{k}$

(C)

$3\hat{i} + 16\hat{j} + \hat{k}$

(D)

$3\hat{i} + 16\hat{j}$

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q15

NAT[2 Marks]

The value of $\int_{0}^{2} f(x) dx$ using Simpson's $1/3^{rd}$ rule with only one interior point is (round off to nearest integer) for $f(x) = 2 + 20x + 30x^5$.

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q16

NAT[1 Mark]

Consider a vector field $\vec{V} = x^3 \hat{i} + 2y^2 x \hat{j} + 0.5z \hat{k}$. The divergence of $\vec{V}$ at the point $(1, 2, 1)$ is (rounded to one decimal place).

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q17

MCQ[2 Marks]

The value of $\int_{0}^{\pi} \int_{-1}^{1} r^2 \sin^2 \theta \, dr \, d\theta$ is

(A)

$\frac{\pi}{4}$

(B)

$\frac{\pi}{8}$

(C)

$\frac{\pi}{16}$

(D)

$\frac{\pi}{3}$

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q18

NAT[2 Marks]

The directional derivative of $f = x^3 + 4y^2 + z^2$ at the point $P(2, 1, 3)$ in the direction of the vector $\vec{V} = 3\hat{i} - 4\hat{k}$ is (rounded to one decimal place).

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q19

MCQ[1 Mark]

The value of $\lim_{x \to 0} \left[ \frac{1}{x} \ln(1+x) \right]$ is

(A)

e

(B)

1

(C)

0

(D)

$\frac{1}{e}$

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q20

MCQ[1 Mark]

Let $\vec{F}(x,y) = e^x \sin x \hat{i} + x \hat{j}$. If $C$ is the circle $x^2 + y^2 = 4$ oriented anticlockwise, then $\int_{C} \vec{F} \cdot d\vec{R}$ equals

(A)

$4\pi$

(B)

$6\pi$

(C)

$7\pi$

(D)

$8\pi$

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q21

NAT[2 Marks]

In 3D xyz-space, if $\vec{v} = 3z\hat{i} + 2z\hat{j} + z\hat{k}$ and $\text{curl}(\vec{v}) = a\hat{i} + b\hat{j} + c\hat{k}$, then the value of $(a+b+c)$ is (in integer).

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q22

NAT[2 Marks]

The local minimum value of the real function $f(x) = 2x^3 - 21x^2 + 36x - 20$ is (in integer).

Explanation

Differential Equations

6 Questions

GATE→2025→Engineering Mathematics→Differential Equations

Q23

MCQ[2 Marks]

If $\frac{dy}{dx} + y = x$ and $y(0) = 0$, then the value of $y(1)$ is

(A)

e

(B)

$\frac{1}{e}$

(C)

1

(D)

-1

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q24

MCQ[1 Mark]

The solution of the initial value problem given by $y'' + y' - 2y = 0$; $y(0) = 3$, $y'(0) = 6$ is

(A)

$4e^x + e^{-2x}$

(B)

$4e^x - e^{-2x}$

(C)

$4e^x + 3e^{-2x}$

(D)

$4e^{-2x} - 3e^x$

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q25

MCQ[2 Marks]

Which ONE of the following is the implicit form of the solution for the differential equation given below? $$\frac{dy}{dx} + \frac{2x + 3y}{3x + 5y} = 0$$

(A)

$x^2 - 3xy - \frac{5y^2}{2} - C = 0$

(B)

$x^2 - 3xy + \frac{5y^2}{2} - C = 0$

(C)

$x^2 + 3xy - \frac{5y^2}{2} - C = 0$

(D)

$x^2 + 3xy + \frac{5y^2}{2} - C = 0$

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q26

MCQ[1 Mark]

The second order ordinary differential equation $\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 5y$ has boundary conditions $y(0) = 0$ and $y(1) = 1$. The type of these boundary conditions is

(A)

Neumann

(B)

Dirichlet

(C)

Cauchy

(D)

Robin

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q27

MCQ[2 Marks]

A real gas is produced at constant temperature $30^\circ\text{C}$ with $(\partial Z / \partial P)_T = Z^2$. The difference in compressibility between the real gas and an ideal gas is

(A)

Z

(B)

$Z^2$

(C)

$\sqrt{Z}$

(D)

$\frac{1}{Z^2}$

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q28

NAT[2 Marks]

Given $\frac{dy}{dx} = x^2 y$ with $y(0) = 1$ and step size $h = 0.1$. Using Euler's first-order method, the value of $y(0.2)$ is (rounded to three decimal places).

Explanation

Complex Variables

4 Questions

GATE→2026→Engineering Mathematics→Complex Variables

Q29

MCQ[2 Marks]

Which of the following is the correct representation of the given expression? $$\frac{(1+i)^8}{(1-i)^6}$$ (A) $2e^{-i\frac{\pi}{2}}$ (B) -1 (C) $2e^{i\frac{\pi}{2}}$ (D) 1

(A)

$2e^{-i\frac{\pi}{2}}$

(B)

-1

(C)

$2e^{i\frac{\pi}{2}}$

(D)

1

Explanation

GATE→2025→Engineering Mathematics→Complex Variables

Q30

NAT[1 Mark]

The product of the roots of the equation $x^4 + 1 = 0$ is (answer in integer).

Explanation

GATE→2024→Engineering Mathematics→Complex Variables

Q31

MCQ[1 Mark]

A complex number is defined as $z = x + iy$ with $i = \sqrt{-1}$. $\overline{z}$ is the complex conjugate of $z$. The imaginary part of $(2z + 4\overline{z} + 4iy)$ is

(A)

6

(B)

2

(C)

2y

(D)

3y

Explanation

GATE→2023→Engineering Mathematics→Complex Variables

Q32

MCQ[1 Mark]

Let $z_1$ and $z_2$ be two arbitrary complex numbers with non-zero modulus. Which of the following conditions is FALSE?

(A)

$|z_1 + z_2| > |z_1| + |z_2|$

(B)

$0 \le |z_1 + z_2| < \infty$

(C)

$|z_1 + z_2| \le |z_1| + |z_2|$

(D)

$|z_1 z_2| = |z_1| |z_2|$

Explanation

Probability and Statistics

6 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q33

MCQ[1 Mark]

Two identical cubic dice are rolled simultaneously. Which of the following is the probability that at least one of the face values of the two dice is greater than 3?

(A)

$\frac{1}{2}$

(B)

$\frac{3}{4}$

(C)

$\frac{1}{4}$

(D)

$\frac{3}{8}$

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q34

MCQ[2 Marks]

Consider a random continuous variable X and its distribution function $f(x)$. Which of the following options describes the Probability Density Function (PDF)?

(A)

$\frac{d}{dx}f(x)$

(B)

$\frac{d^2}{dx^2}f(x)$

(C)

$\int f(x) dx$

(D)

$\int \frac{1}{f(x)} dx$

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q35

MCQ[1 Mark]

Four fair coins are tossed simultaneously. The probability that at least one tail turns up is

(A)

$\frac{1}{16}$

(B)

$\frac{15}{16}$

(C)

$\frac{7}{8}$

(D)

$\frac{1}{2}$

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q36

NAT[1 Mark]

The number of ways in which a supervisor can choose four workers out of 10 equally competent workers is

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q37

NAT[2 Marks]

A switch-over event in a producing well occasionally results in a reportable oil leak with probability 1 in 500. With 10 switch-over events per day following a Poisson distribution, the number of days in a 365-day year with no reportable oil leaks is (rounded to nearest integer).

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q38

NAT[2 Marks]

If $Z$ is standard normal variable, the probability $P(-3 \le Z \le 3)$ is (rounded off to three decimal places).

Explanation

Numerical Methods

5 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q39

MCQ[2 Marks]

The values of a function $f(x)$ at three different $x$ are given as: $f(0.90) = 7.75$, $f(1.00) = 9.00$, $f(1.10) = 10.35$ Using the central difference approximation method, which of the following is the CORRECT estimate of first differential, $f'(1.00)$ and second differential, $f''(1.00)$, respectively?

(A)

10.00 and 13.00

(B)

13.00 and 10.00

(C)

10.00 and 10.00

(D)

13.00 and 13.00

Explanation

GATE→2025→Engineering Mathematics→Numerical Methods

Q40

MCQ[2 Marks]

Let $f(x) = \ln x$. The first derivative $f'(x)$ is to be calculated at $x=1$ using numerical differentiation. $f'(1)$ is calculated using first order forward difference ($f_{FD}'$), first order backward difference ($f_{BD}'$), and second order central difference ($f_{CD}'$), using interval width $h=0.1$. The CORRECT order of the values of $f_{FD}'$, $f_{BD}'$, and $f_{CD}'$ is

(A)

$f_{FD}' > f_{CD}' > f_{BD}'$

(B)

$f_{CD}' > f_{BD}' > f_{FD}'$

(C)

$f_{BD}' > f_{FD}' > f_{CD}'$

(D)

$f_{BD}' > f_{CD}' > f_{FD}'$

Explanation

GATE→2023→Engineering Mathematics→Numerical Methods

Q41

MCQ[1 Mark]

In the $4^{th}$ order Runge-Kutta method for solving ordinary differential equations with step size $h < 1$, the ratio of the order of local truncation error to the order of global error is

(A)

h

(B)

$h^2$

(C)

$\frac{1}{h}$

(D)

$\frac{1}{h^2}$

Explanation

GATE→2023→Engineering Mathematics→Numerical Methods

Q42

NAT[2 Marks]

Using Simpson's one-third rule (with step size $h = 0.25$), the area under the curve $y = e^{-x^3}$, from $x = 0$ to $x = 1$ is (rounded to two decimal places).

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q43

MSQ[1 Mark]

Simpson's one-third rule will give the exact value of polynomial integrals up to what degree $n$?

(A)

1

(B)

2

(C)

3

(D)

4

Explanation