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Linear Algebra(7)Calculus(16)Differential Equations(5)Complex Variables(5)Probability and Statistics(5)Numerical Methods(6)

Linear Algebra

7 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

Which one of the following is the pair of eigenvalues of the matrix $\begin{bmatrix} -3 & 4 \\ 4 & 3 \end{bmatrix}$?

(A)

-7, 7

(B)

-5, 5

(C)

-4, 3

(D)

-3, 4

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

Consider a Cartesian coordinate system defined over a 3-dimensional vector space with orthogonal unit basis vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$. Let vector $a=\sqrt{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\hat{k}$, and vector $b=\frac{1}{\sqrt{2}}\hat{i}+\sqrt{2}\hat{j}-\hat{k}$. The inner product of these vectors $(a\cdot b)$ is

(A)

0

(B)

1

(C)

-1

(D)

2

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q3

NAT[2 Marks]

Consider the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$. The eigenvalues of the matrix are $0.27$ and ______ (rounded off to 2 decimal places).

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q4

MSQ[1 Mark]

Consider a linear homogeneous system of equations $Ax = 0$. For a non-trivial solution to exist, which of the following conditions is/are satisfied?

(A)

Determinant of A = 0

(B)

r = n

(C)

r < n

(D)

Determinant of A != 0

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q5

NAT[1 Mark]

Consider a matrix $A = \begin{bmatrix} -5 & a \\ -2 & -2 \end{bmatrix}$. If eigenvalues are $-1$ and $-6$, value of $a$ is.

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q6

NAT[2 Marks]

Matrix $M = \begin{bmatrix} 10 & 6 \\ 6 & 10 \end{bmatrix}$. Sum of all eigenvalues of $M^3$ is.

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

Given matrix $A = \begin{bmatrix} x & 1 & 3 \\ y & 2 & 6 \\ 3 & 5 & 7 \end{bmatrix}$, ordered pair $(x, y)$ for which $\det(A)=0$ is

(A)

(1,1)

(B)

(1,2)

(C)

(2,2)

(D)

(2,1)

Explanation

Calculus

16 Questions

GATE→2026→Engineering Mathematics→Calculus

Q8

MCQ[1 Mark]

A two-dimensional temperature profile is given as $T = 2x^2 + 3xy + y^2$. If $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ directions, respectively, which one of the following is the directional derivative of $T$ at the location $x = 2$, $y = 2$?

(A)

8\hat{i} + 12\hat{j}

(B)

14\hat{i} + 10\hat{j}

(C)

8\hat{i} - 12\hat{j}

(D)

14\hat{i} - 10\hat{j}

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q9

MCQ[1 Mark]

Which one of the following is the value of $\lim_{x\to 0} \frac{e^{x} - x - 1}{\cos x - 1}$?

(A)

-1

(B)

0

(C)

1

(D)

\infty

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q10

MCQ[2 Marks]

Consider the following curve in polar coordinates. $r = 2 - 2 \sin \theta$ Which one of the following is the area enclosed by the curve for $0 \le \theta \le 2\pi$?

(A)

3\pi

(B)

4\pi

(C)

5\pi

(D)

6\pi

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q11

MCQ[2 Marks]

Consider a Cartesian coordinate system... Choose the condition for which the divergence of the vector field $v = ax\hat{i} - by\hat{j}$ is zero.

(A)

a - b = 0

(B)

a < b

(C)

a > b

(D)

a + b = 0

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q12

MCQ[1 Mark]

The first non-zero term in the Taylor series expansion of $(1-x)-e^{-x}$ about $x=0$ is

(A)

1

(B)

x^2 / 2

(C)

-x^2 / 2

(D)

-x^2 / 2

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q13

MCQ[2 Marks]

Area of cardioid $r = a(1 - \cos \theta)$, $0 \le \theta \le 2\pi$ is.

(A)

3\pi a^2 / 2

(B)

2\pi a^2 / 3

(C)

2\pi a^2

(D)

3\pi a^2

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q14

NAT[2 Marks]

Line integral $\int_C F(r) \cdot dr$ for $F(r) = x\hat{i} + y\hat{j} + z\hat{k}$ along path $r(t) = \cos(t)\hat{i} + \sin(t)\hat{j} + t\hat{k}$ ($0 \le t \le \pi$) is.

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q15

NAT[2 Marks]

Function $f(x,y,z) = x^4 + 2y^3 + z^2$. Directional derivative at point $P(-1,1,-1)$ along $(\hat{i} + \hat{j})$ is.

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q16

MCQ[1 Mark]

Which one of the following is the CORRECT value of $y$, as defined by $y = \lim_{x\to 0} \frac{2x}{e^x - 1}$?

(A)

1

(B)

2

(C)

0

(D)

\infty

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q17

MCQ[1 Mark]

The vector $\vec{v}$ is defined as $\vec{v} = zx\hat{i} + 2xy\hat{j} + 3yz\hat{k}$. Value of divergence of $\vec{v}$ evaluated at $(3,2,1)$ is

(A)

0

(B)

3

(C)

14

(D)

13

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q18

MCQ[1 Mark]

For a two-dimensional plane, polar and cartesian unit vectors are related. Which option is the CORRECT value of $\frac{\partial(\hat{e}_r + \hat{e}_\theta)}{\partial \theta}$?

(A)

1

(B)

\hat{e}_\theta

(C)

\hat{e}_r + \hat{e}_\theta

(D)

-\hat{e}_r + \hat{e}_\theta

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q19

NAT[2 Marks]

First derivative of $U(r) = 4\left(\frac{1}{r^{12}} - \frac{1}{r^6}\right)$ evaluated at $r=1$ is.

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q20

MCQ[1 Mark]

Let $f(x) = e^{-|x|}$. The value of $\frac{df}{dx}$ at $x = -1$ is

(A)

-e

(B)

e

(C)

1/e

(D)

-1/e

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q21

MCQ[1 Mark]

The value of real variable $x \ge 0$ which maximizes $f(x) = x^e e^{-x}$ is

(A)

e

(B)

0

(C)

1/e

(D)

1

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q22

NAT[2 Marks]

Directional derivative of $f(x,y,z) = 4x^2 + 2y^2 + z^2$ at $(1,1,1)$ along vector $\vec{v} = \hat{i} - \hat{k}$ is.

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q23

NAT[2 Marks]

Surface integral $\iint_S \left(\frac{2x\hat{i} + 3y\hat{j} + 4z\hat{k}}{4\pi}\right) \cdot \hat{n} dA$ over sphere of radius $4$ is.

Explanation

Differential Equations

5 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q24

NAT[2 Marks]

Consider a differential equation: $x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} - 3y = 0$ with boundary conditions $\left. \begin{matrix} y = 3 \\ \frac{dy}{dx} = -5 \end{matrix} \right\}$ at $x = 1$ The value of $y$ at $x = 2$ is ______ (rounded off to two decimal places).

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q25

MSQ[2 Marks]

Consider the differential equation $\frac{dy}{dx} + \frac{y}{x} = 0$. Choose the CORRECT option(s) for the solution $y$.

(A)

$y = x + c$

(B)

$y = c/x$

(C)

$y = -x + c$

(D)

$y = 0$

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q26

NAT[2 Marks]

Consider ODE $x^2 y'' - x y' - 3y = 0$ with $y(1)=2$ and $y(2)=17/2$. Solution $y(x)$ at $x=3/2$ is.

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q27

NAT[1 Mark]

Particle position $x(t)$ governed by $\frac{d^2x}{dt^2} = -\omega^2 x$ with $x(0)=1, x'(0)=0$. Position at $t = 3\pi/\omega$ is.

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q28

MCQ[2 Marks]

PDE heat conduction equation with given initial/boundary conditions. Value of $t$ at which $\frac{u(0.5,t)}{u(0.5,0)} = \frac{1}{e}$ is

(A)

1

(B)

e

(C)

\pi

(D)

1/e

Explanation

Complex Variables

5 Questions

GATE→2026→Engineering Mathematics→Complex Variables

Q29

MCQ[2 Marks]

Consider the following complex numbers $z_{1} = r_{1}(\cos \theta_{1} + i \sin \theta_{1})$ $z_{2} = r_{2}(\cos \theta_{2} + i \sin \theta_{2})$ where $r_{1}, r_{2}$ are real numbers, $0 \le \theta_{1} \le \frac{\pi}{2}$, $0 \le \theta_{2} \le \frac{\pi}{2}$, and $i = \sqrt{-1}$. If $|z_{1} + z_{2}| = |z_{1}| + |z_{2}|$, which one of the following conditions is necessarily CORRECT?

(A)

$\theta_{1} = 0, \theta_{2} = \frac{\pi}{2}$

(B)

$\theta_{1} = \frac{\pi}{2}, \theta_{2} = 0$

(C)

$r_{1} = r_{2}$

(D)

$\theta_{1} = \theta_{2}$

Explanation

GATE→2025→Engineering Mathematics→Complex Variables

Q30

MCQ[1 Mark]

Consider two complex numbers $z_{1}=1-i$ and $z_{2}=i$. The argument of $z_{1}z_{2}$ is

(A)

0

(B)

\pi/4

(C)

\pi/2

(D)

\pi

Explanation

GATE→2024→Engineering Mathematics→Complex Variables

Q31

MCQ[1 Mark]

If $z_{1}=-1+i$ and $z_{2}=2i$, then $\text{Arg}(z_{1}/z_{2})$ is

(A)

3\pi/4

(B)

\pi/4

(C)

\pi/2

(D)

\pi/3

Explanation

GATE→2023→Engineering Mathematics→Complex Variables

Q32

MCQ[1 Mark]

Given $F = \frac{|z_1 + z_2|}{|z_1| + |z_2|}$ where $z_1 = 2+3i$ and $z_2 = -2+3i$, which option is correct?

(A)

F < 0

(B)

F < 1

(C)

F > 1

(D)

F = 1

Explanation

GATE→2022→Engineering Mathematics→Complex Variables

Q33

MCQ[1 Mark]

The value of $(1+i)^{12}$, where $i=\sqrt{-1}$, is

(A)

-64i

(B)

64i

(C)

64

(D)

-64

Explanation

Probability and Statistics

5 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q34

NAT[1 Mark]

An experiment is performed by rolling an unbiased die once. Events A and B are defined as follows: Event A: The outcome of the rolled die is a prime number. Event B: The outcome of the rolled die is an odd number less than 4. The conditional probability of A given B, denoted by $P(A|B)$, is ______ (rounded off to one decimal place).

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q35

NAT[2 Marks]

The data given in the table are fitted to the equation $y = mx$ using the method of least squares. The value of $m$ is ______ (rounded off to one decimal place). $x$: 2, 4, 1, 3 $y$: 10, 7, 2, 6

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q36

MCQ[1 Mark]

A box contains 3 identical green balls and 7 identical blue balls. Two balls are randomly drawn without replacement from the box. The probability of drawing 1 green and 1 blue ball is

(A)

(3P1 * 7P1) / 10P2

(B)

(10P3 * 10P7) / 10P2

(C)

(10C3 * 10C7) / 10C2

(D)

(3C1 * 7C1) / 10C2

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q37

MCQ[2 Marks]

A probability distribution function is given as $p(x) = 1/a$ for $x \in (0, a)$. For a function $f(x) = x^2$, the expectation of $f(x)$ is

(A)

a^2 / 3

(B)

a^3 / 3

(C)

2a^2 / 3

(D)

2a^3 / 3

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q38

MCQ[1 Mark]

Consider normal probability distribution function $f(x)=\frac{4}{\sqrt{2\pi}}e^{-8(x+3)^{2}}$. Ordered pair $(\mu, \sigma)$ is

(A)

(3, 1/4)

(B)

(-3, 1/4)

(C)

(3, 4)

(D)

(-3, 4)

Explanation

Numerical Methods

6 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q39

NAT[1 Mark]

Using single step trapezoidal rule, the value of $\int_{0}^{0.5} (1 + 16x - 20x^2) dx$ is ______ (rounded off to two decimal places).

Explanation

GATE→2025→Engineering Mathematics→Numerical Methods

Q40

NAT[2 Marks]

The Newton-Raphson method is used to find the root of $f(x) \equiv x^2 - x - 1 = 0$. Starting with initial guess $x_{(0)} = 1$, the second iterate $x_{(2)}$ is ______ (rounded off to 2 decimal places).

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q41

NAT[2 Marks]

Newton-Raphson method for $f(x) = e^x - 5x = 0$ with initial guess $x^{(0)} = 1.0$, value of next iterate $x^{(1)}$ rounded off to 2 decimal places is.

Explanation

GATE→2023→Engineering Mathematics→Numerical Methods

Q42

MCQ[2 Marks]

Simpson's one-third rule estimate for $\int_{-1}^{1} \sqrt{1-x^2} dx$ with interval length $0.5$ is

(A)

1/3 - 1/\sqrt{3}

(B)

1/3 + 2/\sqrt{3}

(C)

1/3 + 1/\sqrt{3}

(D)

1/3 - 2/\sqrt{3}

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q43

NAT[1 Mark]

Area of circular field is $25 \text{ m}^2$. Radius $r$ determined using Newton-Raphson method with initial guess $r = 2.500$ m. Revised estimate after one iteration is m.

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q44

NAT[2 Marks]

ODE solved using predictor (implicit Euler) and corrector (trapezoidal rule) with step size $0.5$. Value of $y$ at $x = 0.5$ is.

Explanation