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Linear Algebra(10)Calculus(10)Differential Equations(6)Analysis of Complex Variables(9)Probability and Statistics(9)Numerical Methods(2)

Linear Algebra

10 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

A matrix $P \in \mathbb{R}^{3 \times 3}$ satisfies $P^{3} - 4P^{2} + 5P - 2I = 0$ where I is a $3 \times 3$ identity matrix. The set of all possible eigenvalues of matrix P is

(A)

{1, 2, 3}

(B)

{1}

(C)

{1, 2}

(D)

{2}

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

Consider a matrix $A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & 0 \\ 0 & 1 & 2 \end{bmatrix}$. Let $b = \begin{bmatrix} 1 \\ 3 \\ 0 \end{bmatrix}$ and $x = \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}$. The number of solutions to the linear system of equations $Ax = b$ is

(A)

0

(B)

1

(C)

6

(D)

infinitely many

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q3

MCQ[1 Mark]

A $2n \times 2n$ matrix $A = [a_{ij}]$ has its elements as $a_{ij} = \begin{cases} \beta & \text{if } (i+j) \text{ is odd}, \\ -\beta & \text{if } (i+j) \text{ is even} \end{cases}$, where n is any integer greater than 2 and $\beta$ is any non-zero real number. The rank of A is

(A)

1

(B)

2

(C)

n

(D)

2n

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q4

NAT[1 Mark]

If one of the eigenvectors of the matrix $A = \begin{bmatrix} -1 & -1 \\ x & -4 \end{bmatrix}$ is along the direction $\begin{bmatrix} \alpha \\ 2\alpha \end{bmatrix}$, where \alpha is any non-zero real number, then the value of x is (in integer).

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q5

MCQ[1 Mark]

A matrix M is constructed by stacking three column vectors $v_{1}, v_{2}, v_{3}$ as $M = \begin{bmatrix} v_{1} & v_{2} & v_{3} \end{bmatrix}$. Choose the set of vectors from the following options such that $rank(M) = 3$.

(A)

v_{1} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, v_{2} = \begin{bmatrix} 0 \\ -1 \\ 0 \end{bmatrix}, v_{3} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}

(B)

v_{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, v_{2} = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}, v_{3} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

(C)

v_{1} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, v_{2} = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}, v_{3} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}

(D)

v_{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, v_{2} = \begin{bmatrix} -1 \\ 1 \\ -1 \end{bmatrix}, v_{3} = \begin{bmatrix} 0 \\ -1 \\ 0 \end{bmatrix}

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q6

NAT[1 Mark]

A 3 x 3 matrix P with all real elements has eigenvalues $\frac{1}{4}$, 1, and -2. The value of $|P^{-1}|$ is (rounded off to nearest integer).

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

Choose solution set S corresponding to the systems of two equations $x - 2y + z = 0$, $x - z = 0$

(A)

$S = \{\alpha \begin{bmatrix} 1 \\ 1 \end{bmatrix} | \alpha \in \mathbb{R}\}$

(B)

$S = \{\alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + \beta \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} | \alpha, \beta \in \mathbb{R}\}$

(C)

$S = \{\alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + \beta \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} | \alpha, \beta \in \mathbb{R}\}$

(D)

$S = \{\alpha \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} | \alpha \in \mathbb{R}\is$

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q8

NAT[2 Marks]

The rank of the matrix A given below is one. The ratio $\frac{\alpha}{\beta}$ is (rounded off to the nearest integer).

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q9

MSQ[1 Mark]

Given $M = \begin{bmatrix} 2 & 3 & 7 \\ 6 & 4 & 7 \\ 4 & 6 & 14 \end{bmatrix}$, which of the following statement(s) is/are correct?

(A)

The rank of M is 2

(B)

The rank of M is 3

(C)

The rows of M are linearly independent

(D)

The determinant of M is 0

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q10

MSQ[2 Marks]

The matrix $A = \begin{bmatrix} 4 & 3 \\ 9 & -2 \end{bmatrix}$ has eigenvalues -5 and 7. The eigenvector(s) is/are

(A)

\begin{bmatrix} 1 \\ 1 \end{bmatrix}

(B)

\begin{bmatrix} 3 \\ 4 \end{bmatrix}

(C)

\begin{bmatrix} 2 \\ -6 \end{bmatrix}

(D)

\begin{bmatrix} 2 \\ 8 \end{bmatrix}

Explanation

Calculus

10 Questions

GATE→2026→Engineering Mathematics→Calculus

Q11

MCQ[1 Mark]

Which one among the following statements is true for the function $f(x) = x|x|$?

(A)

$f(x)$ is discontinuous but not differentiable at $x=0$.

(B)

$f(x)$ is discontinuous and differentiable at $x=0$.

(C)

$f(x)$ is continuous but not differentiable at $x=0$.

(D)

$f(x)$ is continuous and differentiable at $x=0$.

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q12

MSQ[2 Marks]

Given $f(x,y) = x^{2} - 2xy + y^{2}$. The complete contour of the equation $f(x,y) = 1$ is described by the option(s)

(A)

A line, $x - y = 1$

(B)

A circle with radius 1

(C)

An ellipse with length of major axis equal to 1

(D)

A line, $x - y = -1$

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q13

NAT[2 Marks]

Let $f(t) = \begin{cases} 1, & t \in [0,2] \\ -t+3, & t \in [2,3] \\ 0, & \text{otherwise} \end{cases}$. Then $\int_{-\infty}^{\infty} f(\tau) d\tau = \dots$ (rounded off to one decimal place)

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q14

MCQ[2 Marks]

The value of the surface integral $\iint_{S} (2x+z) dy dz + (2x+z) dx dz + (2z+y) dx dy$ over the sphere $S: x^{2} + y^{2} + z^{2} = 9$ is

(A)

$72\pi$

(B)

$144\pi$

(C)

$36\pi$

(D)

$432\pi$

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q15

MCQ[2 Marks]

The value of the integral $\int_{-\pi}^{\pi} (\cos^{6}x + \cos^{4}x) dx$ is

(A)

\frac{\pi}{2}

(B)

\frac{5\pi}{8}

(C)

\frac{11\pi}{8}

(D)

\frac{9\pi}{8}

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q16

MCQ[2 Marks]

Let C be the closed curve in the xy-plane, traversed in the counterclockwise direction along the boundary of the rectangle with vertices at (0,0), (2,0), (2,1), (0,1). The value of the line integral $\oint_{C} (-e^{y} dx + e^{x} dy)$ is

(A)

e^{2} + 2e - 3

(B)

e^{2} - 2e - 3

(C)

e^{2} + e - 1

(D)

e^{2} + e + 1

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q17

MCQ[1 Mark]

What is $\lim_{x \to 0} f(x)$ where $f(x) = x \sin\frac{1}{x}$?

(A)

0

(B)

1

(C)

\infty

(D)

Limit does not exist

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q18

NAT[2 Marks]

Consider the real-valued function $g(x) = \max\{(x-2)^{2}, -2x+7\}$, where $x \in (-\infty, \infty)$. The minimum value attained by $g(x)$ is (rounded off to one decimal place).

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q19

MCQ[1 Mark]

A periodic function $f(x)$, with period 2, is defined as... The Fourier series of this function contains

(A)

Both \cos (n\pi x) and \sin (n\pi x) where n=1,2,3,...

(B)

Only \sin (n\pi x) where n=1,2,3,...

(C)

Only \cos (n\pi x) where n=1,2,3,...

(D)

Only \cos (2n\pi x) where n=1,2,3,...

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q20

NAT[1 Mark]

The global minimum of $x^{3}e^{-|x|}$ for $x \in (-\infty, \infty)$ occurs at $x =$ (round off to one decimal place).

Explanation

Differential Equations

6 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q21

MSQ[2 Marks]

Given a differential equation $\frac{d^{2}x}{dt^{2}} + x = 0$ with $x(0) \neq 0$. Which of the following statements is/are true?

(A)

$|x(t)|^{2} + |\frac{dx(t)}{dt}|^{2} = 0$ for all $t \ge 0$

(B)

$|x(t)|^{2} + |\frac{dx(t)}{dt}|^{2} = c$ for all $t \ge 0$, for some real constant $c > 0$

(C)

$|x(t)|^{2} + |\frac{dx(t)}{dt}|^{2} = e^{jt}$ for all $t \ge 0$

(D)

$|x(t)|^{2} + |\frac{dx(t)}{dt}|^{2} = \sin t$ for all $t \ge 0$

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q22

MCQ[1 Mark]

The solution of the differential equation $\frac{dy}{dx} = 9 \frac{x}{y}$ represents

(A)

a hyperbola

(B)

a parabola

(C)

an ellipse

(D)

a circle

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q23

NAT[1 Mark]

Consider a system given by the following first order differential equation: $\frac{dy}{dt} = y + 2t - t^{2}$ where, $y(0) = 1$ and $0 \le t < \infty$. Using a step size $h = 0.1$ for the improved Euler method, the value of $y(t)$ at $t = 0.1$ is (rounded off to two decimal places).

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q24

NAT[2 Marks]

The solution of an ordinary differential equation $y''' + 3y'' + 3y' + y = 30e^{-t}$ is $y(t) = (c_{0} + c_{1}t - c_{2}t^{2} + c_{3}t^{3})e^{-t}$... The value of $(c_{0} + c_{1} + c_{2} + c_{3})$ is (rounded off to nearest integer).

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q25

MCQ[1 Mark]

The solution $x(t), t \ge 0$, to the differential equation with initial conditions $x(0) = 1$ and $\dot{x}(0) = 0$ is

(A)

$x(t) = 2e^{-kt} + 2kt - 1$

(B)

$x(t) = 2e^{-kt} - 1$

(C)

$x(t) = 1$

(D)

$x(t) = 2e^{-kt} - kt - 1$

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q26

MCQ[2 Marks]

Consider the differential equation $\frac{dy}{dx} + y \ln(y) = 0$. If $y(0) = e$ then $y(1)$ is

(A)

e^{e}

(B)

e^{-e}

(C)

e^{(1/e)}

(D)

e^{(-1/e)}

Explanation

Analysis of Complex Variables

9 Questions

GATE→2026→Engineering Mathematics→Analysis of Complex Variables

Q27

MCQ[1 Mark]

Consider a function $f(z) = z^{2} + z + 1$ where $z \in \mathbb{C}$ is a complex variable. A simple closed contour $\gamma$ in z-plane encloses the point $z = 1 + 0j$. The value of integral $\oint_{\gamma} \frac{f(z)}{z-1} dz = \dots$

(A)

$6\pi j$

(B)

$3\pi i$

(C)

$12\pi i$

(D)

$\pi i$

Explanation

GATE→2025→Engineering Mathematics→Analysis of Complex Variables

Q28

NAT[1 Mark]

Consider the function $f(z) = \frac{2z + 1}{z^{2} - z}$, where z is a complex variable. The sum of the residues at singular points of $f(z)$ is (in integer).

Explanation

GATE→2025→Engineering Mathematics→Analysis of Complex Variables

Q29

MSQ[2 Marks]

Choose the correct statement(s) from the following options, regarding Cauchy's theorem on complex integration $\oint_{C} f(z) dz$ where C is a simple closed path in a simply connected domain D.

(A)

Cauchy's theorem cannot be directly applied to conclude that $\oint f(z) dz = 0$ when $f(z) = \frac{1}{z^{2}}$ and C is the unit circle

(B)

If $f(z)$ is analytic in D, then it can be concluded that $\oint_{C} f(z) dz = 0$ for any simple closed path C in D

(C)

The function $f(z)$ must be analytic in D to conclude $\oint_{C} f(z) dz = 0$ for any simple closed path C in D

(D)

$\oint_{C} f(z) dz \neq 0$ when $f(z) = \frac{1}{z^{2}}$, since the function is not analytic at $z=0$

Explanation

GATE→2024→Engineering Mathematics→Analysis of Complex Variables

Q30

MCQ[1 Mark]

Let $z = x + iy$ be a complex variable and $\bar{z}$ be its complex conjugate. The equation $\bar{z}^{2} + z^{2} = 2$ represents a

(A)

parabola

(B)

hyperbola

(C)

ellipse

(D)

circle

Explanation

GATE→2024→Engineering Mathematics→Analysis of Complex Variables

Q31

MSQ[2 Marks]

The complex functions $f(z) = u(x,y) + i v(x,y)$ and $\overline{f(z)} = u(x,y) - i v(x,y)$ are both analytic in a given domain. Choose the correct option(s) from the following.

(A)

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} = 0

(B)

\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \neq 0

(C)

\frac{df(z)}{dz} = 0

(D)

\frac{df(z)}{dz} \neq 0

Explanation

GATE→2023→Engineering Mathematics→Analysis of Complex Variables

Q32

MCQ[1 Mark]

$F(z) = \frac{1}{1-z}$, when expanded as a power series around $z = 2$, would result in $F(z) = \sum_{k=0}^{\infty} a_{k}(z-2)^{k}$, with the region of convergence (ROC) $|z-2| < 1$. The coefficients $a_{k}, k \ge 0$, are given by the expression

(A)

(-1)^{k}

(B)

(-1)^{k+1}

(C)

(\frac{1}{2})^{k}

(D)

(\frac{-1}{2})^{k+1}

Explanation

GATE→2023→Engineering Mathematics→Analysis of Complex Variables

Q33

MCQ[2 Marks]

Let $f(z) = j \frac{1-z}{1+z}$ where z denotes a complex number and j denotes $\sqrt{-1}$. The inverse function $f^{-1}(z)$ maps the real axis to the

(A)

unit circle with centre at the origin

(B)

unit circle with centre not at the origin

(C)

imaginary axis

(D)

real axis

Explanation

GATE→2022→Engineering Mathematics→Analysis of Complex Variables

Q34

MSQ[2 Marks]

For the complex number $z = \frac{a+jb}{a-jb}$, where $a > 0$ and $b > 0$. Which of the following statement(s) is/are true?

(A)

The phase is 2 \tan^{-1}\frac{b}{a}

(B)

The phase is \tan^{-1}\frac{2b}{a}

(C)

The magnitude is 1

(D)

\sqrt{\frac{a^{2}+b^{2}}{a^{2}-b^{2}}}$

Explanation

GATE→2022→Engineering Mathematics→Analysis of Complex Variables

Q35

NAT[2 Marks]

Consider the function $f(z) = \frac{1}{(z+1)(z+2)(z+3)}$. The residue of $f(z)$ at $z = -1$, is

Explanation

Probability and Statistics

9 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q36

NAT[1 Mark]

A data set is given to be [1, 2, 0, -1, -3, 1, 2, 0, 1]. The median of the data set is (rounded off to the nearest integer)

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q37

MCQ[2 Marks]

A box contains three apples and two oranges. Two fruits are removed randomly in succession. The probability that the first is an apple and the second is an orange is

(A)

\frac{3}{5}

(B)

\frac{3}{20}

(C)

\frac{3}{10}

(D)

\frac{2}{5}

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q38

NAT[2 Marks]

The probability of a student missing a class is 0.1. In a total number of 10 classes, the probability that the student will not miss more than one class is (rounded off to two decimal places).

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q39

NAT[1 Mark]

Indian Premier League has divided the sixteen cricket teams into two equal pools... the probability that one team has a blue logo jersey and another has a red logo jersey is (rounded off to one decimal place).

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q40

NAT[2 Marks]

The readings recorded from a 20-psig pressure gauge are given in the Table... The regression coefficient of determination, $R^{2} =$ (rounded off to three decimal places).

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q41

NAT[2 Marks]

A random variable X has a probability density function... The probability of $X > 2$ is (rounded off to three decimal places).

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q42

NAT[1 Mark]

X is a discrete random variable which takes values 0, 1 and 2... The value of $E[X] - E[X^{2}]$ is (rounded off to one decimal place).

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q43

NAT[2 Marks]

How many five-digit numbers can be formed using the integers 3, 4, 5 and 6 with exactly one digit appearing twice?

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q44

NAT[1 Mark]

A bag contains six red balls and four blue balls. If three balls are drawn in succession without replacement, the probability that the second and third balls drawn are red is (round off to two decimal places).

Explanation

Numerical Methods

2 Questions

GATE→2025→Engineering Mathematics→Numerical Methods

Q45

MCQ[2 Marks]

Newton-Raphson method is used to compute the inverse of the number 1.6. Among the following options, the initial guess of the solution that results in non-convergence of the iterative process is

(A)

0.55

(B)

0.75

(C)

1.15

(D)

1.25

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q46

NAT[1 Mark]

The Newton-Raphson method is applied to determine the solution of $f(x) = 0$ where $f(x) = x - \cos(x)$. If the initial guess of the solution is $x_{0} = 0$, the value of the next approximation $x_{1}$ is (round off to two decimal places).

Explanation