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Linear Algebra(9)Calculus(12)Differential Equations(5)Analysis of Complex Variables(6)Probability and Statistics(12)Numerical Methods(3)

Linear Algebra

9 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

The eigenvalues of the matrix $P = \begin{pmatrix} 1 & -2 \\ 3 & -4 \end{pmatrix}$ are

(A)

-1 and -2

(B)

1 and 2

(C)

0 and -1

(D)

0 and 1

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

An invertible $n \times n$ matrix $P$ satisfies $P^3 = I$, where $I$ is the identity matrix of size $n$. Which one of the following options is true?

(A)

$P^5 = P$

(B)

$P^5 = P^{-1}$

(C)

$P^4 = P^{-1}$

(D)

$P^4 = P^2$

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q3

NAT[2 Marks]

If $\begin{pmatrix} 2 \\ a \end{pmatrix}$ is an eigenvector corresponding to the smallest eigenvalue of the matrix $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$, the value of $a$ is _______ (Answer in integer)

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q4

MCQ[1 Mark]

The determinant of a $3 \times 3$ matrix $M$ is $8$. If every element of $M$ is multiplied by $-2$, the determinant of the modified matrix will be ________.

(A)

-64

(B)

-16

(C)

8

(D)

64

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q5

MCQ[1 Mark]

The eigenvalues of the matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ are

(A)

$\cos\theta$ and $\sin\theta$

(B)

$e^{i\theta}$ and $e^{-i\theta}$ where $i = \sqrt{-1}$

(C)

$\cos\theta$ and $-\sin\theta$

(D)

$\cos 2\theta$ and $\sin 2\theta$

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q6

MCQ[2 Marks]

If $A = \begin{pmatrix} 1 & -1 \\ 2 & -2 \end{pmatrix}$, the eigenvalues of $A$ are

(A)

-1 and 0.

(B)

-1 and +1.

(C)

-1 and -1.

(D)

+1 and 0.

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

$Q = \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix}$ is a $2 \times 2$ matrix. Which one of the following statements is TRUE?

(A)

$Q$ is equal to its transpose.

(B)

$Q$ is equal to its inverse.

(C)

$Q$ is of full rank.

(D)

$Q$ has linearly dependent columns.

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q8

MCQ[1 Mark]

Which one of the following vectors is an eigenvector corresponding to the eigenvalue $= 1$ for the matrix $A$? $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & -1 & 0 \\ 1 & -1 & 1 \end{bmatrix}$$

(A)

$\begin{bmatrix} 1 & 0 & 1 \end{bmatrix}^T$

(B)

$\begin{bmatrix} 1 & 1 & 0 \end{bmatrix}^T$

(C)

$\begin{bmatrix} 0 & 1 & 0 \end{bmatrix}^T$

(D)

$\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}^T$

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q9

MCQ[1 Mark]

If the given matrices $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ satisfy $A^2 = kA - 2I$, the value of coefficient $k$ is ________.

(A)

1

(B)

2

(C)

0

(D)

4

Explanation

Calculus

12 Questions

GATE→2026→Engineering Mathematics→Calculus

Q10

MCQ[1 Mark]

Let $f$ be a function of real variables $x$ and $y$, defined as: $$f(x,y) = x^2 y + 3y^2 x$$ The value of $\frac{\partial^2 f}{\partial x^2}$ at $x = 1$, $y = 1$ is

(A)

0

(B)

1

(C)

2

(D)

3

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q11

MCQ[2 Marks]

Let $f$ be a function of two real variables $x$ and $y$, defined as: $$f(x,y) = x^2 + y^2$$ The value of the line integral $\int \vec{\nabla} f \cdot d\vec{l}$ from point $P(1,0)$ to point $Q(0,1)$ along the path $L$ shown in the figure below is _______. Here $\vec{\nabla} = \hat{x}\frac{\partial}{\partial x} + \hat{y}\frac{\partial}{\partial y}$, with $\hat{x}$ and $\hat{y}$ as the unit vectors along X and Y axes, respectively.

(A)

0

(B)

1

(C)

-1

(D)

$\pi$

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q12

MCQ[1 Mark]

For any continuous and differentiable function $f(x)$ with first derivative $f'(x) \neq 0$ for all values of $x$, which one of the following is ALWAYS TRUE for $a \neq b$? Here, $a$ and $b$ are finite and real.

(A)

$f(a) > f(b)$

(B)

$f(a) = f(b)$

(C)

$f(a) \neq f(b)$

(D)

$f(a) < f(b)$

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q13

MCQ[1 Mark]

If $f(x) = x - \frac{1}{x}$, the value of $\lim_{\Delta x \to 0} \frac{f(1 + \Delta x) - f(1)}{\Delta x}$ is

(A)

0

(B)

$\frac{1}{2}$

(C)

1

(D)

2

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q14

NAT[1 Mark]

The trigonometric Fourier series expansion of the periodic function in the figure has coefficients $\{a_n\}$ and $\{b_n\}$ for cosine and sine terms, respectively. The value of $a_1 / a_3$ is ________. Give your answer rounded off to 1 decimal place.

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q15

NAT[2 Marks]

For $\vec{F} = (x + y)\hat{i} + (x + y)\hat{j}$, the value of $\oint \vec{F} \cdot d\vec{r}$ along the path shown in the figure is ________. Give your answer as an integer.

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q16

MCQ[1 Mark]

For the function $f(x,y) = e^x \cos(y)$, what is the value of $\frac{\partial^2 f}{\partial x \partial y}$ at $(x = 0, y = \pi/2)$?

(A)

0

(B)

1

(C)

$-1$

(D)

$e^{\pi/2}$

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q17

MSQ[2 Marks]

For the function $f(x) = x^4 - x^2$, which of the following statements is/are TRUE?

(A)

The function is symmetric about $x = 0$.

(B)

The minimum value of the function is $-0.5$.

(C)

The function has two minima.

(D)

The function is an odd function.

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q18

MCQ[1 Mark]

Evaluation of the integral $\int \frac{dx}{\sqrt{2x - x^2}}$ results in ________.

(A)

$\sin^{-1}(x - 1) + c$

(B)

$\cos^{-1}(x - 1) + c$

(C)

$\sin^{-1}\left(\frac{x}{2}\right) + c$

(D)

$\cos^{-1}\left(\frac{x}{2}\right) + c$

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q19

MCQ[1 Mark]

If $\vec{V} = a\hat{i} + b\hat{j} + c\hat{k}$, identify the INVALID operation on $\vec{V}$.

(A)

$\nabla \cdot \nabla \times (\vec{V})$

(B)

$\nabla \times \nabla \cdot (\vec{V})$

(C)

$\nabla(\nabla \cdot (\vec{V}))$

(D)

$\nabla \times \nabla(\vec{V})$

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q20

MSQ[1 Mark]

Given $x$ is real, identify all the even-functions among the following:

(A)

$x|x|$

(B)

$\frac{\cos(x)}{x}$

(C)

$\sin(x^2)$

(D)

$e^{-|x|}$

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q21

NAT[2 Marks]

Using divergence theorem, evaluate the integral $\iint_S \vec{F} \cdot \vec{n}\,dA$, where $S$ is the surface of the cone $x^2 + y^2 \le z^2$, $0 \le z \le 3$. If $\vec{F} = 4x\hat{i} + 3z\hat{j} + 5y\hat{k}$ is a vector function with outer unit normal vector $\vec{n}$, the value of the integral is ________ (rounded off to the nearest integer).

Explanation

Differential Equations

5 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q22

NAT[2 Marks]

The concentration $p$ (in $\mu\text{g/dL}$ i.e., micrograms/deciliter) of a hormone, as a function of time $t$ (in hours) is governed by the following differential equation for $t \ge 0$: $$\frac{dp}{dt} = e^{-0.1t} - 0.1p$$ If $p(0) = 20\text{ }\mu\text{g/dL}$, then $p(10) =$ _______ $\mu\text{g/dL}$. (Round off to one decimal place)

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q23

MCQ[2 Marks]

Given a function $y(x)$ satisfying the differential equation $$y'' - 0.25y = 0,$$ with initial conditions $y(0) = 1$; $y'(0) = 1$, what is the value of $y(\log_e 100)$? Here, $y'$ and $y''$ are the first and second derivatives of $y$, respectively.

(A)

14.05

(B)

14.25

(C)

14.95

(D)

14.65

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q24

MCQ[2 Marks]

Consider a system of the following two partial differential equations: $$\frac{\partial \alpha}{\partial x} = -2\frac{\partial \beta}{\partial t}$$ $$\frac{\partial \beta}{\partial x} = -2\frac{\partial \alpha}{\partial t}$$ Which one of the following choices is a possible solution for the system?

(A)

$\alpha(t,x) = (x - t)^2 + (x + t)^2$ and $\beta(t,x) = (x - t)^2 - (x + t)^2$.

(B)

$\alpha(t,x) = (x - 2t)^2 + (x + 2t)^2$ and $\beta(t,x) = (x - 2t)^2 - (x + 2t)^2$.

(C)

$\alpha(t,x) = \left(x - \frac{t}{2}\right)^2 + \left(x + \frac{t}{2}\right)^2$ and $\beta(t,x) = \left(x - \frac{t}{2}\right)^2 - \left(x + \frac{t}{2}\right)^2$.

(D)

$\alpha(t,x) = \left(x - \frac{t}{2}\right)^2 + 2\left(x + \frac{t}{2}\right)^2$ and $\beta(t,x) = 2\left(x - \frac{t}{2}\right)^2 - \left(x + \frac{t}{2}\right)^2$.

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q25

MCQ[2 Marks]

The time-dependent growth of a bacterial population is governed by the equation $$\frac{dx}{dt} = x\left(1 - \frac{x}{200}\right),$$\nwhere $x$ is the population size at time $t$. The initial population size is $x_0 = 100$ at $t = 0$. As $t \to \infty$, the population size of bacteria asymptotically approaches ________.

(A)

150

(B)

200

(C)

300

(D)

500

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q26

MCQ[2 Marks]

Solution of the differential equation $\frac{dy}{dx} - y = \cos x$ is ________.

(A)

$y = \frac{\sin x - \cos x}{2} + ce^x$

(B)

$y = \frac{\sin x + \cos x}{2} + ce^x$

(C)

$y = \frac{\sin x - \cos x}{2} + ce^{-x}$

(D)

$y = \frac{\sin x + \cos x}{2} + ce^{-x}$

Explanation

Analysis of Complex Variables

6 Questions

GATE→2025→Engineering Mathematics→Analysis of Complex Variables

Q27

MSQ[2 Marks]

Which of the following is/are CORRECT about the value of $y = \log_e(-e^x)$, where $x$ is a real number?

(A)

$y$ is real valued

(B)

$y$ is complex valued

(C)

$y = x \pm i n\pi$, where $n$ is an odd integer and $i = \sqrt{-1}$

(D)

$y = x \pm i n\pi$, where $n$ is an even integer and $i = \sqrt{-1}$

Explanation

GATE→2024→Engineering Mathematics→Analysis of Complex Variables

Q28

MCQ[1 Mark]

What is the value of the following complex line integral counter-clockwise? $$\oint_{|z|=3} \frac{8}{z(z-2)(z-4)} dz$$

(A)

$+j2\pi$

(B)

$-j2\pi$

(C)

$-j10\pi$

(D)

$+j10\pi$

Explanation

GATE→2024→Engineering Mathematics→Analysis of Complex Variables

Q29

MCQ[2 Marks]

What is the value of the following integral using the residue integration method? $$\int_{-\infty}^{\infty} \frac{dx}{1 + x^4}$$

(A)

$\frac{\pi}{\sqrt{2}}$

(B)

$\frac{\pi}{2\sqrt{2}}$

(C)

$\frac{\pi}{4}$

(D)

$\frac{\pi}{2}$

Explanation

GATE→2023→Engineering Mathematics→Analysis of Complex Variables

Q30

MSQ[2 Marks]

The function $f(Z) = \frac{1}{Z - 1}$ of a complex variable $Z$ is integrated on a closed contour in an anti-clockwise direction. For which of the following contours, does this integral have a non-zero value?

(A)

$|Z - 2| = 0.01$

(B)

$|Z - 1| = 0.1$

(C)

$|Z - 3| = 5$

(D)

$|Z| = 2$

Explanation

GATE→2023→Engineering Mathematics→Analysis of Complex Variables

Q31

NAT[2 Marks]

Calculate the reciprocal of the coefficient of $z^3$ in the Taylor series expansion of the function $f(z) = \sin(z)$ around $z = 0$. (Provide the answer as an integer.)

Explanation

GATE→2022→Engineering Mathematics→Analysis of Complex Variables

Q32

MCQ[2 Marks]

In the complex $z$-domain, the value of the integral $\oint_C \frac{z^3 - 9}{3z - i} dz$ is ________.

(A)

$\frac{2\pi}{81} - 6i\pi$

(B)

$\frac{2\pi}{81} + 6i\pi$

(C)

$-\frac{2\pi}{81} + 6i\pi$

(D)

$-\frac{2\pi}{81} - 6i\pi$

Explanation

Probability and Statistics

12 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q33

MCQ[1 Mark]

$X$ is a random variable whose mean is $3$ and standard deviation is $2$. A new random variable $Y$ is defined as: $$Y = \frac{X - 3}{4}$$ The standard deviation of $Y$ is

(A)

0.5

(B)

1

(C)

2

(D)

8

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q34

MCQ[1 Mark]

A biased coin has a probability of heads equal to $1/3$ and a probability of tails equal to $2/3$. A binary random variable $X$ assumes a value $1$ for heads and $-1$ for tails. The variance of $X$ is

(A)

$1/3$

(B)

$5/9$

(C)

$8/9$

(D)

$1$

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q35

MSQ[2 Marks]

Which of the following conditions result(s) in a higher statistical power for comparing means of two samples using $t$-test?

(A)

An increase in the sizes of both the samples from 100 to 1000

(B)

An increase in the significance level from $\alpha = 0.01$ to $\alpha = 0.2$

(C)

A decrease in the significance level from $\alpha = 0.2$ to $\alpha = 0.01$

(D)

A decrease in the sizes of both the samples from 1000 to 100

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q36

MCQ[1 Mark]

The elements of the dataset $\{-5, 1, a, 5, b\}$ are in ascending order. If both mean and median of the dataset are equal to $3$, what is the value of $b$?

(A)

7

(B)

11

(C)

9

(D)

12

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q37

MCQ[2 Marks]

$N$ independent trials of a binomial random variable yield an expectation value of $16$ and variance of $12$. What is the value of $N$?

(A)

80

(B)

32

(C)

48

(D)

64

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q38

NAT[1 Mark]

A group of four people were residing together when a new virus was detected. If the probability of each person being infected is $0.1$, then the probability that at least two of them are infected is ________. Give your answer rounded off to 3 decimal places.

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q39

NAT[1 Mark]

A random noise signal with Gaussian distribution has a mean of zero and a standard deviation of $1\text{ mV}$. The probability that an instantaneous measurement of this signal is greater than $2\text{ mV}$ or lesser than $-2\text{ mV}$ is ________. Give your answer as a percentage rounded off to the nearest integer.

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q40

MCQ[1 Mark]

What is the magnitude of the difference between the mean and the median of the dataset $\{1, 2, 3, 4, 6, 8\}$?

(A)

0

(B)

1

(C)

0.5

(D)

0.25

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q41

MCQ[1 Mark]

For a Binomial random variable $X$, $E(X)$ and $\text{Var}(X)$ are the expectation and variance, respectively. Which one of the following statements CANNOT be true?

(A)

$E(X) = 20$ and $\text{Var}(X) = 16$

(B)

$E(X) = 6$ and $\text{Var}(X) = 5.4$

(C)

$E(X) = 10$ and $\text{Var}(X) = 15$

(D)

$E(X) = 64$ and $\text{Var}(X) = 12.8$

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q42

MCQ[2 Marks]

A $20\text{ mV}$ DC signal has been superimposed with a $10\text{ mV}$ RMS band-limited Gaussian noise with a flat spectrum up to $5\text{ kHz}$. If an integrating voltmeter is used to measure this DC signal, what is the minimum averaging time (in seconds) required to yield a $99\%$ accurate result with $95\%$ certainty?

(A)

0.1

(B)

1.0

(C)

5.0

(D)

10.0

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q43

NAT[1 Mark]

Due to the current COVID pandemic conditions, assume that positive or negative status of any individual are equally likely. There are $3$ members in a family. If one of the members has tested COVID positive, the conditional probability that at least $2$ members are COVID positive is ________ (rounded off to three decimal places).

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q44

NAT[2 Marks]

In a biological study, the experimental values measured from $6$ subjects are given in the table below. Using this data, the linear regression coefficient for estimating the weight of the heart based on the systolic pressure is ________ (rounded off to two decimal places). | Systolic pressure (in mm Hg) | 110 | 130 | 100 | 90 | 120 | 140 | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | | **Weight of the heart (in g)** | 490 | 390 | 420 | 450 | 300 | 500 |

Explanation

Numerical Methods

3 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q45

MCQ[2 Marks]

The Newton-Raphson method is used to find the roots of the equation $x = e^{-2x}$. If the initial guess $x_0 = 0$, the estimate of the root after the first iteration, $x_1 =$ _______

(A)

$1$

(B)

$1/2$

(C)

$1/3$

(D)

$1/4$

Explanation

GATE→2025→Engineering Mathematics→Numerical Methods

Q46

NAT[2 Marks]

Newton Raphson method is used to solve the equation $x^3 - 2x - 5 = 0$ with an initial guess $x_0 = 3$. What is $x_1$, the value after ONE iteration? (rounded off to two decimal places)

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q47

MCQ[1 Mark]

To solve the equation $x = 2\cos x$ using Newton-Raphson's method, which one of the following iterations should be used?

(A)

$x_{n+1} = x_n - \frac{x_n - 2\cos x_n}{1 + 2\sin x_n}$

(B)

$x_{n+1} = x_n + \frac{x_n - 2\cos x_n}{1 + 2\sin x_n}$

(C)

$x_{n+1} = x_n + \frac{1 + 2\sin x_n}{x_n - 2\cos x_n}$

(D)

$x_{n+1} = x_n - \frac{1 + 2\sin x_n}{x_n - 2\cos x_n}$

Explanation