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Linear Algebra(11)Calculus(10)Differential Equations(8)Complex Variables(4)Probability and Statistics(4)Numerical Methods(5)

Linear Algebra

11 Questions

GATE→2025→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

Let A and B be real symmetric matrices of same size. Which one of the following options is correct?

(A)

A^{T} = A^{-1}

(B)

AB = BA

(C)

(AB)^{T} = B^{T}A^{T}

(D)

A = A^{-1}

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

Consider the system of linear equations $x+2y+z=5$ $2x+ay+4z=12$ $2x+4y+6z=b$\nThe values of $a$ and $b$ such that there exists a non-trivial null space and the system admits infinite solutions are

(A)

a = 8, b = 14

(B)

a = 4, b = 12

(C)

a = 8, b = 12

(D)

a = 4, b = 14

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q3

MCQ[2 Marks]

The matrix $$\begin{bmatrix} 1 & a \\ 8 & 3 \end{bmatrix}$$ (where $a>0$) has a negative eigenvalue if $a$ is greater than

(A)

3/8

(B)

1/8

(C)

1/4

(D)

1/5

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q4

NAT[1 Mark]

A linear transformation maps a point $(x, y)$ in the plane to the point $(\hat{x}, \hat{y})$ according to the rule $$\hat{x}=3y, \quad \hat{y}=2x$$\nThen, the disc $x^{2}+y^{2}\le 1$ gets transformed to a region with an area equal to ____ (Rounded off to two decimals)\nUse $\pi=3.14$.

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q5

MCQ[1 Mark]

Given a function 𝜑= ଵ ଶ(𝑥ଶ+ 𝑦ଶ+ 𝑧ଶ) in three-dimensional Cartesian space, the value of the surface integral ඾𝐧ෝ∙∇𝜑 d𝑆 , ௌ where 𝑆 is the surface of a sphere of unit radius and 𝐧ෝ is the outward unit normal vector on 𝑆, is (A) 4𝜋 (B) 3𝜋 (C) 4𝜋/3 (D) 0

(A)

4𝜋

(B)

3𝜋

(C)

4𝜋/3

(D)

0

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q6

MCQ[1 Mark]

The Fourier series expansion of 𝑥ଷ in the interval −1 ≤𝑥< 1 with periodic continuation has (A) only sine terms (B) only cosine terms (C) both sine and cosine terms (D) only sine terms and a non-zero constant

(A)

only sine terms

(B)

only cosine terms

(C)

both sine and cosine terms

(D)

only sine terms and a non-zero constant

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

If 𝐀= ቂ 10 2𝑘+ 5 3𝑘−3 𝑘+ 5 ቃ is a symmetric matrix, the value of k is ___________. (A) 8 (B) 5 (C) −0.4 (D) 1 + √1561 12

(A)

8

(B)

5

(C)

−0.4

(D)

1 + √1561 12

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q8

MCQ[2 Marks]

The value of the integral ර൬ 6𝑧 2𝑧ସ−3𝑧ଷ+ 7𝑧ଶ−3𝑧+ 5൰d𝑧 evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole 𝑧= 𝑖, where 𝑖 is the imaginary unit, is (A) (−1 + 𝑖) 𝜋 (B) (1 + 𝑖) 𝜋 (C) 2(1 −𝑖)𝜋 (D) (2 + 𝑖)𝜋

(A)

(−1 + 𝑖) 𝜋

(B)

(1 + 𝑖) 𝜋

(C)

2(1 −𝑖)𝜋

(D)

(2 + 𝑖)𝜋

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q9

MSQ[2 Marks]

The system of linear equations in real (𝑥, 𝑦) given by (𝑥 𝑦) ቈ 2 5 −2𝛼 𝛼 1 ቉= (0 0) involves a real parameter 𝛼 and has infinitely many non-trivial solutions for special value(s) of 𝛼. Which one or more among the following options is/are non-trivial solution(s) of (𝑥, 𝑦) for such special value(s) of 𝛼? (A) 𝑥= 2, 𝑦= −2 (B) 𝑥= −1, 𝑦= 4 (C) 𝑥= 1, 𝑦= 1 (D) 𝑥= 4, 𝑦= −2

(A)

𝑥= 2, 𝑦= −2

(B)

𝑥= −1, 𝑦= 4

(C)

𝑥= 1, 𝑦= 1

(D)

𝑥= 4, 𝑦= −2

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q10

NAT[2 Marks]

Let a random variable X follow Poisson distribution such that Prob(𝑋= 1) = Prob(𝑋= 2). The value of Prob(𝑋= 3) is __________ (round off to 2 decimal places).

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q11

NAT[2 Marks]

Consider two vectors: 𝑎⃗= 5 𝐢+ 7 𝐣+ 2 𝐤 𝑏ሬ⃗= 3 𝐢− 𝐣+ 6 𝐤 Magnitude of the component of 𝑎⃗ orthogonal to 𝑏ሬ⃗ in the plane containing the vectors 𝑎⃗ and 𝑏ሬ⃗ is __________ (round off to 2 decimal places).

Explanation

Calculus

10 Questions

GATE→2026→Engineering Mathematics→Calculus

Q12

MCQ[1 Mark]

Domain A is bounded by curve $x^{2}=4y$, ordinate $x=2$, and x-axis. The value of $\iint_{A}y~dxdy$ is

(A)

1/5

(B)

1/3

(C)

5/12

(D)

1/2

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q13

MCQ[1 Mark]

Let $\phi$ be a scalar function. Then, $\nabla\phi$ is

(A)

always perpendicular to the surface of constant $\phi$

(B)

always parallel to the surface of constant $\phi$

(C)

the minimum rate of change of scalar $\phi$

(D)

always zero

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q14

MCQ[1 Mark]

The divergence of the curl of a vector field is

(A)

the magnitude of this vector field

(B)

the argument of this vector field

(C)

the magnitude of the curl of this vector field

(D)

zero

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q15

MCQ[2 Marks]

In the closed interval $[0, 3]$, the minimum value of the function $f$ given below is $$f(x)=2x^{3}-9x^{2}+12x$$

(A)

0

(B)

4

(C)

5

(D)

9

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q16

NAT[2 Marks]

The directional derivative of the function $f$ given below at the point $(1,0)$ in the direction of $\frac{1}{2}(\hat{i}+\sqrt{3}\hat{j})$ is ____ (rounded off to 1 decimal place). $$f(x,y)=x^{2}+xy^{2}$$

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q17

MCQ[1 Mark]

The value of the surface integral $$\oint_{S}z~dx~dy$$\nwhere S is the external surface of the sphere $x^{2}+y^{2}+z^{2}=R^{2}$ is

(A)

0

(B)

4\pi R^3

(C)

\frac{4\pi}{3}R^{3}

(D)

\pi R^3

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q18

MCQ[1 Mark]

Let $f(.)$ be a twice differentiable function from $\mathbb{R}^{2}\rightarrow\mathbb{R}$. If $p,x_{0}\in\mathbb{R}^{2}$ where $||p||$ is sufficiently small, then $$f(x_{0}+p)=f(x_{0})+\nabla f(x_{0})^{T}p+\frac{1}{2}p^{T}\nabla^{2}f(\psi)p$$\nwhere $\psi\in\mathbb{R}^{2}$ is a point on the line segment joining $x_{0}$ and $x_{0}+p$. If $x_{0}$ is a strict local minimum of $f(x)$, then which one of the following statements is TRUE?

(A)

\nabla f(x_{0})^{T}p > 0 \text{ and } p^{T}\nabla^{2}f(\psi)p = 0

(B)

\nabla f(x_{0})^{T}p = 0 \text{ and } p^{T}\nabla^{2}f(\psi)p > 0

(C)

\nabla f(x_{0})^{T}p = 0 \text{ and } p^{T}\nabla^{2}f(\psi)p = 0

(D)

\nabla f(x_{0})^{T}p = 0 \text{ and } p^{T}\nabla^{2}f(\psi)p < 0

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q19

NAT[2 Marks]

If the value of the double integral $$\int_{x=3}^{4}\int_{y=1}^{2}\frac{dy~dx}{(x+y)^{2}}$$\nis $\log_{e}(a/24)$, then $a$ is ____ (answer in integer).

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q20

MCQ[1 Mark]

The figure shows the plot of a function over the interval $[-4, 4]$. Which one of the options given CORRECTLY identifies the function?

(A)

|2-x|

(B)

|2-|x||

(C)

|2+|x||

(D)

2-|x|

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q21

NAT[1 Mark]

A vector field $$\vec{B}(x,y,z)=x\hat{i}+y\hat{j}-2z\hat{k}$$\nis defined over a conical region having height $h=2$, base radius $r=3$ and axis along z, as shown in the figure. The base of the cone lies in the x-y plane and is centered at the origin.\nIf $\hat{n}$ denotes the unit outward normal to the curved surface S of the cone, the value of the integral $$\int_{S}\vec{B}\cdot\hat{n}~dS$$ equals

Explanation

Differential Equations

8 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q22

MCQ[1 Mark]

The order and degree of the following differential equation are $m$ and $n$, respectively. $$\frac{\partial^{3}\varphi}{\partial x^{3}}+\frac{\partial^{2}\varphi}{\partial y^{2}}\frac{\partial\varphi}{\partial x}+(\frac{\partial^{2}\varphi}{\partial x^{2}})^{2}+\frac{\partial\varphi}{\partial y}=0$$\nThe value of $(m-n)$ is

(A)

2

(B)

3

(C)

1

(D)

0

Explanation

GATE→2026→Engineering Mathematics→Differential Equations

Q23

MCQ[2 Marks]

Consider the following differential equation: $$\frac{\partial y}{\partial x}=3\frac{\partial y}{\partial t}+y$$\nIf $y(x,0)=10e^{-2x}$ then the solution of the differential equation is

(A)

y(x,t) = 10e^{-2x-t}

(B)

y(x,t) = 10e^{-2x+t}

(C)

y(x,t) = 10e^{-2x-2t}

(D)

y(x,t) = 10e^{-2x+2t}

Explanation

GATE→2026→Engineering Mathematics→Differential Equations

Q24

MCQ[2 Marks]

The Laplace transform of $f(t)=\sin 2t \sin 4t$ is

(A)

\frac{16s}{(s^{2}+4)(s^{2}+36)}

(B)

\frac{32s}{(s^{2}+4)(s^{2}+36)}

(C)

\frac{16}{(s^{2}+4)(s^{2}+36)}

(D)

\frac{32}{(s^{2}+4)(s^{2}+36)}

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q25

MCQ[1 Mark]

For the differential equation given below, which one of the following options is correct? $$\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0, \quad 0\le x\le 1, \quad 0\le y\le 1$$

(A)

u = e^{x+y} \text{ is a solution for all x and y}

(B)

u = e^{x} \sin y \text{ is a solution for all x and y}

(C)

u = \sin x \sin y \text{ is a solution for all x and y}

(D)

u = \cos x \cos y \text{ is a solution for all x and y}

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q26

NAT[2 Marks]

Let $y$ be the solution of the differential equation with the initial conditions given below. If $y(x=2)=A~\ln 2$, then the value of $A$ is ____ (rounded off to 2 decimal places). $$x^{2}\frac{d^{2}y}{dx^{2}}+3x\frac{dy}{dx}+y=0, \quad y(x=1)=0, \quad \frac{dy}{dx}(x=1)=1$$

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q27

NAT[2 Marks]

If $x(t)$ satisfies the differential equation $$t\frac{dx}{dt}+(t-x)=0$$\nsubject to the condition $x(1)=0$ then the value of $x(2)$ is ____ (rounded off to 2 decimal places).

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q28

MCQ[2 Marks]

Which one of the options given is the inverse Laplace transform of $$\frac{1}{s^{3}-s}?$$ $u(t)$ denotes the unit-step function.

(A)

(-1 + \frac{1}{2}e^{-t} + \frac{1}{2}e^{t})u(t)

(B)

(\frac{1}{3}e^{-t} - e^{t})u(t)

(C)

(-1 + \frac{1}{2}e^{-(t-1)} + \frac{1}{2}e^{(t-1)})u(t-1)

(D)

(-1 - \frac{1}{2}e^{-(t-1)} - \frac{1}{2}e^{(t-1)})u(t-1)

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q29

NAT[2 Marks]

Consider the second-order linear ordinary differential equation $$x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-y=0, \quad x\ge 1$$\nwith the initial conditions $$y(x=1)=6, \quad \frac{dy}{dx}\Big|_{x=1}=2$$\nThe value of $y$ at $x=2$ equals ____ (Answer in integer)

Explanation

Complex Variables

4 Questions

GATE→2026→Engineering Mathematics→Complex Variables

Q30

MCQ[2 Marks]

If $w=\log_{e}z=\log_{e}(x+iy)$, where $i=\sqrt{-1}$, then which one of the following statements is correct?

(A)

w is analytic everywhere except at z = 0

(B)

w is non-analytic everywhere

(C)

The conjugate functions of w are \log_e(x^2+y^2) and \log_e(x^2-y^2)

(D)

The conjugate functions of w are \tan^{-1}(x/y) and \tan^{-1}(y/x)

Explanation

GATE→2025→Engineering Mathematics→Complex Variables

Q31

NAT[2 Marks]

If C is the unit circle in the complex plane with its center at the origin, then the value of $n$ in the equation given below is ____ (rounded off to 1 decimal place). $$\oint_{c}\frac{z^{3}}{(z^{2}+4)(z^{2}-4)}dz=2~\pi~i~n$$

Explanation

GATE→2024→Engineering Mathematics→Complex Variables

Q32

MCQ[1 Mark]

Let $f(z)$ be an analytic function, where $z=x+iy$. If the real part of $f(z)$ is $\cosh x \cos y$, and the imaginary part of $f(z)$ is zero for $y=0$ then $f(z)$ is

(A)

\cosh x \exp(-iy)

(B)

\cosh z \exp z

(C)

\cosh z \cos y

(D)

\cosh z

Explanation

GATE→2023→Engineering Mathematics→Complex Variables

Q33

NAT[1 Mark]

The value of $k$ that makes the complex-valued function $$f(z)=e^{-kx}(\cos 2y-i~\sin 2y)$$\nanalytic, where $z=x+iy$, is

Explanation

Probability and Statistics

4 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q34

MCQ[2 Marks]

A student council of 10 members consists of two students from engineering school, three students from science school and five students from arts school. The university administration selects three students from the council at random. What is the chance that out of the selected students, two belong to the same school and the third belongs to different school?

(A)

\frac{79}{120}

(B)

\frac{2}{120}

(C)

\frac{89}{120}

(D)

\frac{69}{120}

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q35

MCQ[1 Mark]

If two unbiased coins are tossed, then what is the probability of having at least one head?

(A)

0.25

(B)

0.5

(C)

0.6

(D)

0.75

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q36

NAT[2 Marks]

Let X be a continuous random variable defined on $[0,1]$ such that its probability density function $f(x)=1$ for $0\le x\le 1$ and $0$ otherwise. Let $Y=\log_{e}(X+1)$. Then the expected value of Y is ____ (rounded off to 2 decimal places).

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q37

MCQ[1 Mark]

A machine produces a defective component with a probability of $0.015$. The number of defective components in a packed box containing $200$ components produced by the machine follows a Poisson distribution. The mean and the variance of the distribution are

(A)

3 and 3, respectively

(B)

\sqrt{3} \text{ and } \sqrt{3}, \text{ respectively}

(C)

0.015 \text{ and } 0.015, \text{ respectively}

(D)

3 and 9, respectively

Explanation

Numerical Methods

5 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q38

MCQ[1 Mark]

Newton-Raphson method for solving algebraic equations is based on

(A)

Taylor series

(B)

Fourier series

(C)

Laurent series

(D)

Power series

Explanation

GATE→2026→Engineering Mathematics→Numerical Methods

Q39

MCQ[1 Mark]

The exact solution of $\int_{0}^{4}\frac{dx}{1+x}$ is represented as $n$. If $m$ represents numerically evaluated value of the above integral using Trapezoidal rule by considering four equal subintervals in the range of $x$, then $(m-n)$ is

(A)

-0.074

(B)

-0.003

(C)

0.074

(D)

0.003

Explanation

GATE→2025→Engineering Mathematics→Numerical Methods

Q40

NAT[1 Mark]

The values of a function $f(x)$ obtained for different values of $x$ are shown in the table below. | $x$ | 0 | 0.25 | 0.5 | 0.75 | 1.0 | | --- | --- | --- | --- | --- | --- | | $f(x)$ | 0.4 | 1.8 | 0.9 | 1.5 | 2.0 | \nUsing Simpson's one-third rule, $\int_{0}^{1.0} f(x) dx \approx$ _____ (rounded off to 2 decimal places).

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q41

MCQ[1 Mark]

In order to numerically solve the ordinary differential equation $\frac{dy}{dt}=-y$ for $t>0$, with an initial condition $y(0)=1$, the following scheme is employed: $$\frac{y_{n+1}-y_{n}}{\Delta t}=-\frac{1}{2}(y_{n+1}+y_{n})$$\nHere, $\Delta t$ is the time step and $y_{n}=y(n\Delta t)$ for $n=0,1,2,...$ This numerical scheme will yield a solution with non-physical oscillations for $\Delta t>h$. The value of h is

(A)

1/2

(B)

1

(C)

3/2

(D)

2

Explanation

GATE→2023→Engineering Mathematics→Numerical Methods

Q42

NAT[2 Marks]

The initial value problem $$\frac{dy}{dt}+2y=0, \quad y(0)=1$$\nis solved numerically using the forward Euler's method with a constant and positive time step of $\Delta t$.\nLet $y_{n}$ represent the numerical solution obtained after n steps. The condition $|y_{n+1}|\le |y_{n}|$ is satisfied if and only if $\Delta t$ does not exceed ____ (Answer in integer)

Explanation