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Linear Algebra(10)Calculus(11)Vector Calculus(5)Differential Equations(9)Probability and Statistics(5)Numerical Methods(3)

Linear Algebra

10 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

Two square matrices 𝐀 and 𝐁 of the same order are such that, det[𝐀∙𝐁] = −80 and det[𝐀] = 5. Here, 𝐀∙𝐁 is the product of 𝐀 and 𝐁. The value of det[𝐁𝑻] is (A) −16 (B) 1 − 16 (C) −400 (D) 28

(A)

−16

(B)

1 − 16

(C)

−400

(D)

28

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

Consider the simultaneous equations 2𝑥 +4𝑦 = −7; 3𝑥 +5𝑦 = 1 While solving using Gauss elimination without pivoting, after the forward elimination, the diagonal elements of the coefficient matrix are (A) 2 and −1 (B) 3 and 2 (C) 5 and −2 (D) 3 and 4

(A)

2 and −1

(B)

3 and 2

(C)

5 and −2

(D)

3 and 4

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q3

MCQ[2 Marks]

The determinant of a 2×2 square matrix is 1.6. If the sum of the eigen values is −2.8, the eigen values are (A) −2 and −0.8 (B) −1.8 and −1 (C) −3 and 0.2 (D) −1.2 and −1.6

(A)

−2 and −0.8

(B)

−1.8 and −1

(C)

−3 and 0.2

(D)

−1.2 and −1.6

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q4

MCQ[1 Mark]

If $a, b,$ and $c$ are the roots of $2x^{3}-3x^{2}+px-1=0$ and sum of the two roots is 1, the value of $p$ is:

(A)

2

(B)

1

(C)

3

(D)

\frac{1}{3}

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q5

MCQ[1 Mark]

If the matrix $\begin{bmatrix} x & -3 \\ 2 & x-5 \end{bmatrix}$ is singular, sum of the values of $x$ is:

(A)

5

(B)

0

(C)

-1

(D)

2

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q6

MCQ[2 Marks]

Let $[P]=\begin{bmatrix}2 & 3 & 7 \\ 4 & 1 & 5\end{bmatrix}$ and $[Q]=\begin{bmatrix}8 & 11 & 9 \\ 12 & 6 & 13\end{bmatrix}$ be two matrices. The matrix $[P]^{T}[Q]$ is

(A)

matrix A

(B)

matrix B

(C)

matrix C

(D)

matrix D

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

If A and B are square matrices of order 3 such that $|A|=-1$, $|B|=3$ then $|3AB|$ equals

(A)

-81

(B)

-27

(C)

-9

(D)

81

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q8

MCQ[2 Marks]

If $A=\begin{bmatrix}1 & -1 \\ 2 & -1\end{bmatrix}$ and $B=\begin{bmatrix}a & 1 \\ b & -1\end{bmatrix}$ and $(A+B)^{2}=A^{2}+B^{2},$ then the values of a and b are

(A)

$a=4$ $b=1$

(B)

$a=1$ $b=4$

(C)

$a=0$, $b=4$

(D)

$a=2$, $b=4$

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q9

MCQ[1 Mark]

Determinant of a matrix remains unaltered if

(A)

its columns and rows are interchanged

(B)

two parallel lines are identical

(C)

two parallel lines intersect

(D)

each element of a line is multiplied by the same factor

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q10

MCQ[2 Marks]

The matrix $\begin{bmatrix} (3-x) & 2 & 2 \\ 2 & (4-x) & 1 \\ -2 & -4 & (-1-x) \end{bmatrix}$ is singular for the following values of x

(A)

x=0 and x=3

(B)

x=0 and x=-3

(C)

x=0 and x=6

(D)

x=0 and x=-6

Explanation

Calculus

11 Questions

GATE→2026→Engineering Mathematics→Calculus

Q11

MCQ[2 Marks]

1 1 The value of the integral 𝐼 = ∫ ∫ 𝑥cos(𝑥𝑦)𝑑𝑥𝑑𝑦 is 0 0 (A) 1−cos(1) (B) 1+cos(1) (C) 1−sin(1) (D) 1+sin(1)

(A)

1−cos(1)

(B)

1+cos(1)

(C)

1−sin(1)

(D)

1+sin(1)

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q12

MCQ[2 Marks]

The series $\sum_{n=0}^{r}q^{n}=1+q+q^{2}+...$ has the sum:

(A)

\frac{1}{1-q}-\frac{q^{r+1}}{1-q}

(B)

\frac{1}{q-1}+\frac{q^{r+1}}{q-1}

(C)

\frac{1}{1-q}+\frac{q^{r+1}}{1-q}

(D)

\frac{1}{q-1}

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q13

MCQ[2 Marks]

The value of $\lim_{x\rightarrow0}\frac{x~\cos~x-\sin~x}{x^{2}\sin~x}$ is:

(A)

-\frac{1}{3}

(B)

\frac{1}{3}

(C)

3

(D)

-3

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q14

MCQ[2 Marks]

If $u=x^{4}+y^{4}+3x^{2}y^{2}$, the value of $(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y})^{-1}$ is:

(A)

4u

(B)

\frac{1}{4u}

(C)

\frac{1}{2u}

(D)

2u

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q15

MSQ[1 Mark]

A continuously differentiable function $y=f(x)$ satisfies one or more of the following conditions at its point of inflection:

(A)

\frac{d^{2}y}{dx^{2}}\neq0

(B)

\frac{d^{2}y}{dx^{2}}=0

(C)

\frac{d^{3}y}{dx^{3}}=0

(D)

\frac{d^{3}y}{dx^{3}}\neq0

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q16

NAT[2 Marks]

The calculated value of $e^{1/2}$ using Taylor's series with first four terms for $x=\frac{1}{2}$ is ___ (Rounded off to 3 decimal places)

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q17

MCQ[1 Mark]

$\lim_{x\rightarrow0}\frac{\sqrt{1+x}-1}{x}$ is equal to

(A)

0

(B)

\frac{1}{2}

(C)

1

(D)

2

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q18

MCQ[1 Mark]

The value of $I=\int_{0}^{\frac{\pi}{2}}\frac{(\sin x+\cos x)^{2}}{\sqrt{1+\sin 2x}}dx$ is

(A)

0

(B)

1

(C)

2

(D)

3

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q19

MCQ[1 Mark]

Function f(x) by Maclaurin's series (as an infinite series) can be expressed as

(A)

f(x)=f(1)+xf^{\prime}(1)+\frac{x^{2}}{2!}f^{\prime\prime}(1)+\frac{x^{3}}{3!}f^{\prime\prime\prime}(1)+\cdot\cdot\cdot+\infty

(B)

f(x)=f(0)+xf^{\prime}(0)+\frac{x^{2}}{2!}f^{\prime\prime}(0)+\frac{x^{3}}{3!}f^{\prime\prime\prime}(0)+\cdot\cdot\cdot+\infty

(C)

f(x)=f(1)-xf^{\prime}(1)+\frac{x^{2}}{2!}f^{\prime\prime}(1)-\frac{x^{3}}{3!}f^{\prime\prime\prime}(1)+\cdot\cdot\cdot+\infty

(D)

f(x)=f(0)-xf^{\prime}(0)+\frac{x^{2}}{2!}f^{\prime\prime}(0)-\frac{x^{3}}{3!}f^{\prime\prime\prime}(0)+\cdot\cdot\cdot+\infty

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q20

NAT[1 Mark]

The slope of the function $f(x)=2x^{4}-3x^{2}+5x$ at $x=2$ is [Answer in integer]

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q21

MCQ[2 Marks]

The function $(x-2)^{2}(x+2)^{2}$ has

(A)

minima at +2 and maxima at -2

(B)

minima at -2 and maxima at +2

(C)

minima at -2 and +2

(D)

maxima at -2 and +2

Explanation

Vector Calculus

5 Questions

GATE→2026→Engineering Mathematics→Vector Calculus

Q22

NAT[2 Marks]

The following two vectors are adjacent sides of a parallelogram: 𝐴⃗ = 2𝑖̂+3𝑗̂−𝑘̂ and 𝐵⃗⃗ = 5𝑖̂−4𝑘̂ The magnitude of area of the parallelogram is ________ (Rounded off to two decimal places)

Explanation

GATE→2024→Engineering Mathematics→Vector Calculus

Q23

MCQ[1 Mark]

The divergence of the curl of a twice continuously differentiable vector function is

(A)

0

(B)

1

(C)

2

(D)

8

Explanation

GATE→2024→Engineering Mathematics→Vector Calculus

Q24

NAT[2 Marks]

The directional derivative of $u(x,y,z)=x^{2}y+y^{2}z$ at the point (1, 2, 3) in the direction $\hat{i}+2\hat{j}+3\hat{k}$ is ___ (Rounded off to 2 decimal places)

Explanation

GATE→2023→Engineering Mathematics→Vector Calculus

Q25

MCQ[2 Marks]

A vector $\vec{F}=5\hat{i}-10\hat{j}+8\hat{k}$ is passing through the origin of a 3-D frame. Considering the tendency of rotation in the counter clockwise direction as positive, the moment about a point A: (3, 4, 8) is

(A)

$-16\hat{i}+112\hat{j}+50\hat{k}$

(B)

$112\hat{i}+16\hat{j}-50\hat{k}$

(C)

$50\hat{i}-112\hat{j}+16\hat{k}$

(D)

$-112\hat{i}-16\hat{j}+50\hat{k}$

Explanation

GATE→2022→Engineering Mathematics→Vector Calculus

Q26

NAT[2 Marks]

Work done by a moving particle in the force field $\vec{F}=6x^{2}\hat{i}+(3xz+y)\hat{j}+4z\hat{k}$, moving along the straight line from (0,0,0) to (1,2,3) is [Answer in integer]

Explanation

Differential Equations

9 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q27

MCQ[1 Mark]

Identify the following partial differential equations in the given order based on their generic form where 𝑐 is a constant, and 𝑢 and 𝑢 are functions of (𝑥,𝑡) and (𝑥,𝑦), 1 2 respectively: (i) 𝜕2𝑢1 = 𝑐2𝜕2𝑢1 (ii) 𝜕𝑢1 = 𝑐2𝜕2𝑢1 (iii) 𝜕2𝑢2+ 𝜕2𝑢2 = 0 𝜕𝑡2 𝜕𝑥2 𝜕𝑡 𝜕𝑥2 𝜕𝑥2 𝜕𝑦2 (A) (i) Wave equation (ii) Heat equation (iii) Laplace equation (B) (i) Heat equation (ii) Laplace equation (iii) Wave equation (C) (i) Laplace equation (ii) Wave equation (iii) Heat equation (D) (i) Heat equation (ii) Wave equation (iii) Laplace equation

(A)

(i) Wave equation (ii) Heat equation (iii) Laplace equation

(B)

(i) Heat equation (ii) Laplace equation (iii) Wave equation

(C)

(i) Laplace equation (ii) Wave equation (iii) Heat equation

(D)

(i) Heat equation (ii) Wave equation (iii) Laplace equation

Explanation

GATE→2026→Engineering Mathematics→Differential Equations

Q28

MCQ[1 Mark]

𝑑2𝑦 The general solution to the ordinary differential equation = cos2𝑥 is 𝑑𝑥2 (A) cos2𝑥 𝑦 = − +𝐶 𝑥+𝐶 , where 𝐶 and 𝐶 are constants 1 2 1 2 4 (B) sin2𝑥 𝑦 = +𝐶 , where 𝐶 is a constant 1 1 2 (C) cos2𝑥 𝑦 = +𝐶 𝑥+𝐶 , where 𝐶 and 𝐶 are constants 1 2 1 2 4 (D) sin2𝑥 𝑦 = − +𝐶 , where 𝐶 is a constant 1 1 2

(A)

cos2𝑥 𝑦 = − +𝐶 𝑥+𝐶 , where 𝐶 and 𝐶 are constants 1 2 1 2 4

(B)

sin2𝑥 𝑦 = +𝐶 , where 𝐶 is a constant 1 1 2

(C)

cos2𝑥 𝑦 = +𝐶 𝑥+𝐶 , where 𝐶 and 𝐶 are constants 1 2 1 2 4

(D)

sin2𝑥 𝑦 = − +𝐶 , where 𝐶 is a constant 1 1 2

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q29

MCQ[1 Mark]

$\frac{s}{s^{2}+a^{2}}$ is the Laplace transform of:

(A)

cos at

(B)

sin at

(C)

sinh at

(D)

cosh at

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q30

MCQ[1 Mark]

$y^{\prime\prime}+p(x)y^{\prime}+q(x)y=r(x)$ is a ordinary differential equation.

(A)

second order, nonhomogeneous, and linear

(B)

second order, homogeneous, and linear

(C)

second order, homogeneous, and nonlinear

(D)

first order, nonhomogeneous, and linear

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q31

NAT[2 Marks]

The value of $y$ at $x=0.3$ and $y(x=0)=0$ for the differential equation $\frac{dy}{dx}=3y+2e^{x}$ is ___ (Rounded off to 3 decimal places)

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q32

MCQ[1 Mark]

Laplace transform of a function $f(t)=t^{4}$ as a function of 's' is

(A)

\frac{120}{s^{5}}

(B)

\frac{24}{s^{5}}

(C)

\frac{120}{s^{4}}

(D)

\frac{24}{s^{4}}

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q33

MCQ[2 Marks]

The complete solution for the differential equation $\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}-3y=0$ for $y(0)=0$ and $y^{\prime}(0)=8$ is

(A)

2(e^{x}-e^{3x})

(B)

2(e^{x}-e^{-3x})

(C)

2(e^{x}+e^{3x})

(D)

2(e^{x}+e^{-3x})

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q34

MCQ[1 Mark]

$y=ae^{mx}+be^{-mx}$ is the solution of the differential equation

(A)

\frac{dy}{dx}-my=0

(B)

\frac{dy}{dx}+my=0

(C)

\frac{d^{2}y}{dx^{2}}+m^{2}y=0

(D)

\frac{d^{2}y}{dx^{2}}-m^{2}y=0

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q35

NAT[2 Marks]

Given that $\frac{dy}{dx}=2x+y$ and $y=1$ when $x=0$ Using Runge-Kutta fourth order method, the value of y at $x=0.2$ is ___ (rounded off to 3 decimal places).

Explanation

Probability and Statistics

5 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q36

MCQ[1 Mark]

A box contains one red ball and two blue balls. One ball is drawn at random, its colour is noted and the ball is returned to the box. The process is repeated. In the first two draws, blue ball appears each time. The probability of drawing blue ball again in the third draw is (A) 2 3 (B) 1 3 (C) 1 2 (D) 1

(A)

2 3

(B)

1 3

(C)

1 2

(D)

1

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q37

MCQ[1 Mark]

The median of the following set of numbers: 154, 130, 144, 137, 156, 146, 138, 149, 160, 138 is:

(A)

145.0

(B)

145.2

(C)

151.0

(D)

146.0

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q38

NAT[1 Mark]

The mean and variance of a dataset are 16 and 25, respectively. The coefficient of variation of the dataset, in %, is ___ (Rounded off to 2 decimal places)

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q39

NAT[1 Mark]

The chances for passing an endurance test for three threshers P, Q and R are $1/6$, $1/4$ and $1/2$, respectively. The probability of passing the endurance test by at least one thresher, in fraction, is ___ (Rounded off to 2 decimal places)

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q40

MCQ[1 Mark]

The probability of having 53 Sundays in a randomly selected leap year is

(A)

1/7

(B)

1/4

(C)

2/7

(D)

4/7

Explanation

Numerical Methods

3 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q41

NAT[2 Marks]

While solving an equation 𝑒−𝑥 −𝑥 = 0 for 𝑥, using Newton-Raphson method with initial guess 𝑥 = 2, the solution after first iteration (𝑥 ) is ________ (Rounded off 0 1 to two decimal places)

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q42

NAT[1 Mark]

The root of the equation $\sin~x-4x+1=0$ after its first iteration, using Newton-Raphson method with an initial guess of $x_{0}=0.2$, is [round off to three decimal places]

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q43

NAT[2 Marks]

The power consumption readings (in watt) by an instrument at fixed intervals of time (in seconds) are tabulated below: [Table provided in paper]. Using Simpson's $1/3^{rd}$ rule, the energy expenditure of the instrument in joule is [round off to two decimal places]

Explanation