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Linear Algebra(8)Calculus(9)Vector Calculus(7)Differential Equations(6)Probability and Statistics(9)Numerical Methods(4)

Linear Algebra

8 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

Select the correct pair of eigenvectors for the following symmetric matrix: $$\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$$ (A) $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} -2 \\ 1 \end{bmatrix}$ (B) $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ (C) $\begin{bmatrix} -1 \\ -2 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ (D) $\begin{bmatrix} -1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ -1 \end{bmatrix}$

(A)

$\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{and} \begin{bmatrix} -2 \\ 1 \end{bmatrix}$

(B)

$\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{and} \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

(C)

$\begin{bmatrix} -1 \\ -2 \end{bmatrix} \text{and} \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

(D)

$\begin{bmatrix} -1 \\ 2 \end{bmatrix} \text{and} \begin{bmatrix} 2 \\ -1 \end{bmatrix}$

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

Which one of the following matrices has eigenvalues 1 and 6? (A) $\begin{bmatrix} 5 & -2 \\ -2 & 2 \end{bmatrix}$ (B) $\begin{bmatrix} 3 & -1 \\ -2 & 2 \end{bmatrix}$ (C) $\begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$ (D) $\begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$

(A)

$\begin{bmatrix} 5 & -2 \\ -2 & 2 \end{bmatrix}$

(B)

$\begin{bmatrix} 3 & -1 \\ -2 & 2 \end{bmatrix}$

(C)

$\begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$

(D)

$\begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q3

MCQ[2 Marks]

Which one of the following matrices is orthogonal?

(A)

Option A

(B)

Option B

(C)

Option C

(D)

Option D

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q4

MSQ[1 Mark]

Which of the following statements is/are correct for a square matrix $A$ with real number entries? $A^T$ denotes the transpose of $A$ and $A^{-1}$ denotes the inverse of $A$. (A) $A$ is symmetric if $A^T = -A$ (B) $A$ is skew-symmetric if $A^T = -A$ (C) If $A$ is orthogonal, then $A^T = A^{-1}$ (D) If $A$ is orthogonal, then its determinant is zero.

(A)

A is symmetric if A^{T} = -A

(B)

A is skew-symmetric if A^{T} = -A

(C)

If A is orthogonal, then A^{T} = A^{-1}

(D)

If A is orthogonal, then its determinant is zero.

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q5

MCQ[2 Marks]

If $\begin{bmatrix} 1 & 2 \\ 8 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \lambda \begin{bmatrix} x \\ y \end{bmatrix}$ where $x, y$ are not identically zero, then the values of $\lambda$ are (A) $5, -3$ (B) $4, -4$ (C) $3, -5$ (D) $5, -4$

(A)

5,-3

(B)

4,-4

(C)

3,-5

(D)

5,-4

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q6

NAT[1 Mark]

The sum of eigenvalues of the matrix $$\begin{bmatrix} 4 & 3 & 2 \\ 0 & -1 & 2 \\ 0 & 0 & -3 \end{bmatrix}$$\nis ___ (in integer).

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

NAT[2 Marks]

For the equation $$\begin{vmatrix} x+3 & 3x+4 & 4x+5 \\ -2 & -3 & -4 \\ -3 & -4 & -5 \end{vmatrix} = 0$$\nthe value of $x$ is ___ (in integer).

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q8

MCQ[2 Marks]

For a 3x3 matrix, the value of the determinant is $-48$ and the trace is $8$. If one of the eigenvalues is $4$, the other two are (A) $2, -3$ (B) $1, -3$ (C) $6, -2$ (D) $-4, 0$

(A)

2,-3

(B)

1,-3

(C)

6,-2

(D)

-4,0

Explanation

Calculus

9 Questions

GATE→2026→Engineering Mathematics→Calculus

Q9

MCQ[1 Mark]

Choose one correct option. Given $w = f(ax + by)$ where $a$ and $b$ are constants. The value of $\left(b\frac{\partial w}{\partial x} - a\frac{\partial w}{\partial y}\right)$ is: (A) $-a$ (B) $b$ (C) $b-a$ (D) $0$

(A)

-a

(B)

b

(C)

b-a

(D)

0

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q10

MSQ[1 Mark]

Which of the following functions is/are expandable using Maclaurin series? (A) $\ln(1+z)$ (B) $\ln z$ (C) $\frac{1}{z^2}$ (D) $\exp(z)$

(A)

\ln(1+z)

(B)

\ln z

(C)

\frac{1}{z^{2}}

(D)

\exp(z)

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q11

NAT[2 Marks]

The value of $\lim_{x\rightarrow 0}\frac{6(x - \sin x)}{x^3}$ is ___ (in integer).

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q12

MCQ[1 Mark]

Which one of the expansions of $\ln\left(\frac{1+x}{1-x}\right)$ about the origin for $|x|<1$ is correct? (A) $x - \frac{x^2}{2} + \frac{x^3}{3}$ (B) $2\left(x - \frac{x^2}{2} + \frac{x^3}{3} - \dots\right)$ (C) $x + \frac{x^3}{3} + \frac{x^5}{5} + \dots$ (D) $2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + \dots\right)$

(A)

x-\frac{x^{2}}{2}+\frac{x^{3}}{3}

(B)

2(x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdot\cdot\cdot)

(C)

x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdot\cdot\cdot

(D)

2(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdot\cdot\cdot)

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q13

NAT[2 Marks]

The integral $\int_{0}^{1} x e^{-x} dx$ evaluates to ___ (Round off to two decimal places).

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q14

MCQ[1 Mark]

The value of $\lim_{x\rightarrow 1} \frac{7x^{7} - 20x^{5} + 13x}{3x^{3} + x - 4}$ is (A) $-\frac{38}{10}$ (B) $-\frac{51}{10}$ (C) $\frac{38}{10}$ (D) undefined

(A)

-\frac{38}{10}

(B)

-\frac{51}{10}

(C)

\frac{38}{10}

(D)

undefined

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q15

NAT[1 Mark]

The maximum value of function $f(x) = 4x^3 - 24x^2 + 36$ in the domain $[-1, 5]$ is ___ (round off to nearest integer).

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q16

MCQ[2 Marks]

Match the plots in Section I with the corresponding functions in Section II.\nFunctions: (1) $y = \frac{\sin^2 x}{x}$, (2) $y = \frac{\sin x}{x}$, (3) $y = x \sin^2 x$, (4) $y = x \sin x$.

(A)

P-2, Q-3, R-4, S-1

(B)

P-2, Q-3, R-1, S-4

(C)

P-1, Q-4, R-3, S-2

(D)

P-2, Q-3, R-1, S-4

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q17

MCQ[1 Mark]

The Taylor series expansion around $x=0$ of the function $f(x)=\frac{x+1}{e^{x}+1}$ truncated to first two terms is (A) $\frac{1}{2}+\frac{1}{4}x$ (B) $\frac{1}{2}+\frac{1}{2}x$ (C) $\frac{1}{2}+x$ (D) $\frac{1}{2}+2x$

(A)

\frac{1}{2}+\frac{1}{4}x

(B)

\frac{1}{2}+\frac{1}{2}x

(C)

\frac{1}{2}+x

(D)

\frac{1}{2}+2x

Explanation

Vector Calculus

7 Questions

GATE→2026→Engineering Mathematics→Vector Calculus

Q18

NAT[1 Mark]

Value of scalar triple product $\vec{a}\cdot(\vec{b}\times\vec{c})$ is (answer in integer) $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ $\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$ $\vec{c} = 3\hat{i} + 4\hat{j} - \hat{k}$

Explanation

GATE→2026→Engineering Mathematics→Vector Calculus

Q19

NAT[2 Marks]

Magnitude of the vector in the direction of the most rapid increase of $\phi(x,y,z) = x^2 - yz$ at point $P(3, 4, 1)$ is ___.

Explanation

GATE→2025→Engineering Mathematics→Vector Calculus

Q20

NAT[2 Marks]

For a two-dimensional field described by $T(x,y) = \frac{1}{3}xy(x+y)$, the magnitude of its gradient at the point $(1,1)$ is ___ (rounded off to two decimal places).

Explanation

GATE→2024→Engineering Mathematics→Vector Calculus

Q21

NAT[1 Mark]

The divergence of the vector field $$\vec{V} = x^{2}y\hat{i} + y^{3}z\hat{j} + z^{4}\hat{k}$|\nat the point $(1,1,1)$ is ___ (Round off to the nearest integer).

Explanation

GATE→2023→Engineering Mathematics→Vector Calculus

Q22

MCQ[2 Marks]

A fluid flow field is given by the velocity vector $\vec{V} = e^{xyz}(x\hat{i} + z\hat{k})$. The curl of velocity at $(1, 2, 3)$ is (A) $e^{6}(9\hat{i} - 16\hat{j} - 3\hat{k})$ (B) $e^{6}(9\hat{i} - 3\hat{k})$ (C) $e^{6}(9\hat{i} + 16\hat{j} - 3\hat{k})$ (D) $e^{6}(-16\hat{i} + 9\hat{j} - 3\hat{k})$

(A)

e^{6}(9\hat{i}-16\hat{j}-3\hat{k})

(B)

e^{6}(9\hat{i}-3\hat{k})

(C)

e^{6}(9\hat{i}+16\hat{j}-3\hat{k})

(D)

e^{6}(-16\hat{i}+9\hat{j}-3\hat{k})

Explanation

GATE→2023→Engineering Mathematics→Vector Calculus

Q23

MCQ[2 Marks]

Given, $\vec{\varphi} = xy\hat{i} + yz\hat{j} + xz\hat{k}$. $S$ is a surface bounded by the planes $x=0$, $y=0$, $z=0$, $x=3$, $y=2$, and $z=1$. If $\hat{n}$ is the unit normal to $S$, then $\iint \vec{\varphi}\cdot\hat{n}\text{ }ds$ is (A) 18 (B) 9 (C) 36 (D) 3

(A)

18

(B)

9

(C)

36

(D)

3

Explanation

GATE→2022→Engineering Mathematics→Vector Calculus

Q24

MCQ[1 Mark]

Given that $V$ is a closed volume in space bounded by the surface $S$ with unit normal $\vec{n}$. If $\vec{f}$ is any non-zero vector and $\vec{\nabla}$ is the gradient/divergence operator, then the volume integral $\int_{V}(\vec{\nabla}\cdot\vec{f})\text{ }dV$ is equal to the surface integral $\int_{S}(\n\cdot\vec{f})\text{ }dS$ by virtue of (A) Stokes Curl theorem (B) Reynolds transport theorem (C) Buckingham Pi theorem (D) Gauss divergence theorem

(A)

Stokes Curl theorem

(B)

Reynolds transport theorem

(C)

Buckingham Pi theorem

(D)

Gauss divergence theorem

Explanation

Differential Equations

6 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q25

MCQ[1 Mark]

Given a function $f(t) = e^{-at}$, where $a$ is a constant. The Laplace transform of the function is $\mathcal{L}[f(t)] = F(s)$. Which one of the following options is correct? (A) $F(s) = \frac{1}{s-a}$ (B) $F(s) = \frac{s}{s^2-a^2}$ (C) $F(s) = \frac{a}{s^2-a^2}$ (D) $F(s) = \frac{1}{s+a}$

(A)

$F(s)=\frac{1}{(s-a)}$

(B)

$F(s)=\frac{s}{(s^{2}-a^{2})}$

(C)

$F(s)=\frac{a}{s^{2}-a^{2}}$

(D)

$F(s)=\frac{1}{(s+a)}$

Explanation

GATE→2026→Engineering Mathematics→Differential Equations

Q26

NAT[2 Marks]

Solution of the differential equation $10x^2 \frac{d^2y}{dx^2} - 20x \frac{dy}{dx} + 22.4y = 0$ is $y = c_1 x^{m_1} + c_2 x^{m_2}$ where $m_1 \neq m_2$. The value of $m_1 + m_2$ (answer in integer) is

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q27

MCQ[2 Marks]

For two continuous functions $M(x,y)$ and $N(x,y)$, the relation $M\text{ }dx + N\text{ }dy = 0$ describes an exact differential equation if (A) $\frac{\partial M}{\partial x} = \frac{\partial N}{\partial y}$ (B) $\frac{\partial M}{\partial x} = -\frac{\partial N}{\partial y}$ (C) $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ (D) $\frac{\partial M}{\partial y} = -\frac{\partial N}{\partial x}$

(A)

\frac{\partial M}{\partial x}=\frac{\partial N}{\partial y}

(B)

\frac{\partial M}{\partial x}=-\frac{\partial N}{\partial y}

(C)

\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}

(D)

\frac{\partial M}{\partial y}=-\frac{\partial N}{\partial x}

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q28

MCQ[2 Marks]

If $\frac{dy}{dx} = 4xy$, $y(0) = 1$, then (A) $y = 2x^2 + 1$ (B) $y = 2e^{2x^2} - 1$ (C) $y = 2e^{x^2} - 1$ (D) $y = e^{2x^2}$

(A)

y = 2x^{2} + 1

(B)

y = 2e^{2x^{2}} - 1

(C)

y = 2e^{x^{2}} - 1

(D)

y = e^{2x^{2}}

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q29

MCQ[1 Mark]

Order (O) and degree (D) of the differential equation $\left(\frac{dy}{dx}\right)^{3} = \sqrt{\frac{d^{2}y}{dx^{2}} + 10}$ are (A) $O=2$ and $D=1$ (B) $O=1$ and $D=2$ (C) $O=6$ and $D=1$ (D) $O=2$ and $D=6$

(A)

O=2 and D=1

(B)

O=1 and D=2

(C)

O=6 and D=1

(D)

O=2 and D=6

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q30

MCQ[2 Marks]

The general solution to the following differential equation where A and B are constants is $$\frac{d^{2}y}{dt^{2}} - 4\frac{dy}{dt} + 4y = 0$$ (A) $y = A \sin(2t) + B$ (B) $y = A \sin(2t) + B \cos(2t)$ (C) $y = A e^{2t} + B e^{-2t}$ (D) $y = A e^{2t} + B t e^{2t}$

(A)

y=A \sin(2t)+B

(B)

y=A \sin(2t)+B \cos(2t)

(C)

y=A e^{2t}+B e^{-2t}

(D)

y=A e^{2t}+B t e^{2t}

Explanation

Probability and Statistics

9 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q31

NAT[1 Mark]

A box contains 20 thermometers, 3 of which are defective. One person randomly draws 2 thermometers from the box, one-by-one, without replacement. The probability in percent (rounded-off to two decimal places) that none of these TWO thermometers is defective, is ___ %.

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q32

NAT[2 Marks]

Consider a distribution with the following probability density function: $$f(x) = \begin{cases} 0.5, & 0 < x < 2 \\ 0.0, & \text{Otherwise} \end{cases}$$\nGiven that the mean of the above probability distribution is 1, the variance (rounded off to two decimal places) is ___.

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q33

NAT[1 Mark]

A linear regression model was fitted to a set of $(x,y)$ data. The total sum of squares and sum of squares of error are 1200 and 120, respectively. The coefficient of determination ($R^2$) of the fit is ___ (rounded off to one decimal place).

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q34

MCQ[1 Mark]

If $X_{1}$ and $X_{2}$ are independent normally distributed random variables with means $\mu_{1}$ and $\mu_{2}$, and variances $\rho_{1}$ and $\rho_{2}$ (or standard deviations/variances), respectively, then the combination $X = X_{1} + X_{2}$ has mean $\mu$ and variance $\rho$ such that (A) $\mu = \mu_{1} + \mu_{2}$ and $\rho = \rho_{1} + \rho_{2}$ (B) $\mu^{2} = \mu_{1}^{2} + \mu_{2}^{2}$ and $\rho = \rho_{1} + \rho_{2}$ (C) $\mu = \mu_{1} + \mu_{2}$ and $\rho^{2} = \rho_{1}^{2} + \rho_{2}^{2}$ (D) $\mu^{2} = \mu_{1}^{2} + \mu_{2}^{2}$ and $\rho^{2} = \rho_{1}^{2} + \rho_{2}^{2}$

(A)

\mu = \mu_{1} + \mu_{2} and \rho = \rho_{1} + \rho_{2}

(B)

\mu^{2} = \mu_{1}^{2} + \mu_{2}^{2} and \rho = \rho_{1} + \rho_{2}

(C)

\mu = \mu_{1} + \mu_{2} and \rho^{2} = \rho_{1}^{2} + \rho_{2}^{2}

(D)

\mu^{2} = \mu_{1}^{2} + \mu_{2}^{2} and \rho^{2} = \rho_{1}^{2} + \rho_{2}^{2}

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q35

MCQ[1 Mark]

Consider the normal (Gaussian) distributions $a$, $b$, $c$ shown in the figure. $\sigma_p$ and $\mu_p$ are standard deviation and mean of distribution $p$, respectively, and means are positive. Which one of the following deductions is correct? (A) $\sigma_a < \sigma_b < \sigma_c$ (B) $\sigma_a > \sigma_b > \sigma_c$ (C) $\mu_a = \mu_b = \mu_c$ (D) $\mu_a > \mu_b > \mu_c$

(A)

\sigma_{a}<\sigma_{b}<\sigma_{c}

(B)

\sigma_{a}>\sigma_{b}>\sigma_{c}

(C)

\mu_{a}=\mu_{b}=\mu_{c}

(D)

\mu_{a}>\mu_{b}>\mu_{c}

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q36

NAT[2 Marks]

The following data is obtained from an experiment: $x$: 1, 2, 3 $y$: 8, 15, 19\nIf the data is fit using the straight line $y = mx + c$ using the least-squares method, then the value of $m$ is ___ (Round off to one decimal place).

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q37

NAT[1 Mark]

The probability of setting an easy exam paper by three setters are $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$. If all three are setting one paper each, then the probability that at least one of the papers will be easy is ___ (round off to 2 decimal places).

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q38

NAT[1 Mark]

A set of observations with normal distribution of error as $\pm 1.96\sigma$ (where $\sigma$ is standard deviation) is equivalent to the confidence interval of ___ $\%$ (round off to the nearest integer).

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q39

NAT[2 Marks]

A set of 11 $(x, y)$ data points is least-squares fitted to a quadratic polynomial. If the sum of squares of error is $2.4$, the variance of error is ___ (round off to 1 decimal place).

Explanation

Numerical Methods

4 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q40

NAT[2 Marks]

Taking number of intervals $n = 3$, the value of the integral $\int_{0}^{0.3} e^{-x^2} dx$ using Trapezoidal method is (rounded off to two decimal places).

Explanation

GATE→2025→Engineering Mathematics→Numerical Methods

Q41

NAT[2 Marks]

Two consecutive estimates of the root of a function $f(x)$ obtained using the Newton-Raphson method are $x_i = 8.5$ and $x_{i+1} = 13.5$, and the value of the function at $x_i$ is $15$. The numerical value of first derivative of the function evaluated at $x_i$ is ___ (in integer).

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q42

MCQ[2 Marks]

Which one of the following equations will fail to converge to a root with an initial guess value of $x = 0.5$, using the Newton-Raphson method? (A) $x(1-x) = 0$ (B) $e^{x} - 3x^2 = 0$ (C) $x - \ln(3x) = 0$ (D) $\tan(x) - x = 0$

(A)

x(1-x)=0

(B)

e^{x}-3x^{2}=0

(C)

x-\ln(3x)=0

(D)

\tan(x)-x=0

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q43

NAT[2 Marks]

The integral of the function $f(x) = 0.2 + 10x^2$ estimated by the trapezoidal rule with a single segment from $x = 0$ to $x = 1$ is ___ (round off to 1 decimal place).

Explanation