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Linear Algebra(10)Calculus(8)Ordinary Differential Equations (ODE)(5)Partial Differential Equations (PDE)(7)Probability and Statistics(5)Numerical Methods(8)

Linear Algebra

10 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

Matrix $P$ is given as $P = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$. The TRUE option is

(A)

Trace of P is equal to the sum of the Eigen values of P.

(B)

$P^T P$ is an identity matrix.

(C)

P is a skew-symmetric matrix.

(D)

Absolute magnitude of each Eigen value is 1.

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

MCQ[1 Mark]

Given $\begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{Bmatrix} x_1 \\ x_2 \\ x_3 \end{Bmatrix} = \begin{Bmatrix} 0 \\ 0 \end{Bmatrix}$. The above system of equations represents a

(A)

plane

(B)

line

(C)

volume

(D)

point

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q3

NAT[1 Mark]

A matrix is given as $\begin{bmatrix} 9 & 15 \\ 15 & 50 \end{bmatrix}$. By performing Cholesky decomposition, $|l_{22}|$ of the lower triangular matrix is

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q4

MCQ[1 Mark]

Suppose $\lambda$ is an eigenvalue of matrix $A$ and $x$ is the corresponding eigenvector. For matrix $B = A - 2I$, the eigenvalue corresponding to $x$ is

(A)

\lambda

(B)

\lambda+2

(C)

2\lambda

(D)

\lambda-2

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q5

MCQ[1 Mark]

Let $A = \begin{bmatrix} 1 & 1 \\ 1 & 3 \\ -2 & -3 \end{bmatrix}$ and $b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$. For $Ax = b$ to be solvable, the correct condition on $b_1, b_2, b_3$ is

(A)

$b_1+b_2+b_3=1$

(B)

$3b_1+b_2+2b_3=0$

(C)

$b_1+3b_2+b_3=2$

(D)

$b_1+b_2+b_3=2$

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q6

MCQ[2 Marks]

What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix}$?

(A)

1, 2, 5

(B)

1, 3, 4

(C)

-5, 1, 2

(D)

-5, -1, 2

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

MSQ[1 Mark]

If $M$ is an arbitrary real $n \times n$ matrix, which of the following matrices will have non-negative eigenvalues?

(A)

M^2

(B)

MM^T

(C)

M^T M

(D)

(M^T)^2

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q8

MSQ[2 Marks]

For the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{bmatrix}$, which of the following statements is/are TRUE?

(A)

Eigenvalues of A^T are same as A

(B)

Eigenvalues of A^-1 are reciprocals of A

(C)

Eigenvectors of A^T are same as A

(D)

Eigenvectors of A^-1 are same as A

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q9

MSQ[1 Mark]

Matrix M has eigenvalues 5 and 2. Form $Q = M^3 - 4M^2 - 2M$. Which of the following is/are eigenvalue(s) of Q?

(A)

15

(B)

25

(C)

-20

(D)

-30

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q10

MCQ[2 Marks]

Cartesian coordinates of point P are $(1,1,1)$. Transformed coordinates due to $45^{\circ}$ clockwise rotation about positive x-axis are

(A)

(1, 0, \sqrt{2})

(B)

(1, 0, -\sqrt{2})

(C)

(-1, 0, \sqrt{2})

(D)

(-1, 0, -\sqrt{2})

Explanation

Calculus

8 Questions

GATE→2026→Engineering Mathematics→Calculus

Q11

NAT[1 Mark]

It is given that $x$ and $y$ are integers in the equation $(x + y - 7)^2 + (y + 3x - 13)^2 = 0$. The value of $(x^3 + y^3)$ is

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q12

MCQ[2 Marks]

Let $f(x)$ be a continuous function defined in $[0,2] \to \mathbb{R}$ satisfying $\int_{0}^{2} f(x)[x - f(x)] dx = \frac{2}{3}$. The value of $f(1)$ is

(A)

1

(B)

2

(C)

1/2

(D)

0

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q13

MCQ[2 Marks]

The value of $\lim_{x \to \infty} (x - \sqrt{x^2 + x})$ is equal to

(A)

-1

(B)

-0.5

(C)

-2

(D)

0

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q14

NAT[2 Marks]

The maximum value of function $h(x) = -x^3 + 2x^2$ in the interval $[-1, 1.5]$ is equal to

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q15

MCQ[2 Marks]

Vector field $\vec{p}$ and scalar field $r$ given. Consider statements P (curl of $\vec{p}$ is zero) and Q (gradient of $r$ is null vector). Which option is CORRECT?

(A)

Both P and Q are FALSE

(B)

P is TRUE and Q is FALSE

(C)

P is FALSE and Q is TRUE

(D)

Both P and Q are TRUE

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q16

MCQ[1 Mark]

For the integral $I = \int_{-1}^{1} \frac{1}{x^2} dx$, which of the following statements is TRUE?

(A)

I = 0

(B)

I = 2

(C)

I = -2

(D)

The integral does not converge

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q17

MSQ[2 Marks]

For the function $f(x) = e^x |\sin x|; x \in \mathbb{R}$, which of the following statements is/are TRUE?

(A)

Function is continuous at all x

(B)

Function is differentiable at all x

(C)

Function is periodic

(D)

Function is bounded

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q18

MSQ[2 Marks]

For the function $f(x) = \max\{3-x, x-1\}$, which statement(s) is/are TRUE?

(A)

It is continuous on its domain

(B)

It has a local minimum at x=2

(C)

It has a local maximum at x=2

(D)

It is differentiable on its domain

Explanation

Ordinary Differential Equations (ODE)

5 Questions

GATE→2026→Engineering Mathematics→Ordinary Differential Equations (ODE)

Q19

MCQ[2 Marks]

An Euler-Cauchy ODE is given by $x^2 \frac{d^2y}{dx^2} = 6y$. The general solution is

(A)

$y(x) = ax^3 + \frac{b}{x^2}$

(B)

$y(x) = ax^2 + \frac{b}{x^3}$

(C)

$y(x) = ax^2 + b \ln x$

(D)

$y(x) = ax^3 + b \ln x$

Explanation

GATE→2025→Engineering Mathematics→Ordinary Differential Equations (ODE)

Q20

NAT[2 Marks]

Let $y$ be the solution of initial value problem $y'' + 0.8y' + 0.16y = 0$, $y(0)=3$, $y'(0)=4.5$. Then $y(1)$ is equal to

Explanation

GATE→2023→Engineering Mathematics→Ordinary Differential Equations (ODE)

Q21

NAT[1 Mark]

In differential equation $\frac{dy}{dx} + \alpha x y = 0$ with $\alpha > 0$, $y=1.0$ at $x=0$, and $y=0.8$ at $x=1.0$, the value of $\alpha$ is

Explanation

GATE→2023→Engineering Mathematics→Ordinary Differential Equations (ODE)

Q22

NAT[2 Marks]

ODE solved by backward difference scheme. Given $u_{n-1} = 1.75$, $t = 3.14$ s, $\Delta t = 0.01$ s. The value of $(u_n - u_{n-1})$ is

Explanation

GATE→2022→Engineering Mathematics→Ordinary Differential Equations (ODE)

Q23

MCQ[1 Mark]

For the equation $\frac{d^3 y}{dx^3} + x \left(\frac{dy}{dx}\right)^{3/2} + x^2 y = 0$, the correct description is

(A)

ordinary differential equation of order 3 and degree 2

(B)

ordinary differential equation of order 3 and degree 3

(C)

ordinary differential equation of order 2 and degree 3

(D)

ordinary differential equation of order 3 and degree 3/2

Explanation

Partial Differential Equations (PDE)

7 Questions

GATE→2025→Engineering Mathematics→Partial Differential Equations (PDE)

Q24

MCQ[1 Mark]

Which one of Participating options is the correct Fourier series of the periodic function $f(x)$ with period 4?

(A)

Option A

(B)

Option B

(C)

Option C

(D)

Option D

Explanation

GATE→2025→Engineering Mathematics→Partial Differential Equations (PDE)

Q25

MSQ[1 Mark]

Which of the following equations belong to the class of second-order, linear, homogeneous partial differential equations?

(A)

Option A

(B)

Option B

(C)

Option C

(D)

Option D

Explanation

GATE→2024→Engineering Mathematics→Partial Differential Equations (PDE)

Q26

MCQ[1 Mark]

Partial differential equation integration and general solution identification.

(A)

$u = x^2 + f(y) + g(x)$

(B)

$u = x^2 + x f(y) + h(y)$

(C)

$u = x^2 + x f(y) + g(x)$

(D)

$u = x^2 + f(y) + y g(x)$

Explanation

GATE→2024→Engineering Mathematics→Partial Differential Equations (PDE)

Q27

MSQ[1 Mark]

For the partial differential equation $x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2}$, which option(s) is/are CORRECT?

(A)

elliptic for x > 0 and y > 0

(B)

parabolic for x > 0 and y > 0

(C)

elliptic for x = 0 and y > 0

(D)

hyperbolic for x < 0 and y > 0

Explanation

GATE→2023→Engineering Mathematics→Partial Differential Equations (PDE)

Q28

MSQ[1 Mark]

For the function $f(x) = px^4 + qx^5$ expressed in Fourier series over $[-L, L]$, which options are true:

(A)

a_n depend on p

(B)

a_n depend on q

(C)

b_n depend on p

(D)

b_n depend on q

Explanation

GATE→2022→Engineering Mathematics→Partial Differential Equations (PDE)

Q29

MCQ[1 Mark]

Consider the expression $z = \sin(y + it) + \cos(y - it)$ where $i = \sqrt{-1}$. The partial differential equation derived from this expression is

(A)

\frac{\partial^2 z}{\partial t^2} + \frac{\partial^2 z}{\partial y^2} = 0

(B)

\frac{\partial^2 z}{\partial t^2} - \frac{\partial^2 z}{\partial y^2} = 0

(C)

\frac{\partial z}{\partial t} - i \frac{\partial z}{\partial y} = 0

(D)

\frac{\partial z}{\partial t} + i \frac{\partial z}{\partial y} = 0

Explanation

GATE→2022→Engineering Mathematics→Partial Differential Equations (PDE)

Q30

NAT[1 Mark]

For $f(x) = \cos^4 x$ expressed as Fourier cosine series $\sum f_n \cos nx$, numerical value of $(f_4 + f_5)$ is

Explanation

Probability and Statistics

5 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q31

NAT[1 Mark]

Bag I contains 4 white and 6 black balls. Bag II contains 4 white and 3 black balls. One ball is drawn at random from any one of the two bags and it is found to be a black ball. The probability that the black ball was drawn from Bag I is

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q32

MCQ[1 Mark]

X is a random variable with given probability mass function. Then, the variance of X is

(A)

20.81

(B)

28.40

(C)

31.70

(D)

10.89

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q33

MCQ[1 Mark]

Probability a student passes only Math is 1/6, only English is 4/9, both is 1/3. Probability of passing in at least one subject is

(A)

17/18

(B)

11/18

(C)

14/18

(D)

1/18

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q34

NAT[1 Mark]

Probabilities of independent events A and B are 0.5 and 0.8. Probability of occurrence of at least A or B is

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q35

NAT[2 Marks]

Linear regression model $y = a + bx$ has intercept $a = 6.1$ for given $(x, y)$ data points. Coefficient $b$ is

Explanation

Numerical Methods

8 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q36

NAT[2 Marks]

Starting with $x = 0.5$, the second approximation for the root of $f(x) = e^{-x} - x$ by Newton-Raphson method is

Explanation

GATE→2026→Engineering Mathematics→Numerical Methods

Q37

NAT[2 Marks]

Tabulated data values used for second-degree Lagrange/Newton interpolation polynomial $P_2(x)$. The value of $P_2(0)$ is

Explanation

GATE→2025→Engineering Mathematics→Numerical Methods

Q38

NAT[2 Marks]

Using Euler method with step size $h = 0.5$ for $\frac{dy}{dx} = y + 2x - x^2$, $y(0)=1$, the value of $y$ at $x = 1.0$ is

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q39

MCQ[1 Mark]

The smallest positive root of the equation $x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 45 = 0$ lies in the range

(A)

0 < x <= 2

(B)

2 < x <= 4

(C)

6 <= x <= 8

(D)

10 <= x <= 100

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q40

NAT[1 Mark]

Consider tabular data for $f(x)$. The value of $f(1.5)$ estimated using second-order Newton's interpolation formula is

Explanation

GATE→2023→Engineering Mathematics→Numerical Methods

Q41

MCQ[2 Marks]

A smooth and convex-shaped function $f(x)$ integrated numerically using trapezoidal rule. The numerical area obtained will be

(A)

less than actual

(B)

more than actual

(C)

exactly equal to actual

(D)

cannot be obtained

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q42

NAT[1 Mark]

Recursive Newton-Raphson iteration for square root with initial guess $x_1=1$ and $P=10$. The numerical value of $x_5$ is

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q43

NAT[2 Marks]

Differential equation $\frac{dy}{dx} = 4(x+2) - y$ with $y=3$ at $x=1$. Value of $y$ at $x=1.4$ using Euler's method with step-size $h=0.2$ is

Explanation