Back to Sections

Type:ALLMCQNATMSQ

Linear Algebra(11)Calculus(17)Differential Equations(7)Probability and Statistics(7)Numerical Methods(4)

Linear Algebra

11 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

Which of the following is one of the eigenvalues for the matrix given below? $$\begin{bmatrix} 3 & 4 \\ 4 & -3 \end{bmatrix}$$

(A)

1

(B)

3

(C)

5

(D)

7

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

MSQ[2 Marks]

For the given matrix, which of the statements given below is/are true? $$\begin{bmatrix} 3 & 1 & 5 \\ 2 & -1 & 0 \\ 5 & 2 & 9 \end{bmatrix}$$

(A)

The matrix is full rank

(B)

The matrix is invertible

(C)

The matrix has a determinant

(D)

The matrix has a transpose

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q3

NAT[2 Marks]

If the straight lines given by the following two equations are parallel to each other, the value of a is $$4x+7y=6; \quad 3ax+42y=24$$ (answer in integer)

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q4

NAT[1 Mark]

The value of k, for which the linear equations $2x+3y=6$ and $4x+6y=3k$ have at least one solution, is (Answer in integer)

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q5

NAT[2 Marks]

Let $$A=\begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}.$$ If the eigenvalues of A are -2, 1, and 2, then the value of k is (Answer in integer)

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q6

MCQ[1 Mark]

A value of k for which the linear equations $(k-1)x+3y=0$ and $2x+ky=0$ have a non-zero solution is

(A)

1

(B)

2

(C)

3

(D)

4

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q7

MCQ[1 Mark]

Let $\vec{OR}$ be the vector that is perpendicular to the vectors $\vec{OP}=2\hat{i}-3\hat{j}+\hat{k}$ and $\vec{OQ}=-2\hat{i}+\hat{j}+\hat{k}$. If the length of the vector $\vec{OR}$ is $\alpha\sqrt{3}$, then $\alpha$ is

(A)

3

(B)

4

(C)

5

(D)

6

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q8

NAT[2 Marks]

If $\vec{v}=\begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$ is an eigenvector of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix}$ corresponding to the non-zero eigenvalue $\lambda$, then the value of $\lambda$ is

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q9

NAT[2 Marks]

If $A=\begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix}$, the value of $|A^{4}+3A^{2}-5A+6I|$ is

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q10

MCQ[1 Mark]

If the eigenvalues of a $2\times2$ matrix P are 4 and 2, then the eigenvalues of the matrix $P^{-1}$

(A)

0, 0

(B)

0.0625, 0.25

(C)

0.25, 0.5

(D)

2, 4

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q11

MCQ[2 Marks]

A $2\times2$ matrix P has an eigenvalue $\lambda_{1}=2$ with eigenvector $x_{1}=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and another eigenvalue $\lambda_{2}=5$, with eigenvector $x_{2}=\begin{pmatrix} 1 \\ 1 \end{pmatrix}$. The matrix P is

(A)

$\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$

(B)

$\begin{pmatrix} 2 & 3 \\ 0 & 5 \end{pmatrix}$

(C)

$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$

(D)

$\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$

Explanation

Calculus

17 Questions

GATE→2026→Engineering Mathematics→Calculus

Q12

MCQ[1 Mark]

Consider the two functions $f_{1}(x)=\frac{x^{2}-4}{x-2}$ and $f_{2}(x)=x^{2}-2x+2.$ Which of the following is the value of $(f_{1}(x)+f_{2}(x))$ as $x\rightarrow2$?

(A)

0

(B)

6

(C)

8

(D)

\infty

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q13

MCQ[1 Mark]

Which of the following functions has the highest area under the curve between $x=0$ and $x=10$?

(A)

$y=x+8$

(B)

$y=3x$

(C)

$y=2x+1$

(D)

$y=x+2$

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q14

NAT[2 Marks]

Consider the equation $\frac{dy}{dx}=\frac{1}{x}$. The value of the integral $\int_{1}^{2}dy$ using trapezoidal method and interval $h=0.25$ (rounded off to two decimal places) is

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q15

NAT[2 Marks]

A straight line $y=x-1$ intersects a circle with center at $x=1$, $y=1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is (rounded off to three decimal places)

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q16

NAT[2 Marks]

From the following plot of $\frac{dy}{dx}$ versus x and if $y(2)=5$, the value of $y(3)$ is (rounded off to one decimal place)

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q17

MCQ[1 Mark]

The minimum value of the function $$f(x)=x+\frac{4}{x}$$ for $x>0$ is

(A)

1

(B)

2

(C)

3

(D)

4

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q18

MCQ[2 Marks]

If the function $$f(x)=\begin{cases} \sin 2x, & \text{for } x>0 \\ a+bx, & \text{for } x\le0 \end{cases}$$ where a and b are constants, is differentiable at $x=0$, then $a+b$ is equal to

(A)

1

(B)

2

(C)

3

(D)

4

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q19

MCQ[1 Mark]

The value of the series at $x = \dots$ [Expression: $1+\sin~x+\cos^{2}x+\sin^{3}x+\cdot\cdot\cdot$]

(A)

1 / (sqrt(2) + 1)

(B)

sqrt(2) / (sqrt(2) + 1)

(C)

1 / (sqrt(2) - 1)

(D)

other matching option D

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q20

NAT[2 Marks]

The value of the limit $\lim_{x\rightarrow\infty}\frac{x}{2}\ln\left(1+\frac{2024}{x}\right)$ is

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q21

NAT[1 Mark]

The value of $\lim_{x\rightarrow0}\left[\frac{\cos~2x-\cos~4x}{x^{2}}\right]$ is

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q22

NAT[1 Mark]

A series (S) is given as $$S=1+3+5+7+9+\dots$$ The sum of the first 50 terms of S is

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q23

NAT[1 Mark]

If $7^{3x}=216$, the value of $7^{-x}$ (rounded off to three decimal places) is

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q24

NAT[1 Mark]

The distance between the two points of intersection of $x^{2}+y=7$ and $x+y=7$ (rounded off to two decimal places) is

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q25

NAT[2 Marks]

If $f(x)=\frac{\sin~x+\cos~x}{\sin~x-\cos~x}$, the value of $f^{\prime}(x)$ at $x=0$ is

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q26

NAT[2 Marks]

If $f(2)=5$ and $(f(x))(f(x+1))=3$ for all real values of x, the value of $f(10)$ is

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q27

NAT[1 Mark]

The maximum value of the function $f(x)=3x^{2}-2x^{3}$ for $x>0$ is

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q28

NAT[2 Marks]

For $x_{1}>0$ and $x_{2}>0$, the value of $\lim_{x_{1}\rightarrow x_{2}}\frac{x_{1}-x_{2}}{x_{2}\ln(\frac{x_{1}}{x_{2}})}$ is

Explanation

Differential Equations

7 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q29

MSQ[1 Mark]

The equation $\frac{d^{2}y}{dx^{2}}-y=0$ has a solution of the form $y=e^{Ax}$. The value(s) of A satisfying this is/are:

(A)

0

(B)

1

(C)

-1

(D)

-\infty

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q30

MCQ[1 Mark]

Let $y(t)$ be a bacterial population whose growth is given by $$\frac{dy}{dt}=\lambda(y+2)$$ where $\lambda$ is the growth rate constant. If $y(0)=1$ and $y(1)=4$, then the value of $\lambda$ is

(A)

\ln 2

(B)

\ln 3

(C)

\ln 4

(D)

\ln 6

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q31

MCQ[2 Marks]

Let m and n be fixed real numbers. If the function $y(t)=C_{1}e^{t}+C_{2}e^{-t}$ is a solution of $$\frac{d^{2}y}{dt^{2}}+m\frac{dy}{dt}+ny=0$$ for any constants $C_{1}$ and $C_{2}$, then $m+n$ is equal to

(A)

-2

(B)

-1

(C)

0

(D)

1

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q32

MCQ[1 Mark]

The solution of the differential equation $\frac{dy}{dx}=y+e^{-x}$ that satisfies $y(0)=-\frac{1}{2}$ is

(A)

$-\frac{1}{2}e^{-\frac{x}{2}}$

(B)

$-\frac{1}{2}e^{x}$

(C)

$-\frac{1}{2}e^{-x}$

(D)

$-\frac{1}{2}e^{\frac{x}{2}}$

Explanation

GATE→2024→Engineering Mathematics→Differential Equations

Q33

NAT[2 Marks]

Let $y(x)=x^{2}\ln~x$ for $x>0$, be a solution of $x^{2}\frac{d^{2}y}{dx^{2}}+4y=\alpha x\frac{dy}{dx}$. Then the value of $\alpha$ is

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q34

MCQ[1 Mark]

What is the order of the differential equation given below? $$\frac{d^{2}y}{dx^{2}}-6x=3x^{4}-2x^{3}+2$$

(A)

1

(B)

2

(C)

3

(D)

4

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q35

MSQ[2 Marks]

Consider the ordinary differential equation $$\frac{dy}{dx}=f(x,y)=2x^{2}-y^{2}.$$ If $y(1)=1$, the value(s) of $y(1.5)$, using the Euler's implicit method $[y_{n+1}=y_{n}+hf(x_{n+1},y_{n+1})]$ with a step size of $h=0.5$, is (are)

(A)

$-1-5\sqrt{0.3}$

(B)

$-1+5\sqrt{0.3}$

(C)

$1+5\sqrt{0.3}$

(D)

$1-5\sqrt{0.3}$

Explanation

Probability and Statistics

7 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q36

NAT[1 Mark]

The mean of the following series is 0.01, 0.02, 0.03, ......, 1 (rounded off to three decimal places)

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q37

NAT[1 Mark]

Two fair six-sided dice, with sides numbered 1 to 6, are thrown once. The probability of getting 7 as the sum of the numbers on the top side of the dice is (Round off to two decimal places)

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q38

MCQ[1 Mark]

The six faces of a cube (die) are numbered as 1, 2, 3, 4, 5 and 6, and it is rolled once. An outcome is the observed number on the top face. If the probability of getting an odd number as an outcome is twice that of an even number, then the probability of getting a number less than 3 is

(A)

1/9

(B)

1/3

(C)

4/9

(D)

other matching option C

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q39

NAT[1 Mark]

Two fair six-sided dice are thrown. The probability of getting 12 as the product of the numbers on the dice (rounded off to two decimal places) is

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q40

NAT[2 Marks]

Ten playing cards numbered $1,2,3,....,10$ are placed face down on a table. One card is drawn at random, its number recorded, and then replaced face down. A card is drawn again at random. The probability that the number on the second draw is greater than the number on the first draw (rounded off to two decimal places) is

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q41

NAT[1 Mark]

The probability density function of a random variable X is $p(x)=2e^{-2x}$. The probability $P(1\le X\le2)$ (rounded off to two decimal places) is

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q42

NAT[2 Marks]

Consider a random variable X with mean $\mu_{x}=0.1$ and variance $\sigma_{X}^{2}=0.2$. A new random variable $Y=2X+1$ is defined. The variance of the random variable Y (rounded off to one decimal place) is

Explanation

Numerical Methods

4 Questions

GATE→2025→Engineering Mathematics→Numerical Methods

Q43

NAT[1 Mark]

Consider a nonlinear algebraic equation, $e^{x}-2=0$. Using the Newton-Raphson method, with the initial guess of $x_{0}=1$ the approximated value of the root of the equation after one iteration is (Round off to two decimal places)

Explanation

GATE→2025→Engineering Mathematics→Numerical Methods

Q44

NAT[2 Marks]

Let $a_{0}=0$ and define $a_{n}=\frac{1}{2}(1+a_{n-1})$ for all positive integers $n\ge1$. The least value of n for which $|1-a_{n}|<\frac{1}{2^{10}}$ is (Answer in integer)

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q45

NAT[2 Marks]

The absolute relative error in evaluating the integral $\int_{0}^{1}x^{2}dx$ by the trapezoidal rule with the step size 0.25 is $\%$ (rounded off to 2 decimal places).

Explanation

GATE→2022→Engineering Mathematics→Numerical Methods

Q46

NAT[1 Mark]

Consider a nonlinear algebraic equation, $x~\ln x+x-1=0$. Using the Newton-Raphson method, with the initial guess of $x_{0}=3$, the value of x after one iteration (rounded off to one decimal place) is

Explanation