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Linear Algebra(8)Calculus(14)Vector Calculus(6)Differential Equations(2)Probability and Statistics(11)Numerical Methods(3)

Linear Algebra

8 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q1

MCQ[1 Mark]

If $I$ is an identity matrix, the polynomial equation that satisfies the Cayley-Hamilton theorem for the matrix $P = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$ is

(A)

$P^{2}-5P+6I=0$

(B)

$P^{2}-5P+5I=0$

(C)

$P^{2}-6P+5I=0$

(D)

$P^{2}-4P+3I=0$

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q2

MCQ[2 Marks]

Given $x=\begin{bmatrix}1\\2\\1\end{bmatrix}, b=\begin{bmatrix}-16\\-3\\7\end{bmatrix}, P^{-1}=\begin{bmatrix}e&-6&-7\\f&9&11\\-1&-2&8\end{bmatrix}$ and $Px=b$. If $Py=d$ and $d=\begin{bmatrix}-16\\0\\7\end{bmatrix}$, the value of $y_{2}$ is

(A)

29

(B)

27

(C)

25

(D)

5

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q3

MCQ[1 Mark]

The rank of $3\times3$ matrix A is 2. The determinant of the matrix is

(A)

0

(B)

1

(C)

2

(D)

3

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q4

NAT[2 Marks]

The characteristic polynomial of a third order matrix, A, is given by $f(x)=x^{3}-6~x^{2}+11~x-6$. If one of the eigen values of A is 1, then sum of the other two eigen values is (in integer)

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q5

MCQ[2 Marks]

Matrix $A = \begin{bmatrix} 1 & 4 & 3 \\ 5 & 2 & 1 \\ 6 & 4 & 3 \end{bmatrix}$ and $B = A - A^{T}$, then $B$ is

(A)

symmetric.

(B)

skew symmetric.

(C)

diagonal.

(D)

scalar.

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q6

MCQ[1 Mark]

Matrix A is negative definite. Which one of the following is NOT the correct statement about the matrix?

(A)

It is symmetric.

(B)

Determinant of A is always less than zero.

(C)

All the eigen values are less than zero.

(D)

Trace of A is always less than zero.

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q7

MCQ[2 Marks]

The slope and intercept values of three linear equations are given... The above system of equations has

(A)

Trivial solution.

(B)

A single solution.

(C)

Multiple solutions.

(D)

No Solution.

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q8

MCQ[1 Mark]

Two vectors x and y are shown in the figure. The projection vector of x on y is

(A)

\frac{x^{T}y}{y^{T}y}y

(B)

x \times y

(C)

\frac{x \times y}{y^{T}y}

(D)

\frac{x^{T}y}{x^{T}x}x

Explanation

Calculus

14 Questions

GATE→2026→Engineering Mathematics→Calculus

Q9

MSQ[1 Mark]

Which of the following statements about Fourier series is/are correct?

(A)

The Fourier series of an even function contains only cosine terms

(B)

The Fourier series of an odd function contains only cosine terms

(C)

The Fourier series of an odd function contains only sine terms

(D)

The Fourier series of an even function contains only sine terms

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q10

MCQ[2 Marks]

The value of the determinant of the Hessian of $f(x,y)=x^{2}+y^{2}+xy-8x-7y$ at its stationary point is

(A)

2

(B)

3

(C)

4

(D)

5

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q11

MCQ[1 Mark]

If $y=x^{x}$, then $\frac{dy}{dx}$ is

(A)

x^{x}(\ln~x+1)

(B)

x\times x^{x-1}

(C)

x\times(x-1)

(D)

x \ln x

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q12

NAT[1 Mark]

The value of $\lim_{x\to 0}(\frac{\tan 11x}{\tan 5x})$ is

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q13

NAT[2 Marks]

The area bounded by the curves, $y=\sqrt{x}$, and $y=8x^{2}$ is (rounded off to 3 decimal places)

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q14

MCQ[1 Mark]

The value of integral, $I=\int_{0}^{\pi/4}\cos x \sin^3 x dx$ is

(A)

\frac{1}{64}

(B)

\frac{1}{16}

(C)

\frac{1}{4}

(D)

1

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q15

MCQ[1 Mark]

The value of $\lim_{x\to 0}(\frac{n \sin 5x}{\sin 3x})$ is

(A)

2n

(B)

3n/5

(C)

6n/5

(D)

5n/3

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q16

MCQ[1 Mark]

$x$ and $y$ are functions of independent variables $r$ and $\theta$ as given below: $x=r~cos~\theta$, $y=r~sin~\theta$. The Jacobian of $x, y$ is

(A)

\tan~\theta

(B)

r^{2}\sin~\theta~\cos~\theta

(C)

r^{2}

(D)

r

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q17

MCQ[2 Marks]

The evaluation of the integral $I=\int\frac{e^{x-1}+x^{e-1}}{e^{x}+x^{e}}dx$ yields

(A)

ln(e^{x}+x^{e})

(B)

\frac{1}{e}ln(e^{x}-x^{e})

(C)

\frac{1}{e}ln(e^{x}+x^{e})

(D)

ln(e^{x}-x^{e})

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q18

MCQ[2 Marks]

Given the function $f(x)=|x|+|x-1|$. For all the real values of x, which one of the following statements is CORRECT?

(A)

The function is continuous and not differentiable at one point.

(B)

The function is continuous but not differentiable at two points.

(C)

The function is discontinuous.

(D)

The function is continuous and differentiable.

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q19

MCQ[1 Mark]

The value of $\lim_{x \to 0} \frac{(1-x)^n - 1}{x}$ is

(A)

0

(B)

1

(C)

-n

(D)

n

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q20

MCQ[1 Mark]

The constant term of the Fourier coefficients of the periodic function $f(x)$ where $f(x) = \begin{cases} k, & -\pi < x < 0 \\ 0, & 0 < x < \pi \end{cases}$, $f(x+2\pi)=f(x)$ and $k=\text{constant}$ is

(A)

k

(B)

2k

(C)

2\pi

(D)

0

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q21

NAT[2 Marks]

Let $f(x)$ be a continuous and differentiable function on [3, 18]. If $f(3)=-50$ and $f^{\prime}(x)\le20$, then the largest possible value of $f(18)$, is

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q22

NAT[2 Marks]

The value of $\int_{0}^{1}x\log(1+x)dx$ is (round off to two decimal places)

Explanation

Vector Calculus

6 Questions

GATE→2026→Engineering Mathematics→Vector Calculus

Q23

MCQ[1 Mark]

Let $\hat{i}$ and $\hat{j}$ be the unit vectors in the $x$ and $y$ direction, respectively. The divergence $(\nabla \cdot \vec{F})$ of $\vec{F} = x^2y\hat{i} + y^3x\hat{j}$ at the point $(3, 2)$ is

(A)

12

(B)

24

(C)

36

(D)

48

Explanation

GATE→2025→Engineering Mathematics→Vector Calculus

Q24

NAT[2 Marks]

The directional derivative of a function $f(x,y,z)=4x^{2}+8y^{2}+9z^{2}$ at the point P (3, 4, 5) in the direction vector $\vec{b}=2\hat{i}-3\hat{j}+4\hat{k}$ is (rounded off to 1 decimal place)

Explanation

GATE→2024→Engineering Mathematics→Vector Calculus

Q25

MCQ[1 Mark]

The magnitude of the curl of the vector $\vec{F} = 2x\hat{i} + 3y\hat{j} + 4z\hat{k}$, is

(A)

0

(B)

4

(C)

9

(D)

24

Explanation

GATE→2024→Engineering Mathematics→Vector Calculus

Q26

MCQ[2 Marks]

Vectors $\mathbf{a}=2\mathbf{i}+3\mathbf{j}-4\mathbf{k}$ and $\mathbf{b}=4\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ represent the two adjacent sides of a triangle. The magnitude of the area of the triangle and the unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ respectively, are

(A)

28.93 and 0.58i - 0.76j - 0.27k

(B)

28.93 and 17.0i - 22.0j - 8.0k

(C)

14.46 and 0.58i - 0.76j - 0.27k

(D)

14.46 and 17.0i - 22.0j - 8.0k

Explanation

GATE→2023→Engineering Mathematics→Vector Calculus

Q27

MCQ[1 Mark]

Given two vectors $\vec{A}=3\hat{i}+2\hat{j}$ and $\vec{B}=\hat{i}+\hat{j}$, the magnitude of projection of $\vec{A}$ along $\vec{B}$ is

(A)

\frac{5}{\sqrt{2}}

(B)

\frac{5}{\sqrt{13}}

(C)

\frac{5}{\sqrt{26}}

(D)

5

Explanation

GATE→2022→Engineering Mathematics→Vector Calculus

Q28

MCQ[1 Mark]

A velocity field in Cartesian coordinate system is expressed as $\vec{v} = x\hat{i} + y\hat{j} + p(z)\hat{k}$, where $p(0)=0$. If $\text{div } \vec{v} = 0$, $p(z)$ is

(A)

0

(B)

-2z

(C)

2

(D)

2z

Explanation

Differential Equations

2 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q29

MCQ[1 Mark]

Which one of the following is a non-linear, second-order, and first-degree differential equation?

(A)

\frac{d^{2}y}{dx^{2}}+\cos~x\frac{dy}{dx}+e^{x}y=0

(B)

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

(C)

\frac{d^{2}y}{dx^{2}}+\sin~x=0

(D)

(\frac{dy}{dx})^{2}+3y=0

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q30

NAT[2 Marks]

Let $\frac{dT}{dt}\propto(T_{R}-T)$, where $T_{R}$ and $T$ are temperatures in degree centigrade of a room and thermometer, respectively, and $t$ denotes time in minutes. A thermometer at a reading of $2^{\circ}\text{C}$ is brought in a room of temperature $40^{\circ}\text{C}$. Two minutes later, the thermometer reads $15^{\circ}\text{C}$. The time elapsed in minutes when the thermometer reads $39.5^{\circ}\text{C}$, is

Explanation

Probability and Statistics

11 Questions

GATE→2026→Engineering Mathematics→Probability and Statistics

Q31

MCQ[1 Mark]

Match the distribution with its corresponding probability density/mass function.

(A)

P\rightarrow 2, Q \rightarrow 3, R \rightarrow 1, S \rightarrow 4

(B)

P\rightarrow 1, Q \rightarrow 3, R \rightarrow 2, S \rightarrow 4

(C)

P\rightarrow 3, Q \rightarrow 2, R \rightarrow 1, S \rightarrow 4

(D)

P\rightarrow 2, Q \rightarrow 4, R \rightarrow 3, S \rightarrow 1

Explanation

GATE→2026→Engineering Mathematics→Probability and Statistics

Q32

NAT[2 Marks]

Using linear regression (least squares), the best-fit line for the given dataset is $y=2.2x+2.3$. Given the residual sum of squares is 25.8, the coefficient of determination $(r^2)$ is

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q33

NAT[1 Mark]

The box plot of a data set is shown below. The interquartile range of the data set is

Explanation

GATE→2025→Engineering Mathematics→Probability and Statistics

Q34

NAT[2 Marks]

The percent Fe content of a random sample consisting of five observations is shown: 62, 64, 63, 60, 61. If mean grade of the stockpile is estimated using the above data, the standard error of mean grade, in %, is (rounded off to 3 decimal places)

Explanation

GATE→2024→Engineering Mathematics→Probability and Statistics

Q35

NAT[2 Marks]

A thermal power plant has an agreement with three mines M1, M2 and M3 to receive 'Grade 1' coal, in the proportion of 60%, 25% and 15%, respectively. The probabilities that a wagon supplied coal to the plant containing below 'Grade 1' from mines M1, M2 and M3 are 0.02, 0.03 and 0.04, respectively. On a random check, a sample wagon is found to carry below 'Grade 1' coal. The probability that the wagon belongs to mine M1, is

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q36

MCQ[1 Mark]

Given two random variables X and Y, the expected value $E(3X-5Y)$ is

(A)

3E(X)-5E(Y)

(B)

3E(X)+5E(Y)

(C)

3E(X)-5E(Y)-15E(XY)

(D)

E(X)-E(Y)-E(XY)

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q37

MCQ[2 Marks]

In a quality control process of coal supplied to a thermal plant, the 3-sigma control limits for fixed carbon (FC) are defined by $40\%\pm15\%$. The process is termed "out of control" if: Rule 1: 4 out of 5 successive values... Rule 2: Any one value crosses any of the 3-sigma control limits. For the continuous data: 49, 51, 56, 20, 46, 48, 47, 49, 45, 41, 42, 40, the process is

(A)

out of control because of both rules 1 & 2.

(B)

out of control because of rule 1 only.

(C)

out of control because of rule 2 only.

(D)

not out of control.

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q38

MCQ[2 Marks]

A regression line is constructed between shovel production rate and shovel swing angle for 50 observations as shown below... If residuals are normally distributed and significance tests of the parameters are conducted at 0.05 significance level, the true statement is

(A)

Both intercept and slope are significant.

(B)

Intercept is significant but slope is not significant.

(C)

Intercept is not significant but slope is significant.

(D)

Both intercept and slope are not significant.

Explanation

GATE→2023→Engineering Mathematics→Probability and Statistics

Q39

NAT[2 Marks]

The time between consecutive accidents in days in an underground coal mine in a year are as follows: 10, 15, 6, 18, 12, 14, 16, 9, 21, 15, 26, 18, 22, 25, 13. Assuming exponential distribution, the probability that there will be no accident over a 10-day period is [rounded off to 2 decimal places]

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q40

MCQ[2 Marks]

The closest match of the scatter plot between the variables X and Y with the approximate attribute is

(A)

P->I, Q->II, R->III, S->IV

(B)

P->II, Q->I, R->IV, S->III

(C)

P->III, Q->IV, R->I, S->II

(D)

P->IV, Q->II, R->III, S->I

Explanation

GATE→2022→Engineering Mathematics→Probability and Statistics

Q41

NAT[2 Marks]

In a health centre, the probability of 'full occupancy' of COVID beds for a day is 0.8. Assuming Binomial probability distribution, the probability of full occupancy exactly for 5 days in a week, is

Explanation

Numerical Methods

3 Questions

GATE→2026→Engineering Mathematics→Numerical Methods

Q42

MCQ[2 Marks]

Given the data in the Table, Newton's divided difference interpolation polynomial is $y=b_{0}+b_{1}(x-2)+b_{2}(x-2)(x-2.5)$. If the values of $b_{0}$ and $b_{1}$ are 3 and 1, respectively, then the value of $b_{2}$ is

(A)

3.0

(B)

3

(C)

4

(D)

5

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q43

MCQ[2 Marks]

Magnitude of error in the determination of the integral, $I$ using Simpson's $1/3$ rule, taking step length as 1.0 is $I=\int_{1}^{3}(x^{3}+6)dx$

(A)

0

(B)

1.0

(C)

1.5

(D)

2.0

Explanation

GATE→2024→Engineering Mathematics→Numerical Methods

Q44

NAT[2 Marks]

The root of the function, $f(x)=x^{3}-2x^{2}+3x-1$ in the interval [0, 1] using bisection method after two iterations, is

Explanation