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Calculus(15)Linear Algebra(22)Special Topics (Engineering Mathematics)(2)Differential Equations(10)Numerical Methods (Special)(2)AE-S05-T01(22)Probability and Statistics(1)Complex Analysis(1)Numerical Methods(2)Fourier Series and Laplace Transforms(2)

Calculus

15 Questions

GATE→2026→Engineering Mathematics→Calculus

Q1

MCQ[1 Mark]

Consider the contour 𝐶 shown in the figure below. For the vector 𝐹⃗ = (𝑥 + 2𝑦)𝑒̂𝑥 + (2𝑥 + 4𝑦)𝑒̂𝑦 , the integral ∮𝐶 𝐹⃗ ⋅ 𝑑𝑙⃗ =________. Here 𝑑𝑙⃗ represents an infinitesimal length along the contour 𝐶.

(A)

0

(B)

2

(C)

4

(D)

6

Explanation

GATE→2026→Engineering Mathematics→Calculus

Q2

NAT[2 Marks]

The minimum value of the function 𝑓(𝑥) = |𝑥| + |2𝑥 + 3| for real 𝑥 is ________ (rounded off to 1 decimal place).

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q3

MCQ[1 Mark]

𝑓(𝑥) = {𝐴+ 𝑥 , if 𝑥< 2 1 + 𝑥2 , if 𝑥≥2 If the function 𝑓(𝑥) is continuous at 𝑥= 2, the value of 𝐴 is _____.

(A)

2

(B)

2.5

(C)

3

(D)

3.5

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q4

NAT[2 Marks]

The maximum value of the function 𝑓(𝑥) = (𝑥−1)(𝑥−2)(𝑥−3) in the domain [0, 3] occurs at 𝑥= ______(rounded off to two decimal places).

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q5

NAT[2 Marks]

lim 𝑥→0 1−cos(2𝑥) 𝑥2 = _____ (answer in integer).

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q6

NAT[2 Marks]

The equation of a closed curve in two-dimensional polar coordinates is given by 𝑟= 2 √𝜋(1 −sin 𝜃) . The area enclosed by the curve is ________ (answer in integer).

Explanation

GATE→2025→Engineering Mathematics→Calculus

Q7

NAT[2 Marks]

𝑖̂ and 𝑗̂ denote unit vectors in the 𝑥 and 𝑦 directions, respectively. The outward flux of the two-dimensional vector field 𝑣⃗= 𝑥𝑖̂ + 𝑦𝑗̂ over the unit circle centered at the origin is ______ (rounded off to two decimal places).

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q8

NAT[1 Mark]

Using Trapezoidal rule with one interval, the approximate value of the definite integral: ∫ 𝑑𝑥 1 + 𝑥2 2 1 = ______ (rounded off to 2 decimal places).

Explanation

GATE→2024→Engineering Mathematics→Calculus

Q9

MSQ[2 Marks]

Consider the function 𝑓(𝑥) = {𝑥2 for 𝑥< 0 𝑥 for 𝑥≥0 where x is real. Which of the following statements is/are correct?

(A)

The function is continuous for all x

(B)

The derivative of the function is discontinuous at x = 0

(C)

The derivative of the function is continuous at x = 1

(D)

The function is discontinuous at x = 0

Explanation

GATE→2023→Engineering Mathematics→Calculus

Q10

MCQ[2 Marks]

Given the function y ( x) = ( x + 3)( x − 2) , for −4 x  4 . What is the value of x at which the function has a minimum?

(A)

3 − 2

(B)

1 − 2

(C)

1 2

(D)

3 2 Aerospace Engineering (AE)

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q11

MCQ[1 Mark]

The equation of the straight line representing the tangent to the curve $y = x^2$ at the point $(1,1)$ is

(A)

$y = 2x - 2$

(B)

$x = 2y - 1$

(C)

$y - 1 = 2(x - 1)$

(D)

$x - 1 = 2(y - 1)$

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q12

NAT[1 Mark]

The arc length of the parametric curve: $x = \cos\θ$, $y = \sin\θ$, $z = \θ$ from $\θ = 0$ to $\θ = 2\pi$ is equal to ______ (round off to one decimal place).

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q13

MCQ[2 Marks]

The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius $R$ is

(A)

R

(B)

5/4 R

(C)

4/3 R

(D)

3/2 R

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q14

MSQ[2 Marks]

The real function $y = \sin^2(|x|)$ is

(A)

continuous for all x

(B)

differentiable for all x

(C)

not continuous at x = 0

(D)

not differentiable at x = 0

Explanation

GATE→2022→Engineering Mathematics→Calculus

Q15

NAT[2 Marks]

Vector field $\vec{v} = x^3\hat{i} + y^3\hat{j} + z^3\hat{k}$. Using the Gauss divergence theorem, the value of the outward flux of the vector field over the surface of a sphere of unit radius centered at the origin is ______ (round off to one decimal place).

Explanation

Linear Algebra

22 Questions

GATE→2026→Engineering Mathematics→Linear Algebra

Q52

MSQ[1 Mark]

An 𝑛 × 𝑛 square matrix 𝐴 satisfies 𝐴𝑇 = 𝐴−1. The determinant of this matrix may take which of the following value(s)?

(A)

+1

(B)

−1

(C)

𝑛

(D)

0

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q53

NAT[1 Mark]

If a matrix can be written as 𝐴 = 𝑢𝑣 𝑇 , where both 𝑢 and 𝑣 are 𝑛-dimensional real- valued non-zero column vectors, then the rank of the matrix 𝐴 is ________ (answer in integer).

Explanation

GATE→2026→Engineering Mathematics→Linear Algebra

Q54

MCQ[2 Marks]

𝑎 𝑏 For the matrix 𝐴 = [ ], if the relation 𝑎 + 𝑏 = 𝑐 + 𝑑 holds and 𝑎, 𝑏, 𝑐, 𝑑 ≠ 0, 𝑐 𝑑 then which one of the following statements about 𝐴 is FALSE?

(A)

1 [ ] is an eigenvector 1

(B)

𝜆 = 𝑎 + 𝑏 is an eigenvalue

(C)

𝜆 = 𝑑 − 𝑏 is an eigenvalue

(D)

𝜆 = 𝑑 + 𝑏 is an eigenvalue

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q55

MCQ[1 Mark]

For any real symmetric matrix 𝐴, the transpose of 𝐴 is ____ .

(A)

inverse of 𝐴

(B)

null matrix

(C)

−𝐴

(D)

𝐴

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q56

MCQ[1 Mark]

2𝑥+ 3𝑦+ 𝑧= 0 𝑥+ 𝑦= 0 𝑦+ 𝑧= 0 The given system of equations has

(A)

a unique solution.

(B)

infinitely many solutions.

(C)

no solution.

(D)

a finite number of solutions.

Explanation

GATE→2025→Engineering Mathematics→Linear Algebra

Q57

MCQ[1 Mark]

The eigenvalues of the matrix [1 2 0 3] are _____.

(A)

0, 2

(B)

2, 3

(C)

1, 3

(D)

1, 2

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q58

MCQ[1 Mark]

The following system of linear equations 7𝑥−3𝑦+ 𝑧= 0 3𝑥−𝑦+ 𝑧= 0 𝑥−𝑦−𝑧= 0 has:

(A)

infinitely many solutions

(B)

a unique solution

(C)

no solution

(D)

three solutions

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q59

MCQ[1 Mark]

The three-dimensional stress-strain relationship for an isotropic material is given as { 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 𝜏𝑦𝑧 𝜏𝑥𝑧 𝜏𝑥𝑦} = [ 𝑃 𝑄 𝑄 𝑄 𝑃 𝑄 𝑄 𝑄 𝑃 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑅 0 0 0 𝑅 0 0 0 𝑅] { 𝜀𝑥𝑥 𝜀𝑦𝑦 𝜀𝑧𝑧 𝛾𝑦𝑧 𝛾𝑥𝑧 𝛾𝑥𝑦} where, P, Q and R are the three elastic constants, 𝜎 and 𝜏 represent normal and shear stresses and 𝜀 and 𝛾 represent normal and engineering shear strains. Which one of the following options is correct?

(A)

𝑅= 𝑃−𝑄 2

(B)

𝑅= 𝑄−𝑃 2

(C)

𝑄= 𝑃−𝑅 2

(D)

𝑄= 𝑅−𝑃 2

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q60

MCQ[2 Marks]

Given 𝑦= 𝑒𝑝𝑥sin 𝑞𝑥, where 𝑝 and 𝑞 are non-zero real numbers, the value of the differential expression 𝑑2𝑦 𝑑𝑥2 −2𝑝𝑑𝑦 𝑑𝑥+ (𝑝2 + 𝑞2)𝑦 is

(A)

0

(B)

1

(C)

𝑝2 + 𝑞2

(D)

𝑝𝑞

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q61

MCQ[2 Marks]

The volume of the solid formed by a complete rotation of the shaded portion of the circle of radius R about the y-axis is 𝑘𝜋R3. The value of 𝑘 is:

(A)

5 12

(B)

5 24

(C)

7 12

(D)

7 24

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q62

MCQ[2 Marks]

At a point in the trajectory of an unpowered space vehicle moving about the Earth, the altitude above the mean sea level is 600 km, and the speed with reference to a coordinate system fixed to the center of mass of the Earth is 9 km/s. Assume that the Earth is a sphere with a radius 6400 km and 𝐺𝑀𝐸𝑎𝑟𝑡ℎ= 3.98 × 1014 m3/s2, where, G is the universal gravitational constant and 𝑀𝐸𝑎𝑟𝑡ℎ is mass of the Earth. The trajectory is:

(A)

Circular

(B)

Elliptic

(C)

Parabolic

(D)

Hyperbolic

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q63

MSQ[2 Marks]

The figure shows plots of two yield loci for an isotropic material, where 𝜎𝐼 and 𝜎𝐼𝐼 are the principal stresses, and 𝜎𝑌 is the yield stress in uniaxial tension. Which of the following statements is/are correct?

(A)

Criterion P represents the von Mises criterion

(B)

Criterion Q represents the Tresca criterion

(C)

Criterion P represents the Tresca criterion

(D)

Criterion Q represents the von Mises criterion

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q64

NAT[2 Marks]

Consider the matrix 𝐴= [5 −4 𝑘 −1], where k is a constant. If the determinant of A is 3, then the ratio of the largest eigenvalue of A to the constant k is ___________ (rounded off to 1 decimal place).

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q65

NAT[2 Marks]

The state of stress at a point is caused by two separate loading cases. One of them produces a pure uniaxial tension along the 𝑥′ direction, and other one produces a pure uniaxial compression along the 𝑦′ direction, as shown in the figure. The sum of maximum and minimum principal stresses for the resultant state of stress caused by both loads acting simultaneously is ___________ N/mm2 (rounded off to 1 decimal place).

Explanation

GATE→2024→Engineering Mathematics→Linear Algebra

Q66

NAT[2 Marks]

For a general aviation airplane, one of the complex conjugate pair of eigenvalues for longitudinal dynamics is given by −0.039 ± 0.0567 𝑖 (in SI units). If the system is disturbed to excite only this mode, the time taken for the amplitude of response to become half in magnitude is _______ s (rounded off to 1 decimal place).

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q67

MCQ[1 Mark]

The direction in which a scalar field φ( x, y, z ) has the largest rate of change at any point with position vector r = xiˆ + yjˆ + zkˆ is the same as that of the vector

(A)

φ

(B)

 (φ)r

(C)

φr

(D)

(𝛻𝜙 • 𝑑𝑟⃗)𝑟⃗

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q68

MSQ[1 Mark]

Which of the following statement(s) is/are true with respect to eigenvalues and eigenvectors of a matrix?

(A)

The sum of the eigenvalues of a matrix equals the sum of the elements of the principal diagonal.

(B)

1 If is an eigenvalue of a matrix A, then is always an eigenvalue of its  transpose (AT).

(C)

1 If is an eigenvalue of an orthogonal matrix A, then is also an eigenvalue of  A.

(D)

If a matrix has n distinct eigenvalues, it also has n independent eigenvectors.

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q69

NAT[1 Mark]

The system of equations x − 2 y + αz = 0 2 x + y − 4 z = 0 x − y + z = 0 has a non-trivial solution for α= ______. (Answer in integer)

Explanation

GATE→2023→Engineering Mathematics→Linear Algebra

Q70

MSQ[2 Marks]

Given the vectors A = 9iˆ − 5 ˆj + 2 kˆ B = 11iˆ + 4 ˆj + kˆ C = − 7iˆ + 14 ˆj − 3kˆ which of the following statement(s) is/are TRUE?

(A)

Vectors A , B and C are coplanar

(B)

The scalar triple product of the vectors A , B and C is zero

(C)

A and B are perpendicular

(D)

C is parallel to A  B Aerospace Engineering (AE)

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q71

MCQ[1 Mark]

Let $\hat{i}, \hat{j}, \hat{k}$ be unit vectors in x, y, z directions. If vector $\hat{i} + \hat{j}$ is rotated about positive $\hat{k}$ by $135^{\circ}$, one gets

(A)

-\hat{i}

(B)

-\hat{i}

(C)

-\frac{1}{\sqrt{2}}\hat{j}

(D)

-\sqrt{2}\hat{i}

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q72

MSQ[1 Mark]

If $\hat{a}$, $\hat{b}$, $\hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s)

(A)

0

(B)

1

(C)

-1

(D)

infinity

Explanation

GATE→2022→Engineering Mathematics→Linear Algebra

Q73

NAT[2 Marks]

The largest eigenvalue of the given matrix $\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$ is

Explanation

Special Topics (Engineering Mathematics)

2 Questions

GATE→2026→Engineering Mathematics→Special Topics (Engineering Mathematics)

Q78

MCQ[1 Mark]

The Fourier series representation of a square wave is shown in the figure below. The fluctuations seen near x = ±1 are named after which one of the following scientists?

(A)

Cauchy

(B)

Fourier

(C)

Gibbs

(D)

Laplace

Explanation

GATE→2022→Engineering Mathematics→Special Topics (Engineering Mathematics)

Q79

MCQ[1 Mark]

Let $x$ be a real number and $i = \sqrt{-1}$. Then the real part of $\cos(ix)$ is

(A)

sinh x

(B)

cosh x

(C)

cos x

(D)

sin x

Explanation

Differential Equations

10 Questions

GATE→2026→Engineering Mathematics→Differential Equations

Q40

MCQ[1 Mark]

The following equation with respect to 𝜑(𝑥, 𝑡), where 𝑎 is a non-zero constant, represents ________. 𝜕𝜑 𝜕𝜑 +𝑎 =0 𝜕𝑡 𝜕𝑥

(A)

linear wave propagation

(B)

transient heat conduction

(C)

Newton’s law of cooling

(D)

radiative transfer

Explanation

GATE→2026→Engineering Mathematics→Differential Equations

Q41

NAT[2 Marks]

Consider the differential equation with the initial conditions given below. If 𝑦(𝑥) 𝑑𝑦 is the solution of the equation, the value of the slope, , at 𝑥 = ln(2) is 𝑑𝑥 ________ (rounded off to three decimal places). 𝑑2 𝑦 𝑑𝑦 𝑑𝑦 2 +2 + 𝑦 = 0 with 𝑦|𝑥=0 = 0 and | =1 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑥=0

Explanation

GATE→2026→Engineering Mathematics→Differential Equations

Q42

NAT[2 Marks]

An object of mass 1 kg is launched with an initial speed of 𝑣𝑜 into a large tank filled with a viscous liquid. The liquid exerts a resistive force (drag) of the form 𝐷 = 𝛼𝑣 on any object that is moving inside it, where 𝑣 is the instantaneous speed of the object and 𝛼 = 1 kg/s. If the effect of gravity is ignored, the time taken by the object to slow down to the speed 𝑣𝑜 /2 is ________ s (rounded off to 2 decimal places). Assume that the tank is sufficiently large for the above deceleration to happen inside the tank.

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q43

MCQ[1 Mark]

The partial differential equation 𝜕2𝑢 𝜕𝑥2 + 4 𝜕2𝑢 𝜕𝑥𝜕𝑦+ 2 𝜕2𝑢 𝜕𝑦2 = 0 is ______.

(A)

elliptic

(B)

hyperbolic

(C)

parabolic

(D)

of mixed type

Explanation

GATE→2025→Engineering Mathematics→Differential Equations

Q44

NAT[2 Marks]

Consider the ordinary differential equation: 1 2 𝑑𝑦 𝑑𝑥+ 𝑦 𝑥= 1. If 𝑦= 2/3 at 𝑥= 1, then the value of 𝑦 at 𝑥= 3 is ______ (rounded off to the nearest integer).

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q45

MCQ[1 Mark]

Consider the one-dimensional wave equation + = 0 for − x , t .0 t x x 2 For an initial condition u ( x,0) = e− , the solution at t = 1 is

(A)

( x −1) 2 u ( x,1) = e−

(B)

1 u ( x,1) = e−

(C)

x 2 u ( x,1) = e−

(D)

( x +1) 2 u ( x,1) = e−

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q46

MSQ[2 Marks]

dy Consider the equation + ay = sin x , where a and ω are constants. Given dx y = 1 at x = 0 , select all correct statement(s) from the following as x →.

(A)

y → 0 if a  0

(B)

y → 1 if a = 0

(C)

y → A exp ( a x ) if a  0 ; A is a constant

(D)

y → B sin (x + C ) if a  0 ; B and C are constants

Explanation

GATE→2023→Engineering Mathematics→Differential Equations

Q47

NAT[2 Marks]

Consider the differential equation 2 d 2 y dy x + 4 x + 2 y = 0 for x  1 2 dx dx dy with initial conditions y = 0 , = 1 at x = 1 . The value of y at x = 2 is ____ . dx (round off to two decimal places) Aerospace Engineering (AE)

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q48

MCQ[2 Marks]

Consider the differential equation $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + y = 0$. The boundary conditions are $y = 0$ and $\frac{dy}{dx} = 1$ at $x = 0$. Then the value of $y$ at $x = 1/2$ is

(A)

0

(B)

sqrt(e)

(C)

sqrt(e)/2

(D)

sqrt(e/2)

Explanation

GATE→2022→Engineering Mathematics→Differential Equations

Q49

MCQ[2 Marks]

Consider the partial differential equation $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 where x, y are real. If$f(x,y) = a(x)b(y)$, where$a(x)$and$b(y)$ are real functions, which one of the following statements can be true?

(A)

a(x) periodic and b(y) linear

(B)

both exponential

(C)

a(x) periodic and b(y) exponential

(D)

both periodic

Explanation

Numerical Methods (Special)

2 Questions

GATE→2025→Engineering Mathematics→Numerical Methods (Special)

Q76

NAT[2 Marks]

An approximate solution of the equation 𝑥3 −17 = 0 is to be obtained using the Newton-Raphson method. If the initial guess is 𝑥𝑜= 2 , the value at the end of the first iteration is 𝑥1 = _____ (rounded off to two decimal places).

Explanation

GATE→2023→Engineering Mathematics→Numerical Methods (Special)

Q77

MCQ[1 Mark]

If a monotonic and continuous function y = f ( x) has only one root in the interval x1  x  x2 , then

(A)

f ( x1 ) f ( x2 )  0

(B)

f ( x1 ) f ( x2 ) = 0

(C)

f ( x1 ) f ( x2 )  0

(D)

f ( x1 ) − f ( x2 ) = 0 Aerospace Engineering (AE) u u

Explanation

AE-S05-T01

22 Questions

GATE→2026→Engineering Mathematics→AE-S05-T01

Q52

MSQ[1 Mark]

An 𝑛 × 𝑛 square matrix 𝐴 satisfies 𝐴𝑇 = 𝐴−1. The determinant of this matrix may take which of the following value(s)?

(A)

+1

(B)

−1

(C)

𝑛

(D)

0

Explanation

GATE→2026→Engineering Mathematics→AE-S05-T01

Q53

NAT[1 Mark]

If a matrix can be written as 𝐴 = 𝑢𝑣 𝑇 , where both 𝑢 and 𝑣 are 𝑛-dimensional real- valued non-zero column vectors, then the rank of the matrix 𝐴 is ________ (answer in integer).

Explanation

GATE→2026→Engineering Mathematics→AE-S05-T01

Q54

MCQ[2 Marks]

𝑎 𝑏 For the matrix 𝐴 = [ ], if the relation 𝑎 + 𝑏 = 𝑐 + 𝑑 holds and 𝑎, 𝑏, 𝑐, 𝑑 ≠ 0, 𝑐 𝑑 then which one of the following statements about 𝐴 is FALSE?

(A)

1 [ ] is an eigenvector 1

(B)

𝜆 = 𝑎 + 𝑏 is an eigenvalue

(C)

𝜆 = 𝑑 − 𝑏 is an eigenvalue

(D)

𝜆 = 𝑑 + 𝑏 is an eigenvalue

Explanation

GATE→2025→Engineering Mathematics→AE-S05-T01

Q55

MCQ[1 Mark]

For any real symmetric matrix 𝐴, the transpose of 𝐴 is ____ .

(A)

inverse of 𝐴

(B)

null matrix

(C)

−𝐴

(D)

𝐴

Explanation

GATE→2025→Engineering Mathematics→AE-S05-T01

Q56

MCQ[1 Mark]

2𝑥+ 3𝑦+ 𝑧= 0 𝑥+ 𝑦= 0 𝑦+ 𝑧= 0 The given system of equations has

(A)

a unique solution.

(B)

infinitely many solutions.

(C)

no solution.

(D)

a finite number of solutions.

Explanation

GATE→2025→Engineering Mathematics→AE-S05-T01

Q57

MCQ[1 Mark]

The eigenvalues of the matrix [1 2 0 3] are _____.

(A)

0, 2

(B)

2, 3

(C)

1, 3

(D)

1, 2

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q58

MCQ[1 Mark]

The following system of linear equations 7𝑥−3𝑦+ 𝑧= 0 3𝑥−𝑦+ 𝑧= 0 𝑥−𝑦−𝑧= 0 has:

(A)

infinitely many solutions

(B)

a unique solution

(C)

no solution

(D)

three solutions

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q59

MCQ[1 Mark]

The three-dimensional stress-strain relationship for an isotropic material is given as { 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 𝜏𝑦𝑧 𝜏𝑥𝑧 𝜏𝑥𝑦} = [ 𝑃 𝑄 𝑄 𝑄 𝑃 𝑄 𝑄 𝑄 𝑃 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑅 0 0 0 𝑅 0 0 0 𝑅] { 𝜀𝑥𝑥 𝜀𝑦𝑦 𝜀𝑧𝑧 𝛾𝑦𝑧 𝛾𝑥𝑧 𝛾𝑥𝑦} where, P, Q and R are the three elastic constants, 𝜎 and 𝜏 represent normal and shear stresses and 𝜀 and 𝛾 represent normal and engineering shear strains. Which one of the following options is correct?

(A)

𝑅= 𝑃−𝑄 2

(B)

𝑅= 𝑄−𝑃 2

(C)

𝑄= 𝑃−𝑅 2

(D)

𝑄= 𝑅−𝑃 2

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q60

MCQ[2 Marks]

Given 𝑦= 𝑒𝑝𝑥sin 𝑞𝑥, where 𝑝 and 𝑞 are non-zero real numbers, the value of the differential expression 𝑑2𝑦 𝑑𝑥2 −2𝑝𝑑𝑦 𝑑𝑥+ (𝑝2 + 𝑞2)𝑦 is

(A)

0

(B)

1

(C)

𝑝2 + 𝑞2

(D)

𝑝𝑞

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q61

MCQ[2 Marks]

The volume of the solid formed by a complete rotation of the shaded portion of the circle of radius R about the y-axis is 𝑘𝜋R3. The value of 𝑘 is:

(A)

5 12

(B)

5 24

(C)

7 12

(D)

7 24

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q62

MCQ[2 Marks]

At a point in the trajectory of an unpowered space vehicle moving about the Earth, the altitude above the mean sea level is 600 km, and the speed with reference to a coordinate system fixed to the center of mass of the Earth is 9 km/s. Assume that the Earth is a sphere with a radius 6400 km and 𝐺𝑀𝐸𝑎𝑟𝑡ℎ= 3.98 × 1014 m3/s2, where, G is the universal gravitational constant and 𝑀𝐸𝑎𝑟𝑡ℎ is mass of the Earth. The trajectory is:

(A)

Circular

(B)

Elliptic

(C)

Parabolic

(D)

Hyperbolic

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q63

MSQ[2 Marks]

The figure shows plots of two yield loci for an isotropic material, where 𝜎𝐼 and 𝜎𝐼𝐼 are the principal stresses, and 𝜎𝑌 is the yield stress in uniaxial tension. Which of the following statements is/are correct?

(A)

Criterion P represents the von Mises criterion

(B)

Criterion Q represents the Tresca criterion

(C)

Criterion P represents the Tresca criterion

(D)

Criterion Q represents the von Mises criterion

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q64

NAT[2 Marks]

Consider the matrix 𝐴= [5 −4 𝑘 −1], where k is a constant. If the determinant of A is 3, then the ratio of the largest eigenvalue of A to the constant k is ___________ (rounded off to 1 decimal place).

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q65

NAT[2 Marks]

The state of stress at a point is caused by two separate loading cases. One of them produces a pure uniaxial tension along the 𝑥′ direction, and other one produces a pure uniaxial compression along the 𝑦′ direction, as shown in the figure. The sum of maximum and minimum principal stresses for the resultant state of stress caused by both loads acting simultaneously is ___________ N/mm2 (rounded off to 1 decimal place).

Explanation

GATE→2024→Engineering Mathematics→AE-S05-T01

Q66

NAT[2 Marks]

For a general aviation airplane, one of the complex conjugate pair of eigenvalues for longitudinal dynamics is given by −0.039 ± 0.0567 𝑖 (in SI units). If the system is disturbed to excite only this mode, the time taken for the amplitude of response to become half in magnitude is _______ s (rounded off to 1 decimal place).

Explanation

GATE→2023→Engineering Mathematics→AE-S05-T01

Q67

MCQ[1 Mark]

The direction in which a scalar field φ( x, y, z ) has the largest rate of change at any point with position vector r = xiˆ + yjˆ + zkˆ is the same as that of the vector

(A)

φ

(B)

 (φ)r

(C)

φr

(D)

(𝛻𝜙 • 𝑑𝑟⃗)𝑟⃗

Explanation

GATE→2023→Engineering Mathematics→AE-S05-T01

Q68

MSQ[1 Mark]

Which of the following statement(s) is/are true with respect to eigenvalues and eigenvectors of a matrix?

(A)

The sum of the eigenvalues of a matrix equals the sum of the elements of the principal diagonal.

(B)

1 If is an eigenvalue of a matrix A, then is always an eigenvalue of its  transpose (AT).

(C)

1 If is an eigenvalue of an orthogonal matrix A, then is also an eigenvalue of  A.

(D)

If a matrix has n distinct eigenvalues, it also has n independent eigenvectors.

Explanation

GATE→2023→Engineering Mathematics→AE-S05-T01

Q69

NAT[1 Mark]

The system of equations x − 2 y + αz = 0 2 x + y − 4 z = 0 x − y + z = 0 has a non-trivial solution for α= ______. (Answer in integer)

Explanation

GATE→2023→Engineering Mathematics→AE-S05-T01

Q70

MSQ[2 Marks]

Given the vectors A = 9iˆ − 5 ˆj + 2 kˆ B = 11iˆ + 4 ˆj + kˆ C = − 7iˆ + 14 ˆj − 3kˆ which of the following statement(s) is/are TRUE?

(A)

Vectors A , B and C are coplanar

(B)

The scalar triple product of the vectors A , B and C is zero

(C)

A and B are perpendicular

(D)

C is parallel to A  B Aerospace Engineering (AE)

Explanation

GATE→2022→Engineering Mathematics→AE-S05-T01

Q71

MCQ[1 Mark]

Let $\hat{i}, \hat{j}, \hat{k}$ be unit vectors in x, y, z directions. If vector $\hat{i} + \hat{j}$ is rotated about positive $\hat{k}$ by $135^{\circ}$, one gets

(A)

-\hat{i}

(B)

-\hat{i}

(C)

-\frac{1}{\sqrt{2}}\hat{j}

(D)

-\sqrt{2}\hat{i}

Explanation

GATE→2022→Engineering Mathematics→AE-S05-T01

Q72

MSQ[1 Mark]

If $\hat{a}$, $\hat{b}$, $\hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s)

(A)

0

(B)

1

(C)

-1

(D)

infinity

Explanation

GATE→2022→Engineering Mathematics→AE-S05-T01

Q73

NAT[2 Marks]

The largest eigenvalue of the given matrix $\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$ is

Explanation

Probability and Statistics

1 Question

GATE→2024→Engineering Mathematics→Probability and Statistics

Q74

NAT[1 Mark]

Two fair dice with numbered faces are rolled together. The faces are numbered from 1 to 6. The probability of getting odd numbers on both the dice is ________________ (rounded off to 2 decimal places).

Explanation

Complex Analysis

1 Question

GATE→2025→Engineering Mathematics→Complex Analysis

Q75

MCQ[1 Mark]

If 𝑖= √−1, (𝑖+1)3 𝑖−1 = ________.

(A)

2

(B)

−2

(C)

(𝑖+ 1)

(D)

(𝑖−1)

Explanation

Numerical Methods

2 Questions

GATE→2025→Engineering Mathematics→Numerical Methods

Q76

NAT[2 Marks]

An approximate solution of the equation 𝑥3 −17 = 0 is to be obtained using the Newton-Raphson method. If the initial guess is 𝑥𝑜= 2 , the value at the end of the first iteration is 𝑥1 = _____ (rounded off to two decimal places).

Explanation

GATE→2023→Engineering Mathematics→Numerical Methods

Q77

MCQ[1 Mark]

If a monotonic and continuous function y = f ( x) has only one root in the interval x1  x  x2 , then

(A)

f ( x1 ) f ( x2 )  0

(B)

f ( x1 ) f ( x2 ) = 0

(C)

f ( x1 ) f ( x2 )  0

(D)

f ( x1 ) − f ( x2 ) = 0 Aerospace Engineering (AE) u u

Explanation

Fourier Series and Laplace Transforms

2 Questions

GATE→2026→Engineering Mathematics→Fourier Series and Laplace Transforms

Q78

MCQ[1 Mark]

The Fourier series representation of a square wave is shown in the figure below. The fluctuations seen near x = ±1 are named after which one of the following scientists?

(A)

Cauchy

(B)

Fourier

(C)

Gibbs

(D)

Laplace

Explanation

GATE→2022→Engineering Mathematics→Fourier Series and Laplace Transforms

Q79

MCQ[1 Mark]

Let $x$ be a real number and $i = \sqrt{-1}$. Then the real part of $\cos(ix)$ is

(A)

sinh x

(B)

cosh x

(C)

cos x

(D)

sin x

Explanation