Mechanical Gate Subject

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Mathematics Mechanical previous year questions with answer

Ques 21 Gate 2022 Set-1

Solution of ∇²𝑇 = 0 in a square domain (0 < 𝑥 < 1 and 0 < 𝑦 < 1) with boundary conditions:

𝑇(𝑥, 0) = 𝑥; 𝑇(0, 𝑦) = 𝑦; 𝑇(𝑥, 1) = 1 + 𝑥; 𝑇(1, 𝑦) = 1 + 𝑦

is

T(x, y) = x − xy+ y

T(x,y)=x+y

T(x,y)=-x+y

T(x,y)=x+xy+y

Laplace's Equation:
Laplace's equation in two dimensions is given by:
    ∂²T/∂x² + ∂²T/∂y² = 0
We will test each given option by first checking if it satisfies Laplace's equation, and then if it satisfies all four boundary conditions.
Boundary Conditions:
• T(x, 0) = x
• T(0, y) = y
• T(x, 1) = 1 + x
• T(1, y) = 1 + y
Option A: T(x, y) = x − xy + y
• Check PDE:
    ∂T/∂x = 1 − y
    ∂²T/∂x² = 0
    ∂T/∂y = −x + 1
    ∂²T/∂y² = 0
    ∇²T = 0 + 0 = 0. The PDE is satisfied.
• Boundary Conditions:
    T(x, 0) = x − x(0) + 0 = x. (Matches)     T(0, y) = 0 − (0)y + y = y. (Matches)     T(x, 1) = x − x(1) + 1 = x − x + 1 = 1. (Does NOT match 1 + x) Since this option fails one boundary condition, it is not the correct solution.
Option B: T(x, y) = x + y
• Check PDE:
    ∂T/∂x = 1
    ∂²T/∂x² = 0
    ∂T/∂y = 1
    ∂²T/∂y² = 0
    ∇²T = 0 + 0 = 0. The PDE is satisfied.
• Boundary Conditions:
    T(x, 0) = x + 0 = x. (Matches T(x, 0) = x)     T(0, y) = 0 + y = y. (Matches T(0, y) = y)     T(x, 1) = x + 1. (Matches T(x, 1) = 1 + x)     T(1, y) = 1 + y. (Matches T(1, y) = 1 + y) All boundary conditions are satisfied. This option satisfies both Laplace's equation and all boundary conditions. Therefore, this is the correct solution.
For completeness, let's briefly check the PDE for the remaining options.
Option C: T(x, y) = -x + y
• PDE:
    ∂T/∂x = -1 &implies; ∂²T/∂x² = 0
    ∂T/∂y = 1 &implies; ∂²T/∂y² = 0
    ∇²T = 0 + 0 = 0. The PDE is satisfied.
• Boundary Conditions:
    T(x, 0) = -x + 0 = -x. (Does NOT match x) This option fails the first boundary condition.
Option D: T(x, y) = x + xy + y
• PDE:
    ∂T/∂x = 1 + y
    ∂²T/∂x² = 0
    ∂T/∂y = x + 1
    ∂²T/∂y² = 0
    ∇²T = 0 + 0 = 0. The PDE is satisfied.
• Check Boundary Conditions:
    T(x, 0) = x + x(0) + 0 = x. (Matches)     T(0, y) = 0 + (0)y + y = y. (Matches)     T(x, 1) = x + x(1) + 1 = x + x + 1 = 2x + 1. (Does NOT match 1 + x) This option fails one boundary condition.

Based on the analysis, only T(x, y) = x + y satisfies both Laplace's equation and all the given boundary conditions.

Ques 22 Gate 2022 Set-1

The Fourier series expansion of x³ in the interval −1 ≤ 𝑥 < 1 with periodic continuation has

only sine terms

only cosine terms

both sine and cosine terms

only sine terms and a non-zero constant

Ques 23 GATE 2022 SET-1

The limit has a finite value for a real α. The value of α and the corresponding limit p are?

α = −3π, p = π

α = −2π, p = 2π

α = π, p = π

α = 2π, p = 3π

Ques 24 GATE 2022 SET-1

Given a function 𝜑 = 1/2(x² + y² + z²) in three-dimensional Cartesian space, the value of the surface integral

where 𝑆 is the surface of a sphere of unit radius and 𝐧' is the outward unit normal vector on 𝑆, is

4π

3π

4π/3

Ques 25 GATE 2022 SET-1

If is a symmetric matrix, the value of k is?

−0.4

(1 + √1561)/12

Ques 26 GATE 2022 SET-1

The value of the integral

evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole z = i, where i is the imaginary unit, is?

(-1 + i)π

(1 + i)π

2(1 - i)π

(2 + i)π

Ques 27 GATE 2022 SET-1

The system of linear equations in real (x, y) given by

involves a real parameter 𝛼 and has infinitely many non-trivial solutions for special value(s) of 𝛼. Which one or more among the following options is/are non-trivial solution(s) of (𝑥,𝑦) for such special value(s) of 𝛼?

x = 2, y = -2

x = -1, y = 4

x = 1, y = 1

x = 4, y = -2

Ques 28 GATE 2022 SET-1

Let a random variable X follow Poisson distribution such that

The value of Prob(X = 3) is __________ (round off to 2 decimal places).

0.18 is the correct answer.

Ques 29 GATE 2022 SET-1

Consider two vectors
A⃗ = 5i + 7j + 2k
B⃗ = 3i - j + 6k.
Magnitude of the component of 𝑎⃗ orthogonal to 𝑏⃗ in the plane containing the vectors 𝑎⃗ and 𝑏⃗ is __________ (round off to 2 decimal places).

5.92 is the correct answer.

Ques 30 Gate 2020 Set-2

The sum of two normally distributed random variables X and Y is

always normally distributed

normally distributed, only if X and Y have the same standard deviation

normally distributed, only if X and Y have the same mean

normally distributed, only if X and Y are independent