What is the solution of Laplace equation in the square domain in GATE 2022 Mechanical Engineering Set 1?

The correct answer is Option B: T(x, y) = x + y. The Laplace equation ∇²T = 0 requires ∂²T/∂x² + ∂²T/∂y² = 0. For T = x+y: ∂²T/∂x² = 0 and ∂²T/∂y² = 0, so ∇²T = 0 ✓. Checking boundary conditions: T(x,0)=x ✓, T(0,y)=y ✓, T(x,1)=x+1 ✓, T(1,y)=1+y ✓. All boundary conditions are exactly satisfied, confirming T = x+y is the correct solution.

What is the Fourier series property of x³ on [-1,1) in GATE 2022 Mechanical Engineering Set 1?

The correct answer is Option A: only sine terms. The function f(x) = x³ is an odd function because f(-x) = (-x)³ = -x³ = -f(x). The Fourier series of an odd periodic function contains only sine terms (no cosine terms and no constant term). This is because the coefficients aₙ = (1/L)∫f(x)cos(nπx/L)dx vanish for odd functions, while bₙ (sine coefficients) are generally non-zero.

What is the contour integral value in GATE 2022 Mechanical Engineering Set 1 complex analysis?

The correct answer is Option C: 2(1-i)π. The integral is evaluated over a counter-clockwise circular contour enclosing only the pole z = i. Using the residue theorem, the integral equals 2πi times the residue at z = i. Computing the residue of the given function at z = i and multiplying by 2πi yields 2(1-i)π as the final answer.

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Ques 40 Gate 2022 Set-1

The average of the monthly salaries of M, N and S is ₹ 4000. The average of the monthly salaries of N, S and P is ₹ 5000. The monthly salary of P is ₹ 6000.What is the monthly salary of M as a percentage of the monthly salary of P?

75%

100%

125%

(a) is the correct answer.

The average salary of M, N and S is ₹4000, while the average salary of N, S and P is ₹5000. P's salary is ₹6000. We need to find M's salary as a percentage of P's salary.
Solution:
From the first average:
M + N + S = 4000 × 3 = ₹12,000
From the second average:
N + S + P = 5000 × 3 = ₹15,000
Since P = ₹6,000:
N + S = ₹15,000 − ₹6,000 = ₹9,000
Therefore:
M = ₹12,000 − ₹9,000 = ₹3,000
M's salary as a percentage of P's salary:
(3000 ÷ 6000) × 100 = 50%
A) 50: Correct. M's salary is 50% of P's salary.
Correct Answer: A) 50%

Ques 41 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

(b) is the correct answer.

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 42 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

(a) is the correct answer.

Ques 43 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

(b) is the correct answer.

Ques 44 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

(a) is the correct answer.

Ques 45 GATE 2022 SET-1

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

−0.4

(1 + √1561)/12

(b) is the correct answer.

Ques 46 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 47 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

(a, b) is the correct answer.

Ques 48 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 49 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 50 GATE 2022 SET-1

In a linear programming problem, if a resource is not fully utilized, the shadow price of that resource is?

Positive

Negative

Zero

Infinity

Ques 51 GATE 2022 SET-1

Which one of the following is NOT a form of inventory?

Raw materials

Work-in-process materials

Finished goods

CNC Milling Machines

(d) is the correct answer.

Ques 52 GATE 2022 SET-1

Assuming the material considered in each statement is homogeneous, isotropic, linear elastic, and the deformations are in the elastic range, which one or more of the following statement(s) is/are TRUE?

A body subjected to hydrostatic pressure has no shear stress.

If a long solid steel rod is subjected to tensile load, then its volume increases.

Maximum shear stress theory is suitable for failure analysis of brittle materials.

If a portion of a beam has zero shear force, then the corresponding portion of the elastic curve of the beam is always straight.

(a, b, c) is the correct answer.

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