Mathematics Mechanical previous year questions with answer

Ques 1 Gate 2024

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How many combinations of non-null sets A, B, C are possible from the subsets of {2, 3, 5} satisfying the conditions: (i) A is a subset of B, and (ii) B is a subset of C?

A

28

B

27

C

18

D

19

See Answer Video Solution

Ques 2 Gate 2024

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The bar chart gives the batting averages of VK and RS for 11 calendar years from 2012 to 2022. Considering that 2015 and 2019 are world cup years, which one of the following options is true?

A

RS has a higher yearly batting average than that of VK in every world cup year.

B

VK has a higher yearly batting average than that of RS in every world cup year.

C

VK’s yearly batting average is consistently higher than that of RS between the two world cup years.

D

RS’s yearly batting average is consistently higher than that of VK in the last three years.

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Ques 3 Gate 2024

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Four equilateral triangles are used to form a regular closed three-dimensional object by joining along the edges. The angle between any two faces is

A

30°

B

60°

C

45°

D

90°

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Ques 4 Gate 2024

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In order to numerically solve the ordinary differential equation 𝑑𝑦/𝑑𝑡 = −𝑦 for 𝑡 > 0, with an initial condition 𝑦(0) = 1, the following scheme is employed

Here, Δ𝑡 is the time step and 𝑦𝑛 = 𝑦(𝑛Δ𝑡) for 𝑛 = 0, 1, 2, …. This numerical scheme will yield a solution with non-physical oscillations for Δ𝑡 > ℎ. The value of ℎ is

A

1/2

B

1

C

3/2

D

2

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Ques 5 Gate 2024

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The value of the surface integral

where 𝑆 is the external surface of the sphere 𝑥² + 𝑦² + 𝑧² = 𝑅² is

A

0

B

4𝜋𝑅³

C

(4𝜋/3) 𝑅³

D

𝜋𝑅³

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Ques 6 Gate 2024

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Let 𝑓(𝑧) be an analytic function, where 𝑧 = 𝑥 + 𝑖𝑦. If the real part of 𝑓(𝑧) is cosh 𝑥 cos 𝑦, and the imaginary part of 𝑓(𝑧) is zero for 𝑦 = 0, then 𝑓(𝑧) is

A

cosh 𝑥 exp (−𝑖𝑦)

B

cosh 𝑧 exp 𝑧

C

cosh 𝑧 cos 𝑦

D

cosh z

See Answer Video Solution

Ques 7 Gate 2024

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Consider the system of linear equations
𝑥 + 2𝑦 + 𝑧 = 5
2𝑥 + 𝑎𝑦 + 4𝑧 = 12
2𝑥 + 4𝑦 + 6𝑧 = 𝑏
The values of 𝑎 and 𝑏 such that there exists a non-trivial null space and the system admits infinite solutions are

A

𝑎 = 8, 𝑏 = 14

B

𝑎 = 4, 𝑏 = 12

C

𝑎 = 8, 𝑏 = 12

D

𝑎 = 4, 𝑏 = 14

See Answer Video Solution

Ques 8 Gate 2024

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Let f(⋅) be a twice differentiable function from ℝ² → ℝ. If p, x₀ ∈ ℝ² where ‖ p ‖ is sufficiently small (here ‖ ⋅ ‖ is the Euclidean norm or distance function), then:

f(x₀ + p) = f(x₀) + ∇f(x₀)^T p + (1/2) p^T ∇² f(ψ) p

where ψ ∈ ℝ² is a point on the line segment joining x₀ and x₀ + p. If x₀ is a strict local minimum of f(x), then which one of the following statements is TRUE?

A

∇f(x₀)^T p > 0 and p^T ∇² f(ψ) p = 0

B

∇f(x₀)^T p = 0 and p^T ∇² f(ψ) p > 0

C

∇f(x₀)^T p = 0 and p^T ∇² f(ψ) p = 0

D

∇f(x₀)^T p = 0 and p^T ∇² f(ψ) p < 0

See Answer Video Solution

Ques 9 Gate 2024

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In the following series, identify the number that needs to be changed to form the Fibonacci series
1, 1, 2, 3, 6, 8, 13, 21,…

A

8

B

21

C

6

D

13

See Answer Video Solution

Ques 10 Gate 2024

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The velocity field of a two-dimensional, incompressible flow is given by
𝑉⃗ = 2 sinh 𝑥 𝑖̂+ 𝑣(𝑥, 𝑦) 𝑗̂
where 𝑖̂ and 𝑗̂ denote the unit vectors in 𝑥 and 𝑦 directions, respectively. If 𝑣(𝑥, 0) = cosh 𝑥, then 𝑣(0, −1) is

A

1

B

2

C

3

D

4

See Answer Video Solution