Mathematics Mechanical Gate Previous Year Questions

The value of k that makes the complex valued function. f(z) = e^–kx [cos2y – i sin2y] analytic:

If f(x) = 2ln(√e), what is the area bounded by f(x) for the interval [0, 2] on the x-axis?

a) 1/2
b) 1
c) 2
d) 4

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?

a) 50
b) 75%
c) 100%
d) 125%

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

a) T(x, y) = x − xy+ y
b) T(x,y)=x+y
c) T(x,y)=-x+y
d) T(x,y)=x+xy+y

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

a) only sine terms
b) only cosine terms
c) both sine and cosine terms
d) only sine terms and a non-zero constant

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

a) always normally distributed
b) normally distributed, only if X and Y have the same standard deviation
c) normally distributed, only if X and Y have the same mean
d) normally distributed, only if X and Y are independent

Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value appears more than once in E) real eigenvalues. The number of elements in E is ______.

A fair coin is tossed 20 times. The probability that 'head' will appear exactly 4 times in the first ten tosses, and ‘tail’ will appear exactly 4 times in the next ten tosses is ______ (round off to 3 decimal places).

The directional derivative of the function f(x, y) =x²+y² along a line directed from (0,0) to (1,1), evaluated at the point x = 1, y = 1 is

a) √2
b) 2
c) 2√2
d) 4√2

The lengths of a large stock of titanium rods follow a normal distribution with a mean (𝜇) of 440 mm and a standard deviation (𝜎) of 1 mm. What is the percentage of rods whose lengths lie between 438 mm and 441 mm?

a) 81.85%
b) 68.4%
c) 99.75%
d) 86.64%

Four red balls, four green balls and four blue balls are put in a box. Three balls are pulled out of the box at random one after another without replacement. The probability that all the three balls are red is

a) 1/72
b) 1/55
c) 1/36
d) 1/27

You are given three coins: one has heads on both faces, the second has tails on both faces, and the third has a head on one face and a tail on the other. You choose a coin at random and toss it, and it comes up heads. The probability that the other face is tails is

a) 1/4
b) 1/3
c) 1/2
d) 2/3