GATE IN 2014 Question Paper (02-Feb-2014) (Shift 2)

Given
𝑥(𝑡) = 3 sin(1000 𝜋t) and 𝑦(𝑡) = 5 cos $\left(1000\pi t+\frac{\pi }{4}\right)$

The x-y plot will be

Given that x is a random variable in the range [0, ∞] with a probability density function $\frac{e^{-}\frac{x}{2}}{K}$ the value of the constant K is___________.

The figure shows the plot of y as a function of x



The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is :

A vector is defined as

f = yî + xĵ + zk̂
where î, ĵ and k̂ are unit vectors in Cartesian (x, y, z) coordinate system.

The surface integral ∯ 𝐟. ds over the closed surface S of a cube with vertices having the following coordinates: (0,0,0), (1,0,0), (1,1,0), (0,1,0), (0,0,1), (1,0,1), (1,1,1), (0,1,1) is
___________.

The figure shows the schematic of a production process with machines A, B and C. An input job needs to be pre-processed either by A or by B before it is fed to C, from which the final finished product comes out. The probabilities of failure of the machines are given as:

PA = 0.15, PB = 0.05, PC = 0.1.



Assuming independence of failures of the machines, the probability that a given job is successfully processed (up to the third decimal place) is ___________.