GATE EC 2015 Question Paper (31-Jan-2015) (Shift 2)

Consider the differential equation $\frac{\mathrm{dx}}{\mathrm{dt}}=10-0.2\mathrm{x}$ with initial condition $\mathrm{x}\left(0\right)=1$. The response x(t) for $\mathrm{t}>0$ is

The value of the integral $\underset{-\infty }{\overset{\infty }{\int }}12\cos \left(2\pi \mathrm{t}\right)\frac{\sin \left(4\pi \mathrm{t}\right)}{4\pi \mathrm{t}}\mathrm{dt}$ is__________.

If C denotes the counter clockwise unit circle, the value of the contour integral $\frac{1}{2\pi \mathrm{j}}\underset{\mathrm{c}}{\oint }\mathrm{Re}\left\{\mathrm{z}\right\}\mathrm{dz}$ is______.

Let the random variable X represent the number of times of fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of X is ________.

An LC tank circuit consists of an ideal capacitor C connected in parallel with a coil of inductance L having an internal resistance R. The resonant frequency of the tank circuit is