GATE EC 2016 Question Paper (30-Jan-2016) (Shift 1)

The integral 1/2π _D ^ ≤ft( x + y + 10 )dxdy, where D denotes the disc 𝑥² + 𝑦² ≤ 4, evaluates to__________.

A sequence x[n] is specified as:

≤ft[ {*20c {≤ft[ ]}\\ {≤ft[ {- 1} ]} } ] = {≤ft[ {*20c 1&1\\ 1&0 } ]^}≤ft[ {*20c 1\\ 0 } ],for n ≥ 2.

The initial conditions are x[0] = 1, x[1] = 1, and x[n] = 0 for n < 0. The value of x[12] is ________.

In the following integral, the contour C encloses the points 2πj and −2πj

$-\frac{1}{2\pi }\underset{\mathrm{c}}{\oint }\frac{\sin \mathrm{z}}{{\left(\mathrm{z}-2\pi \mathrm{j}\right)}^3}\mathrm{dz}$

The value of the integral is ________

The region specified by $\left\{\left(\rho ,\phi ,\mathrm{z}\right):3\le \rho \le 5,\frac{\pi }{8}\le \phi \le \frac{\pi }{4},3\le \mathrm{z}\le 4.5\right\}$ in cylindrical coordinates has volume of _______.

The Laplace transform of the causal periodic square wave of period T shown in the figure below is