GATE EC 2016 Question Paper (31-Jan-2016) (Shift 2)

The particular solution of the initial value problem given below is

$\frac{d^{2}y}{d{x}^2}+12\frac{dy}{dx}+36y=0$ with $y(0)=3$ and $\frac{dy}{dx}{|}_{x=0}=-36$

If the vectors $e_1=(1,0,2)$, $e_2=(0,1,0)$ and $e_3=(-2,0,1)$ form an orthogonal basis of the three-dimensional real space $R^3$, then the vector $u=(4,3,-3)\in R^3$ can be expressed as

A triangle in the xy-plane is bounded by the straight lines $2x=3y$, $y=0$ and $x=3$. The volume above the triangle and under the plane $x+y+z=6$ is ____________

The values of the integral $\frac{1}{2\pi j}$${\oint }_C\frac{e^z}{z-2}dz$ along a closed contour c in anti-clockwise direction for
(i) the point $z_0=2$ inside the contour c, and
(ii) the point $z_0=2$ outside the contour c, respectively, are

A signal $2\cos \left(\frac{2\pi }{3}t\right)-\cos (\pi t)$ is the input to an LTI system with the transfer function

$H(s)=e^s+e^{-s}$.

If $C_k$ denotes the $k^{th}$ coefficient in the exponential Fourier series of the output signal, then $C_3$ is equal to