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

Consider a $2\times 2$ square matrix

$\left[\begin{array}{cc}\sigma & x\\ \omega & \sigma \end{array}\right]$,

where $x$ is unknown. If the eigenvalues of the matrix A are $(\sigma +j\omega )$ and $(\sigma -j\omega )$, then $x$ is equal to

For $f(z)=\frac{\sin (z)}{z^2}$, the residue of the pole at $z=0$ is ___________

The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting "head" for the first time in the fifth toss is ___________

The integral ${\int }_0^1\frac{dx}{\sqrt{(1-x)}}$ is equal to ___________

Consider the first order initial value problem

$y^{\prime }=y+2x-x^2$, $y(0)=1$, $(0\le x<\infty )$

with exact solution $y(x)=x^2+e^x$. For $x=0.1$, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runge-Kutta method with step-size $h=0.1$ is ___________