GATE IN 2012 Question Paper (12-Feb-2012) (Shift-2)

If $x=\sqrt{-1}$, then the value of $x^x$ is

With initial condition $x(1)=0.5$, the solution of the differential equation,

$t\frac{dx}{dt}+x=t$ is

Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[-1,1]$. The probability that $\max [X,Y]$ is less than $1/2$ is

The unilateral Laplace transform of $f(t)$ is $\frac{1}{s^2+s+1}$. The unilateral Laplace transform of $tf(t)$ is

Given $f(z)=\frac{1}{z+1}-\frac{2}{z+3}$. If $C$ is a counterclockwise path in the $z$ -plane such that $|z+1|=1$, the value of $\frac{1}{2\pi j}{\oint }_{C}f(z)dz$ is