Let $f(x)$ be a real-valued function such that $f^{\prime}\left(x_0\right)=0$ for some $x_0 \in(0,1)$ and $f^{\prime \prime}\left(x_0\right)>0$ for all $x \in(0,1)$. Then $f(x)$ has

Let $(-1-j),(3-j),(3+j)$ and $(-1+j)$ be the vertices of rectangle $C$ in the complex plane. Assuming that $C$ is traversed in counter-clockwise direction, the value of contour integral $\oint_C \frac{d z}{z^2(z-4)}$ is

Let $P(z)=z^3+(1+j) z^2+(2+j) z+3$, where $z$ is complex number. Which one of the following is true?

Suppose the probability that a coin toss shows "head" is $p$, where $0 < p < 1$. The coin is tossed repeatedly until the first "head" appears. The expected number of tosses required is :