GATE EE 2018 Question Paper (10-Feb-2018) (Shift 2)

Let f be a real-valued function of a real variable defined as f(x) = x²for x ≥ 0, and f(x) = -x² for x < 0.

Which one of the following statements is true?

The value of the directional derivative of the function θ (x, y, z) = xy² + yz² + zx² at the point (2, -1, 1) in the direction of the vector p = i + 2j + 2k is

The value of the integral $\underset{C}{\oint }\frac{z+1}{z^2-4}dz$ in counter clockwise direction around a circle C of radius 1 with center at the point z = −2 is

In the logic circuit shown in the figure, Y is given by

The op-amp shown in the figure is ideal. The input impedance $\frac{v_{in}}{i_{in}}$ is given by