GATE IN 2015 Question Paper (01-Feb-2015) (Shift 2)

Let A be an n × n matrix with rank r(0 < r < n). Then Ax = 0 has p independent solutions, where p is

The value of $\oint \frac{1}{z^2}dz,$ where the contour is the unit circle traversed clockwise, is

The double integral $\underset{0}{\overset{a}{\int }}\underset{0}{\overset{y}{\int }}f\left(x,y\right)$ dx dy is equivalent to

The magnitude of the directional derivative of the function f(x, y) = x² + 3y² in a direction normal to the circle x² + y² = 2, at the point (1, 1), is

The figure shows a half-wave rectifier circuit with input voltage V(t) = 10 sin (100 π t) volts. Assuming ideal diode characteristics with zero forward voltage drop and zero reverse current, the average power consumed in watts by the load resistance R${}_L$ is __________.