GATE EC 2019 Question Paper (09-Feb-2019) (Shift 1)

Consider a differentiable function f(x) on the set of real numbers such that f(−1) = 0 and |f′(x)| ≤ 2. Given these conditions, which one of the following inequalities is necessarily true for all x ∈ [−2 , 2] ?

Consider the line integral $\underset{c}{\int }\left(xdy-ydx\right)$ the integral being taken in a counter clockwise direction over the closed curve C that forms the boundary of the region R shown in the figure below. The region R is the area enclosed by the union of a 2 × 3 rectangle and a semi-circle of radius 1. The line integral evaluates to

Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[1] is shown in the figure. Let $W_6=\exp \left(-\frac{j2\pi }{6}\right)$. In the figure, what should be the values of the coefficients a₁, a₂, a₃ in terms of W6 so that X[1] is obtained correctly?

It is desired to find a three-tap causal filter that gives zero signal as an output to an input of the form $x\left[n\right]=c_1\exp \left(-\frac{j\pi n}{2}\right)+c_2\exp \left(\frac{j\pi n}{2}\right)$, where c₁ and c₂ are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n when x[n] is as given above?

In the circuit shown, if v(t) = 2 sin(1000 t) volts, R = 1 kΩ and C = 1 μF, then the steady-state current i(t), in milliamperes (mA), is