GATE Electronics & Communication

Let F1, F2, and F3 be functions of (x, y, z). Suppose that for every given pair of points A and B in space, the line integral ∫(F₁dx+ F₂dy + F₃dz) evaluates to the same value along...

The value of the line integral $\int_P^Q(z^2 dx + 3y^2 dy + 2xz dz)$ along the straight line joining the points P (1,1,2) and Q (2,3,1) is

Let $\mathbf{x}$ be an $n \times 1$ real column vector with length $l = \sqrt{\mathbf{x}^T\mathbf{x}}$. The trace of the matrix $P = \mathbf{x}\mathbf{x}^T$ is

Which one of the following functions is analytic over the entire complex plane?

The families of curves represented by the solution of the equation dy/dx = -(x/y)^n for n = -1 and n = +1, respectively, are

Let H(z) be the z-transform of a real-valued discrete-time signal h[n]. If P(z) = H(z)H(1/z) has a zero at z = 1/2 + 1/2j, and P(z) has a total of four zeros, which one of the foll...

Consider the two-port resistive network shown in the figure. When an excitation of 5 V is applied across Port 1, and Port 2 is shorted, the current through the short circuit at Por...

Let Y(s) be the unit-step response of a causal system having a transfer function $G(s) = \frac{3-s}{(s+1)(s+3)}$. That is, $Y(s) = \frac{G(s)}{s}$. The forced response of the syste...

For an LTI system, the Bode plot for its gain is as illustrated in the figure shown. The number of system poles Np and the number of system zeros Nz in the frequency range 1 Hz ≤ f...

In the table shown, List I and List II, respectively, contain terms appearing on the left-hand side and the right-hand side of Maxwell's equations (in their standard form). Match t...

If X and Y are random variables such that E[2X + Y] = 0 and E[X + 2Y] = 33, then E[X] + E[Y] = _________.

Let Z be an exponential random variable with mean 1. That is, the cumulative distribution function of Z is given by $F_Z(x) = \begin{cases} 1 - e^{-x} & \text{if } x \ge 0 \\ 0 & \...

Consider the signal $f(t) = 1 + 2 \cos(\pi t) + 3 \sin(\frac{2\pi}{3} t) + 4 \cos(\frac{\pi}{2} t + \frac{\pi}{4})$, where t is in seconds. Its fundamental time period, in seconds,...

The baseband signal m(t) shown in the figure is phase-modulated to generate the PM signal $\varphi(t) = \cos(2\pi f_c t + k m(t))$. The time t on the x-axis in the figure is in mil...

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 necessar...

Consider the line integral ∫_C (xdy - ydx) the integral being taken in a counterclockwise direction over the closed curve C that forms the boundary of the region R shown in the fig...

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

Consider a causal second-order system with the transfer function G(s) = \frac{1}{1 + 2s + s^2} with a unit-step R(s) = \frac{1}{s} as an input. Let C(s) be the corresponding output...

The block diagram of a system is illustrated in the figure shown, where X(s) is the input and Y(s) is the output. The transfer function H(s) = \frac{Y(s)}{X(s)} is

Consider a unity feedback system, as in the figure shown, with an integral compensator $\frac{K}{s}$ and open-loop transfer function $G(s) = \frac{1}{s^2 + 3s + 2}$ where K > 0. Th...

In the circuit shown, Vs is a 10 V square wave of period, T = 4 ms with R = 500 Ω and C = 10 µF. The capacitor is initially uncharged at t=0, and the diode is assumed to be ideal....

A CMOS inverter, designed to have a mid-point voltage V_I equal to half of Vdd, as shown in the figure, has the following parameters: Vdd = 3 V μn Cox = 100 μA/V^2 ; Vtn = 0.7 V fo...