GATE EC question library

The value of the contour integral $$\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$$ evaluated over the unit circle $\mid z \mid=1$ is_______.

The number of distinct eigenvalues of the matrix $$A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix}$$ is equal to ____________.

If $X$ and $Y$ are random variables such that $E\left[2X+Y\right]=0$ and $E\left[X+2Y\right]=33$, then $E\left[X\right]+E\left[Y\right]=$___________.

The value of the integral $\displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.

Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$ is given by $$F_{Z}(x)= \left\{\begin{matrix} 1-e^{-x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.

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

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 milliseconds. If the carrier frequency is $f_{c}=50\:kHz$ and $k=10 \pi,$ then the ratio of the minimum instantaneous frequency (in kHz) to the maximum …

Radiation resistance of a small dipole current element of length $l$ at a frequency of $3$ GHz is $3$ ohms. If the length is changed by $1\%$, then the percentage change in the radiation resistance, rounded off to two decimal places, is ________ $\%.$

In the circuit shown, $V_{s}$ is square wave of period $T$ with maximum and minimum values of $8\: V$ and $-10\: V$, respectively. Assume that the diode is ideal and $R_{1}=R_{2}=50\: \Omega.$ The average value of $V_{L}$ is _______ volts (rounded off to $1$ decimal place).

In the circuit shown, the clock frequency, i.e., the frequency of the ClK signal, is $12\:kHz$. The frequency of the signal at $Q_{2}$ is _______ kHz.

Consider a differentiable function $f(x)$ on the set of real numbers, such that $f(-1)=0$ and $\mid f’(x) \mid \leq 2.$ Given these conditions, which one of the following inequalities is necessarily true for all $x \in[-2,2]?$

Consider the line integral $$\int_{c} (xdy-ydx)$$ the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2 \times 3$ 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 $W_{6}=exp\left(-\:\dfrac{j2\pi}{6}\right).$ In the figure, what should be the values of the coefficients $a_{1},a_{2},a_{3}$ in terms of $W_{6}$ so that $X[1]$ is obtained …

It is desired to find a three-tap casual filter which gives zero signal as an output to an input of the form $$x[n]= c_{1}exp\left(-\dfrac{j\pi n}{2}\right)+c_{2}\left(\dfrac{j\pi n}{2}\right),$$ where $c_{1}$ and $c_{2}$ are arbitrary real numbers. The desired three-tap filter is given by $$h[0]=1,\quad h[1]=a,\quad h[2]=b$$ and …

In the circuit shown, if $v(t)=2 \sin(1000\: t)$ volts, $R=1\:k \Omega$ and $C=1\:\mu F,$ then the steady-state current $i(t)$, milliamperes (mA), is

Consider a causal second-order system with the transfer function $$G(s)=\dfrac{1}{1+2s+s^{2}}$$ with a unit-step $R(s)=\dfrac{1}{s}$ as an input. Let $C(s)$ be the corresponding output. The time taken by the system output $c(t)$ to reach $94\%$ of its steady-state value $\underset{t\rightarrow \infty}{\lim}\:c(t),$ rounded off to two decimal …

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 $H(s)=\dfrac{Y(s)}{X(s)}$ is

Let the state-space representation of an LTI system be $x(t)=A x(t)+B u(t), y(t)=Cx(t)+du(t)$ where $A,B,C$ are matrices, $d$ is a scalar, $u(t)$ is the input to the system, and $y(t)$ is its output. Let $B=[0\quad0\quad1]^{T}$ and $d=0$ .Which one of the following options for $A$ and $C$ will ensure that the transfer function of this LTI …

A single bit, equally likely to be $0$ and $1$, is to be sent across an additive white Gaussian noise (AWGN) channel with power spectral density $N_{0}/2.$ Binary signaling with $0 \mapsto p(t),$ and $1 \mapsto q(t),$ is used for the transmission, along with an optimal receiver that minimizes the bit-error probability. Let …

The quantum efficiency $(\eta)$ and responsivity $(R)$ at wavelength $\lambda \:(\text{in}\: \mu m)$ in a p-i-n photodetector are related by