GATE EC question library

The state variable representation of a system is given as $\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$ $y=\begin{bmatrix} 0 &1 \end{bmatrix} x$ The response $y(t)$ is

The output of a standard second-order system for a unit step input is given as $y(t) = 1-\dfrac{2}{\sqrt{3}}e^{-t}\cos \left(\sqrt{3t}-\dfrac{\pi}{6}\right)$. The transfer function of the system is

The transfer function of a mass-spring-damper system is given by $$G(S) = \dfrac{1}{Ms^{2}+Bs+K}$$ The frequency response data for the system are given in the following table. …

A zero mean white Gaussian noise having power spectral density $\dfrac{N_{0}}{2}$ is passed through an LTI filter whose impulse response $h(t)$ is shown in the figure. The variance of the filtered noise at $t = 4$ is

$\begin{Bmatrix} X_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty}$ is an independent and identically distributed (i.i.d.) random process with ܺ$X_{n}$ equally likely to be $+1$ or $−1.\:\:\begin{Bmatrix} Y_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty}$ is another random process obtained as ܻ$Y_{n}=X_{n} + 0.5X_{n-1}.$ The autocorrelation function of …

Consider a binary, digital communication system which uses pulses $g(t)$ and $-g(t)$ for transmitting bits over an AWGN channel. If the receiver uses a matched filter, which one of the following pulses will give the minimum probability of bit error?

Let $X\in \{0,1\}$ and $Y\in \{0,1\}$ be two independent binary random variables. If $P(X=0)=p$ and $P(Y=0)=q,$ then $P(X+Y\geq 1)$ is equal to

An air-filled rectangular waveguide of internal dimensions $a\: cm \times b\: cm\:(a>b)$ has a cutoff frequency of $6\:GHz$ for the dominant $TE_{10}$ mode. For the same waveguide, if the cutoff frequency of the $TM_{11}$mode is $15\: GHz,$ the cutoff frequency of the $TE_{01}$ mode in $GHz$ is __________.

Two half-wave dipole antennas placed as shown in the figure are excited with sinusoidally varying currents of frequency $3\: MHz$ and phase shift of $\frac{\pi}{2}$ between them (the element at the origin leads in phase). If the maximum radiated $E$-field at the point $P$ in the $x-y$ plane occurs at an azimuthal angle of $60^{\circ},$ the …

The electric field of a plane wave propagating in a lossless non-magnetic medium is given by the following expression $$\text{E}(z,t) = \textbf{a}_{x} 5 \cos(2\pi \times 10^{9}t + \beta z) + \textbf{a}_{y} 3 \cos\left(2\pi \times 10^{9} t + \beta z – \frac{\pi}{2}\right)$$ The type of the polarization is

Choose the most appropriate word from the options given below to complete the following sentence. If the athlete had wanted to come first in the race, he ________________ several hours every day.

Choose the most suitable one word substitute for the following expression: Connotation of a road or way

Choose the correct verb to fill in the blank below: Let us __________

Find the missing sequence in the letter series below: $A, CD, GHI,?, UVWXY$

If $x>y>1,$ which of the following must be true? $(i)$ and $(ii)$ $(i)$ and $(iii)$ $(iii)$ and $(iv)$ $(ii)$ and $(iv)$

Ram and Shyam shared a secret and promised to each other that it would remain between them. Ram expressed himself in one of the following ways as given in the choices below. Identify the correct way as per standard English.

In the following question, the first and the last sentence of the passage are in order and numbered $1$ and $6.$ The rest of the passage is split into $4$ parts and numbered as $2,3,4,$ and $5.$ These $4$ parts are not arranged in proper order. Read the sentences and arrange them in a logical sequence to make a passage and choose the correct …

From a circular sheet of paper of radius $30$ cm, a sector of $10\%$ area is removed. If the remaining part is used to make a conical surface, then the ratio of the radius and height of the cone is ________.

$\log \tan1^{\circ}+ \log \tan2^{\circ}+ \dots + \log \tan89^{\circ}$ is

Ms. $X$ will be in Bagdogra from $01/05/2014$ to $20/05/2014$ and from $22/05/2014$ to $31/05/2014.$ On the morning of $21/05/2014,$ she will reach Kochi via Mumbai. Which one of the statements below is logically valid and can be inferred from the above sentences?