GATE EC question library

An unforced linear time invariant (LTI) system is represented by $$\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0& -2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$$ If the initial conditions are $x_1(0)= 1$ and $x_2(0)= -1$, the solution of the state equation is

The Bode asymptotic magnitude plot of a minimum phase system is shown in the figure. If the system is connected in a unity negative feedback configuration, the steady state error of the closed loop system, to a unit ramp input, is _________.

Consider the state space system expressed by the signal flow diagram shown in the figure. The corresponding system is

The input to a $1$ – bit quantizer is a random variable $X$ with pdf $f_{X}( x )= 2e^{-2x}$ for $x\geq 0$ and $f_{X} (x )= 0$ for $x< 0$. For outputs to be of equal probability, the quantizer threshold should be ______.

Coherent orthogonal binary FSK modulation is used to transmit two equiprobable symbol waveforms $s_{1}( t )= \alpha \cos 2\pi f_1 t$, and $s_2(t) = \alpha \cos 2 \pi f_2 t$ where $\alpha = 4$ mV. Assume an AWGN channel with two-sided noise power spectral density $\frac{N_o}{2}= 0.5 \times 10^{-12}$ W/Hz. Using an optimal receiver and the relation …

The power spectral density of a real stationary random process $X(t)$ is given by $$S_X (f) = \begin{cases} \frac{1}{W}, & \mid f \mid \leq W \\ 0, & \mid f \mid > W \end{cases}$$ The value of the expectation $E \left [ \pi X(t)X\left ( t-\frac{1}{4W} \right ) \right ]$ is __________.

In the figure, $M(f)$ is the Fourier transform of the message signal $m(t)$ where $A = 100$ Hz and $B = 40$ Hz. Given $v(t)= \cos (2\pi f_{c}t)$ and $w(t)= \cos (2\pi (f_{c}+A)t)$, where $f_{c}>A$. The cutoff frequencies of both the filters are $f_{c}$. The bandwidth of the signal at the output of the modulater (in Hz) is ______.

If the electric field of a plane wave is $$\overrightarrow{E}(z,t)= \hat{x}3 \cos(\omega t-kz+30^{\circ}) – \hat{y} 4 \sin(\omega t – kz+45^{\circ}) (mV/m),$$ the polarization state of the plane wave is

In the transmission line shown, the impedance $Z_{\text{in}}$ (in ohms) between node A and the ground is _________.

For a rectangular waveguide of internal dimension $a\times b \: (a> b)$, the cut-off frequency for the $TE_{11}$ mode is the arithmetic mean of the cut-off frequencies for $TE_{10}$ mode and $TE_{20}$ mode. If $a= \sqrt{5}$ cm, the value of $b$ (in cm) is ______.

“India is a country of rich heritage and cultural diversity.” Which one of the following facts best supports the claim made in the above sentence?

The value of one U.S. dollar is $65$ Indian Rupees today, compared to $60$ last year. The Indian Rupee has ____________.

'Advice' is ________________.

The next term in the series $81, 54, 36, 24, \dots$ is ________

In which of the following options will the expression $P < M$ be definitely true?

Find the next term in the sequence: $7G, 11K, 13M$,____

The multi-level hierarchical pie chart shows the population of animals in a reserve forest. The correct conclusions from this information are: (i) and (ii) only (i), (ii), (iii) and (iv) (i), (iii) and (iv) only (i), (ii) and (iii) only

A man can row at $8$ $km$ per hour in still water. If it takes him thrice as long to row upstream, as to row downstream, then find the stream velocity in $km$ per hour.

A firm producing air purifiers sold $200$ units in $2012$. The following pie chart presents the share of raw material, labour, energy, plant & machinery, and transportation costs in the total manufacturing cost of the firm in $2012$. The expenditure on labour in $2012$ is Rs. $4,50,000$. In $2013$, the raw material expenses increased by …

A batch of one hundred bulbs is inspected by testing four randomly chosen bulbs. The batch is rejected if even one of the bulbs is defective. A batch typically has five defective bulbs. The probability that the current batch is accepted is ________