GATE EC question library

The output $(Y)$ of the circuit shown in the figure is

A Fourier transform pair is given by $$\left ( \frac{2}{3} \right ) \: u[n+3] \overset{FT}{\Leftrightarrow} \frac{Ae^{-j6 \pi f}}{1- (\frac{2}{3} ) e^{-j2 \pi f}}$$ where $u[n]$ denotes the unit step sequence. The values of $A$ is ____________

A real-valued signal $x(t)$ limited to the frequency band $\mid f \mid \leq \frac{W}{2}$ is passed through a linear time invariant system whose frequency response is $$H(f) = \begin{cases} e^{-j 4 \pi f}, & \mid f \mid \leq \frac{W}{2} \\ 0, & \mid f \mid > \frac{W}{2} \end{cases}.$$ The output of the system is

The sequence $x[n] = 0.5^n \: u[n]$, where $u[n]$ is the unit step sequence, is convolved with itself to obtain $y[n]$. Then $\Sigma_{n= -\infty}^{+ \infty} y[n]$ is _________.

In a Bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a $4$th order all-pole system?

For the second order closed-loop system shown in the figure, the natural frequency (in rad/s) is

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be

In a double side-band (DSB) full carrier AM transmission system, if the modulation index is doubled, then the ratio of total sideband power to the carrier power increases by a factor of __________.

For an antenna radiating in free space, the electric field at a distance of $1$ km is found to be $12 \: mV/m$. Given that intrinsic impedance of the free space is $120 \pi \: \Omega$, the magnitude of average power density due to this antenna at a distance of $2$ km from the antenna (in $nW/m^2$) is _________.

Match column A with column B. $\begin{array}{ll} \textbf{Column A} & \textbf{Column B} \\ \text{1. Point electromagnetic source} & \text{P. Highly directional} \\ \text{2. Dish antenna} & \text{Q. End fire} \\ \text{3. Yagi-Uda antenna} & \text{R. Isotrophic} \end{array}$

With initial values $y(0) =y’(0)=1$, the solution of the differential equation $$\frac{d^2y}{dx^2}+4 \frac{dy}{dx}+4y=0$$ at $x=1$ is ________

Parcels from sender S to receiver R pass sequentially through two-post offices. Each post-office has a probability $\frac{1}{5}$ of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post office is _________

The unilateral Laplace transform of $f(t)$ is $\frac{1}{s^2+s+1}$. Which one of the following is the unilateral Laplace transform of $g(t) = t \cdot f(t)$?

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the triangle, the angle between the hypotenuse and the side is

The steady state output of the circuit shown in the figure is given by $y(t)=A(\omega) \sin (\omega t + \phi ( \omega))$. If the amplitude $\mid A (\omega ) \mid =0.25$, then the frequency $\omega$ is

In the circuit shown in the figure, the value of $v_0(t)$ (in Volts) for $t \to \infty$ is ___________

The equivalent resistance in the infinite ladder network shown in the figure, is $R_e$. The value of $R_e/R$ is __________

For the two-port network shown in the figure, the impedance (Z) matrix (in $\Omega$) is

Consider a silicon sample doped with $N_D = 1 \times 10^{15}/cm^3$ donor atoms. Assume that the intrinsic carrier concentration $n_i = 1.5 \times 10^{10}/cm^3$. If the sample is additionally doped with $N_A = 1 \times 10^{18}/cm^3$ acceptor atoms, the approximate number of electrons$/cm^3$ in the sample, at $T=300 \:K$, will be ________.

Consider two BJTs biased at the same collector current with area $A_1=0.2 \mu m \times 0.2 \mu m$ and $A_2 = 300 \mu m \times 300 \mu m$. Assuming that all other device parameters are identical, $kT/q=26 \: mV$, the intrinsic carrier concentration is $1 \times 10^{10} cm^{-3}$, and $q=1.6 \times 10^{-19} \:C$, the difference between the …