List of practice Questions

A simple closed path $C$ is shown in the complex plane. If \[ \oint_C \frac{2z}{z^2 - 1}\, dz = -i\pi A, \] where $i = \sqrt{-1}$, find $A$ (rounded to two decimals).

Let $x_1(t)=e^{-t}u(t)$ and $x_2(t)=u(t)-u(t-2)$. If $y(t)$ is the convolution $x_1(t)x_2(t)$, find \[ \lim_{t\to\infty} y(t) \] (rounded to one decimal place).

Select the correct statement(s) regarding CMOS implementation of NOT gates.

Let $H(X)$ denote the entropy of a discrete random variable $X$ taking $K$ possible distinct real values. Which of the following statements is/are necessarily true?

Consider the following wave equation \[ \frac{\partial^2 f(x,t)}{\partial t^2} = 10000 \frac{\partial^2 f(x,t)}{\partial x^2} \] Which of the given options is/are solution(s) to the given wave equation?

Consider the PDE \[ a\,\frac{\partial^{2} f(x,y)}{\partial x^{2}} \;+\; b\,\frac{\partial^{2} f(x,y)}{\partial y^{2}} \;=\; f(x,y), \] where $a$ and $b$ are distinct positive real numbers. Select the combinations of real parameters $\xi$ and $\eta$ such that \[ f(x,y)=e^{\,(\xi x+\eta y)} \] is a solution of the PDE.

An ideal OPAMP circuit with a sinusoidal input is shown in the figure. The 3 dB frequency is the frequency at which the magnitude of the voltage gain decreases by 3 dB from the maximum value. Which of the options is/are correct?

Select the Boolean function(s) equivalent to $x + yz$, where $x$, $y$, and $z$ are Boolean variables, and $+$ denotes logical OR.

Consider a closed-loop control system with unity negative feedback and $K G(s)$ in the forward path, where the gain $K = 2$. The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G(s)$ has no poles on the closed right-half of the complex plane. The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is ____________.

The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure. Note that $K$ is varied from 0 to $\infty$. Select the transfer function $G(s)$ that results in the shown root-locus.

The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure.

Statement I: The system is necessarily a pure delay system for inputs which are bandlimited to $-\alpha \le f \le \alpha$.
Statement II: For any wide-sense stationary input process with power spectral density $S_X(f)$, the output power spectral density $S_Y(f)$ obeys $S_Y(f) = S_X(f)$ for $-\alpha \le f \le \alpha$.
Which one of the following combinations is true?

In a circuit, there is a series connection of an ideal resistor and an ideal capacitor. The conduction current (in Amperes) through the resistor is $2\sin(t + \pi/2)$. The displacement current (in Amperes) through the capacitor is ________________.

Consider the CMOS circuit shown in the figure (substrates are connected to their respective sources). The gate width $(W)$ to gate length $(L)$ ratios of the transistors are as shown. Both the transistors have the same gate oxide capacitance per unit area. For the pMOSFET, the threshold voltage is $-1\ \text{V}$ and the mobility of holes is $40\ \text{cm}^2/(\text{V·s})$. For the nMOSFET, the threshold voltage is $1\ \text{V}$ and the mobility of electrons is $300\ \text{cm}^2/(\text{V·s})$. The steady state output voltage $V_O$ is ________________.

Consider the 2-bit multiplexer (MUX) shown in the figure. For OUTPUT to be the XOR of C and D, the values for $A_0, A_1, A_2,$ and $A_3$ are ________________.

The ideal long-channel nMOSFET and pMOSFET devices shown have threshold voltages of $1\text{ V}$ and $-1\text{ V}$, respectively. The MOSFET substrates are connected to their sources. Ignore leakage currents and assume that the capacitors are initially discharged. For the applied voltages as shown, the steady-state voltages are ________________.

Consider the circuit shown in the figure. The current $I$ flowing through the 10 $\Omega$ resistor is ________________.

The Fourier transform $X(j\omega)$ of the signal \[ x(t)=\frac{t}{(1+t^{2})^{2}} \] is ________________.

Consider a long rectangular bar of direct bandgap $p$-type semiconductor. The equilibrium hole density is $10^{17}\,\text{cm}^{-3}$ and the intrinsic carrier concentration is $10^{10}\,\text{cm}^{-3}$. Electron and hole diffusion lengths are $2\,\mu$m and $1\,\mu$m, respectively. The left side of the bar ($x=0$) is uniformly illuminated with a laser having photon energy greater than the bandgap of the semiconductor. Excess electron-hole pairs are generated ONLY at $x=0$ because of the laser. The steady state electron density at $x=0$ is $10^{14}\,\text{cm}^{-3}$ due to laser illumination. Under these conditions and ignoring electric field, the closest approximation (among the given options) of the steady state electron density at $x=2\,\mu\text{m}$ is ______________.

In a non-degenerate bulk semiconductor with electron density $n = 10^{16}$ cm$^{-3}$, the value of $(E_C - E_{Fn}) = 200$ meV, where $E_C$ and $E_{Fn}$ denote the conduction band edge and electron Fermi level, respectively. Assume thermal voltage as 26 meV and intrinsic carrier concentration as $10^{10}$ cm$^{-3}$. For $n = 0.5 \times 10^{16}$ cm$^{-3}$, the closest approximation of $(E_C - E_{Fn})$, among the given options, is ________________.

Consider a system of linear equations $Ax = b$, where \[ A = \begin{bmatrix} 1 & -\sqrt{2} & 3 \\ -1 & \sqrt{2} & -3 \end{bmatrix}, \qquad b = \begin{bmatrix} 1 \\ 3 \end{bmatrix}. \] This system of equations admits ________________.

The current $I$ in the circuit shown is ________________.

\(\int_{0}^{2} \left( |2x^2 - 3x| + \left[x - \frac{1}{2}\right] \right) \, dx,\)
where [t] is the greatest integer function, is equal to

Mr. Thomas invested an amount of Rs. 13,900 divided in two different schemes A and B at the simple interest rate of 14% p.a. and 11% p.a. respectively. If the total amount of simple interest earned in 2 years is Rs. 3508, what was the amount invested in Scheme B ?

The work done by a force on body of mass 5 kg to accelerate it in the direction of force from rest to 20 m/s² in 10 second is

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98/25. Then the sum of the first 21 terms of an AP, whose first term is 10ar, n^th term is a_n and the common difference is 10ar², is equal to