List of practice Questions

Solution of \( \nabla^2 T = 0 \) in a square domain \( (0<x<1 \) and \( 0<y<1) \) with boundary conditions: \[ T(x, 0) = x; \quad T(0, y) = y; \quad T(x, 1) = 1 + x; \quad T(1, y) = 1 + y \] is

Given a function \( \varphi = \frac{1}{2} (x^2 + y^2 + z^2) \) in three-dimensional Cartesian space, the value of the surface integral \[ \int \int_S \hat{n} \cdot \nabla \varphi \, dS, \] where \( S \) is the surface of a sphere of unit radius and \( \hat{n} \) is the outward unit normal vector on \( S \), is

The Fourier series expansion of \( x^3 \) in the interval \( -1 \leq x \leq 1 \) with periodic continuation has

If \( A = \begin{bmatrix} 10 & 2k + 5 \\ 3k - 3 & k + 5 \end{bmatrix} \) is a symmetric matrix, the value of \( k \) is ________________.

A uniform light slender beam AB of section modulus \( EI \) is pinned by a frictionless joint A to the ground and supported by a light inextensible cable CB to hang a weight \( W \) as shown. If the maximum value of \( W \) to avoid buckling of the beam AB is obtained as \( \beta \pi^2 EI \), where \( \pi \) is the ratio of circumference to diameter of a circle, then the value of \( \beta \) is

The figure shows a schematic of a simple Watt governor mechanism with the spindle \( O_1O_2 \) rotating at an angular velocity \( \omega \) about a vertical axis. The balls at P and S have equal mass. Assume that there is no friction anywhere and all other components are massless and rigid. The vertical distance between the horizontal plane of rotation of the balls and the pivot \( O_1 \) is denoted by \( h \). The value of \( h = 400 \, \text{mm} \) at a certain \( \omega \). If \( \omega \) is doubled, the value of \( h \) will be ________________ mm.

The limit \[ p = \lim_{x \to \pi} \left( \frac{x^2 + \alpha x + 2\pi^2}{x - \pi + 2 \sin x} \right) \] has a finite value for a real \( \alpha \). The value of \( \alpha \) and the corresponding limit \( p \) are

A discrete memoryless channel with three inputs and outputs has transition probabilities as shown. The parameter $\alpha \in [0.25,1]$. Find the value of $\alpha$ that maximizes channel capacity (rounded to two decimals).

A (7,4) Hamming code is used over a BSC with flip probability $\epsilon = 0.1$. A transmitted codeword is decoded correctly if at most one bit error occurs. Find the probability of correct decoding (rounded to two decimals).

Consider a channel where either symbol $x_A$ or symbol $x_B$ is transmitted.
The conditional PDFs of $Y$ given $x_A$ and $x_B$ are:
\[ f_{Y|x_A}(y) = e^{-(y+1)}u(y+1), \qquad f_{Y|x_B}(y) = e^{(y-1)}\left(1 - u(y-1)\right), \] where $u(\cdot)$ is the unit step function.
The probability of symbol error is ____________ (rounded off to two decimal places).

In an electrostatic field, the displacement vector is \[ \vec{D}(x,y,z) = (x^3\,\vec{i} + y^3\,\vec{j} + xy^2\,\vec{k})\ \text{C/m}^2. \] A cube of side 1 m is centered at the origin with vertices at \((\pm 0.5, \pm 0.5, \pm 0.5)\). Find the electric charge enclosed within the cube (rounded to two decimals).

A p-type semiconductor in steady state has excess minority carriers zero at $x=\pm 2L_n$, with $L_n = 10^{-4}$ cm. Electron charge $q = -1.6\times 10^{-19}$ C. The recombination rate profile is \[ R = 10^{20} \exp\left(-\frac{|x|}{L_n}\right)\ \text{cm}^{-3}\text{s}^{-1}. \] The magnitude of electron current density at $x=+2L_n$ is __________ mA/cm$^2$ (rounded to two decimals).

A circuit with a diode is shown. The ratio of the minimum to maximum small-signal voltage gain \[ \frac{\partial V_{out}}{\partial V_{in}} \] is to be found (rounded to two decimals).

Consider the shown ideal OPAMP circuit. Find the output voltage $V_0$ (rounded to two decimals).

For the circuit shown with an ideal long-channel nMOSFET biased in saturation, $v_A$ and $v_B$ are the small-signal voltages at nodes A and B. The value of $\dfrac{v_A}{v_B}$ is ____________ (rounded off to one decimal place).

For the closed-loop control system shown, with $r(t)=0$, the steady-state error $e(\infty)$ due to a unit step disturbance $d(t)$ is ____________ (rounded off to two decimal places).

The value of the integral \[ \iint_D 3(x^2 + y^2)\, dx\, dy, \] where $D$ is the shaded triangular region, is __________ (rounded off to the nearest integer).

A linear 2-port network is shown. From the $V_2$–$(-I_2)$ plot: when $V_2 = 5$ V, $I_2 = 0$ mA, and when $V_2 = 4$ V, $I_2 = -4$ mA. Port 2 is now connected to the load in Fig. (c). Find the current $I_2$ (rounded to one decimal place).

Consider a Boolean gate (D) where the output $Y$ is related to the inputs $A$ and $B$ as \[ Y = A + \overline{B}, \] where $+$ denotes logical OR. The Boolean inputs ‘0’ and ‘1’ are also separately available. Using only D gates and inputs ‘0’ and ‘1’, ________________ (select the correct option(s)).

Two linear time-invariant systems with transfer functions \[ G_1(s) = \frac{10}{s^2 + s + 1}, \qquad G_2(s) = \frac{10}{s^2 + s\sqrt{10} + 10} \] have unit step responses $y_1(t)$ and $y_2(t)$, respectively. Which of the following statements is/are true?

Consider an FM broadcast that employs the pre-emphasis filter with frequency response \[ H_{pe}(\omega) = 1 + \frac{j\omega}{\omega_0}, \] where $\omega_0 = 10^{4}$ rad/s. For the network shown in the figure to act as a corresponding de-emphasis filter, the appropriate pair(s) of $(R, C)$ values is/are ________________.

A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of $10^{-4}\,\text{cm}$, with air as the dielectric. Assume the speed of light in air to be $3\times10^{8}\,\text{m/s}$. The frequency/frequencies of TM waves which can propagate in this waveguide is/are ________________.

The outputs of four systems $(S_1, S_2, S_3, S_4)$ corresponding to the input signal $\sin(t)$, for all time $t$, are shown. Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time-invariant)?

Select the CORRECT statement(s) regarding semiconductor devices.