List of practice Questions

In the following two-dimensional momentum equation for natural convection over a surface immersed in a quiescent fluid at temperature \( T_\infty \) (g is the gravitational acceleration, \( \beta \) is the volumetric thermal expansion coefficient, \( \nu \) is the kinematic viscosity, \( u \) and \( v \) are the velocities in \( x \) and \( y \) directions, respectively, and \( T \) is the temperature) \[ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = g \beta (T - T_\infty) + \nu \frac{\partial^2 u}{\partial y^2}, \] the term \( g \beta (T - T_\infty) \) represents

Assuming the material considered in each statement is homogeneous, isotropic, linear elastic, and the deformations are in the elastic range, which one or more of the following statement(s) is/are TRUE?

Which of the following heat treatment processes is/are used for surface hardening of steels?

The type of fit between a mating shaft of diameter \( 25.0 + 0.010 \) mm and a hole of diameter \( 25.015 - 0.015 \) mm is ________________.

In a linear programming problem, if a resource is not fully utilized, the shadow price of that resource is

Which one of the following is NOT a form of inventory?

The Clausius inequality holds good for

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).