List of practice Questions

Given vectors \[ \vec{A} = 9\hat{i} - 5\hat{j} + 2\hat{k}, \vec{B} = 11\hat{i} + 4\hat{j} + \hat{k}, \vec{C} = -7\hat{i} + 14\hat{j} - 3\hat{k} \] which of the following statements are TRUE?

Consider a one-dimensional inviscid supersonic flow in a diverging duct with heat addition ($Q_{in}$). Which of the following statement(s) is/are always TRUE?

Consider the International Standard Atmosphere (ISA) with $h$ being the geopotential altitude (in km) and $\dfrac{dT}{dh}$ being the temperature gradient (in K/m). Which of the following combination(s) of $\Big(h, \dfrac{dT}{dh}\Big)$ is/are correct as per ISA?

A rigid bar $AB$ of length 3 m is subjected to a uniformly distributed load of $100 \,\text{N/m}$. The bar is supported at $A$ (pin) and by rod $CD$ connected at $D$. The rod $CD$ has axial stiffness $40 \,\text{N/mm}$, and $C$ is pinned. Find the vertical deflection at point $D$ (in mm).

A cantilever beam of length $2a$ is loaded at the tip with force $F$. The beam is supported in the middle by a roller (pin). Find the reaction moment at the built-in end of the beam as $\alpha Fa$, where $\alpha =$ ....................... (round off to one decimal place).

The shear flow distribution in a single cell, thin-walled beam under a shear load $S_y$ is shown in the figure. The cell has horizontal symmetry with booms marked $1$ to $4$. The shear modulus $G$ is same for all walls, and the area of the cell is $135000 \,\text{mm}^2$. With respect to point $O$, find the distance of shear centre $S$ (in mm). (round off to nearest integer)

The range of f(x) = $4\sin^{-1}(\frac{x^2}{x^2+1})$ is

A specimen containing maximum initial surface crack of size 1.5 mm is subjected to cyclic loading with $ \sigma_{max} = 300 $ MPa and $ \sigma_{min} = 0 $ MPa. Assuming specimen geometric factor of 1, and referring to the given figure, the crack growth rate in $\mu$m cycle$^{-1}$ is ................... (round off to nearest integer).

The alloy A (given in the phase diagram) is cooled slowly from the liquid state to just below the eutectic temperature. The ratio of weight fractions of pro-eutectic $\alpha$ to eutectic $\alpha$ is ................... (round off to 1 decimal place).

A chimney as shown in the figure requires to have a natural draft (pressure difference between the furnace and the bottom of chimney, $ P_o - P_1 $) of $ 1.0133 \times 10^3 $ Pa.
Given: acceleration due to gravity, g = 9.81 m s$^{-2}$
Assume densities of air and flue do not change along the chimney height. Neglect frictional energy loss and kinetic energy difference at the bottom and top of the chimney.
If the density difference between the air and flue is 0.5 kg m$^{-3}$, the minimum height (h) of the chimney in meters is ................... (round off to nearest integer).

For the given schematic TTT diagram of an eutectoid steel, the following statement(s) is/are true for the heat treatment schedules HT-1, HT-2, and HT-3.

The mechanism of creep for a single crystal as depicted in the schematic is

An elutriator is used to separate particles based on their sizes in flowing air as shown in the figure. Assuming spherical particles, the diameter ($D_{50}$) of the suspended particles which have 50% chance to report to overflow by turbulent air flow is expressed as

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line.
The given figure consists of 16 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both PQ (anti-diagonal) and MN (main diagonal) form lines of symmetry? (The figure is representative)

In the given figure, PQRS is a parallelogram with PS = 7 cm, PT = 4 cm and PV = 5 cm. What is the length of RS in cm? (The diagram is representative.)

For the equation $ \begin{vmatrix} x+3 & 3x+4 & 4x+5 \\ -2 & -3 & -4 \\ -3 & -4 & -5 \end{vmatrix} = 0 $ the value of $ x $ is ................... (in integer).

The sum of eigenvalues of the matrix $ \begin{bmatrix} 4 & 3 & 2 \\ 0 & -1 & 2 \\ 0 & 0 & -3 \end{bmatrix} $ is ................... (in integer).

Order (O) and degree (D) of the differential equation $ \left(\frac{dy}{dx}\right)^3 = \sqrt{\frac{d^2y}{dx^2} + 10} $ are

Match the plots in Section I with the corresponding functions in Section II.
Section I

Section II
(1) $ y = \frac{\sin^2x}{x} $
(2) $ y = x \sin^2x $
(3) $ y = \frac{\sin x}{x} $
(4) $ y = x \sin x $

A 200 mm thick slab is rolled using 500 mm diameter rolls under cold rolling and hot rolling conditions, separately. The coefficient of friction is 0.04 in cold rolling and 0.4 in hot rolling. The ratio of maximum possible thickness reduction in cold rolling to that in hot rolling is ................... (round off to 2 decimal places).

A thin plate is loaded in plane stress condition with $ \sigma_{xx} = 110 $ MPa, $ \sigma_{yy} = -50 $ MPa, $ \tau_{xy} = -70 $ MPa. The maximum principal stress in MPa is ................... (round off to nearest integer).

Two circular surfaces A and B with the values of emissivity $\epsilon$, temperature T, and respective view factors are shown in the figure. Consider heat radiation only between surfaces A and B.
Given: Stefan-Boltzmann constant, $ \sigma = 5.67 \times 10^{-8} $ Wm$^{-2}$K$^{-4}$
Net heat flow rate by radiation from surface A in kW is ................... (round off to 1 decimal place).

Copper ore assaying 10 wt.% Cu is fed to a concentration plant at the rate of 100 tons/h. If the grades of concentrate and tailing are 30 wt.% Cu and 1 wt.% Cu, respectively, the percentage recovery of copper in concentrate is ................... (round off to nearest integer).
Given: 1 ton = 1000 kg

Diffraction pattern of a polycrystalline BCC metal is obtained using monochromatic X-rays of wavelength 0.25 nm. If the first peak occurs at Bragg angle ($\theta$) of 30$^{\circ}$, then the radius of the metal atom in nm is ................... (round off to 2 decimal places).

In an aqueous solution of Fe$^{2+}$ ions with concentration of 10$^{-4}$ M at 298 K and atmospheric pressure, the reduction potential of Fe in volt is ................... (round off to 2 decimal places).
Given: Standard reduction potential, $E^{\circ}_{\text{Fe}^{2+}/\text{Fe}} = -0.44$ V
Faraday's constant, F = 96500 C per mole of electrons
Universal gas constant, R = 8.314 J mol$^{-1}$K$^{-1}$