List of practice Questions

A ‘frabjous’ number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?

For any y∈R, let cot$^{-1}$(y) ∈ (0,π) and tan$^{-1}$(y) ∈ ($-\frac{\pi}{2},\frac{\pi}{2}$). Then the sum of all the solutions of the equation$tan^{-1}(\frac{6y}{9-y^2})+cot^{-1}(\frac{9-y^2}{6y})=\frac{2\pi}{3}\, for\,0<|y|<3,$ is equal to

A supersonic aircraft has an air intake ramp that can be rotated about the leading edge $O$ such that the shock from the leading edge meets the cowl lip as shown. Select all the correct statement(s) as per oblique shock theory when flight Mach number $M$ increases.

A perfect gas stored in a reservoir exhausts through a convergent nozzle. The jet emerges at choked conditions with average velocity $u$. If reservoir pressure $p_0$ increases while $T_0$ remains constant, determine effect on $M, u, T, p, \rho$.

A scramjet engine features an intake, isolator, combustor, and a nozzle, as shown. Station 3 indicates the combustor entry point. Assume stagnation enthalpy is constant between Stations 1 and 3, and air is a calorically perfect gas with specific heat ratio $\gamma$. Select the correct expression for Mach number $M_3$ at the inlet to the combustor from the options given.

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)

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).