GATE AE 2022 Question Paper & Solutions

Quantitative Aptitude

Writing too many things on the ________ while teaching could make the students get _________.

Quantitative Aptitude

Which one of the following is a representation (not to scale and in bold) of all values of $x$ satisfying the inequality $2 - 5x \le -\frac{6x - 5}{3}$ on the real number line?

Quantitative Aptitude

If $f(x) = 2 \ln\left(\sqrt{e^x}\right)$, what is the area bounded by $f(x)$ for the interval $[0, 2]$ on the $x$-axis?

Quantitative Aptitude

A person was born on the fifth Monday of February in a particular year. Which one of the following statements is correct based on the above information?

Quantitative Aptitude

Which one of the groups given below can be assembled to get the shape that is shown above using each piece only once without overlapping with each other? (rotation and translation operations may be used).

Verbal Aptitude

Fish belonging to species S in the deep sea have skins that are extremely black (ultra-black skin). This helps them not only to avoid predators but also sneakily attack their prey. However, having this extra layer of black pigment results in lower collagen on their skin, making their skin more fragile. Which one of the following is the CORRECT logical inference based on the information in the above passage?

Quantitative Aptitude

For the past m days, the average daily production at a company was 100 units per day. If today's production of 180 units changes the average to 110 units per day, what is the value of m?

Quantitative Aptitude

Consider the following functions for non-zero positive integers, $p$ and $q$: $f(p, q) = p \times p \times \cdots \times p \quad (q \text{ terms}) = p^q; \quad f(p, 1) = p$ $g(p, q) = p^{p^{p^{\cdot^{\cdot^{\cdot}}}}} \quad (\text{up to } q \text{ terms}); \quad g(p, 1) = p$ Which one of the following options is correct based on the above?

Quantitative Aptitude

Four cities P, Q, R and S are connected through one-way routes as shown in the figure. The travel time between any two connected cities is one hour. The boxes beside each city name describe the starting time of first train of the day and their frequency of operation. For example, from city P, the first trains of the day start at 8 AM with a frequency of 90 minutes to each of R and S. A person does not spend additional time at any city other than the waiting time for the next connecting train. If the person starts from R at 7 AM and is required to visit S and return to R, what is the minimum time required?

Analytical & Spatial Aptitude

Equal sized circular regions are shaded in a square sheet of paper of 1 cm side length. Two cases, case M and case N, are considered as shown in the figures below. In the case M, four circles are shaded in the square sheet and in the case N, nine circles are shaded in the square sheet as shown. What is the ratio of the areas of unshaded regions of case M to that of case N? case M case N

Basic Fluid Mechanics

The equation of the straight line representing the tangent to the curve $y = x^2$ at the point $(1, 1)$ is:

Linear Algebra

Let $\hat{i}, \hat{j}$, and $\hat{k}$ be the unit vectors in the $x, y$ and $z$ directions, respectively. If the vector $\hat{i} + \hat{j}$ is rotated about positive $\hat{k}$ by $135^\circ$, one gets:

Basic Fluid Mechanics

Let x be a real number and i = √-1. Then the real part of cos(ix) is

Compressible Flow & Shocks

The point of maximum entropy on a Fanno-curve in a Temperature-Entropy (T-s) diagram represents the

Basic Fluid Mechanics

Consider a two-dimensional potential flow over a cylinder. If the freestream speed is $U_\infty$, the maximum speed on the cylinder surface is:

Strength of Materials & Elasticity

Consider steady, two-dimensional, incompressible flow over a non-porous flat plate as shown in the figure. For the control volume $PQRS$, the speed, $u_\infty$, at section $PQ$ is uniform and the speed at section $RS$ is given by $u(y) = A_0 \left(\frac{y}{h}\right)^n$, where $n$ is a positive integer. The value of $A_0$ for which the flow through section $PS$ will vanish is:

Basic Fluid Mechanics

Consider the velocity distribution, u(y) shown in the figure. For two adjacent fluid layers L1 and L^2, the viscous force exerted by L1 on L2 is

Atmospheric Flight Mechanics

The service ceiling of an airplane is the altitude

Strength of Materials & Elasticity

Regarding the horizontal tail of a conventional airplane, which one of the following statements is true?

Strength of Materials & Elasticity

A beam with a symmetrical T-shaped cross-section, as shown in the figure, is subjected to pure bending. The maximum magnitude of the normal stress is realised:

Strength of Materials & Elasticity

A three-member truss is simply supported at $Q$ and $R$, and loaded at $P$ by a horizontal force $F$ as shown. The force in $QR$ is:

Thin-Walled Structures

The closed thin-walled rectangular channel shown in figure (i) is opened by introducing a sharp cut at the center of the bottom edge, as shown in figure (ii). Which one of the following statements is correct?

Rocket Propulsion

The region of highest static temperature in a rocket engine and the region of highest heat flux are ____________, respectively.

Strength of Materials & Elasticity

If $\hat{a}, \hat{b}, \hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s):

Compressible Flow & Shocks

Across an oblique shock wave in a calorifically perfect gas,

Airfoil & Wing Theory

NACA 2412 airfoil has

Atmospheric Flight Mechanics

For International Standard Atmosphere (ISA) up to $11\text{ km}$, which of the following statement(s) is/are true?

Strength of Materials & Elasticity

Let $\sigma$ and $\tau$ represent the normal stress and shear stress on a plane, respectively. The Mohr circle(s) that may possibly represent the state of stress at points in a beam of rectangular cross-section under pure bending is/are:

Strength of Materials & Elasticity

An isotropic linear elastic material point under plane strain condition in the $x$-$y$ plane always obeys:

Compressors and Turbines

A high-pressure-ratio multistage axial compressor encounters an extreme loading mismatch during starting. Which of the following technique(s) can be used to alleviate this problem?

Strength of Materials & Elasticity

The arc length of the parametric curve: x = cos $\theta$ , y = sin $\theta$ , z = $\theta$ from $\theta$ = 0 to $\theta$ = 2$\pi$ is equal to ________ (round off to one decimal place).

Atmospheric Flight Mechanics

An unpowered glider is flying at a glide angle of 10 degrees. Its lift-to-drag ratio is (round off to two decimal places).

Strength of Materials & Elasticity

The two-dimensional plane-stress state at a point is: $\sigma_{xx} = 110\text{ MPa}; \sigma_{yy} = 30\text{ MPa}; \tau_{xy} = 40\text{ MPa}$. The normal stress, $\sigma_n$, on a plane inclined $45^\circ$ as shown in the figure is ______ $\text{MPa}$ (round off to the nearest integer).

Thermodynamic Cycles of Gas Turbines

In a static test, a turbofan engine with bypass ratio of 9 has core hot exhaust speed 1.5 times that of fan exhaust speed. The engine is operated at a fuel to air ratio of f = 0.03. Both the fan and the core streams have no pressure thrust. The ratio of fan thrust to thrust from the core engine is _____ (round off to one decimal place).

Inlets, Combustors & Nozzles

In a single stage turbine, the hot gases come out of stator/ nozzle at a speed of 500 m/s and at an angle of 70 degrees with the turbine axis as shown. The design speed of the rotor blade is 250 m/s at the mean blade radius. The rotor blade angle, $\beta$, at the leading edge is ________ degrees (round off to one decimal place).

Linear Algebra

The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius R is

Differential Equations

Consider the differential equation $\frac{d^2 y}{dx^2} - 2 \frac{dy}{dx} + y = 0$. The boundary conditions are $y = 0$ and $\frac{dy}{dx} = 1$ at $x = 0$. Then the value of $y$ at $x = 1/2$ is:

Differential Equations

Consider the partial differential equation $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$ where $x, y$ are real. If $f(x, y) = a(x)b(y)$, where $a(x)$ and $b(y)$ are real functions, which one of the following statements can be true?

Differential Equations

A cylindrical object of diameter $900\text{ mm}$ is designed to move axially in air at $60\text{ m/s}$. Its drag is estimated on a geometrically half-scaled model in water. Coefficients of dynamic viscosity and densities for air and water are $1.86 \times 10^{-5}\text{ Pa}\cdot\text{s}, 1.2\text{ kg/m}^3$ and $1.01 \times 10^{-3}\text{ Pa}\cdot\text{s}, 1000\text{ kg/m}^3$ respectively. Drag measured for the model is $2280\text{ N}$. Drag experienced by the full-scale object is _____ $\text{N}$ (rounded off to the nearest integer).

Airfoil & Wing Theory

Consider a conventional subsonic fixed-wing airplane. $e$ is the Oswald efficiency factor and $AR$ is the aspect ratio. Corresponding to the minimum $\left( \frac{C_D}{C_L^{3/2}} \right)$, which of the following relations is true?

Strength of Materials & Elasticity

A horizontal load $F$ is applied at point $R$ on a two-member truss, as shown in the figure. Both the members are prismatic with cross-sectional area, $A_0$, and made of the same material with Young's modulus $E$. The horizontal displacement of point $R$ is:

Strength of Materials & Elasticity

Which of the following is NOT always true for a combustion process taking place in a closed system?

Strength of Materials & Elasticity

The real function $y = \sin^2(|x|)$ is:

Compressible Flow & Shocks

A convergent nozzle fed from a constant pressure, constant temperature reservoir, is discharging air to atmosphere at 1 bar (absolute) with choked flow at the exit (marked as Q). Flow through the nozzle can be assumed to be isentropic. If the exit area of the nozzle is increased while all the reservoir parameters and ambient conditions remain the same, then at steady state

Strength of Materials & Elasticity

For a conventional airplane in straight, level, constant velocity flight condition, which of the following condition(s) is/are possible on Euler angles $(\phi, \theta, \psi)$, angle of attack $(\alpha)$ and the sideslip angle $(\beta)$?

Space Dynamics & Orbital Mechanics

Consider a high Earth-orbiting satellite of angular momentum per unit mass $h$ and eccentricity $e$. The mass of the Earth is $M$ and $G$ is the universal gravitational constant. The distance between the satellite's center of mass and the Earth's center of mass is $r$, the true anomaly is $\theta$, and the phase angle is zero. Which of the following statements is/are true?

Rocket Propulsion

A rocket operates at an absolute chamber pressure of $20\text{ bar}$ to produce thrust, $F_1$. The hot exhaust is optimally expanded to $1\text{ bar}$ (absolute pressure) using a convergent-divergent nozzle with exit to throat area ratio $\left(\frac{A_e}{A_t}\right)$ of $3.5$ and thrust coefficient, $C_{F,1} = 1.42$. The same rocket when operated at an absolute chamber pressure of $50\text{ bar}$ produces thrust $F_2$ and the thrust coefficient is $C_{F,2}$. Which of the following statement(s) is/are correct?

Calculus

$\vec{v} = x^3 \hat{i} + y^3 \hat{j} + z^3 \hat{k}$ is a vector field where $\hat{i}, \hat{j}, \hat{k}$ are the base vectors of a Cartesian coordinate system. Using the Gauss divergence theorem, the value of the outward flux of the vector field over the surface of a sphere of unit radius centered at the origin is ______ (rounded off to one decimal place).

Linear Algebra

The largest eigenvalue of the given matrix is _____: $\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$

Strength of Materials & Elasticity

A rotational velocity field in an air flow is given as ˆ ˆV ayi bxj   , with a = 10 s-1, b = 20 s-1. The air density is 1.0 kg/m3 and the pressure at (x, y) = (0 m, 0 m) is 100 kPa. Neglecting gravity, the pressure at (x, y) = (6 m, 8 m) is _____ kPa (rounded off to nearest integer).

Airfoil & Wing Theory

Consider a circulation distribution over a finite wing given by the equation below: $\Gamma(y) = \begin{cases} \Gamma_0 \left(1 - \frac{2y}{b}\right) & \text{if } 0 \le y \le \frac{b}{2} \\[6pt] \Gamma_0 \left(1 + \frac{2y}{b}\right) & \text{if } -\frac{b}{2} \le y \le 0 \end{cases}$ The wingspan $b$ is $10\text{ m}$, the maximum circulation $\Gamma_0$ is $20\text{ m}^2/\text{s}$, density of air is $1.2\text{ kg/m}^3$ and the free stream speed is $80\text{ m/s}$. The lift over the wing is ______ $\text{N}$ (rounded off to the nearest integer).

Strength of Materials & Elasticity

Consider a solid cylinder housed inside another cylinder as shown in the figure. Radius of the inner cylinder is 1 m and its height is 2 m. The gap between the cylinders is 5 mm and is filled with a fluid of viscosity 10-4 Pa-s. The inner cylinder is rotating at a constant angular speed of 5 rad/s while the outer cylinder is stationary. Friction at the bottom surfaces can be ignored. Velocity profile in the vertical gap between the cylinders can be assumed to be linear. The driving moment required for the rotating motion of the inner cylinder is ____________ Nm (rounded off to two decimal places).

Compressible Flow & Shocks

In a converging duct, area and velocity at section P are 1 m 2 and 15 m/s, respectively. The temperature of the fluid is 300 K. Air flow through the nozzle can be assumed to be inviscid and isothermal. Characteristic gas constant is 287 J/(kg-K) and ratio of specific heats is for air. To ensure that the air flow remains incompressible (Mach number, M ≤ 0.3) in the duct, the minimum area required at section Q is _____ m^2 (rounded off to two decimal places).

Airfoil & Wing Theory

Consider a thin symmetric airfoil at 2 degree angle of attack in a uniform flow at 50 m/s. The pitching moment coefficient about its leading edge is _____ (rounded off to three decimal places).

Inlets, Combustors & Nozzles

A convergent-divergent nozzle with adiabatic walls is designed for an exit Mach number of 2.3. It is discharging air to atmosphere under the conditions indicated in the figure. Flow through the nozzle is inviscid, the characteristic gas constant for air is 287 J/(kg-K), and $\gamma$ =1.4. When the reservoir pressure is 25 bar (absolute), and temperature is 300 K, Prandtl- Meyer expansion waves appear at the nozzle exit as shown. The minimum percentage change in the reservoir pressure required to eliminate the wave system at the nozzle exit under steady state is _______%.

Compressors and Turbines

A conventional airplane of mass 5000 kg is doing a level turn of radius 1000 m at a constant speed of 100 m/s at sea level. Taking the acceleration due to gravity as 10 m/s^2, the bank angle of the airplane is degrees.

Strength of Materials & Elasticity

Given: The tip deflection and tip slope for a tip loaded cantilever of length $L$ are $\frac{NL^3}{3EI}$ and $\frac{NL^2}{2EI}$, respectively, where $N$ is the tip force and $EI$ is the flexural rigidity. A cantilever $PQ$ of rectangular cross-section is subjected to transverse load, $F$, at its mid-point. Two cases are considered as shown in the figure. In Case I, the end $Q$ is free and in Case II, $Q$ is simply supported. The ratio of the magnitude of the maximum bending stress at $P$ in Case I to that in Case II is ______ (rounded off to one decimal place).

Strength of Materials & Elasticity

A simply supported Aluminium column of length 1 m and rectangular cross- section w × t with t ≤ w, is subjected to axial compressive loading. Young's modulus is 70 GPa. Yield stress under uniaxial compression is 120 MPa. The value of t at which the failure load for yielding and buckling coincide is _______mm.

Thin-Walled Structures

A 0.5 m long thin-walled circular shaft of radius 2 cm is to be designed for an axial load of 7.4 kN and a torque of 148 Nm applied at its tip, as shown in the figure. The allowable stress under uniaxial tension is 100 MPa. Using maximum principal stress criterion, the minimum thickness, t, of the shaft so that it does not fail is _____ mm (rounded off to the nearest integer).

Strength of Materials & Elasticity

A $10\text{ kN}$ axial load is applied eccentrically on a rod of square cross-section ($1\text{ cm} \times 1\text{ cm}$) as shown in the figure. The strains measured by the two strain gages attached to the top and bottom surfaces at a distance of $0.5\text{ m}$ from the tip are $\epsilon_1 = 0.0016$ and $\epsilon_2 = 0.0004$, respectively. The eccentricity in loading, $e$, is ______ $\text{mm}$.

Thin-Walled Structures

For a thin-walled I section, the width of the two flanges as well as the web height are the same, i.e., 2b = 20 mm. Thickness is 0.6 mm. The second moment of area about a horizontal axis passing through the centroid is ______ mm4.

Thin-Walled Structures

A damper with damping coefficient, c, is attached to a mass of 5 kg and spring of stiffness 5 kN/m as shown in figure. The system undergoes under-damped oscillations. If the ratio of the 3rd amplitude to the 4th amplitude of oscillations is 1.5, the value of c is ____ Ns/m (rounded off to the nearest integer).

Atmospheric Flight Mechanics

A uniform rigid prismatic bar of total mass $m$ is suspended from a ceiling by two identical springs as shown in the figure. Let $\omega_1$ and $\omega_2$ be the natural frequencies of mode I and mode II respectively ($\omega_1 < \omega_2$). The value of $\omega_2 / \omega_1$ is ______ (rounded off to one decimal place).

Inlets, Combustors & Nozzles

An ideal ramjet is to operate with exhaust gases optimally expanded to ambient pressure at an altitude where temperature is 220 K. The exhaust speed at the nozzle exit is 1200 m/s at a temperature of 1100 K. Given: $\gamma$ = 1.4 at 220 K; R = 287 J/(kg-K) for air $\gamma$ = 1.33 at 1100 K; R = 287 J/(kg-K) for exhaust gases. The cruise speed of this ramjet is ________ m/s (rounded off to nearest integer).

Compressors and Turbines

A multistage axial compressor takes in air at $1\text{ atm}, 300\text{ K}$ and compresses it to a minimum of $5\text{ atm}$. The mean blade speed is $245\text{ m/s}$ and work coefficient, $\frac{\Delta C_\theta}{U}$ is $0.55$ for each stage. For air, use $C_p = 1005\text{ J/(kg}\cdot\text{K)}, R = 287\text{ J/(kg}\cdot\text{K)}$ and $\gamma = 1.4$. If the compression is isentropic, the number of stages required is ______ (rounded off to the next higher integer).