GATE AE 2014 Question Paper & Solutions

Quantitative Aptitude

For a real symmetric matrix [A], which of the following statements is true:

Quantitative Aptitude

The series ( )∑ ∞ = -= 1 23 2 m m m xms converges for all x with R x≤ -2 given by

Quantitative Aptitude

The function given by { 0),/ 1 sin( 0, 0 ≠ == x x xf(x) is

Quantitative Aptitude

Given the boundary-value problem 1 0 , 0< < = +     xkydx dyxdx d , with 0 ) 1 ( ) 0 (= =y y . Then the solutions of the boundary-value problem for 1=k (given by 1y ) and 5=k (given by 5y ) satisfy:

Quantitative Aptitude

The value of ∫= 1 0 41000 dx x I, obtained by using Simpson's rule with 2 equally spaced intervals is,

Quantitative Aptitude

For a NACA 5-digit airfoil of chord c, the designed lift coefficient and location of maximum camber along the chord from the leading edge are denoted by CL and Xm respectively. For NACA12018 airfoil, which combination of CL and Xm given below are correct?

Quantitative Aptitude

For inviscid, supersonic flow over a diamond shaped airfoil, shown in the figure, which statement is correct among the following?

Quantitative Aptitude

Consider supersonic flow near a corner (at an angle $\theta$from the horizontal) with an attached oblique shock (at an angle $\beta$ with horizontal) as shown in figure. If Mach number M decreases gradually from a high supersonic value, which of the following statements is correct?

Analytical & Spatial Aptitude

The streamlines of a potential line vortex is concentric circles with respect to the vortex center as shown in figure. Velocity along these streamlines, outside the core of the vortex can be written as, vv$\theta \theta$ = $\Gamma$ 2$\pi \pi \pi \pi$, where strength of the vortex is $\Gamma$ 2$\pi \pi$ and r is radial direction. The value of circulation along the curve shown in the figure is:

Quantitative Aptitude

The monthly rainfall chart based on 50 years of rainfall in Agra is shown in the following figure. Which of the following are true? (k percentile is the value such that k percent of the data fall below that value) (i) On average, it rains more in July than in December (ii) Every year, the amount of rainfall in August is more than that in January (iii) July rainfall can be estimated with better confidence than February rainfall (iv) In August, there is at least 500 mm of rainfall

Aerospace Materials & Fatigue

It is desired to measure the Young's modulus and the Poisson's ratio of a given homogeneous, isotropic material. A bar of length 20cm and square cross-section (10mm× 10mm) of this material is subjected to a tensile load of 40kN. Under this load, length increases to 20.1cm while the cross- section reduces to 9.98mm× 9.98mm. Young's modulus and Poisson's ratio of the material are:

Basic Fluid Mechanics

In general, for any given solid subjected to arbitrary loading, which of the following statements is always true:

Strength of Materials & Elasticity

Which one of the following objects with inclined face at 45oo , subjected to the given stresses, are in static equilibrium:

Differential Equations

A damped single degree of freedom system whose undamped natural frequency,𝝎𝝎𝒏𝒏=10Hz, is subjected to sinusoidal external force. Power is half of the maximumfor the two frequencies of 60.9469 rad/s and 64.7168 rad/s. The damping factor associated with the vibrating system (in %) is _______.

Structural Dynamics & Aeroelasticity

The boundary conditions for a rod with circular cross-section, under torsional vibration,are changed from fixed-free to fixed-fixed. The fundamental natural frequency of the fixed-fixed rod is k times that of fixed-free rod. The value of k is

Compressible Flow & Shocks

Match the appropriate engine (in right column) with the corresponding aircraft (in left column) for most efficient performance of the engine. a. Low speed transport aircraft i. Ramjet b. High subsonic civilian aircraft ii. Turboprop c. Supersonic fighter aircraft iii. Turbojet d. Hypersonic aircraft iv. Turbofan

Compressors and Turbines

For a given fuel flow rate and thermal efficiency, the take-off thrust for a gas turbine engine burning aviation turbine fuel (considering fuel-air ratiof<<1) is

Compressors and Turbines

For a fifty percent reaction axial compressor stage, following statements are given: I. Velocity triangles at the entry and exit of the rotor are symmetrical II. The whirl or swirl component of absolute velocity at the entry of rotor and entry of stator are same. Which of the following options are correct?

Rocket Propulsion

A small rocket having a specific impulse of 200s produces a total thrust of 98kN, out of which 10kN is the pressure thrust. Considering the acceleration due to gravity to be 9.8m/s^2, the propellant mass flow rate in kg/s is

Compressors and Turbines

The thrust produced by a turbojet engine

Atmospheric Flight Mechanics

The moment coefficient measured about the centre of gravity and about aerodynamic centre of a given wing-body combination are 0.0065 and -0.0235 respectively. The aerodynamic centre lies 0.06 chord lengths ahead of the centre of gravity. The lift coefficient for this wing-body is _____.

Atmospheric Flight Mechanics

The vertical ground load factor on a stationary aircraft parked in its hangar is:

Atmospheric Flight Mechanics

Under what conditionshould a glider be operated to ensure minimum sink rate?

Dynamic Stability

In most airplanes, the Dutch roll mode can be excited by applying

Space Dynamics & Orbital Mechanics

Considering R as the radius of the moon, the ratio of the velocities of two spacecraft orbiting moon in circular orbit at altitudes R and 2R above the surface of the moon is ________. Q. 26 - Q. 55 carry two marks each.

Compressors and Turbines

If [ ]     - -= 4 3 3 3A . Then ( )] [ 3 ] [ 7 ] [det 2 I A A- + - is

Compressors and Turbines

For the periodic function given by { 0, 2 0 , 2) ( < < - - < <= x xx f $\pi \pi$ with ) ( ) 2 (x fx f = +$\pi$ , using Fourier series, the sum + - + - =7 1 5 1 3 11s converges to

Inlets, Combustors & Nozzles

Let $\Gamma$ be the boundary of the closed circular region A given by 12 2≤ +y x . Then ( )∫ $\Gamma$ - = dsxyx I 23 9 3 (where ds means integration along the bounding curve) is

Basic Fluid Mechanics

Solution to the boundary-value problem 3 0 , 59 2 2 < < = + -x x udx u d with 0 , 0 ) 0 ( 3 = = =xdx duu is

Calculus

The Laplace transform L(u(t))=U(s), for the solution u(t) of the problem 0 , 1 22 2 > = + +t udt du dt u d with initial conditions . 5) 0 (, 0 ) 0 ( = =dt duu is given by:

Strength of Materials & Elasticity

For a steady, incompressible two-dimensional flow, represented in Cartesian co-ordinates (xx, yy), a student correctly writes the equation of pathline of any arbitrary particle as, ddxx dddd = aaxx and ddyy dddd = bbyy, wherea and bare constants having unit of (second)-1. If value of a is 5, the value of b is _____. AE 5/11 AE (GATE 2014) - AE

Basic Fluid Mechanics

Figures (a) - (d) below show four objects. Dimensions and surface conditions of the objects are shown in the respective figures. All four objects are placed independently in a steady, uniform flow of same velocity and the direction of flow is from left to right as shown in (a). The flow field can be considered as 2-D, viscous and incompressible. Following statements are made regarding the drag that these objects experience. (i) Drag of object (a) is more than the drag of object (d) (ii) Drag of object (a) is less than the drag of object (d) (iii) Drag of object (b) is more than the drag of object (c) (iv) Drag of object (c) is more than the drag of object (b) (v) Drag of object (a) is more than the drag of object (b) Choose the correct combination of statements from the options given above:

Compressible Flow & Shocks

A student needs to find velocity across a stationary normal shock. He measures density and pressure across the shock as shown in the figure below. 1bar = 105Pa. (No shock table is needed for the calculations). The value of u1 in m/s is ____.

Airfoil & Wing Theory

For inviscid, compressible flow past a thin airfoil, shown in the figure, free-stream Mach number and pressure are denoted by 𝑴𝑴∞ and 𝒑𝒑∞respectively. Ratio of pressure at point A and 𝒑𝒑∞is 0.8 and specific heat ratio is 1.4. If the Mach number at point A is 1.0 and rest of the flow field is subsonic, the value of 𝑴𝑴∞is

Wind Tunnel Testing & Instrumentation

A student can measure free-stream velocity of a low-speed wind tunnel using a i. Pitot tube alone aligned with the flow direction. ii. Pitot tube aligned with the flow direction with static pressure measurement at an appropriate position on the tunnel wall. iii. Pitot tube aligned with the flow direction along with barometer pressure reading of the outside ambient. iv. Pitot static tube alone aligned with the flow direction. Considering the above statements, which of the following options is correct?

Airfoil & Wing Theory

Induced velocity w at a point zz = zz1 along the lifting line can be calculated using the formula ww(zz1) = - 1 4$\pi \pi$ ∫ dd$\Gamma$ ddzz ss -ss 1 zz-zz1 ddzz. Given $\Gamma$2 $\Gamma$oo2 + zz2 ss2 = 1, wherezz, $\Gamma$oo and sare given in figure below. For theabove semi-elliptic distribution of circulation, $\Gamma$, the downwash velocity at any point zz1,for symmetric flight can be obtained as, ww(zz1) = $\Gamma$oo 4$\pi \pi$ss [$\pi \pi$ + zz1II], where II = zz1 ∫ ddzz �ss2-zz2�zz-zz1� ss -ss . Which of the following options is correct if the induced drag is DDii (given ∫ �1 - zz2 ss2 ss -ss ddzz = $\pi \pi$ss 2 )

Calculus

Two overflowing water reservoirs are connected with a100m long pipe of circular cross-section (of radius, R = 0.02m), such that height difference h remains constant as shown in the figure below. The centerline velocity in the pipe is 10m/s. The velocity profile inside the pipe over the entire length is uu = - RR2 4$\mu \mu$ dddd dd xx �1 - $\pi \pi$2 RR2�, where, dddd dd xx is a constant pressure gradient along the pipe length, x is measured from theleft end of the pipe along its central axis and r is radial location inside the pipe with respect to its axis.(Given data: Density and kinematic viscosity of water are 1000kg/m3 and 1x10-6m2/srespectively; acceleration due to gravity is 10m/s^2). If all other losses except the frictional losses at the pipe wall are neglected, the value of h in meter is ____. AE 7/11 AE (GATE 2014) - AE

Differential Equations

A 1.8m long steel beam of rectangular cross section (10mm× 6mm) is simply supported with a length of 1.2m between the supports and an overhang of 0.3m on either side. Young's modulus for the material of the beam is 200GPa. For a 50N load applied at the center of the beam, magnitude of the slopeof the beam at tip S is _______.

Thin-Walled Structures

There are 2 designs proposed for a shaft of length l, with a torque carrying capacity of T.Design I is a solid circular cross-section shaft of diameter30mm. Design II is a thin-walled circular shaft of average diameter 40mm. Thickness of the wall in Design II has to be determined such that maximum shear stress is the same in both the designs for the given torque T(so that same material can be used for manufacturing both the shafts). Ratio of mass of shaft using Design Ito the mass of shaft using Design II is

Strength of Materials & Elasticity

A structural member of rectangular cross-section 10mm× 6mm and length 1m is made of steel (Young's modulus is 200GPa and coefficient of thermal expansion is 12 × 10-6 / CCoo ). It is rigidly fixed at both the ends and then subjected to a gradual increase in temperature. Ignoring the three dimensional effects, the structural member will buckle if the temperature is increased by ∆TT CCoo which is

Thin-Walled Structures

A gas cylinder (closed thin-walled cylindrical pressure vessel) of diameter 30cm and wall thickness 1mm is subjected to a design maximum internal pressure of 5bar (0.5MPa). The material used for manufacturing this cylinder has a failure stress of 260MPa. Assuming von Mises failure criterion, the factor of safety (with respect to maximum allowable stress) for this cylinder is

Strength of Materials & Elasticity

A cantilevered beam is subjected to a parabolic distribution of shear traction at the right edge while the top and bottom surfaces are traction free. To solve this problem, following Airy's stress function is proposed: $\phi \phi$ = CC1xxyy + CC2xxyy3 + CC3xx2yy2 + CC4xx3yy. This is an admissible Airy's function that would satisfy the bi-harmonic equation as well as the boundary conditions if and only if

Structural Dynamics & Aeroelasticity

1kg mass is hanging from a spring of stiffness 500N/m attached to a massless,symmetric beam of length 0.6m, moment of inertia about the bending axisII = 8.33 × 10-10 mm4 and Young's modulus E=210GPa as shown in the figure. The fundamental natural frequency (in rad/s) of the system is

Basic Fluid Mechanics

A single degree of freedom system is vibrating with initial (first cycle) amplitude of 5cm. The viscous damping factor associated with the vibrating system is 2%. Vibration amplitude of the fifth cycle (in cm) is

Inlets, Combustors & Nozzles

A cruise missile with an ideal ramjet engine is flying at Mach 4.0 at an altitude where the ambient temperature is 100K. Considerratio of specific heats$\gamma$ = 1.4 and specific gas constant R = 287J/kgK. If the stagnation temperature in the combustion chamber is equal to 2310K, the speed of the exhaust gases (in m/s) is _____________.

Airfoil & Wing Theory

A gas turbine engine is operating under the following conditions: Stagnation temperature at turbine inlet 1350 K Stagnation pressure at the turbine inlet 10 bar Static temperature at turbine exit 800 K Velocity at turbine exit 200 m/s Total-to-total efficiency of turbine 0.96 $\gamma$ (ratio of specific heats) 1.33 CP(specific heat at constant pressure) 1.147 kJ/kgK The stagnation pressure (in bar) in the nozzle (considering isentropic nozzle) is equal to _______. AE 9/11 AE (GATE 2014) - AE

Inlets, Combustors & Nozzles

Air at a stagnation temperature of 300K (ratio of specific heats, $\gamma$ = 1.4 and specific gas constant R = 287 J/kgK) enters the impeller of a centrifugal compressor in axial direction. The stagnation pressure ratio between the diffuser outlet and impeller inlet is 4.0. The impeller blade radius is 0.3 m and it is rotating at 15000 rev/min. If the slip factor $\sigma$s (Ratio of tangential component of air velocity at the blade tip to the blade tip speed) is 0.88, the overall efficiency (total-to-total) of the compressor (in %) is ___________.

Rocket Propulsion

A stationary two stage rocket with initial mass of 16000kg, carrying a payload of 1000kg, is fired in a vertical trajectory from the surface of the earth. Both the stages of the rocket have same specific impulse, Isp, of 300s and same structural coefficient of 0.14. The acceleration due to gravity is 9.8m/s^2. Neglecting drag and gravity effects and considering both the stages with same payload ratio, the terminal velocity attained by the payload in m/s is __________________.

Inlets, Combustors & Nozzles

An aircraft is flying at Mach 3.0 at an altitude where the ambient pressure and temperature are 50 kPa and 200 K respectively. If the converging-diverging diffuser of the engine (considered isentropic with ratio of specific heats, $\gamma$ = 1.4 and specific gas constant R = 287 J/kgK) has a throat area of 0.05 m^2, the mass flow rate through the engine in kg/s is

Rocket Propulsion

A cryogenic rocket has a specific impulse of 455s and characteristic velocity of 2386 m/s. The value of thrust coefficient for this rocket is

Atmospheric Flight Mechanics

For a given airplane with a given wing loading executing a turn in the vertical plane, under what conditions will the turn radius be minimum and the turn rate be maximum?

Space Dynamics & Orbital Mechanics

Lift-off distance for a given aircraft of weight W is SLO. If the take-off weight is reduced by 10%, then the magnitude of percentage change in the lift-off distance (assume all other parameters to remain constant) is _______.

Rocket Propulsion

Which of the following design parameters influence the maximum rate-of-climb for a jet-propelled airplane? P. Wing loading Q. Maximum thrust-to-weight ratio R. Zero-lift drag coefficient S. Maximum lift-to-drag ratio

Static Stability & Control

Consider the following four statements regarding aircraft longitudinal stability: P. CM,cg at zero lift must be positive Q. $\partial$CM,cg/$\partial \alpha$a must be negative ($\alpha$a is absolute angle of attack) R. CM,cg at zero lift must be negative S. Slope of CL versus $\alpha$a must be negative Which of the following combination is the necessary criterion for stick fixed longitudinal balance and static stability?

Atmospheric Flight Mechanics

Data for a light, single-engine, propeller driven aircraft in steady level flight at sea-level is as follows: velocity V∞ = 40m/s, weight W = 13000N, lift coefficient CL = 0.65, drag coefficient CD = 0.025 + 0.04CL 2 and power available Pav = 100,000J/s. The rate of climb possible for this aircraft under the given conditions (in m/s) is

Atmospheric Flight Mechanics

An aerospace system operating under standard sea level conditions is governed by Compressors and Turbines (Centrifugal Compressors). Compute the required parameter in standard SI units (rounded off to the acceptable range).

Atmospheric Flight Mechanics

An aerospace system operating under standard sea level conditions is governed by Inlets, Combustors & Nozzles (Combustor Design & Flame Stabilization). Compute the required parameter in standard SI units (rounded off to the acceptable range).

Atmospheric Flight Mechanics

An aerospace system operating under standard sea level conditions is governed by Rocket Propulsion (Solid, Liquid & Hybrid Rocket Engines). Compute the required parameter in standard SI units (rounded off to the acceptable range).

Basic Fluid Mechanics

In an aerospace engineering analysis of Compressors and Turbines (Axial Flow Gas Turbines & Blade Cooling), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Basic Fluid Mechanics

In an aerospace engineering analysis of Strength of Materials & Elasticity (Euler Buckling of Columns & Energy Methods), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Basic Fluid Mechanics

In an aerospace engineering analysis of Thin-Walled Structures (Torsion of Multi-Cell Thin-Walled Tubes), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Atmospheric Flight Mechanics

An aerospace system operating under standard sea level conditions is governed by Thin-Walled Structures (Tension Field Beams & Wagner Beams). Compute the required parameter in standard SI units (rounded off to the acceptable range).

Basic Fluid Mechanics

In an aerospace engineering analysis of Structural Dynamics & Aeroelasticity (Free & Forced Vibrations of Continuous Beams), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Basic Fluid Mechanics

In an aerospace engineering analysis of Aerospace Materials & Fatigue (Aluminum, Titanium & Aerospace Alloys), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Basic Fluid Mechanics

In an aerospace engineering analysis of Aerospace Materials & Fatigue (Damage Tolerance & Non-Destructive Testing), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?