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# GATE 2011 : Aerospace Engineering

11 pages, 65 questions, 29 questions with responses, 31 total responses
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2011 AEROSPACE ENGINEERING AE Q. 1 Q. 25 carry one mark each. Q.1 Consider x, y, z to be right-handed Cartesian coordinates. A vector function is defined in this coordinate system as v 3 xi 3 xyj yz 2k , where i, j and k are the unit vectors along x, y and z axes, respectively. The curl of v is given by (A) z 2i 3 yk Q.2 (B) z 2 j 3 yk (B) f ( x) log x (B) (0, 0, 0) (C) (3, 2, 1) 0.73 6.73 1 5 (B) (C) 2 4 (D) Two simultaneous equations given by y x and y x have (A) a unique solution (C) no solution (B) infinitely many solutions (D) a finite number of multiple solutions In three-dimensional linear elastic solids, the number of non-trivial stress-strain relations, straindisplacement equations and equations of equilibrium are, respectively, (A) 3, 3 and 3 Q.7 (D) (1, 1, 3) x x 2 5 Q.6 (D) f ( x) const. 2 Consider the function f ( x1 , x2 ) x12 2 x2 e 1 2 . The vector pointing in the direction of maximum increase of the function at the point (1, -1) is (A) Q.5 (C) f ( x) e x 2 2 The function f ( x1 , x2 , x3 ) x12 x2 x3 2 x1 4 x2 6 x3 14 has its minimum value at (A) (1, 2, 3) Q.4 (D) z 2 i 3 yk Which of the following functions is periodic? (A) f ( x) x 2 Q.3 (C) z 2i 3 yj (B) 6, 3 and 3 (C) 6, 6 and 3 (D) 6, 3 and 6 An Euler-Bernoulli beam in bending is assumed to satisfy (A) both plane stress as well as plane strain conditions (B) plane strain condition but not plane stress condition (C) plane stress condition but not plane strain condition (D) neither plane strain condition nor plane stress condition Q.8 A statically indeterminate frame structure has (A) same number of joint degrees of freedom as the number of equilibrium equations (B) number of joint degrees of freedom greater than the number of equilibrium equations (C) number of joint degrees of freedom less than the number of equilibrium equations (D) unknown number of joint degrees of freedom, which cannot be solved using laws of mechanics Q.9 Consider a single degree of freedom spring-mass-damper system with mass, damping and stiffness of m , c and k , respectively. The logarithmic decrement of this system can be calculated using (A) AE 2 c 4mk c 2 (B) c 4mk c 2 (C) 2 c mk c 2 (D) 2 c mk 4c 2

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2011 Q.10 AEROSPACE ENGINEERING AE Consider a single degree of freedom spring-mass system of spring stiffness k1 and mass m which has a natural frequency of 10 rad/s. Consider another single degree of freedom spring-mass system of spring stiffness k 2 and mass m which has a natural frequency of 20 rad/s. The spring stiffness k 2 is equal to (B) 2k1 (A) k1 Q.11 (C) (D) 4k1 k1 4 Consider a simply supported two-dimensional beam If the beam is converted into a fixed-fixed beam as then the degree of static indeterminacy will (A) increase by 3 Q.12 (B) increase by 2 (C) decrease by 1 (D) decrease by 3 An impulsive launch of a rocket minimizes the loss of burn-out velocity due to (A) aerodynamic drag force only (B) gravitational force only (C) both aerodynamic drag and gravitational forces (D) reaction jet control force Q.13 Multi-staging in rockets improves the burn-out performance by increasing mainly stage-wise (A) payload mass ratios (C) propellant masses Q.14 In an un-powered glide of an aircraft having weight W, lift L and drag D, the equilibrium glide angle is defined as L D (A) tan 1 Q.15 (B) structural mass efficiencies (D) control system masses D L (B) tan 1 (B) zero (D) equal to the thrust If an aircraft is performing a positive yawing manoeuvre, the side slip angle (A) is always zero Q.17 (B) is never zero (D) could be any value (B) aft of the wing aerodynamic centre (D) aft of neutral point It is seen that the drag polar of a certain aerofoil is symmetric about the Cd axis. This drag polar could refer to (A) NACA 0012 AE (C) is always negative For an airplane to be statically stable, its centre of gravity must always be (A) ahead of wing aerodynamic centre (C) ahead of neutral point Q.18 W L (D) tan 1 Lift on an aircraft climbing vertically up is (A) equal to its weight (C) equal to the drag Q.16 L W (C) tan 1 (B) NACA 4415 (C) NACA 23012 (D) None of the above

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2011 Q.19 AEROSPACE ENGINEERING AE The aerodynamic centre of a supersonic aerofoil, with chord c, is located at (A) the leading edge Q.20 (B) profile drag R2 r ln 2 r 2 R R2 r (C) V r sin 1 2 2 ln R r (C) wave drag (D) induced drag (B) V r cos 1 R2 r ln 2 r 2 R R2 r (D) V r sin 1 2 2 ln R r A main objective of by-pass in a turbo-fan engine is to increase (A) mass flow rate through engine inlet (C) mass flow rate through exhaust nozzle Q.23 (D) 0.75c Consider a potential flow with free stream velocity V , over a spinning circular cylinder of radius R and circulation . The stream function, , where = 0 on the cylinder surface, in cylindrical coordinates (r, ) is given by (A) V r cos 1 Q.22 (C) 0.5c Winglets are used on wings to minimize (A) skin friction drag Q.21 (B) 0.25c (B) turbine inlet temperature (D) compressor pressure ratio The pressure ratio in any one stage of a jet engine compressor is limited by (A) entry stagnation temperature in that stage (B) entry Mach number in that stage (C) pressure gradient induced separation in that stage (D) mass flow rate in that stage Q.24 Thermodynamic cycle on which the jet engine operates can be (A) open Rankine cycle only (C) open Brayton cycle only Q.25 (B) either open or closed Rankine cycle (D) either open or closed Brayton cycle Propulsion efficiency of a jet engine is (A) directly proportional to both the thrust power and the air mass flow rate (B) inversely proportional to both the thrust power and the air mass flow rate (C) directly proportional to the thrust power and inversely proportional to the air mass flow rate (D) inversely proportional to the thrust power and directly proportional to the air mass flow rate Q. 26 to Q. 55 carry two marks each. Questions Q.26 to Q. 30 are numerica l answer type. The answ er to each of these questio ns is either a positive whole number, or a positive real number with maximum of 2 decimal places. Q.26 Consider a cantilever beam having length L=1 m, square cross-section (width = depth = 0.01 m) and Young s modulus 50 GPa. The beam is subjected to a transverse load P = 1 N at the mid-span (L/2) at the center of the cross-section. Under the small deformation theory, the transverse deflection of the beam (in mm) at its free-end is P L/2 L AE

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2011 AEROSPACE ENGINEERING AE Q.27 Consider a beam in bending with a solid circular cross-section of 1 mm2, which is subjected to a transverse shear force of 1 N. The shear stress at the center of the cross-section (in N/mm2) is Q.28 A simply supported slender column of square cross section (width=depth=d) has to be designed such that it buckles at the same instant as it yields. Length of the column is given to be 1.57 m and it is made of a material whose Young s modulus is 200 GPa and yield stress is 240 MPa. The width, d, of the column (in cm) should be Q.29 A turbojet powered aircraft is flying at Mach number 0.8 at an altitude of 10 km. The inlet and exit areas of the engine are 0.7 m2 and 0.4 m2 respectively. The exhaust gases have velocity of 500 m/s and pressure of 60 kPa. The free stream pressure, density and speed of sound are 26.5 kPa, 0.413 kg/m3 and 299.5 m/s respectively. The thrust of the engine (in kN) is Q.30 A low speed wind tunnel has a contraction ratio of 14:1 and the cross-sectional area of the test section is 1 m2. The static pressure difference between the settling chamber and the test section is 40 cm of water column. Assume g 9.81 m/s 2 , air 1.2 kg/m 3 and water 1000 kg/m 3 . The speed of air in the test section (in m/s) is Questions Q.31 to Q.55 are multiple choice type. Q.31 Consider the function f ( x) x sin( x) . The Newton-Raphson iteration formula to find the root of the function starting from an initial guess x ( 0 ) at iteration k is sin x ( k ) x ( k ) cos x k 1 cos x ( k ) sin x ( k ) x ( k ) cos x k 1 cos x ( k ) sin x ( k ) x ( k ) cos x k 1 cos x ( k ) sin x ( k ) x ( k ) cos x k 1 cos x ( k ) (A) x ( k 1) (C) x ( k 1) Q.32 (B) x ( k 1) (D) x ( k 1) 2 a where a and b are real numbers. The two eigenvalues of this matrix b 2 1 and 2 are real and distinct 1 2 when Consider the matrix (A) a 0 and b 0 Q.33 The solution of 1 9 (B) a 0 and b 0 (C) a 0 and b 0 dy y 3et t 2 with initial condition y (0) 1 is given by dt (A) et (t 3) 2 (B) 4et (D) (C) AE (t 2) 2 (D) a 0 and b 0 9 t (t 2 2t 2) 5 2e 1 5 2et (t 2 2t 2)

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2011 Q.34 AEROSPACE ENGINEERING AE A jet engine is operating at a Mach number of 0.8 at an altitude of 10 km. The efficiency of the air intake is 0.8 and that of the compressor is 0.87. The total temperatures (in K) at the exits of the air intake and the compressor respectively are (Ambient pressure = 26.5 kPa; Ambient temperature = 223.3 K; Gas constant, = 1.4; prc 8 ) (A) 251.9 and 458.2 Q.35 (B) 234.9 and 486.8 (D) 234.9 and 458.2 A rocket engine is tested on a test bed under the ideal condition of fully expanded jet. The exhaust velocity is 2 km/s through a nozzle of area 2.5 m2. The mass flow rate is 200 kg/s. The specific impulse of the propellant and the thrust developed respectively are (assume g = 9.81 m/s2) (A) 175.87 s and 200 kN (C) 231.87 s and 200 kN Q.36 (C) 251.9 and 486.8 (B) 203.87 s and 400 kN (D) 280.87 s and 400 kN A body undergoes deformation under plane strain conditions when subjected to the following stresses (in MPa): xx 450, yy 450, xy 75, xz 0, yz 0 . What are the remaining components of stresses (in MPa) and strains? Assume the material to be isotropic and linear-elastic with Young s modulus E = 200 GPa and Poisson s ratio = 1/3. (A) zz 0, xx 0.00225, yy 0.00225, xy 0.002, xz 0, yz 0 (B) zz 300, xx 0.001, yy 0.001, xy 0.001, xz 0, yz 0 (C) zz 300, xx 0.00225, yy 0.00225, xy 0.001, xz 0, yz 0 (D) zz 0, xx 0.001, yy 0.001, xy 0.002, xz 0, yz 0 Q.37 Which of the following Airy s stress functions could satisfy the given boundary conditions, assuming constant values of xx = P, yy = Q and xy = R, along the boundary? y x x2 y2 Q Rxy 2 2 2 y x2 (C) P Q Rxy 2 2 (A) P Q.38 An aircraft is performing a coordinated turn manoeuvre at a bank angle of 30o and forward speed of 100 m/s. Assume g = 9.81 ms-2. The load factor and turn radius respectively are (A) (2/ 3) and 1.76 km (C) 2 and 0.18 km Q.39 y2 x2 Q Rxy 2 2 2 x y2 (D) P Q Rxy 2 2 (B) P (B) 3 and 17.6 km (D) (2/ 3) and 17.6 km An aircraft in a steady level flight at forward speed of 50 m/s suddenly rolls by 180o and becomes inverted. If no other changes are made to the configuration or controls of the aircraft, the nature of the subsequent flight path taken by the aircraft and its characteristic parameter(s) (assume g = 9.81 ms-2) are (A) straight line path with a speed of 50 m/s (B) upward circular path with a speed of 50 m/s and radius of 127.4 m (C) downward circular path with a speed of 50 m/s and radius of 127.4 m/s (D) downward circular path with a speed of 25 m/s and radius of 254.8 m/s AE

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2011 Q.40 AEROSPACE ENGINEERING AE An aircraft with a mass of 5000 kg takes off from sea level with a forward speed of 50 m/s and starts to climb with a climb angle of 15o. The rate of climb and excess thrust available at the start of the climb respectively (assume g = 9.81 ms-2) are (A) 13.40 m/s and 13146.0 N (C) 13.40 m/s and 12694.1 N Q.41 A glider having a mass of 500 kg is taken to an altitude of 1000 m with a jeep moving on ground at 54 kmph. Upon reaching the required altitude in 50 s, the glider is released and starts its descent. Under the assumption of equilibrium glide, the range and endurance of the glider for a constant liftto-drag ratio of 15 are (A) 15.0 km and 1002.2 s respectively (C) 1.0 km and 601.3 s respectively Q.42 (B) 0.18 and 1900 km (D) 0.22 and 1900 km An aircraft in level flight encounters a vertical gust, which excites the phugoid mode. The phugoid motion completes 10 cycles in 50 s and its amplitude reduces to half of its maximum value in 25 s. The eigenvalues of the phugoid mode are (A) 0.05 0.02i Q.44 (B) 15.0 km and 601.3 s respectively (D) 1.0 km and 50 s respectively An elliptic orbit has its perigee at 400 km above the Earth s surface and apogee at 3400 km above the Earth s surface. For this orbit, the eccentricity and semi-major axis respectively are (assume radius of Earth = 6400 km) (A) 0.18 and 8300 km (C) 0.22 and 8300 km Q.43 (B) 12.94 m/s and 12694.1 N (D) 12.94 m/s and 13146.0 N (B) 0.5 0.2i (C) 0.028 1.26i (D) 0.028 1.26i Consider the inviscid, adiabatic flow of air at free stream conditions, M1 = 2, p1 = 1 atm and T1 = 288 K around a sharp expansion corner ( = 20 ) as shown below. The Prandtl-Meyer function, , is given as a function of Mach number, M, as (M) 1 tan 1 1 (M 2 1) tan 1 M 2 1 1 1 Assume air to be calorically perfect with = 1.4. The Mach number, M2, downstream of the expansion corner is approximately M1, p1, T1 (A) 2.00 Q.45 (C) 2.83 (D) 3.14 Consider a steady two dimensional zero-pressure gradient laminar flow of air over a flat plate as shown below. The free stream conditions are U = 100 ms-1, = 1.2 kg m-3, p = 1 atm and 1.8 10 5 kg m-1s-1. The ratio of displacement thickness to momentum thickness of the boundary layer at a distance of 2 m from the leading edge is Boundary layer U p 2m (A) 7.53 AE (B) 1.76 M2 (B) 2.59 (C) 2.91 (D) 0.39

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2011 Q.46 AEROSPACE ENGINEERING AE In the context of Prandtl s lifting line theory for a finite wing, which of the following combinations of statements is TRUE? P: The bound vortex is responsible for the lift force Q: The trailing vortices are responsible for the induced drag R: The bound vortex is responsible for the induced drag S: The trailing vortices are responsible for the lift force (A) P,Q Q.47 (B) Q,R (C) R,S (D) P,S Consider flow over a thin aerofoil at Mach number, M = 0.5 at an angle of attack, . Using the Prandtl-Glauert rule for compressibility correction, the formula for lift coefficient, cl, can be written as (A) 5.44 (B) 6.28 (C) 7.26 (D) 14.52 Common Data Questions Common Data for Questions 48 and 49: The partial differential equation (PDE) governing free vibrations of a uniform Euler-Bernoulli beam is 4w 2w m 2 0 , where EI is the flexural stiffness, m is the mass per unit length, x 4 t w( x, t ) is the bending displacement, x is the coordinate along the beam length, t is time and L is the beam given by EI length. x 0 x L x L Q.48 To solve the PDE, the number of boundary conditions (BC) and initial conditions (IC) needed are (A) 4 BC, 3 IC (B) 2 BC, 2 IC (C) 2 BC, 4 IC (D) 4 BC, 2 IC Q.49 For the cantilever beam shown in the figure, which of the following CANNOT be a possible boundary condition? (A) w(0, t ) 0 (B) 2w ( L, t ) 0 x 2 (C) 2w (0, t ) 0 x 2 (D) 3w ( L, t ) 0 x 3 Common Data for Questions 50 and 51: Consider an inviscid, adiabatic flow of air at free stream Mach Number, M = 2, across a compression corner ( = 20 ) as shown. The free stream total enthalpy is h0 = 810 kJ kg-1. Assume that air is calorically perfect with = 1.4, R = 287 J kg-1 K-1. M P Q.50 (B) > 20 and < 30 (C) = 30 (D) > 30 (C) 1612.74 K (D) 2257.84 K The total temperature at point P is (A) 806.37 K AE 1m The shock angle is (A) = 20 Q.51 (B) 1128.92 K

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2011 AEROSPACE ENGINEERING AE General Aptitude (GA) Questions Q. 56 Q. 60 carry one mark each. Q.56 Choose the word from the options given below that is most nearly opposite in meaning to the given word: Deference (A) aversion (B) resignation (C) suspicion (D) contempt Q.57 Choose the most appropriate word(s) from the options given below to complete the following sentence. We lost confidence in him because he never __________ the grandiose promises he had made. (A) delivered (B) delivered on (C) forgot (D) reneged on Q.58 Choose the word or phrase that best completes the sentence below. ______________ in the frozen wastes of Arctic takes special equipment. (A) To survive (B) Surviving (C) Survival (D) That survival Q.59 In how many ways 3 scholarships can be awarded to 4 applicants, when each applicant can receive any number of scholarships? (A) 4 Q.60 (B) 12 (C) 64 (D) 81 Choose the most appropriate word from the options given below to complete the following sentence. The _________ of eviden ce was on the side of the plaintiff since all but one witness testified that his story was correct. (A) paucity (B) propensity (C) preponderance (D) accuracy Q. 61 to Q. 65 carry two marks each. Q.61 If (2y+1)/(y+2) < 1, then which of the following alternatives gives the CORRECT range of y? (A) - 2 < y < 2 Q.62 (C) - 3 < y < 1 (D) - 4 < y < 1 A student attempted to solve a quadratic equation in x twice. However, in the first attempt, he incorrectly wrote the constant term and ended up with the roots as (4, 3). In the second attempt, he incorrectly wrote down the coefficient of x and got the roots as (3, 2). Based on the above information, the roots of the correct quadratic equation are (A) (-3, 4) AE (B) - 2 < y < 1 (B) (3, -4) (C) (6, 1) (D) (4, 2)

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2011 Q.63 AEROSPACE ENGINEERING AE L, M and N are waiting in a queue meant for children to enter the zoo. There are 5 children between L and M, and 8 children between M and N. If there are 3 children ahead of N and 21 children behind L, then what is the minimum number of children in the queue? (A) 28 Q.64 (B) 27 (C) 41 (D) 40 Four archers P, Q, R and S try to hit a bull s eye during a tournament consisting of seven rounds. As illustrated in the figure below, a player receives 10 points for hitting the bulls eye, 5 points for hitting within the inner circle and 1 point for hitting within the outer circle. Outer circle 1 point Inner circle 5 points Bull s eye 10 points The final scores received by the players during the tournament are listed in the table below. Round 1 2 3 4 5 6 7 P 1 5 1 10 1 10 5 Q 5 10 1 10 5 5 10 R 1 10 1 1 5 1 1 S 10 1 5 1 10 1 1 The most accurate and the most consistent players during the tournament are respectively (A) P and S Q.65 (B) Q and R (C) Q and Q (D) R and Q Nimbus clouds are dark and ragged, stratus clouds appear dull in colour a nd cover the entire sky. Cirrus clouds are thin and delicate, whereas cumulus clouds look like cotton balls. It can be inferred from the passage that (A) A cumulus cloud on the ground is called fog (B) It is easy to predict the weather by studying clouds (C) Clouds are generally of very different shapes, sizes and mass (D) There are four basic cloud types: stratus, nimbus, cumulus and cirrus END OF THE QUESTION PAPER AE

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