## Q 31-35

## Q 36-40

## Q. 26 β Q. 55 carry two marks each.

Q.26 The following system of equations

2π₯−π¦−π§=0,

−π₯+2π¦−π§=0,

−π₯−π¦+2π§=0

Options | (A) has no solution. | (B) has a unique solution. | (C) has three solutions. | (D) has an infinite number of solutions. |

Q.27 A supersonic flow in a constant area duct at Mach number M_{1} encounters a ramp of angle θ_{1} (see Figure 1). The resulting oblique shock with shock angle β_{1} is then reflected from the top wall. For the reflected shock, the turn angle is θ_{2} and the shock angle is β_{2}.

Use the weak shock solution from the θ-β-M plot shown in Figure 2 to choose the correct option from the following.

Options | (A) β_{1} > β_{2} | (B) β_{1} < β_{2} | (C) θ_{1} > θ_{2} | (D) θ_{1} < θ_{2} |

Q.28 Which of the following statements about adverse yaw of an airplane is/are correct?

P. It is caused by flow separation resulting from large rudder deflection.

Q. It is caused by dissimilar drag forces acting on the two halves of the wing resulting from aileron deflections of same magnitude.

R. It can be eliminated by ensuring that the upward deflection of one aileron is greater than the downward deflection of the opposite aileron.

Options | (A) P only | (B) Q only | (C) P and R | (D) Q and R |

Q.29 In a turbojet engine, the compressor outlet temperature increases with decreasing efficiency of the compressor. If the turbine inlet temperature remains constant, with decreasing efficiency of the compressor, the thrust specific fuel consumption of the engine

(A) decreases, as the heat input is lower.

(B) remains unchanged.

(C) increases, as the compressor needs more work input from the turbine.

(D) decreases, as the thrust produced is higher.

Q.30 For a 1 m long simply supported beam with a concentrated vertical load of 200 N and a concentrated bending moment of 100 Nm at the center as shown in the figure, the correct bending moment diagram is:

## Q 41-45

Q.31 For real x, the number of points of intersection between the curves π¦=π₯ and π¦=cosπ₯ is _________.

Q.33 The curve π¦=π(π₯) is such that its slope is equal to π¦² for all real x. If the curve passes through (1, -1), the value of y at π₯= −2 is _______ (*round off to 1 decimal place*).

Q.34 T he inviscid, incompressible flow field resulting from a uniform flow past a circular cylinder of radius *R* centered at the origin is given by:

where π’_{π} and π’_{π} are the radial and azimuthal velocity components in polar coordinates, (r, ο±), as shown in the figure. U is the free stream speed. Ignore the effects of gravity. The azimuthal location (in the first quadrant) on the cylinder at which the pressure coefficient is zero is _______ degrees (*round off to the nearest integer*).

Q.35 A cylindrical container of radius *R* = *50* cm is filled with water up to a height h_{o}. Upon rotating the cylinder about its central axis at a constant angular speed, the free surface takes a parabolic shape (see figure), and is displaced upwards by *h _{1}* = 10 cm at

*r*=

*R*. The magnitude of the downward displacement

*h*of the free surface at

_{2}*r*= is ________ cm (

*round off to the nearest integer*).

## Q 46-50

Q.36 A two-dimensional, incompressible fluid flow is described by the stream function Ψ = π₯π¦³ m²/s on the Cartesian *x-y* plane. If the density and dynamic viscosity of the fluid are 1 kg/m³ and 0.1 kg/m-s, respectively, the magnitude of the pressure gradient in the x direction at *x*=1 *m* and *y*=1 *m* is _______ N/m³ (*round off to 1 decimal place*).

Q.37 The static pressure ratio across a stationary normal shock is given by

where *M _{1}* is the upstream Mach number. For a stationary normal shock in air (πΎ=1.4,π
=287 J/kg-K) with upstream flow conditions given by: speed 800 m/s, static temperature 300 K and static pressure 1 atm., the static pressure downstream of the shock is __________ atm. (

*round off to 2 decimal places*).

Q.38 For a symmetric airfoil at an angle of attack of 10°, assuming thin airfoil theory, the magnitude of the pitching moment coefficient about the leading edge is__________ (*round off to 2 decimal places*).

Q.39 The span-wise distribution of circulation over a finite wing of span *b* = 10 m is

If Γ_{}=20 π²/π and the free stream density and speed are 1.2 kg/m³ and 100 m/s, respectively, the total lift is __________ kN (*round off to 2 decimal places*).

Q.40 The airplane shown in figure starts executing a symmetric pull-up maneuver from steady level attitude with a constant nose-up pitch acceleration of 20 deg/s². The vertical load factor measured at this instant at the centre of gravity (CG) is 2. Given that the acceleration due to gravity is 9.81 m/s², the vertical load factor measured at point P on the nose of the airplane, which is 2 m ahead of the CG, is _____ (*round off to 2 decimal places*).

## Q 51-55

Q.41 Consider an airplane with a weight of 8000 N, wing area of 16 m², wing zero-lift drag coefficient of 0.02, Oswald’s efficiency factor of 0.8, and wing aspect ratio of 6, in steady level flight with wing lift coefficient of 0.375. Considering the same flight speed and ambient density, the ratio of the induced drag coefficient during steady level flight to that during a 30° climb is _______ (*round off to 2 decimal places*).

Q.42 The product of earth’s mass (*M*) and the universal gravitational constant (*G*) is GM = 3.986×10^{14} m³/s². The radius of earth is 6371 km. The minimum increment in the velocity to be imparted to a spacecraft flying in a circular orbit around the earth at an altitude of 4000 km to make it exit earth’s gravitational field is _______ km/s (*round off to 2 decimal places*).

Q.43 A propeller driven airplane has a gross take-off weight of 4905 N with a wing area of 6.84 m². Assume that the wings are operating at the maximum of 13, the propeller efficiency is 0.9 and the specific fuel consumption of the engine is 0.76 kg/kW-hr. Given that the density of air at sea level is 1.225 kg/m³ and the acceleration due to gravity is 9.81 m/s², the weight of the fuel required for an endurance of 18 hours at sea level is _______N (*round off to the nearest integer*).

Q.44 The design of an airplane is modified to increase the vertical tail area by 20% and decrease the moment arm from the aerodynamic centre of the vertical tail to the airplane centre of gravity by 20%. Assuming all other factors remain unchanged, the ratio of the modified to the original directional static stability ( due to tail fin) is _______ (*round off to 2 decimal places*).

Q.45 For a rocket engine, the velocity ratio r is *V _{a}/V_{e}*, where

*V*is the vehicle velocity and

_{a}*V*is the exit velocity of the exhaust gases. Assume the flow to be optimally expanded through the nozzle. For

_{e}*= 2, if F is the thrust produced and πΜ is the mass flow rate of exhaust gases, then, πΉ/(πΜ π*

**r**_{e}) is ______.

## Q 56-60

Q.46 The specific impulse of a rocket engine is 3000 Ns/kg. The mass of the rocket at burnout is 1000 kg. The propellant consumed in the process is 720 kg. Assume all factors contributing to velocity loss to be negligible. The change in vehicle velocity Δu is _____ km/s (*round off to 2 decimal places*).

Q.47 The combustion products of a gas turbine engine can be assumed to be a calorically perfect gas with γ = 1.2. The pressure ratio across the turbine stage is 0.14. The measured turbine inlet and exit stagnation temperatures are 1200 K and 900 K, respectively. The total-tototal turbine efficiency is ________ % (*round off to the nearest integer*).

Q.48 The figure shows the velocity triangles for an axial compressor stage. The specific work input to the compressor stage is ____________ kJ/kg (*round off to 2 decimal places*).

Q.49 As shown in the figure, a rigid slab CD of weight *W* (distributed uniformly along its length) is hung from a ceiling using three cables of identical length and cross-sectional area. The central cable is made of steel (Young’s modulus = 3E) and the other two cables are made of aluminium (Young’s modulus = E). The percentage of the total weight taken by the central cable is ______ % (*round off to the nearest integer*).

Q.50 All the bars in the given truss are elastic with Young’s modulus 200 GPa, and have identical cross-sections with moment of inertia 0.1 cm^{4}. The lowest value of the load *P* at which the truss fails due to buckling is _______ kN (*round off to the nearest integer*).

## Q 61-65

Q.51 A solid circular shaft is designed to transmit a torque *T* with a factor of safety of 2. It is proposed to replace the solid shaft by a hollow shaft of the same material and identical outer radius. If the inner radius is half the outer radius, the factor of safety for the hollow shaft is _____ (*round off to 1 decimal place*).

Q.52 In the structure shown in the figure, bars AB and BC are made of identical material and have circular cross-sections of 10 mm radii. The yield stress of the material under uniaxial tension is 280 MPa. Using the von Mises yield criterion, the maximum load along the z-direction (perpendicular to the plane of paper) that can be applied at C, such that AB does not yield is _____ N (*round off to the nearest integer*).

Q.53 A thin-walled tube, with the cross-section shown in the figure, is subjected to a torque of *T* = 1 kN-m. The walls have uniform thickness *t* = 1 mm and shear modulus *G* = 26 GPa. Assume that the curved portion is semi-circular. The shear stress in the wall is _____ MPa (*round off to 1 decimal place*).

Q.54 For a damped spring-mass system, mass * m*= 10 kg, stiffness

*= 10³ N/m, and damping coefficient*

**k***= 20 kg/s. The ratio of the amplitude of oscillation of the first cycle to that of the fifth cycle is ______ (*

**c***round off to 1 decimal place*).

Q.55 For the system of springs and masses shown below, * k*= 1250 N/m and

*= 10 kg. The highest natural frequency,*

**m***of the system is _______ radians/s (*

**ω***round off to the nearest integer*).