# GATE 2013 ME – Complete Solutions Q1. The partial differential equation $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac{\partial^2 u}{\partial x^2}$$ is a

(A) linear equation of order 2                   (B) non-linear equation of order

(C) linear equation of order 1                    (D) non-linear equation of order 2

Solution: (D)

Q2. The eigenvalues of a symmetric matrix are all

(A) complex with non-zero positive imaginary part.

(B) complex with non-zero negative imaginary part.

(C) real.

(D) pure imaginary.

Solution: (C)

Q3. Match the CORRECT pairs. (A) P-2, Q-1, R-3                   (B) P-3, Q-2, R-1

(C) P-1, Q-2, R-3                   (D) P-3, Q-1, R-2

Solution: (D)

Q4. A rod of length L having uniform cross-sectional area A is subjected to a tensile force P as shown in the figure below. If the Young’s modulus of the material varies linearly from E1 to E2 along the length of the rod, the normal stress developed at the section-SS is (A) $$\frac{P}{A}$$                                     (B) $$\frac{P(E_{1}-E_{2})}{A(E_{1}+E_{2})}$$

(C) $$\frac{PE_{2}}{AE_{1}}$$               (D) $$\frac{PE_{1}}{AE_{2}}$$

Solution: (A)

Q5. Two threaded bolts A and B of same material and length are subjected to identical tensile load. If the elastic strain energy stored in bolt A is 4 times that of bolt B and the mean diameter of bolt A is 12 mm, the mean diameter of bolt B in mm is

(A) 16                (B) 24                   (C) 36                  (D) 48

Solution: (B)

Q6. A link OB is rotating with a constant angular velocity of 2 rad/s in counter clockwise direction and a block is sliding radially outward on it with an uniform velocity of 0.75 m/s with respect to the rod, as shown in the figure below. If OA = 1 m, the magnitude of the absolute acceleration of the block at location A in m/s2 is (A) 3                (B) 4                  (C) 5                  (D) 6

Solution: (C)

Q7. For steady, fully developed flow inside a straight pipe of diameter D, neglecting gravity effects, the pressure drop ∆pover a length L and the wall shear stress τw are related by

(A) (B) (C) (D) Solution: (A)

Q8. The pressure, dry bulb temperature and relative humidity of air in a room are 1 bar, 30°C and 70%, respectively. If the saturated steam pressure at 30°C is 4.25 kPa, the specific humidity of the room air in kg water vapour/kg dry air is

(A) 0.0083                 (B) 0.0101                  (C) 0.0191                 (D) 0.0232

Solution: (C)

Q9. Consider one-dimensional steady state heat conduction, without heat generation, in a plane wall; with boundary conditions as shown in the figure below. The conductivity of the wall is given by k = k0 + bT; where k0 and b are positive constants, and T is temperature. As x increases, the temperature gradient ( dT/ dx ) will

(A) remain constant                (B) be zero             (C) increase               (D) decrease

Solution: (D)

Q10. In a rolling process, the state of stress of the material undergoing deformation is

(A) pure compression         (B) pure shear        (C) compression and shear      (D) tension and shear

Solution: (C)

Q11. Match the CORRECT pairs. (A) P-4, Q-3, R-1, S-2                (B) P-4, Q-2, R-3, S-1

(C) P-2, Q-3, R-4, S-1                (D) P-2, Q-4, R-1, S-3

Solution: (A)

Q12. A metric thread of pitch 2 mm and thread angle 60° is inspected for its pitch diameter using 3-wire method. The diameter of the best size wire in mm is

(A) 0.866                 (B) 1.000              (C) 1.154                (D) 2.000

Solution: (C)

Q13. Customers arrive at a ticket counter at a rate of 50 per hr and tickets are issued in the order of their arrival. The average time taken for issuing a ticket is 1 min. Assuming that customer arrivals form a Poisson process and service times are exponentially distributed, the average waiting time in queue in min is

(A) 3               (B) 4               (C) 5                (D) 6

Solution: (C)

Q14. In simple exponential smoothing forecasting, to give higher weightage to recent demand information, the smoothing constant must be close to

(A) -1              (B) zero             (C) 0.5              (D) 1.0

Solution: (D)

Q15. A steel bar 200 mm in diameter is turned at a feed of 0.25 mm/rev with a depth of cut of 4 mm. The rotational speed of the workpiece is 160 rpm. The material removal rate in mm3/s is

(A) 160              (B) 167.6             (C) 1600              (D) 1675.5

Solution: (D)

Q16. A cube shaped casting solidifies in 5 min. The solidification time in min for a cube of the same material, which is 8 times heavier than the original casting, will be

(A) 10                (B) 20                (C) 24                 (D) 40

Solution: Marks to all

Q17. For a ductile material, toughness is a measure of

(A) resistance to scratching                                       (B) ability to absorb energy up to fracture

(C) ability to absorb energy till elastic limit           (D) resistance to indentation

Solution: (B)

Q18. In order to have maximum power from a Pelton turbine, the bucket speed must be

(A) equal to the jet speed.                         (B) equal to half of the jet speed.

(C) equal to twice the jet speed.              (D) independent of the jet speed.

Solution: (B)

Q19. Consider one-dimensional steady state heat conduction along x-axis (0 ≤ x ≤ L), through a plane wall with the boundary surfaces (x=0 and x=L) maintained at temperatures of 0°C and 100°C. Heat is generated uniformly throughout the wall. Choose the CORRECT statement.

(A) The direction of heat transfer will be from the surface at 100°C to the surface at 0°C.

(B) The maximum temperature inside the wall must be greater than 100°C.

(C) The temperature distribution is linear within the wall.

(D) The temperature distribution is symmetric about the mid-plane of the wall.

Solution: (B)

Q20. A cylinder contains 5 m3 of an ideal gas at a pressure of 1 bar. This gas is compressed in a reversible isothermal process till its pressure increases to 5 bar. The work in kJ required for this process is

(A) 804.7                (B) 953.2               (C) 981.7              (D) 1012.2

Solution: (A)

Q21. A long thin walled cylindrical shell, closed at both the ends, is subjected to an internal pressure. The ratio of the hoop stress (circumferential stress) to longitudinal stress developed in the shell is

(A) 0.5                 (B) 1.0                (C) 2.0                (D) 4.0

Solution: (C)

Q22. If two nodes are observed at a frequency of 1800 rpm during whirling of a simply supported long slender rotating shaft, the first critical speed of the shaft in rpm is

(A) 200                (B) 450                 (C) 600                (D) 900

Solution: (A)

Q23. A planar closed kinematic chain is formed with rigid links PQ = 2.0 m, QR = 3.0 m, RS = 2.5 m and SP = 2.7 m with all revolute joints. The link to be fixed to obtain a double rocker (rocker-rocker) mechanism is

(A) PQ             (B) QR              (C) RS             (D) SP

Solution: (C)

Q24. Let X be a normal random variable with mean 1 and variance 4. The probability P{X < 0} is

(A) 0.5                                                                  (B) greater than zero and less than 0.5

(C) greater than 0.5 and less than 1.0           (D) 1.0

Solution: (B)

Q25. Choose the CORRECT set of functions, which are linearly dependent.

(A) $$\sin x$$, $$\sin ^{2}x$$ and $$\cos ^{2}x$$         (B) $$\cos x$$, $$\sin x$$ and $$\tan x$$

(C) $$\cos 2x, \sin ^{2}x$$ and $$\cos ^{2}x$$                (D) $$\cos 2x,\sin x$$ and $$\cos x$$

Solution: (C)

Q26. The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors. where S in the sphere, x2 + y2 + z2 = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is

(A) π            (B) 2π            (C) 3π/4            (D) 4π

Solution: (A)

Q27. The function f (t) satisfies the differential equation and the auxiliary conditions, The Laplace transform of f(t) is given by

(A) $$\frac{2}{s+1}$$                        (B) $$\frac{4}{s+1}$$

(C) $$\frac{4}{s^{2}+1}$$               (D) $$\frac{2}{s^{4}+1}$$

Solution: (C)

Q28. Specific enthalpy and velocity of steam at inlet and exit of a steam turbine, running under steady state, are as given below:

Specific enthalpy (kJ/kg)        Velocity (m/s)

Inlet steam condition                 3250                                 180

Exit steam condition                  2360                                   5

The rate of heat loss from the turbine per kg of steam flow rate is 5 kW. Neglecting changes in potential energy of steam, the power developed in kW by the steam turbine per kg of steam flow rate, is

(A) 901.2                  (B) 911.2               (C) 17072.5                 (D) 17082.5

Solution: (A)

Q29. Water is coming out from a tap and falls vertically downwards. At the tap opening, the stream diameter is 20 mm with uniform velocity of 2 m/s. Acceleration due to gravity is 9.81 m/s2. Assuming steady, inviscid flow, constant atmospheric pressure everywhere and neglecting curvature and surface tension effects, the diameter in mm of the stream 0.5 m below the tap is approximately

(A) 10            (B) 15              (C) 20               (D) 25

Solution: (B)

Q30. A steel ball of diameter 60 mm is initially in thermal equilibrium at 1030°C in a furnace. It is suddenly removed from the furnace and cooled in ambient air at 30°C, with convective heat transfer coefficient h = 20 W/m2K. The thermo-physical properties of steel are: density ρ = 7800 kg/m3, conductivity k = 40 W/mK and specific heat c = 600 J/kgK. The time required in seconds to cool the steel ball in air from 1030°C to 430°C is

(A) 519                (B) 931              (C) 1195               (D) 2144

Solution: (D)

Q31. A flywheel connected to a punching machine has to supply energy of 400 Nm while running at a mean angular speed of 20 rad/s. If the total fluctuation of speed is not to exceed ±2%, the mass moment of inertia of the flywheel in kg-m2 is

(A) 25                (B) 50                  (C) 100                  (D) 125

Solution: (A)

Q32. A compound gear train with gears P, Q, R and S has number of teeth 20, 40, 15 and 20, respectively. Gears Q and R are mounted on the same shaft as shown in the figure below. The diameter of the gear Q is twice that of the gear R. If the module of the gear R is 2 mm, the center distance in mm between gears P and S is (A) 40              (B) 80                  (C) 120                  (D) 160

Solution: (B)

Q33. A pin jointed uniform rigid rod of weight W and length is supported horizontally by an external force F as shown in the figure below. The force F is suddenly removed. At the instant of force removal, the magnitude of vertical reaction developed at the support is (A) zero                (B) W/4               (C) W/2              (D) W

Solution: (B)

Q34. Two cutting tools are being compared for a machining operation. The tool life equations are:

Car bide Tool:      VT1.6 = 3000

HSS tool:             VT0.6 = 200

where V is the cutting speed in m/min and T is the tool life in min. The carbide tool will provide higher tool life if the cutting speed in m/min exceeds

(A) 15.0                 (B) 39.4                (C) 49.3                (D) 60.0

Solution: (B)

Q35. In a CAD package, mirror image of a 2D point P(5,10) is to be obtained about a line which passes through the origin and makes an angle of 45° counterclockwise with the X-axis. The coordinates of the transformed point will be

(A) (7.5, 5)                (B) (10, 5)                 (C) (7.5, −5)               (D) (10, −5)

Solution: (B)

Q36. A linear programming problem is shown below.

Maximize    3x + 7y

Subject to   3x + 7y ≤ 10

4x + 6y ≤ 8

x, y ≥ 0

It has

(A) an unbounded objective function.                (B) exactly one optimal solution.

(C) exactly two optimal solutions.                       (D) infinitely many optimal solutions.

Solution: (B)

Q37. Cylindrical pins of diameter are electroplated in a shop. Thickness of the plating is 30±2.0 micron. Neglecting gage tolerances, the size of the GO gage in mm to inspect the plated components is

(A) 25.042               (B) 25.052                   (C) 25.074              (D) 25.084

Solution: (D)

Q38. During the electrochemical machining (ECM) of iron (atomic weight = 56, valency = 2) at current of 1000 A with 90% current efficiency, the material removal rate was observed to be 0.26 gm/s. If Titanium (atomic weight = 48, valency = 3) is machined by the ECM process at the current of 2000 A with 90% current efficiency, the expected material removal rate in gm/s will be

(A) 0.11                  (B) 0.23                    (C) 0.30                  (D) 0.52

Solution: (C)

Q39. A single degree of freedom system having mass 1 kg and stiffness 10 kN/m initially at rest is subjected to an impulse force of magnitude 5 kN for 10−4 seconds. The amplitude in mm of the resulting free vibration is

(A) 0.5                   (B) 1.0                     (C) 5.0                   (D) 10.0

Solution: (C)

Q40. A bar is subjected to fluctuating tensile load from 20 kN to 100 kN. The material has yield strength of 240 MPa and endurance limit in reversed bending is 160 MPa. According to the Soderberg principle, the area of cross-section in mm2of the bar for a factor of safety of 2 is

(A) 400                      (B) 600                   (C) 750                   (D) 1000

Solution: (D)

Q41. A simply supported beam of length L is subjected to a varying distributed load sin (3p x/L) Nm−1, where the distance x is measured from the left support. The magnitude of the vertical reaction force in N at the left support is

(A) zero                   (B) L/3π                    (C) L/π                  (D) 2L/π

Solution: (B)

Q42. Two large diffuse gray parallel plates, separated by a small distance, have surface temperatures of 400 K and 300 K. If the emissivities of the surfaces are 0.8 and the Stefan-Boltzmann constant is 5.67 × 10−8W/m2K4, the net radiation heat exchange rate in kW/m2 between the two plates is

(A) 0.66                (B) 0.79                    (C) 0.99                    (D) 3.96

Solution: (A)

Q43. A hinged gate of length 5 m, inclined at 30° with the horizontal and with water mass on its left, is shown in the figure below. Density of water is 1000 kg/m3. The minimum mass of the gate in kg per unit width (perpendicular to the plane of paper), required to keep it closed is (A) 5000                    (B) 6600                   (C) 7546                  (D) 9623

Solution: (D)

Q44. The pressure, temperature and velocity of air flowing in a pipe are 5 bar, 500 K and 50 m/s, respectively. The specific heats of air at constant pressure and at constant volume are 1.005 kJ/kgK and 0.718 kJ/kgK, respectively. Neglect potential energy. If the pressure and temperature of the surroundings are 1 bar and 300 K, respectively, the available energy in kJ/kg of the air stream is

(A) 170                   (B) 187                  (C) 191                (D) 213

Solution: (B)

Q45. The probability that a student knows the correct answer to a multiple choice question is 2/3. If the student does not know the answer, then the student guesses the answer. The probability of the guessed answer being correct is 1/4. Given that the student has answered the question correctly, the conditional probability that the student knows the correct answer is

(A) 2/3                (B) 3/4                (C) 5/6                    (D) 8/9

Solution: (D)

Q46. The solution to the differential equation where k is a constant, subjected to the boundary conditions u(0) = 0 and U(L) = U, is

(A) (B) (C) (D) Solution: (B)

Q47. The value of definite integral is

(A) (B) (C) (D) Solution: (C)

Q48. Common Data for Questions 48 and 49:

A single riveted lap joint of two similar plates as shown in the figure below has the following geometrical and material details.

width of the plate w = 200 mm, thickness of the plate t = 5 mm, number of rivets n = 3, diameter of the rivet dr = 10 mm, diameter of the rivet hole dh = 11 mm, allowable tensile stress of the plate σp = 200 MPa, allowable shear stress of the rivet σs = 100 MPa and allowable bearing stress of the rivet σc = 150 MPa.

If the rivets are to be designed to avoid crushing failure, the maximum permissible load P in kN is

(A) 7.50               (B) 15.00                (C) 22.50                 (D) 30.00

Solution: (C)

Q49. If the plates are to be designed to avoid tearing failure, the maximum permissible load P in kN is

(A) 83                 (B) 125                 (C) 167                 (D) 501

Solution: (C)

Q50. Common Data for Questions 50 and 51:

Water (specific heat, cp = 4.18 kJ/kgK) enters a pipe at a rate of 0.01 kg/s and a temperature of 20°C. The pipe, of diameter 50 mm and length 3 m, is subjected to a wall heat flux in W/m2:

If q”w = 2500 x, where x is in m and in the direction of flow (x = 0 at the inlet), the bulk mean temperature of the water leaving the pipe in °C is

(A) 42                 (B) 62                (C) 74                   (D) 104

Solution: (B)

Q51. If q”w = 5000 and the convection heat transfer coefficient at the pipe outlet is 1000 W/m2K, the temperature in °C at the inner surface of the pipe at the outlet is

(A) 71                 (B) 76                 (C) 79                     (D) 81

Solution: (D)

In orthogonal turning of a bar of 100 mm diameter with a feed of 0.25 mm/rev, depth of cut of 4 mm and cutting velocity of 90 m/min, it is observed that the main (tangential) cutting force is perpendicular to the friction force acting at the chip-tool interface. The main (tangential) cutting force is 1500 N.

The orthogonal rake angle of the cutting tool in degree is

(A) zero                 (B) 3.58                  (C) 5                   (D) 7.16

Solution: (A)

Q53. The normal force acting at the chip-tool interface in N is

(A) 1000                 (B) 1500                  (C) 2000                  (D) 2500

Solution: (B)

In a simple Brayton cycle, the pressure ratio is 8 and temperatures at the entrance of compressor and turbine are 300 K and 1400 K, respectively. Both compressor and gas turbine have isentropic efficiencies equal to 0.8. For the as, assume a constant value of cp (specific heat at constant pressure) equal to 1 kJ/kgK and ratio of specific heats as 1.4. Neglect changes in kinetic and potential energies.

The power required by the compressor in kW/kg of gas flow rate is

(A) 194.7               (B) 243.4                  (C) 304.3              (D) 378.5

Solution: (C)

Q55. The thermal efficiency of the cycle in percentage (%) is

(A) 24.8                (B) 38.6                     (C) 44.8                 (D) 53.1

Solution: (A)