# GATE 2014 ME – Complete Solutions

Q1. Given that the determinant of the matrix

is −12, the determinant of the matrix  is

(A) -96                  (B) -24                  (C) 24                   (D) 96

Solution: (A)

Q2.

(A) 0                 (B) 1                 (C) 3                 (D) not defined

Solution: (A)

Q3. The argument of the complex number $$\frac{1+i}{1-i}$$ where i = √−1 is

(A) $$-\pi$$             (B) $$-\frac{\pi }{2}$$             (C)  $$\frac{\pi }{2}$$              (D) $$\pi$$

Solution: (C)

Q4. The matrix form of the linear system

and  is

(A)                   (B)

(C)                   (D)

Solution: (A)

Q5. Which one of the following describes the relationship among the three vectors, and

(A) The vectors are mutually perpendicular

(B) The vectors are linearly dependent

(C) The vectors are linearly independent

(D) The vectors are unit vectors

Solution: (B)

Q6. A circular rod of length ‘L’ and area of cross-section ‘A’ has a modulus of elasticity ‘E’ and coefficient of thermal expansion ‘α’. One end of the rod is fixed and other end is free. If the temperature of the rod is increased by ΔT, then

(A) stress developed in the rod is E α ΔT and strain developed in the rod is α ΔT

(B) both stress and strain developed in the rod are zero

(C) stress developed in the rod is zero and strain developed in the rod is α ΔT

(D) stress developed in the rod is E α ΔT and strain developed in the rod is zero

Solution: (C)

Q7. A metallic rod of 500 mm length and 50 mm diameter, when subjected to a tensile force of 100 kN at the ends, experiences, an increase in its length by 0.5 mm and a reduction in its diameter by 0.015 mm. The Poisson’s ratio of the rod material is _________

Solution: Key = 0.29 to 0.31

Q8. Critical damping is the

(A) largest amount of damping for which no oscillation occurs in free vibration

(B) smallest amount of damping for which no oscillation occurs in free vibration

(C) largest amount of damping for which the motion is simple harmonic in free vibration

(D) smallest amount of damping for which the motion is simple harmonic in free vibration

Solution: (B)

Q9. A circular object of radius r rolls without slipping on a horizontal level floor with the center having velocity V. The velocity at the point of contact between the object and the floor is

(A) zero

(B) V in the direction of motion

(C) V opposite to the direction of motion

(D) V vertically upward from the floor

Solution: (A)

Q10. For the given statements:

I. Mating spur gear teeth is an example of higher pair

II. A revolute joint is an example of lower pair

(A) Both I and II are false                    (B) I is true and II is false

(C) I is false and II is true                    (D) Both I and II are true

Solution: (D)

Q11. A rigid link PQ is 2 m long and oriented at 20° to the horizontal as shown in the figure. The magnitude and direction of velocity VQ, and the direction of velocity VP are given. The magnitude of VP(in m/s) at this instant is

(A) 2.14                (B) 1.89               (C) 1.21                (D) 0.96

Solution: (D)

Q12. Biot number signifies the ratio of

(A) convective resistance in the fluid to conductive resistance in the solid

(B) conductive resistance in the solid to convective resistance in the fluid

(C) inertia force to viscous force in the fluid

(D) buoyancy force to viscous force in the fluid

Solution: (B)

Q13. The maximum theoretical work obtainable, when a system interacts to equilibrium with a reference environment, is called

(A) Entropy             (B) Enthalpy            (C) Exergy              (D) Rothalpy

Solution: (C)

Q14. Consider a two-dimensional laminar flow over a long cylinder as shown in the figure below.

The free stream velocity is U and the free stream temperature T is lower than the cylinder surface temperature Ts. The local heat transfer coefficient is minimum at point

(A) 1                (B) 2                 (C) 3                 (D) 4

Solution: (B)

Q15. For a completely submerged body with centre of gravity ‘G’ and centre of buoyancy ‘B’, the condition of stability will be

(A) G is located below B                        (B) G is located above B

(C) G and B are coincident                   (D) independent of the locations of G and B

Solution: (A)

Q16. In a power plant, water (density = 1000 kg/m3) is pumped from 80 kPa to 3 MPa. The pump has an isentropic efficiency of 0.85. Assuming that the temperature of the water remains the same, the specific work (in kJ/kg) supplied to the pump is

(A) 0.34                 (B) 2.48                  (C) 2.92                  (D) 3.43

Solution: (D)

Q17. Which one of the following is a CFC refrigerant?

(A) R744               (B)  R290                (C) R502                 (D) R718

Solution: (C)

Q18. The jobs arrive at a facility, for service, in a random manner. The probability distribution of number of arrivals of jobs in a fixed time interval is

(A) Normal              (B) Poisson              (C) Erlang             (D) Beta

Solution: (B)

Q19. In exponential smoothening method, which one of the following is true?

(A) 0 ≤ α ≤ 1 and high value of α is used for stable demand

(B) 0 ≤ α ≤ 1 and high value of α is used for unstable demand

(C) α ≥ 1 and high value of α is used for stable demand

(D) α ≤ 0 and high value of α is used for unstable demand

Solution: (B)

Q20. For machining a rectangular island represented by coordinates P(0,0), Q(100,0), R(100,50) and S(0,50) on a casting using CNC milling machine, an end mill with a diameter of 16 mm is used. The trajectory of the cutter centre to machine the island PQRS is

(A) (−8, −8), (108, −8), (108,58), (−8,58) , (−8, −8)

(B) (8,8), (94,8), (94,44), (8,44), (8,8)

(C) (−8,8), (94,0), (94,44), (8,44), (−8,8)

(D) (0,0), (100,0), (100,50), (50,0), (0,0)

Solution: (A)

Q21. Which one of the following instruments is widely used to check and calibrate geometric features of machine tools during their assembly?

(A) Ultrasonic probe                        (B) Coordinate Measuring Machine (CMM)

(C) Laser interferometer                 (D) Vernier calipers

Solution: (C)

Q22. The major difficulty during welding of aluminium is due to its

(A) high tendency of oxidation                   (B) high thermal conductivity

(C) low melting point                                   (D) low density

Solution: (A)

Q23. The main cutting force acting on a tool during the turning (orthogonal cutting) operation of a metal is 400 N. The turning was performed using 2 mm depth of cut and 0.1 mm/rev feed rate. The specific cutting pressure (in N/mm2) is

(A) 1000               (B) 2000              (C) 3000              (D) 4000

Solution: (B)

Q24. The process of reheating the martensitic steel to reduce its brittleness without any significant loss in its hardness is

(A) normalising             (B) annealing             (C) quenching         (D) tempering

Solution: (A)

Q25. In solid-state welding, the contamination layers between the surfaces to be welded are removed by

(A) alcohol              (B) plastic deformation                (C) water jet              (D) sand blasting

Solution: (B)

Q26. The integral ∮C(ydx – xdy) is evaluated along the circle $$x^{2}+y^{2}=\frac{1}{4}$$ traversed in counter clockwise direction. The integral is equal to

(A) 0            (B) $$\frac{-\pi }{4}$$             (C) $$\frac{-\pi }{2}$$             (D) $$\frac{\pi }{4}$$

Solution: (C)

Q27. If y = f(x) is the solution of $$\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=0$$ with the boundary conditions y = 5 at x = 0, and $$\frac{\mathrm{d} y}{\mathrm{d} x}=2$$ x = 10, f(15) = _________

Solution: Key = 34 to 36

Q28. In the following table, x is a discrete random variable and p(x) is the probability density. The standard deviation of x is

(A) 0.18                (B) 0.36                (C) 0.54                   (D) 0.6

Solution: (D)

Q29. Using the trapezoidal rule, and dividing the interval of integration into three equal subintervals, the definite integral  is _________

Solution: Key = 1.10 to 1.12

Q30. The state of  stress at a point is given by σx = −6 Mpa, σy = 4 MPa, and τxy = −8 Mpa. The maximum tensile stress (in MPa) at the point is ________

Solution: Key = 8.4 to 8.5

Q31. A block R of mass 100 kg is placed on a block S of mass 150 kg as shown in the figure. Block R is tied to the wall by a massless and inextensible string PQ. If the coefficient of static friction for all surfaces is 0.4, the minimum force F (in kN) needed to move the block S is

(A) 0.69              (B) 0.88             (C) 0.98               (D) 1.37

Solution: (D)

Q32. A pair of spur gears with module 5 mm and a center distance of 450 mm is used for a speed reduction of 5:1. The number of teeth on pinion is _______

Solution: Key = 29 to 31

Q33. Consider a cantilever beam, having negligible mass and uniform flexural rigidity, with length 0.01 m. The frequency of vibration of the beam, with a 0.5 kg mass attached at the free tip, is 100 Hz. The flexural rigidity (in N.m2) of the beam is _______

Solution: Key = 0.064 to 0.067

Q34. An ideal water jet with volume flow rate of 0.05 m3/s strikes a flat plate placed normal to its path and exerts a force of 1000 N. Considering the density of water as 1000 kg/m3, the diameter (in mm) of the water jet is _______

Solution: Key = 56

Q35. A block weighing 200 N is in contact with a level plane whose coefficients of static and kinetic friction are 0.4 and 0.2, respectively. The block is acted upon by a horizontal force (in newton) P=10t, where t denotes the time in seconds. The velocity (in m/s) of the block attained after 10 seconds is _______

Solution: Key = 4.8 to 5.0

Q36. A slider crank mechanism has slider mass of 10 kg, stroke of 0.2 m and rotates with a uniform angular velocity of 10 rad/s. The primary inertia forces of the slider are partially balanced by a revolving mass of 6 kg at the crank, placed at a distance equal to crank radius. Neglect the mass of connecting rod and crank. When the crank angle (with respect to slider axis) is 30°, the unbalanced force (in newton) normal to the slider axis is _______

Solution: Key = 29 to 31

Q37. An offset slider-crank mechanism is shown in the figure at an instant. Conventionally, the Quick Return Ratio (QRR) is considered to be greater than one. The value of QRR is _______

Solution: Key = 1.2 to 1.3

Q38. A rigid uniform rod AB of length L and mass m is hinged at C such that AC = L/3, CB = 2L/3. Ends A and B are supported by springs of spring constant k. The natural frequency of the system is given by

(A) $$\sqrt{\frac{k}{2m}}$$             (B) $$\sqrt{\frac{k}{m}}$$

(C) $$\sqrt{\frac{2k}{m}}$$             (D) $$\sqrt{\frac{5k}{m}}$$

Solution: (D)

Q39. A hydrodynamic journal bearing is subject to 2000 N load at a rotational speed of 2000 rpm. Both bearing bore diameter and length are 40 mm. If radial clearance is 20 μm and bearing is lubricated with an oil having viscosity 0.03 Pa.s, the Sommerfeld number of the bearing is _________

Solution: Key = 0.75 to 0.85

Q40. A 200 mm long, stress free rod at room temperature is held between two immovable rigid walls. The temperature of the rod is uniformly raised by 250°C. If the Young’s modulus and coefficient of thermal expansion are 200 GPa and 1×10−5 /°C, respectively, the magnitude of the longitudinal stress (in MPa) developed in the rod is _______

Solution: Key = 499 to 501

Q41. 1.5 kg of water is ion saturated liquid state at 2 bar (vf = 0.001061 m3/kg, uf = 504.0 kJ/kg, hf = 505 kJ/kg). Heat is added in a constant pressure process till the temperature of water reaches 400℃ (v = 1.5493 m3/kg, u = 2967.0 kJ/kg, h = 3277.0 kJ/kg). The heat added (in kJ) in the process is ________

Solution: Key = 4155 to 4160

Q42. Consider one dimensional steady state heat conduction across a wall (as shown in figure below) of thickness 30 mm and thermal conductivity 15 W/m.K. At x = 0, a constant heat flux, q” = 1×105 W/m2 is applied. On the other side of the wall, heat is removed from the wall by convection with a fluid at 25°C and heat transfer coefficient of 250 W/m2.K. The temperature (in °C), at x = 0 is _______

Solution: Key = 620 to 630

Q43. Water flows through a pipe having an inner radius of 10 mm at the rate of 36 kg/hr at 25°C. The viscosity of water at 25°C is 0.001 kg/m.s. The Reynolds number of the flow is _______

Solution: Key = 635 to 638

Q44. For a fully developed flow of water in a pipe having diameter 10 cm, velocity 0.1 m/s and kinematic viscosity 10−5 m2/s, the value of Darcy friction factor is _______

Solution: Key = 0.06 to 0.07

Q45. In a simple concentric shaft-bearing arrangement, the lubricant flows in the 2 mm gap between the shaft and the bearing. The flow may be assumed to be a plane Couette flow with zero pressure gradient. The diameter of the shaft is 100 mm and its tangential speed is 10 m/s. The dynamic viscosity of the lubricant is 0.1 kg/m.s. The frictional resisting force (in newton) per 100 mm length of the bearing is _______

Solution: Key =15 to 16

Q46. The non-dimensional fluid temperature profile near the surface of a convectively cooled flat plate is given by  where y is measured perpendicular to the plate, L is the plate length, and a, b and c are arbitrary constants. TW and T are wall and ambient temperatures, respectively. If the thermal conductivity of the fluid is k and the wall heat flux is q”, the Nusselt number  is equal to

(A) a                (B) b                (C) 2c               (D) (b+2c)

Solution: Key = (B)

Q47. In an air-standard Otto cycle, air is supplied at 0.1 MPa and 308 K. The ratio of the specific heats (γ) and the specific gas constant (R) of air are 1.4 and 288.8 J/kg.K, respectively. If the compression ratio is 8 and the maximum temperature in the cycle is 2660 K, the heat (in kJ/kg) supplied to the engine is _______

Solution: Key = 1400 to 1420

Q48. A reversible heat engine receives 2 kJ of heat from a reservoir at 1000 K and a certain amount of heat from a reservoir at 800 K. It rejects 1 kJ of heat to a reservoir at 400 K. The net work output (in kJ) of the cycle is

(A) 0.8               (B) 1.0               (C) 1.4               (D) 2.0

Solution: (C)

Q49. An ideal reheat Rankine cycle operates between the pressure limits of 10 kPa and 8 MPa, with reheat being done at 4 MPa. The temperature of steam at the inlets of both turbines is 500°C and the enthalpy of steam is 3185 kJ/kg at the exit of the high pressure turbine and 2247 kJ/kg at the exit of low pressure turbine. The enthalpy of water at the exit from the pump is 191 kJ/kg. Use the following table for relevant data.

Disregarding the pump work, the cycle efficiency (in percentage) is _______

Solution: Key = 40 to 42

Q50. Jobs arrive at a facility at an average rate of 5 in an 8 hour shift. The arrival of the jobs follows Poisson distribution. The average service time of a job on the facility is 40 minutes. The service time follows exponential distribution. Idle time (in hours) at the facility per shift will be

(A) 5/7               (B) 14/3               (C) 7/5                  (D) 10/3

Solution: (B)

Q51. A metal rod of initial length L0 is subjected to a drawing process. The length of the rod at any instant is given by the expression,    L(t) = L0(1 + t2), where t is the time in minutes. The true strain rate (in min−1) at the end of one minute is ________

Solution: Key = 0.9 to 1.1

Q52. During pure orthogonal turning operation of a hollow cylindrical pipe, it is found that the thickness of the chip produced is 0.5 mm. The feed given to the zero degree rake angle tool is 0.2 mm/rev. The shear strain produced during the operation is _______

Solution: Key = 2.8 to 3.0

Q53. For the given assembly: 25 H7/g8, match Group A with Group B

(A) P-I, Q-III, R-IV, S-II              (B) P-I, Q-IV, R-III, S-II

(C) P-II, Q-III, R-IV, S-I              (D) P-II, Q-IV, R-III, S-I

Solution: (D)

Q54. If the Taylor’s tool life exponent n is 0.2, and the tool changing time is 1.5 min, then the tool life (in min) for maximum production rate is _______

Solution: Key = 5.9 to 6.1

Q55. An aluminium alloy (density 2600 kg/m3) casting is to be produced. A cylindrical hole of 100 mm diameter and 100 mm length is made in the casting using sand core (density 1600 kg/m3). The net buoyancy force (in newton) acting on the core is _______

Solution: Key = 7 to 8