# GATE 2016 ME – SET 3 – Complete Solutions

Q1. A real square matrix A is called skew-symmetric if

(A) $$A^{T}=A$$                   (B) $$A^{T}=A^{-1}$$

(C) $$A^{T}=-A$$                  (D) $$A^{T}=A+A^{-1}$$

Solution: (C)

Q2.  is equal to

(A) 0                  (B) 1/12                  (C) 4/3                    (D) 1

Solution: (C)

Q3. Solutions of Laplace’s equation having continuous second-order partial derivatives are called

(A) biharmonic functions                      (B) harmonic functions

(C) conjugate harmonic functions       (D) error functions

Solution: (B)

Q4. The area (in percentage) under standard normal distribution curve of random variable Z within limits from −3 to +3 is ________

Solution: Key = 99.6 to 99.8

Q5. The root of the function f(x) = x3 + x − 1 obtained after first iteration on application of Newton-Raphson scheme using an initial guess of x0 = 1 is

(A) 0.682                 (B) 0.686                 (C) 0.750                 (D) 1.000

Solution : Key = (C)

Q6. A force F is acting on a bent bar which is clamped at one end as shown in the figure.

The CORRECT free body diagram is

(A)               (B)

(C)            (D)

Solution: (A)

Q7. The cross-sections of two solid bars made of the same material are shown in the figure. The square cross-section has flexural (bending) rigidity I1, while the circular cross-section has flexural rigidity I2. Both sections have the same cross-sectional area. The ratio I1/I2 is

(A) 1/π                    (B) 2/π

(C) π/3                    (D) π/6

Solution: (C)

Q8. The state of stress at a point on an element is shown in figure (a). The same state of stress is shown in another coordinate system in figure (b).

The components (τxx, τyy, τxy) are given by

(A) (p/√2, −p/√2, 0)           (B) (0, 0, p)             (C) (p, −p, p/√2)                (D) (0, 0, p/√2)

Solution: (B)

Q9. A rigid link PQ is undergoing plane motion as shown in the figure (VP and VQ are non-zero). VQP is the relative velocity of point Q with respect to point P.

Which one of the following is TRUE?

(A) $$V_{QP}$$ has components along and perpendicular to PQ

(B) $$V_{QP}$$ has only one component directed from P to Q

(C) $$V_{QP}$$ has only one component directed from Q to P

(D) $$V_{QP}$$ has only one component perpendicular to PQ

Solution: (D)

Q10. The number of degrees of freedom in a planar mechanism having n links and j simple hinge joints is

(A) 3(n − 3) − 2j            (B) 3(n − 1) − 2j              (C) 3n − 2j              (D) 2j − 3n + 4

Solution: (B)

Q11. The static deflection of a spring under gravity, when a mass of 1 kg is suspended from it, is 1 mm. Assume the acceleration due to gravity g =10 m/s2. The natural frequency of this spring-mass system (in rad/s) is_____________

Solution: Key = 99 to 101

Q12. Which of the bearings given below SHOULD NOT be subjected to a thrust load?

(A) Deep groove ball bearing                                (B) Angular contact ball bearing

(C) Cylindrical (straight) roller bearing              (D) Single row tapered roller bearing

Solution: (C)

Q13. A channel of width 450 mm branches into two sub-channels having width 300 mm and 200 mm as shown in figure. If the volumetric flow rate (taking unit depth) of an incompressible flow through the main channel is 0.9 m3/s and the velocity in the sub-channel of width 200 mm is 3 m/s, the velocity in the sub-channel of width 300 mm is _____________ m/s.

Assume both inlet and outlet to be at the same elevation.

Solution: Key = 0.99 to 1.01

Q14. For a certain two-dimensional incompressible flow, velocity field is given by $$2xy\hat{i}-y^{2}\hat{j}$$. The streamlines for this flow are given by the family of curves

(A) $$x^{2}y^{2}$$ = constant                    (B) $$xy^{2}$$ = constant

(C) $$2xy-y^{2}$$ = constant                       (D) xy = constant

Solution: (B)

Q15. Steady one-dimensional heat conduction takes place across the faces 1 and 3 of a composite slab consisting of slabs A and B in perfect contact as shown in the figure, where kA, kBdenote the respective thermal conductivities. Using the data as given in the figure, the interface temperature T2 (in ℃) is _______

Solution: Key = 67 to 68

Q16. Grashof number signifies the ratio of

(A) inertia force to viscous force             (B) buoyancy force to viscous force

(C) buoyancy force to inertia force         (D) inertia force to surface tension force

Solution: (B)

Q17. The INCORRECT statement about the characteristics of critical point of a pure substance is that

(A) there is no constant temperature vaporization process

(B) it has point of inflection with zero slope

(C) the ice directly converts from solid phase to vapor phase

(D) saturated liquid and saturated vapor states are identical

Solution: (C)

Q18. For a heat exchanger, ∆max is the maximum temperature difference and ∆Tmin is the minimum temperature difference between the two fluids. LMTD is the log mean temperature difference. Cmin and Cmax are the minimum and the maximum heat capacity rates. The maximum possible heat transfer (Qmax) between the two fluids is

(A)              (B)              (C)             (D)

Solution: (B)

Q19. The blade and fluid velocities for an axial turbine are as shown in the figure.

The magnitude of absolute velocity at entry is 300 m/s at an angle of 65° to the axial direction, while the magnitude of the absolute velocity at exit is 150 m/s. The exit velocity vector has a component in the downward direction. Given that the axial (horizontal) velocity is the same at entry and exit, the specific work (in kJ/kg) is__________

Solution: Key =  50 to 54

Q20. Engineering strain of a mild steel sample is recorded as 0.100%. The true strain is

(A) 0.010%            (B) 0.055%               (C) 0.099%              (D) 0.101%

Solution: (C)

Q21. Equal amounts of a liquid metal at the same temperature are poured into three moulds made of steel, copper and aluminum. The shape of the cavity is a cylinder with 15 mm diameter. The size of the moulds are such that the outside temperature of the moulds do not increase appreciably beyond the atmospheric temperature during solidification. The sequence of solidification in the mould
from the fastest to slowest is

(Thermal conductivities of steel, copper and aluminum are 60.5, 401 and 237 W/m-K, respectively. Specific heats of steel, copper and aluminum are 434, 385 and 903 J/kg-K, respectively.

Densities of steel, copper and aluminum are 7854, 8933 and 2700 kg/m3, respectively.)

(A) Copper – Steel – Aluminum                       (B) Aluminum – Steel – Copper

(C) Copper – Aluminum – Steel                       (D) Steel – Copper – Aluminum

Solution: (C)

Q22. In a wire-cut EDM process the necessary conditions that have to be met for making a successful cut are that

(A) wire and sample are electrically non-conducting

(B) wire and sample are electrically conducting

(C) wire is electrically conducting and sample is electrically non-conducting

(D) sample is electrically conducting and wire is electrically non-conducting

Solution: (B)

Q23. Internal gears are manufactured by

(A) hobbing                                        (B) shaping with pinion cutter

(C) shaping with rack cutter           (D) milling

Solution: (B)

Q24.

Match the following part programming codes with their respective functions

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

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

Solution: (C)

Q25. In PERT chart, the activity time distribution is

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

Solution: (D)

Q26. The number of linearly independent eigenvectors of matrix

is ______

Solution: Key = 2

Q27. The value of the line integral $$\oint_{c}\bar{F}\cdot \bar{r}’ds$$ where C is a circle of radius $$\frac{4}{\sqrt{\pi }}$$ units is __________

Here, $$\bar{F}(x,y)=y\hat{i}+2x\hat{j}$$ and $$\bar{r}’$$ is the UNIT tangent vector on the curve C at an arc length s from a reference point on the curve. $$\hat{i}$$ and $$\hat{j}$$ are the basis vectors in the x-y Cartesian reference. In evaluating the line integral, the curve has to be traversed in the counter-clockwise direction.

Solution: Key = 15.9 to 16.1

Q28.  is

(A) 0                  (B) $$\infty$$                 (C) 1/2                  (D) $$-\infty$$

Solution: (C)

Q29. Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is

(A) 16/5525               (B) 64/2197              (C) 3/13               (D) 8/16575

Solution: (A)

Q30. An inextensible massless string goes over a frictionless pulley. Two weights of 100 N and 200 N are attached to the two ends of the string. The weights are released from rest, and start moving due to gravity. The tension in the string (in N) is __________

Solution: Key = 130 to 135

Q31. A circular disc of radius 100 mm and mass 1 kg, initially at rest at position A, rolls without slipping down a curved path as shown in figure. The speed v of the disc when it reaches position B is _________ m/s.

Acceleration due to gravity g = 10 m/s2

Solution: Key = 19.9 to 20.1

Q32. A rigid rod (AB) of length L = √2 m is undergoing translational as well as rotational motion in the x-y plane (see the figure). The point A has the velocity  The end B is constrained to move only along the x direction.

The magnitude of the velocity V2 (in m/s) at the end B is _______

Solution: Key = 2.9 to 3.1

Q33. A square plate of dimension L × L is subjected to a uniform pressure load p = 250 MPa on its edges as shown in the figure. Assume plane stress conditions. The Young’s modulus E = 200 GPa.

The deformed shape is a square of dimension L − 2δ. If L = 2m and δ = 0.001 m, the Poisson’s ratio of the plate material is _________

Solution: Key = 0.18 to 0.22

Q34. Two circular shafts made of same material, one solid (S) and one hollow (H), have the same length and polar moment of inertia. Both are subjected to same torque. Here, θs is the twist and τs is the maximum shear stress in the solid shaft, whereas θH is the twist and τH is the maximum shear stress in the hollow shaft. Which one of the following is TRUE?

(A)                 (B)

(C)                 (D)

Solution: (D)

Q35. A beam of length L is carrying a uniformly distributed load w per unit length. The flexural rigidity of the beam is EI. The reaction at the simple support at the right end is

(A) wL/2             (B) 3wL/8                (C) wL/4                (D) wL/8

Solution: (B)

Q36. Two masses m are attached to opposite sides of a rigid rotating shaft in the vertical plane. Another pair of equal masses m1 is attached to the opposite sides of the shaft in the vertical plane as shown in figure. Consider m = 1 kg, e = 50 mm, e1 = 20 mm, b = 0.3 m, a = 2m and a1 = 2.5 m. For the system to be dynamically balanced, m1 should be _________kg.

Solution: Key = 1.9 to 2.1

Q37. A single degree of freedom spring-mass system is subjected to a harmonic force of constant amplitude. For an excitation frequency of  the ratio of the amplitude of steady state the ratio of the amplitude of steady state

Solution: Key = 0.49 to 0.51

Q38. A bolted joint has four bolts arranged as shown in figure. The cross sectional area of each bolt is 25 mm2. A torque T = 200 N-m is acting on the joint. Neglecting friction due to clamping force, maximum shear stress in a bolt is ______ MPa.

Solution: Key = 39.9 to 40.1

Q39. Consider a fully developed steady laminar flow of an incompressible fluid with viscosity μ through a circular pipe of radius R. Given that the velocity at a radial location of R/2 from the centerline of the pipe is U1, the shear stress at the wall is kμU1/R, where K is __________

Solution: Key = 2.6 to 2.7

Q40. The water jet exiting from a stationary tank through a circular opening of diameter 300 mm impinges on a rigid wall as shown in the figure. Neglect all minor losses and assume the water level in the tank to remain constant. The net horizontal force experienced by the wall is ___________ kN.

Density of water is 1000 kg/m3.

Acceleration due to gravity g = 10 m/s2.

Solution: Key = 8.76 to 8.78

Q41.  For a two-dimensional flow, the velocity field is  where  are the basis vectors in the x-y Cartesian coordinate system. Identify the CORRECT statements from below.

(1)

(2)

(3)

(4)

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

Solution: (B)

Q42. Two large parallel plates having a gap of 10 mm in between them are maintained at temperatures T1 = 1000 K and T2 = 400 K. Given emissivity values, ε1 = 0.5, ε2 = 0.25 and Stefan-Boltzmann constant σ = 5.67 × 10−8 W/m2-K4, the heat transfer between the plates (in kW/m2) is __________

Solution: Key = 10.9 to 11.2

Q43. A cylindrical steel rod, 0.01 m in diameter and 0.2 m in length is first heated to 750℃ and then immersed in a water bath at 100℃. The heat transfer coefficient is 250 W/m2-K. The density, specific heat and thermal conductivity of steel are ρ = 7801 kg/m3, c = 473 J/kg-K, k =43 W/m-K, respectively. The time required for the rod to reach 300℃  is ________ seconds.

Solution: Key =  42 to 45

Q44. Steam at an initial enthalpy of 100 kJ/kg and inlet velocity of 100 m/s, enters an insulated horizontal nozzle. It leaves the nozzle at 200 m/s. The exit enthalpy (in kJ/kg) is __________

Solution: Key = 84 to 86

Q45. In a mixture of dry air and water vapor at a total pressure of 750 mm of Hg, the partial pressure of water vapor is 20 mm of Hg. The humidity ratio of the air in grams of water vapor per kg of dry air (gw/kgda) is _______

Solution: Key = 16.9 to 17.1

Q46. In a 3-stage air compressor, the inlet pressure is p1, discharge pressure is p4 and the intermediate pressures are p2 and p3(p2 < p3). The total pressure ratio of the compressor is 10 and the pressure ratios of the stages are equal. If p1 = 100 kPa, the value of the pressure p3 (in kPa) is ________

Solution: Key = 460 to 470

Q47. In the vapour compression cycle shown in the figure, the evaporating and condensing temperatures are 260 K and 310 K, respectively. The compressor takes in liquid-vapour mixture (state 1) and isentropically compresses it to a dry saturated vapour condition (state 2). The specific heat of the liquid refrigerant is 4.8 kJ/kg-K and may be treated as constant. The enthalpy of evaporation for the refrigerant at 310 K is 1054 kJ/kg.

The difference between the enthalpies at state points 1 and 0 (in kJ/kg) is ____________.

Solution: Key = 1095 to 1130

Q48. S pot welding of two steel sheets each 2 mm thick is carried out successfully by passing 4 kA of current for 0.2 seconds through the electrodes. The resulting weld nugget formed between the sheets is 5 mm in diameter. Assuming cylindrical shape for the nugget, the thickness of the nugget is________mm.

Solution: Key = 2.85 to 2.95

Q49. For an orthogonal cutting operation, tool material is HSS, rake angle is 22°, chip thickness is 0.8 mm, speed is 48 m/min and feed is 0.4 mm/rev. The shear plane angle (in degrees) is

(A) 19.24              (B) 29.70               (C) 56.00                (D) 68.75

Solution: (B)

Q50. In a sheet metal of 2 mm thickness a hole of 10 mm diameter needs to be punched. The yield strength in tension of the sheet material is 100 MPa and its ultimate shear strength is 80 MPa. The force required to punch the hole (in kN) is _________

Solution: Key = 4.9 to 5.1

Q51. In a single point turning operation with cemented carbide tool and steel work piece, it is found that the Taylor’s exponent is 0.25. If the cutting speed is reduced by 50% then the tool life changes by ______ times.

Solution: Key =  14 to 18

Q52. Two optically flat plates of glass are kept at a small angle θ as shown in the figure. Monochromatic light is incident vertically.

If the wavelength of light used to get a fringe spacing of 1 mm is 450 nm, the wavelength of light (in nm) to get a fringe spacing of 1.5 mm is _________

Solution: Key = 674 to 676

Q53. A point P (1, 3, −5) is translated by $$2\hat{i}+3\hat{j}-4\hat{k}$$ and then rotated counter clockwise by 90° about the z-axis. The new position of the point is

(A) (−6, 3, −9)           (B) (−6, −3, −9)            (C) (6, 3, −9)             (D) (6, 3, 9)

Solution: (A)

Q54. The demand for a two-wheeler was 900 units and 1030 units in April 2015 and May 2015, respectively. The forecast for the month of April 2015 was 850 units. Considering a smoothing constant of 0.6, the forecast for the month of June 2015 is

(A) 850 units              (B) 927 units              (C) 965 units                (D) 970 units

Solution: (D)

Q55.

A firm uses a turning center, a milling center and a grinding machine to produce two parts. The table below provides the machining time required for each part and the maximum machining time available on each machine. The profit per unit on parts I and II are Rs. 40 and Rs. 100, respectively. The maximum profit per week of the firm is Rs._________

Solution: Key = 39000 to 41000