# GATE 2015 ME – SET 1 – Complete Solutions

Q1. If any two columns of a determinant

are interchanged, which one of the following statements regarding the value of determinant is CORRECT?

(A) Absolute value remains unchanged but sign will change.

(B) Both absolute value and sign will change.

(C) Absolute value will change but sign will not change.

(D) Both absolute value and sign will remain unchanged.

Solution: (A)

Q2. Among the four normal distributions with probability density functions as shown below, which one has the lowest variance?

(A) I                (B) II                (C) III                  (D) IV

Solution: (D)

Q3. Simpson’s $$\frac{1}{3}$$ rule is used to integrate the function $$f(x)=\frac{3}{5}x^{2}+\frac{9}{5}$$ x= 0 and x = 1 using the least number of equal sub-intervals. The value of the integral is __________

Solution: Key = 2

Q4. The value of

is

(A) 0                   (B) 1/2                 (C) 1/4                 (D) undefined

Solution: (C)

Q5. Given two complex numbers $$z_{1}=5+\frac{5}{\sqrt{3}}i$$ and $$z_{2}=\frac{2}{\sqrt{3}}+2i$$, the argument of  $$\frac{z_{1}}{z_{2}}$$in degrees is

(A) 0                   (B) 30                   (C) 60                  (D) 90

Solution: (A)

Q6. Consider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate, density and friction factor to be constant, if the length of the pipe is doubled and the diameter is halved, the head loss due to friction will increase by a factor of

(A) 4                   (B) 16                    (C) 32                  (D) 64

Solution: (D)

Q7. The Blasius equation related to boundary layer theory is a

(A) third-order linear partial differential equation

(B) third-order nonlinear partial differential equation

(C) second-order nonlinear ordinary differential equation

(D) third-order nonlinear ordinary differential equation

Solution:(D)

Q8. For flow of viscous fluid over a flat p late, if the fluid temperature is the same as the plate temperature, the thermal boundary layer is

(A) thinner than the velocity boundary layer              (B) thicker than the velocity boundary layer

(C) of the same thickness as the velocity boundary layer               (D) not formed at all

Solution: (D)

Q9. For an ideal gas with constant values of specific heats, for calculation of the specific enthalpy

(A) it is sufficient to know only the temperature

(B) both temperature and pressure are required to be known

(C) both temperature and volume are required to be known

(D) both temperature and mass are required to be known

Solution: (A)

Q10. A Carnot engine (CE-1) works between two temperature reservoirs A and B, where TA = 900 K and TB = 500 K. A second Carnot engine (CE-2) works between temperature reservoirs B and C, where TC = 300 K. In each cycle of CE-1 and CE-2, all the heat rejected by CE-1 to reservoir B is used by CE2. For one cycle of operation, if the net Q absorbed by CE-1 to reservoir B is used by CE-2. For one cycle of operation, if the net Q absorbed by CE-1 from reservoir A is 150 MJ, the net heat rejected to reservoir C by CE-2 (in MJ) is _________

Solution: Key = 50

Q11. Air enters a diesel engine with a density of 1.0 kg/m3. The compression ratio is 21. At steady state, the air intake is 30 × 10−3 kg/s and the net work output is 15 kW. The mean effective pressure (in kPa) is ______

Solution: Key = 525

Q12. A steam of moist air (mass flow rate = 10.1 kg/s) with humidity ratio of 0.01 (kg/kg fry air) mixes with a second stream of superheated water vapour flowing at 0.1 kg/s. Assuming proper and uniform mixing with no condensation, the humidity ratio of the final stream (in kg/kg dry air) is __________.

Solution: Key = 0.02

Q13. A wheel of radius r rolls without slipping on a horizontal surface shown below. If the velocity of point P is 10 m/s in the horizontal direction, the magnitude of velocity of point Q (in m/s) is _______

Solution: Key = 20

Q14. Consider a slider crank mechanism with nonzero masses and inertia. A constant torque τ is applied on the crank as when in the figure, Which of the following plots best resembles variation of crank angle, θ versus time.

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

Solution: (D)

Q15. Consider a stepped shaft subjected to a twisting moment applied at B as shown in the figure. Assume shear modulus, G = 77 GPa. The angle of twist at C (in degrees) as _______

Solution: Key = 0.22 ti 0.25

Q16. Two identical trusses support a load of 100 N as shown in the figure. The length of each truss is 1.0 m; cross-sectional area is 200 mm2; Young’s modulus E = 200 GPa. The force in the truss AB (in N) is ______

Solution: Key = 100

Q17. Consider a steel (Young’s modulus E = 200 GPa) column hinged on both sides. Its height is 1.0 m and cross-section is 10 mm × 20 mm. The lowest Euler critical buckling load (in N) is _____

Solution: Key = 3285 to 3295

Q18. A swimmer can swim 10 km in 2 hours when swimming along the flow of a river. While swimming against the flow, she takes 5 hours for the same distance. her speed in still water (in km/h) is _______

Solution: Key = 3.5

Q19. Which one of the following is the most conservative fatigue failure criterion?

(A) Soderberg           (B) Modified Goodman                (C) ASME Elliptic                 (D) ASME Elliptic

Solution: (A)

Q20. Which one of the following types of stress-stain relationship best describes the behavior of brittle materials, such as ceramics and thermosetting plastics, (σ = stress and ε = strain)?

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

Solution: (D)

Q21. Match the following products with preferred manufacturing processes:

(A) P-4, Q-3, R-1, S-2                (B) P-4, Q-3, R-2, S-1

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

Solution: (B)

Q22. Holes of diameter are assembled interchangeably with the pins of diameter The minimum clearance in the assembly will be

(A) 0.048 mm                 (B) 0.015 mm                  (C) 0.005 mm                   (D) 0.008 mm

Solution: (B)

Q23. Under certain cutting conditions, doubling the cutting speed reduces the tool life to (1/16)th of the original. Taylor’s stool life index(n) for this tool-work piece combination will be ________

Solution: Key = 0.25

Q24. In a linear arc welding process, the heat input per  unit length is inversely proportional to

(A) welding current             (B) welding voltage

(C) welding speed                (D) duty cycle of the power source

Solution: (C)

Q25. The function of interpolator ins a CNC machine controller is to

(A) control spindle speed

(B) coordinate feed rates of axes

(C) control tool rapid approach speed

(D) perform Miscellaneous (M) function (tool change, coolant control etc.)

Solution: (B)

Q26. Consider a spatial curve in three-dimensional space given in parametric form by x(t) = cost t, y(t) = sin t,

The length of the curve is _______

Solution: Key = 1.85 to 1.87

Q27. Consider an ant crawling along the curve (x – 2)2 + y2 = 4, where x and y are in meters. The ant starts at the point (4, 0) and moves counter-clockwise with a speed of 1.57 meters per second. The time taken by the ant to reach the point (2, 2) is (in seconds) _______

Solution: Key = 1.9 to 2.1

Q28. Find the solution of $$\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=y$$ which passes through the origin and the point (ln2, 3/4).

(A) $$y=\frac{1}{2}e^{x}-e^{-x}$$                          (B) $$y=\frac{1}{2}(e^{x}+e^{-x})$$

(C) $$y=\frac{1}{2}(e^{x}-e^{-x})$$                       (D) $$y=\frac{1}{2}e^{x}+e^{-x}$$

Solution: (C)

Q29. The probability of obtaining at least two “SIX” in throwing a fair dice 4 times is

(A) 425/432                  (B) 19/144                 (C) 13/144                 (D) 125/432

Solution: (B)

Q30. In the assembly shown below, the part dimensions are:

L1 – 22.0±0.01 mm,

L2 = L3 = 10.0±0.005 mm.

Assuming the normal distribution of part dimensions, the dimension L4 = in mm for assembly condition would be:

(A) $$2.0^{\pm 0.008}$$           (B) $$2.0^{\pm 0.012}$$

(C) $$2.0^{\pm 0.016}$$            (D) $$2.0^{\pm 0.008}$$

Solution: (B)

Q31. A DC welding power source has a linear voltage-current (V-I) characteristic with open circuit voltage of 80 V and a short circuit current of 300 A> For maximum arc power, the current (in Amperes) should be set as _______

Solution: Key = 149 to 151

Q32. A triangular facet in a CAD model has vertices: P1(0, 0, 0); P1(0, 0, 0); P2(1, 1, 0) and P3(1, 1, 1). The area of the facet is

(A) 0.500                (B) 0.707               (C) 1.414                (D) 1.732

Solution: (B)

Q33. Following data refers to the activities of a project, where, node 1 refers to the start and node 5 refers to the end of the project.

The critical path (CP) in the network is

(A) 1-2-3-5             (B) 1-4-3-5             (C) 1-2-3-4-5               (D) 1-4-5

Solution: (B)

Q34. For a canteen, the actual demand for disposable curs was 500 units in January and 600 units in February. The forecast for the month of January was 400 units. The forecast for the month of March considering smoothing coefficients as 0.75 is ______

Solution: Key = 568 to 570

Q35. An orthogonal turning operation is carried out under the following conditions: rake angle = 5°; spindle rotational speed = 400 rpm; axial feed = 0.4 m/min and radial depth of cut = 5 mm. The chip thickness, tc, is found to be 3 mm. The shear angle (in degrees) in this turning process is _______

Solution: Key = 18.5 to 19.0

Q36. The solidification time of a casting is proportional to $$\left ( \frac{V}{A} \right )^{2}$$, where V is the volume of the casting and A is the total casting surface area losing heat. Two cubes of same material and size are cast using and casting process. The top face of one of the cubes is completely insulated The ratio of the solidification time for the cube with top face insulated to that of the other cube is

(A) 35/36                 (B) 36/25                (C) 1                (D) 6/5

Solution: (B)

Q37. In a slab rolling operation, the maximum thickness reduction (∆hmax) is given by ∆hmax = μ2R, where R is the radius of the roll and μ is the coefficient of friction between the roll and the sheet. If μ = 0.1, the maximum angle subtended by the deformation zone at the centre of the roll (bite angle in degrees) is _______

Solution: Key = 5.6 to 5.8

Q38. Considering massless rigid rod and small oscillations, the natural frequency (in rad/s) of vibration of the system shown in the figure is

(A) $$\sqrt{\frac{400}{1}}$$                     (B) $$\sqrt{\frac{400}{2}}$$

(C) $$\sqrt{\frac{400}{3}}$$                     (D) $$\sqrt{\frac{400}{4}}$$

Solution: (D)

Q39. For the truss shown in figure, the magnitude of the force in member PR and the support reaction at R are respectively

(A) 122.47kN and 50 kN

(B) 70.71 kN and 100 kN

(C) 70.71 kN and 50 kN

(D) 81.65 and 100 kN

Solution: (C)

Q40. A ball of mass 0.1 kg, initially at rest, is dropped from height of 1 m. Ball hits the ground and bounces off the ground. Upon impact with the ground, the velocity reduces by 20%. The height (in m) to which the ball will rise is ______

Solution: Key = 0.64

Q41. A pinion with radius r1, and inertia I1 is driving a gear with radius r2 and inertia I2. Torque τ1 is applied on pinion. The following are free body diagrams of pinion and gear showing important forces (F1 and F2) of interaction. Which of the following relations hold true?

(A)                    (B)

(C)                  (D)

Solution: (B)

Q42. A mobile phone has a small motor with an eccentric mass used for vibrator mode. The location of the eccentric mass on motor with respect to center of gravity (CG) of the mobile and the rest of the dimensions of the mobile phone are shown. The mobile is kept on a flat horizontal surface.

Given in addition that the eccentric mass = 2 grams, eccentricity = 2.19 mm, mass of the mobile = 90 grams, g = 9.81 m/s2. Uniform speed of the motor in RPM for which the mobile will get jut lifted off the ground at the end Q is approximately

(A) 3000                   (B) 3500                   (C) 4000                 (D) 4500

Solution: (B)

Q43. A machine element is subjected to the following bi-axial state of stress: σx = 80 MPa; σy = 20 MPa; τxy = 40 MPa. If the shear strength of the material is 100 MPa, the factor of safety as per Tresca’s maximum shear stress theory is

(A) 1.0                (B) 1.0                (C) 2.5                  (D) 2.5

Solution: (B)

Q44. A cantilever beam with flexural rigidity of 200 N.m2 is loaded as shown in the figure. The deflection (in mm) at the tip of the beam is ________

Solution: Key = 0.24 to 0.28

Q45. A precision instrument package (m = 1 kg) needs to be mounted on a surface vibrating at 60 Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to the instrument. Assume that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of the required mounting pad is __________

Solution: Key = 6750 to 7150

Q46. A horizontal plate has been joined to a vertical post using four rivers arranged as shown in the figure. The magnitude of the load on the worst loaded rivet (in N) is ___________

Solution: Key = 1835 to 1845

Q47. Four flow through a pipe of radius R, the velocity and temperature distribution are as follows  :

u(r, x) = C1, and $$T(r,x)=C_{2}\left [ 1-\left ( \frac{r}{R} \right )^{3} \right ]$$where C1 and C2 are constants.

The bulk mean temperature is given by $$T_{m}=\frac{2}{U_{m}R^{2}}\int_{0}^{R}u(r,x)T(r,x)$$, with Um being the mean velocity of flow. The value of Tm is

(A) $$\frac{0.5C_{2}}{U_{m}}$$                     (B) $$0.5C_{2}$$

(C) $$0.6C_{2}$$                                                  (D) $$\frac{0.6C_{2}}{U_{m}}$$

Solution: (C)

Q48. Match the following pairs:

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

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

Solution: (C)

Q49. The velocity field of an incompressible flow is given by

V = (a1x + a2y + a3z)i + (b1x + b2y + b3z)j + (c1x + c­2y+ c3z)k, where a1 = 2 and c3 = −4. The value of b2 is _________

Solution: Key = 1.9 to 2.1

Q50. A 10 mm diameter electrical conductor is covered by an insulation of 2 mm thickness. The conductivity of the insulation is 0.08 W/m-k and the convection coefficient at the insulation surface is 10 W/m2-K. Addition of further insulation of the same material will

(A) increase heat loss continuously                      (B) decrease heat loss continuously

(C) increase heat loss to a maximum and then decrease heat loss

(D) decrease heat loss to a minimum and then increase heat loss

Solution: (C)

Q51. Temperature of nitrogen in a vessel of volume 2 m3 is 288 K. A U-tube manometer connected to the vessel shows a reading of 70 cm of mercury (level higher in the end open to atmosphere). The universal gas constant is 8314 J/kmol-K, atmospheric pressure is 1.01325 bar, acceleration due to gravity is 9.81 m/s2 and density of mercury is 13600 kg/m3. The mass of nitrogen (in kg) in the vessel is ________

Solution: Key = 4.4 to 4.6

Q52. Air (ρ = 1.2 kg/m3 and the kinematic viscosity, v = 2 × 10−5 m2/s) with a velocity o f 2 m/s flows over the top surface of a flat plate of length 2.5 m. If the average value of friction coefficient is $$C_{f}=\frac{1.328}{\sqrt{Re_{x}}}$$ the total drag force (in N) per unit width of the plate is ________

Solution: Key = 0.0158 t0 0.0162

Q53. Water (ρ = 1000 kg/m3) flows through a venturimeter with inlet diameter 80 mm and throat diameter 40 mm. The inlet and throat gauge pressures are measured to be 400 kP and 130 kPa respectively. Assuming the venturimeter to be horizontal and neglecting friction, the inlet velocity (in m/s) is ________

Solution: Key = 6

Q54. A well insulated rigid container of volume 1 m3 contains 1.0 kg of an ideal gas [Cp = 1000 J/(kg.K) and Cv = 800 J(kg.K)] at a pressure of 105 Pa. A stirrer is rotated at constant rpm in the container for 1000 rotations and the applied torque is 100 N-m. The final temperature of the gas (in K) is __________

Solution: Key = 1283.4 to 1287.4

Q55. Steam enters a well insulated turbine and expands isentropically throughout. At an intermediate pressure, 20 percent of the mass is extracted for process heating and the remaining steam expands isentropically to 9 kPa.

Inlet to turbine: P = 14 MPa, T = 560℃, h = 3486 kJ/kg, s = 6.6 kJ/(kg.K)

Intermediate stage: h = 2776 kJ/kg

Exit of turbine: P = 9kPa, hf = 174 kJ/kg, hg = 2574 kJ/kg, sf = 0.6 kJ/(kg.K), sg = 8.1 kJ/(kg.K)

If the flow rate of steam entering the turbine is 100 kg/s, then the work output (in MW) is ____________

Solution: Key = 123.56 to 127.56