Firstranker's choice
GATE 2015
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Q. 1 – Q. 25 carry one mark each.
Q.1
If any two columns of a determinant P= are interchanged, which one of the following statements regarding the value of the determinant is CORRECT?
(A) Absolute value remains unchanged but sign will change.
(B) Both absolute value and sign will change.
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(C) Absolute value will change but sign will not change.
(D) Both absolute value and sign will remain unchanged.
Q.2 Among the four normal distributions with probability density functions as shown below, which one has the lowest variance?
(A) I
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(B) II
(C) III
(D) IV
Q.3 Simpson's rule is used to integrate the function f(x) = x² + 3⁄5 between x = 0 and x = 1 using the least number of equal sub-intervals. The value of the integral is
Q.4 The value of limx?0 1-cos(x²)⁄2x4
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(A) 0
(B) 1⁄2
(C) 1⁄4
(D) undefined
Q.5 Given two complex numbers z1 = 5 + (5v3)i and z2 = 2⁄v3 + 2i, the argument of z1⁄z2 in degrees is
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(A) 0
(B) 30
(C) 60
(D) 90
Q.6 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
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(A) 4
(B) 16
(C) 32
(D) 64
Q.7 The Blasius equation related to boundary layer theory is a
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(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
Q.8 For flow of viscous fluid over a flat plate, if the fluid temperature is the same as the plate temperature, the thermal boundary layer is
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(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
Q.9 For an ideal gas with constant values of specific heats, for calculation of the specific enthalpy,
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(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
Q.10 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 CE-2. For one cycle of operation, if the net heat absorbed by CE-1 from reservoir A is 150 MJ, the net heat rejected to reservoir C by CE-2 (in MJ) is
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Q.11 Air enters a diesel engine with a density of ____ 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
Q.12 A stream of moist air (mass flow rate = 10.1 kg/s) with humidity ratio of 0.01 kg⁄kg dry 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 is ____ kg⁄kg dry air
Q.13 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
Q.14 Consider a slider crank mechanism with nonzero masses and inertia. A constant torque T is applied on the crank as shown in the figure. Which of the following plots best resembles variation of crank angle, ? versus time
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(A)
(B)
(C)
(D)
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Q.15 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) is
Q.16 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
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Q.17 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
Q.18 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
Q.19 Which one of the following is the most conservative fatigue failure criterion?
(A) Soderberg
(B) Modified Goodman
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(C) ASME Elliptic
(D) Gerber
Q.20 Which one of the following types of stress-strain relationship best describes the behaviour of brittle materials, such as ceramics and thermosetting plastics, (s = stress and e = strain)?
(A)
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(B)
(C)
(D)
Q.21 Match the following products with preferred manufacturing processes:
| Product | Process |
|---|---|
| P Rails | 1 Blow molding |
| Q Engine crankshaft | 2 Extrusion |
| R Aluminium channels | 3 Forging |
| S PET water bottles | 4 Rolling |
(A) P-4, Q-3, R-1, S-2
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(B) P-4, Q-3, R-2, S-1
(C) P-2, Q-4, R-3, S-1
(D) P-3, Q-4, R-2, S-1
Q.22 Holes of diameter 25.0+0.020-0.000 mm are assembled interchangeably with the pins of diameter 25.0+0.008-0.005 mm. The minimum clearance in the assembly will be
(A) 0.048 mm
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(B) 0.015 mm
(C) 0.005 mm
(D) 0.008 mm
Q.23 Under certain cutting conditions, doubling the cutting speed reduces the tool life to 1⁄16th of the original. Taylor's tool life index (n) for this tool-workpiece combination will be
Q.24 In a linear arc welding process, the heat input per unit length is inversely proportional to
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(A) welding current
(B) welding voltage
(C) welding speed
(D) duty cycle of the power source
Q.25 The function of interpolator in a CNC machine controller is to
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(A) control spindle speed
(B) coordinate feed rates of axes
(C) control tool rapid approach speed
(D) perform Miscellaneous (M) functions (tool change, coolant control etc.)
Q. 26 – Q. 55 carry two marks each.
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Q.26 Consider a spatial curve in three-dimensional space given in parametric form by x(t) = cos t, y(t) = sin t, z(t) = 2⁄pt, 0 < t < p⁄2. The length of the curve is
Q.27 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)
Q.28 Find the solution of d²y/dx² = e-x which passes through the origin and the point (ln2,2).
(A) y = ex - e-x - 1⁄2
(B) y = 1⁄2(ex + e-x)
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(C) y = 1⁄2(ex – e-x)
(D) y = 1⁄2ex + e-x
Q.29 The probability of obtaining at least two “SIX” in throwing a fair dice 4 times is
(A) 425⁄432
(B) 19⁄144
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(C) 13⁄144
(D) 125⁄432
Q.30 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 ± 0.008
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(B) 2.0 ±0.012
(C) 2.0 ±0.016
(D) 2.0 ±0.020
Q.31 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
Q.32 A triangular facet in a CAD model has vertices: P1(0,0,0); P2(1,1,0) and P3(1,1,1). The area of the facet is
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(A) 0.500
(B) 0.707
(C) 1.414
(D) 1.732
Q.33 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.
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| Activity | Duration (days) |
|---|---|
| 1-2 | 2 |
| 2-3 | 1 |
| 4-3 | 3 |
| 1-4 | 3 |
| 2-5 | 3 |
| 3-5 | 2 |
| 4-5 | 4 |
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
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Q.34 For a canteen, the actual demand for disposable cups 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 coefficient as 0.75 is
Q.35 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
Q.36 The solidification time of a casting is proportional to (V⁄A)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 sand 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) 25⁄36
(B) 36⁄25
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(C) 1
(D) 6⁄5
Q.37 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
Q.38 Considering massless rigid rod and small oscillations, the natural frequency (in rad/s) of vibration of the system shown in the figure is
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(A) 100
(B) 1⁄400
(C) 400⁄3
(D) 400⁄4
Q.39 For the truss shown in figure, the magnitude of the force in member PR and the support reaction at R are respectively
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(A) 122.47 kN and 50 kN
(B) 70.71 kN and 100 kN
(C) 70.71 kN and 50 kN
(D) 81.65 kN and 100 kN
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Q.40 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
Q.41 A pinion with radius r1, and inertia I1 is driving a gear with radius r2 and inertia I2. Torque t1 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) F1 ? F2; t1 = I1?1; F2 = I2?1
(B) F1 = F2; t1 = [I1 + I2 (r1⁄r2)²]?1; F2 = I2?1
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(C) F1 = F2; t1 = I1?1; F2 = I2?2
(D) F1 ? F2; t1 = [I1 + I2 (r1⁄r2)²]?1; F2 = I2?2
Q.42 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 just lifted off the ground at the end Q is approximately
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(A) 3000
(B) 3500
(C) 4000
(D) 4500
Q.43 A machine element is subjected to the following bi-axial state of stress: sx = 80 MPa; sy = 20 MPa; txy = 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
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(A) 1.0
(B) 2.0
(C) 2.5
(D) 3.3
Q.44 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
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Q.45 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
Q.46 A horizontal plate has been joined to a vertical post using four rivets arranged as shown in the figure. The magnitude of the load on the worst loaded rivet (in N) is
Q.47 For flow through a pipe of radius R, the velocity and temperature distribution are as follows: u(r, x) = C1(1 - r²⁄R²), and T(r,x) = C2(1 - r²⁄R²), where C1 and C2 are constants. The bulk mean temperature is given by Tm = ?⁄A u(r, x)T(r, x) dA / Um, with Um being the mean velocity of flow. The value of Tm is
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(A) 0.5C2
(B) 0.5C2
(C) 0.6C2
(D) 0.6C2
Q.48 Match the following pairs:
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| Equation | Physical Interpretation |
|---|---|
| P ? × v = 0 | I Incompressible continuity equation |
| Q ? . v = 0 | II Steady flow |
| R DV⁄Dt = 0 | III Irrotational flow |
| S ?v⁄?t = 0 | IV Zero acceleration of fluid particle |
(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
Q.49 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
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Q.50 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
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Q.51 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
Q.52 Air (? = 1.2 kg/m³and kinematic viscosity, ? = 2 × 10?5 m²/s) with a velocity of 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 Cf = 1.328⁄vRex, the total drag force (in N) per unit width of the plate is
Q.53 Water (? = 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 kPa and 130 kPa respectively. Assuming the venturimeter to be horizontal and neglecting friction, the inlet velocity (in m/s) is
Q.54 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
(A) 500.0
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(B) 773.0
(C) 785.4
(D) 1285.4
Q.55 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°C, h = 3486 kJ/kg, s = 6.6 kJ/(kg.K)
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Intermediate stage: h = 2776 kJ/kg
Exit of turbine: P = 9 kPa, 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
END OF THE QUESTION PAPER
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