## Monday, August 26, 2013

### GATE 2012 Questions and Answer key

Q.(1-25) Carry one marks each
Q.1 The estimate obtained of the definite integral by the Simpson's rule with three-point function evaluation exceeds the exact value by

(a) 0.235   (b) 0.068    (c) 0.024    (d) 0.012
Sol : (d)
Q.2 The annual precipitation data of a city is normally distributed with mean and standard deviation as 1000 mm and 200 mm, respectively. The probability that the annual precipitation will be more than 1200 mm is
(a) <50%   (b) 50%    (c) 75%    (d) 100%                                        Sol: (a)

Q.3 The infinite series 1+x+x^2/2! + x^3/3! + x^4/4! +.........
(a) sec x    (b) e^x     (c) cos x    (d) 1 + sin^2 x                              Sol: (b)

Q.4 The Poisson's ratio is defined as
(a) mode[axial stress/ Lateral Stress]        (b) mode[lateral strain/ axial strain]
(c) mode[lateral stress/axial stress]          (d) mode[axial strain/lateral strain]       Sol: (b)

Q.5 The following statements are related to bending of beams:
I. The slope of the bending moment diagram is equal to the shear force
II. The slope of the shear force diagram is equal to the load intensity
III. The slope of the curvature is equal to the flexural rotation
IV. The second derivative of the deflection is equal to the curvature.
The only False statement is
(a) I        (b) II         (c) III        (d) IV                                                                         Sol. (c)

Q.6 If a small concrete cube is submerged deep in still water in such a way that the pressure exerted on all faces of the cube is p, then the maximum shear stress developed inside the cube is
(a) 0    (b) p/2    (c) p     (d) 2p                                                                            Sol: (a)

Q.7 As per IS:456:2000, in limit state Design of a flexural member, the strain in reinforcing bars under tension at ultimate state should not be less than
(a) fy/Es      (b) fy/Es + 0.002      (c) fy/(1.15Es)      (d)  fy/1.15 Es +0.002         Sol:(d)

Q.8 Which one of the following is categorized as a long term loss of pre-stress in a  pre-stressed concrete member?
(a) Loss due to elastic shortening
(b) Loss due to friction
(c) Loss due to relaxation of strands
(d) Loss due to anchorage slip                                                                               Sol:(c)

Q.9 In a steel plate with bolted connections, the rupture of the net section is a mode of failure under
(a) Tension    (b) Compression     (c) flexure    (d) shear                                    Sol:(a)

Q.10 The ratio of theoritical critical buckling load for a column with fixed ends to that of another column with the same dimensions and material but with pinned ends is equal to:
(a) 0.5   (b) 1.0    (c) 2.0    (d) 4.0                                                      Sol: (d)

Q.11 The effective stress friction angle of a saturated, cohesion-less soil is 38 degrees.  The ratio of shear stress to normal effective stress on the failure plane is
(a) 0.781    (b) 0.616    (c) 0.488     (d) 0.438                        Sol: (a) T= c + Eff. stress *tan(fr. ang.)

Q.12 Two series of compaction tests were performed in the laboratory on an inorganic clayey soil employing two different levels of compaction energy per unit volume of soil. With regard to the above tests, the following statements are made.
I. The optimum moisture content is expected to be more for the tests with higher energy.
II. The maximum dry density is expected to be more for the tests with higher energy.
(a) Only I is TRUE          (b)  Only II is TRUE
(c) Both I and II are TRUE  (d) Neither I and II are TRUE                                    Sol: (b)

Q.13 As per the Indian Standard soil classification system, a sample of silty clay with liquid limit of 40% and plasticity index of 28% is classified as:
(a) CH   (b) CI    (c) CL    (d) CL-ML                                                         Sol: (b) Ip= 0.73(Wl - 20)

Q.14 A smooth rigid retaining wall moves as shown in the sketch causing the back-fill material to fail. The back-fill material is homogeneous and isotropic, and obeys Mohr - Coulomb failure criterion. The major principal stress is
(a) parallel to the wall face and acting downwards
(b) normal to the wall face
(c) oblique to the wall face acting downwards
(d) oblique to the wall face acting upwards.                                                        Sol: (b)

Q.15 An embankment is to be constructed with a granular soil(bulk unit weight = 20kN/m^3) on a saturated clayey silt deposit (un-drained shear strength = 25kPa). Assuming un-drained general shear failure and bearing capacity factor of 5.7, the maximum height (in m) of the embankment at the point of failure is
(a) 20.0    (b) 3.5    (c) 3.0    (d) 2.5                                                                 Sol: (a)

Q.16 A trapezoidal channel is 10.0 m wide at the base and has a side slope of 4 horizontal to 3 vertical. The bed slope is 0.002. The channel is lined with smooth concrete(Manning's n =0.012). The hydraulic radius (in m) for a depth of flow of 3.0 m is
(a) 20.0       (b) 3.5    (c) 3.0    (d) 2.1                                                            Sol: (d) Rh= A/P

Q.17 A rectangular open channel of width 5.0 m is carrying a discharge of 100 m^3/Sec. The Froude number of the flow is 0.8. The depth of flow (in m) in the channel is
(a) 4    (b) 5    (c) 16    (d) 20                                                                       Sol: (a) Fr = V/ (g.d)^0.5

Q.18 The circular water pipes shown in the sketch are flowing full. The velocity of flow (in m/sec) in the branch pipe "R" is
(a) 3    (b) 4    (c) 5   (d) 6                                                 Sol: (b)  Qp = Qq+ Qr (Q=A.V)

Q.19 The ratio of evapo- transpiration to potential evapo- transpiration is in the range
(a) 0.0  to 0.4                    (b) 0.6 to 0.9
(c) 0.0 to 1.0                     (d) 1.0 to 2.0                            Sol: (c)

Q.20 A sample of domestic sewage is digested with silver sulphate, sulphuric acid, potassium dichromate and mercuric sulphate in chemical oxygen demand (COD) test. The digested sample is then titrated with standard ferrous ammonium sulphate(FAS) to determine the un-reacted amount of
(a) mercuric sulphate
(b) potassium dichromate
(c) silver sulphate
(d) sulphuric acid                                                              Sol: (b)

Q.21 Assertion [A]: At a manhole, the crown of the outgoing sewer should not be higher than the crown of incoming sewer
Reason [A]: Transition from a larger diameter incoming sewer to a small diameter outgoing sewer at a manhole should not be made.
The CORRECT option evaluating the above statement is:
(a) Both [A] and [R] are TRUE and [R] is the correct reason for [A]
(b) Both [A] and R are TRUE but [R] is not correct reason for [A].
(c) Both [A] and [R] are false.
(d)[A] is true but [R] is false.                                           Sol: (b)

Q.22 Two major roads with two lanes each are crossing in an urban area to form an un-controlled intersection . The number of conflict points when both roads are one way is 'X' and when both roads are two way is 'Y'. The ratio of X and Y is
(a) 0.25   (b) 0.33   (c) 0.50    (d) 0.75                    Sol: (a) 0.25    X = 6  Y= 24; X/Y = 0.25

Q.23 Two bitumen samples 'X' and 'Y' have softening points 45 degrees Celsius and 60 degrees Celsius respectively. Consider the following statements:
I. Viscosity of 'X' will be higher than that of 'Y' at the same temperature.
II. Penetration value of 'X' will be lesser than that of 'Y' under standard condition:
The CORRECT option evaluating the above statements is
(a) Both I and II are TRUE
(b) I is FALSE and II is TRUE
(c) Both I and II are FALSE
(d) I is TRUE and II is FALSE.                                            Sol: (c)

Q.24 Road roughness is measured using
(a) Benkelman beam
(b) Bump indicator
(c) Dynamic cone penetrometer
(d)Falling weight deflectometer.                                     Sol: (b)

Q.25 Which of the following errors can be eliminated by reciprocal measurements in differential leveling?
I. Errors due to earth's curvature
II. Error due to atmospheric refraction
(a) Both I and II     (b) I only
(c) II only               (d) Neither I or II                          Sol: (a)

Q.26 - Q.55 carry two marks each
Q.27 In an experiment, positive and negative values are equally likely to occur. The probability of obtaining at most one negative value in five trials is
(a) 1/32    (b) 2/32   (c) 3/32   (d) 6/32                      Sol:(d)  P= nCr p^r q^n-r

Q.28 The eigen values of matrix [9  5 ; 5  8] (where 9  5 is first row and 5  8 is second) are
(a)  -2.42 and 6.86
(b) 3.48  and 13.53
(c) 4.70  and 6.86
(d) 6.86 and 9.50                                                                      Sol: (b)

Q.29 For the parallelogram OPQR shown in the sketch,  vect.OP = ai + bj  and vetc.OR = ci + dj. i and j are unit vectors. The area of the parallelogram is
(a) ad - bc        (b) ac - bd         (c) ad - ba         (d) ab - cd        Sol: (a)

Q.30 The solution of the ordinary differential equation  dy/dx + 2y = 0 for the boundary condition, y=5 at x = 1 is
(a) y = e ^(-2x)    (b) y = 2.e^(-2x)     (c) y = 10.95.e^(-2y)    (d) y = 36.95.e^(-2x)
Sol: (d)  y = c.e^(-2x)
Q.31 A simply supported beam is subjected to a uniformly distributed load of intensity w per unit length, on half of the span from one end. The length of the span and the flexural stiffness are denoted as l and EI respectively. The deflection at mid span of the beam is
(a) 5/6144. wl^2/EI    (b) 5/768. w.l^4/ EI
(c) 5/384.wl^4/EI        (d) 5/192. wl^4/EI                              Sol: (b)

Q.32 The sketch shows a column with a pin at the base and rollers at the top. It is subjected to an axial force P and a moment M at mid height. The reaction(s) at R is/are:
(a) A vertical force equal to P
(b) A force equal to P/2
(c) A vertical force equal to P and a horizontal force equal to M/h
(d) A vertical force equal to P/2 and a horizontal force equal to M/h   Sol: (c)

Q.33 A concrete beam pre-stressed with a parabolic tendon is shown in the sketch. The eccentricity of the tendon is measured from the centroid of the cross section. The applies pre-stressing force at service is 1620 kN. The uniformly distributed load of 45 kN/m includes the self weight.
The stress (in N/mm2) in the bottom fibre at mid-span is
(a) tensile 2.90   (b) compressive 2.90
(c) tensile 4.32   (d) compressive 4.32                                Sol: (b)  stress= -P/A-P.e/z+M/z

Q.34 A symmetric frame PQR consists of two inclined members PQ and QR, connected at 'Q' with a rigid joint, and hinged at 'P' and 'R'. The horizontal length PR is l. If a weight W is suspended at 'Q', the bending moment at 'Q' is
(a) Wl/2    (b) Wl/4     (c) Wl/8      (d) zero                            Sol: (d)

Q.35 Two plates are connected by fillet welds of size 10mm and subjected to tension as shown in the figure. The thickness of each plate is 12mm. The yield stress and the ultimate tensile stress of steel are 250 MPa and 410 MPa respectively. The welding is done in the workshop (Ymw= 1.25). As per the Limit State Method of IS 800: 2007, the minimum length(rounded off to the nearest higher multiple of 5 mm) of each weld to transmit a force P equal to 270 kN is
(a) 100 mm    (b) 105 mm    (c) 110 mm    (d) 115 mm                    Sol: (b)

Q.36 Two soil specimens with identical geometric dimensions were subjected to falling head permeability tests in the laboratory under identical conditions. The fall of water head was measured after an identical time interval. The ratio of initial to final water heads for the test involving the first specimen was 1.25. If the co-efficient of permeability  of the specimen is 5 times that of the first, the ratio of initial to final water heads in the test involving the second specimen is
(a) 3.05  (b) 3.80   (c) 4.00    (d) 6.25                                  Sol: (a)   k = aL/At.[ln(h1/h2)]

Q.37 A layer of normally consolidated, saturated silty clay of 1 m thickness is subjected to one dimensional consolidation under a pressure increment of 20 kPa. The properties of the soil are: specific gravity = 2.7, natural moisture content = 45%, compression index = 0.45, and recompression index = 0.05. The initial average effective stress within the layer is 100 kPa. Assuming Terzaghi's theory to be applicable, the primary consolidation settlement(rounded off to the nearest mm) is
(a) 2 mm   (b) 9 mm    (c) 14 mm    (d) 16 mm                    Sol: (d)

Q.38 Steady state seepage is taking place through a soil element at Q,  2 m below the ground surface immediately downstream of the toe of an earthen dam as shown in the sketch. The water level in a piezometer installed at P, 500 mm above Q, is at the ground surface. The water level in a piezometer installed at R, 500 mm below Q, is 100 mm above the ground surface. The bulk saturated unit weight of the soil is 18 kN/m^3 and the unit weight of water is 9.81 kN.m3. The vertical effective stress (in kPa) at Q is
(a) 14.42   (b) 15.89   (c) 16.38    (d) 18.34                  Sol: (b)  Eff Stres = Ysat.h - Yw.hw

Q.39 The top width and depth of flow in a triangular channel were measured as 4 m and 1 m, respectively. The measured velocities on the centre line at the water surface, 0.2 m and 0.8 m below the surface are 0.7 m/sec, 0.6 m/sec and 0.4 m/sec, respectively. Using two- point method of velocity measurement, the discharge (in m^3/sec) in the channel is
(a) 1.4     (b) 1.2   (c) 1.0    (d) 0.8                      Sol: (c)  Vmean = (V0.2+V0.8)/2; Q= A.Vmean

Q.40 Group I contains parameters and Group II lists methods/instruments.
Group I                                                            Group II
P. Stream flow velocity                                                 1. Anemometer
Q. Evapo-transpiration rate                                           2. Penman's method
R. Infiltration rate                                                          3. Horton's method
S. Wind velocity                                                           4. Current meter
The CORRECT match of Group I with Group II is
(a) P-1, Q-2, R-3, S-4
(b) P-4, Q-3, R-2, S-1
(c) P-4, Q-2, R-3, S-1
(d) P-1, Q-3, R-2, S-4                                              Sol: (c)

Q.41 Wheat crop requires 55 cm of water during 120 days of base period. The total rainfall during this period is 100mm. Assume the irrigation efficiency to be 60%. The area(in ha) of the land which can be irrigated with a canal flow of 0.01 m^3/sec is
(a) 13.82     (b) 18.85    (c) 23.04     (d) 230.40             Sol:(a)

Q.42  A water sample has pH of 9.25. The concentration of hydroxyl ions in the water sample is
(a) 10^(-9.25) moles/L     (b) 10^(-4.75) moles/L   (c) 0.302 mg/L   (d) 3.020 mg/L
Sol: (c)
Q.43 A town is required to treat 4.2 m^2/min of water for daily domestic supply. Flocculating particles are to be produced by chemical coagulation. A column analysis indicated that an overflow rate of 0.2 mm/sec will produce satisfactory particle removal in a settling basin at a depth of 3.5 m. The required surface area (in m^2) for settling is
(a) 210   (b) 350     (c) 1728   (d) 21000                  Sol: (b) A= Q/V; Q=4.2 m^2/min;V=0.2mm/s

Q.44 A pavement designer has arrived at a design traffic of 100 million standard axles for a newly developing national highway as per IRC: 37 guidelines using the following data:
Design life = 15 years; commercial vehicle count before pavement construction = 4500 vehicles/day ; annual traffic growth rate = 8%. The vehicle damage factor used in the calculation was
(a) 1.53    (b) 2.24    (c) 3.66     (d) 4.14              Sol: (b) mSA= 365.A[(1+r)^n-1]*Vdf/r

Q.45 The following data are related to a horizontal curved portion of a two-lane highway: length of curve= 200 m, radius of curve = 300 m and width of pavement = 7.5 m. In order to provide a stopping sight distance(SSD) of 80 m, the set back distance (in m) required from the centre line of the inner lane of the pavement is
(a) 2.54   (b) 4.55   (c) 7.10   (d) 7.96                      Sol: (*)

Q.46 A two-lane urban road with one-way traffic has a maximum capacity of 1800 vehicles/hour. Under the jam condition, the average length occupied by the vehicles is 5.0 m. The speed versus density relationship is linear. For a traffic volume of 1000 vehicles/hour, the density (in vehicles/km) is
(a) 52   (b) 58   (c) 67   (d) 75                                 Sol: (c)

Q.47 The horizontal distance between two stations P and Q is 100 m. The vertical angles from P and Q to the top of a vertical tower at T are 3 Deg. and 5 Deg. above horizontal, respectively. The vertical angles from P and Q to the base of the tower are 0.1 D and 0.5 D below horizontal, respectively. Stations P, Q and the tower are in the same vertical plane with P and Q being on the same side of T. Neglecting earth's curvature and atmospheric refraction, the height (in m) of the tower is
(a) 6.972     (b) 12.387     (c) 12.540     (d) 128.745                Sol:(b)

Directions(Q.48- 49) : The flow net around a sheet pile wall is shown in the sketch. The properties of the soil are: permeability co-efficient = 0.09 m/day(isotropic), specific gravity = 2.70 and void ratio =0.85. The sheet pile wall and the bottom of the soil are impermeable.

Q.48 The seepage loss( in m^3 per day per unit length of the wall) of water is
(a) 0.33   (b) 0.38    (c) 0.43    (d) 0.54                                 Sol: (b)

Q.49 The factor of safety against occurrence of piping failure is
(a) 3.55   (b) 2.93     (c) 2.60     (d)  0.39                                  Sol:(c)

Directions(Q.50- 51): An activated sludge system(sketched below) is operating at equilibrium with the following information.
Waste water related data: flow rate = 500 m^3/hour, influent BOD = 150 mg/L, effluent BOD = 10 mg/L. Aeration tank related data: hydraulic retention time = 8 hours, mean-cell-residence time = 240 hours, volume = 4000 m^3, mixed liquor suspended solids = 200 mg/L

Q.50 The food to biomass (F/M) ratio (in kg BOD per kg biomass per day) for the aeration tank is
(a) 0.015    (b) 0.210    (c) 0.225    (d) 0.240                   Sol: (c) F/M = 500*150/ (200*4000) kg/hr

Q.51 The mass (in kg/day) of solids wasted from the system is
(a) 24000    (b) 10000   (c) 800    (d) 33                          Sol: (a) M= 200*500*1000*24 kg/day

Directions (Q.52-53) : The cross section at mid span of a beam at the edge of a slab is shown in the sketch. A portion of the slab is considered as the effective flange width for the beam. The grades of concrete and reinforcing steel are M25 and Fe415 respectively. The total area of reinforcing bars (A) is 4000 mm^2. At the ultimate limit state, xu denotes the depth of the neutral axis from the top fibre. Treat the section as under reinforced and flanged (xu>100 mm).

Q.52 The value of xu(in mm) computed as per the Limit State Method of IS:456:2000 is
(a) 200.0  (b) 223.3   (c) 236.3    (d) 273.6                Sol: (c)

Q.53 The ultimate moment capacity (in kNm) of the section as per the Limit State Method of IS:456:2000 is
(a) 475.2    (b) 717.0    (c) 756.4    (d) 762.5           Sol: (b)

Directions (Q.54-55): The drainage area of a watershed is 50 km^2. The phi index is 0.5 cm/hour and the base flow at the outlet is 10 m^3/sec. One hour unit hydrograph (unit depth = 1cm) of the watershed is triangular in shape with a time base of 15 hours. The peak ordinate occurs at 5 hours.

Q.54 The peak ordinate (in m^3/s/cm) of the unit hydro-graph is
(a) 10.00    (b) 18.52    (c) 37.03    (d) 185.20                Sol: (b)

Q.55 For a storm of depth of 5.5 cm and duration of 1 hour, the peak ordinate (in m^3/sec) of the hydro graph is

(a) 55.00    (b) 82.60     (c) 92.60       (d) 102.60           Sol: (c)

GENERAL APTITUDE (GA) QUESTIONS
Q.56-Q.60 carry one mark each
Q.56 Choose the most appropriate alternative from the options given below to complete  the following sentence:
Suresh's dog is the one ............ was hurt in the stampede.
(a) that   (b) which    (c) who    (d) whom                   Sol:(*)

Q.57 The cost function for a product in a firm is given by 5.q^2, where q is the amount of production. The firm can sell the product at a market price of Rs. 50 per unit. The number of units to be produced by the firm such that the profit is maximized is
(a) 5    (b) 10    (c) 15    (d) 25                            Sol: (a)

Q.58 Choose the most appropriate alternative from the options given below to complete the following sentence:
Despite several ........ the mission succeeded in its attempt to resolve the conflict.
(a) Alternative   (b) setbacks    (c) meetings     (d) delegations   Sol: (b)

Q.59 Which one of the following options is closest in meaning to the word given below?
Mitigate
(a) Diminish      (b) Divulge     (c) Dedicate   (d) Denote             Sol:(a)

Q.60 Choose the grammatically INCORRECT sentence:
(a) They gave us the money back less the service charges of Three Hundred rupees.
(b) This country's expenditure is not less than that of Bangladesh.
(c) The committee initially asked for a funding of Fifty Lakh rupees, but later settled for a lesser sum.
(d) This country's expenditure on educational reforms is very less. Sol: (d)

Q. 61 - Q. 65 carry two marks each.

Q.61 Given the sequence of terms, AD CG FK JP, the next term is
(A) OV (B) OW (C) PV (D) PW                                     Sol: (a)

Q.62 Wanted Temporary, Part-time persons for the post of Field Interviewer to conduct personal
interviews to collect and collate economic data. Requirements: High School-pass, must be
available for Day, Evening and Saturday work. Transportation paid, expenses reimbursed.
Which one of the following is the best inference from the above advertisement?
(A) Gender-discriminatory
(B) Xenophobic
(C) Not designed to make the post attractive

(D) Not gender-discriminatory                                             Sol: (d)

Q.63 A political party orders an arch for the entrance to the ground in which the annual convention is
being held. The profile of the arch follows the equation y = 2x – 0.1x2 where y is the height of the
arch in meters. The maximum possible height of the arch is
(A) 8 meters (B) 10 meters (C) 12 meters (D) 14 meters        Sol:(b)

Q.64  An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X supplies
60% and Y supplies 40% of the shock absorbers. All shock absorbers are subjected to a quality test.
The ones that pass the quality test are considered reliable. Of X’s shock absorbers, 96% are reliable.
Of Y’s shock absorbers, 72% are reliable.
The probability that a randomly chosen shock absorber, which is found to be reliable, is made by Y
is
(A) 0.288 (B) 0.334 (C) 0.667 (D) 0.720   Sol: (b)

Q.65 Which of the following assertions are CORRECT?
P: Adding 7 to each entry in a list adds 7 to the mean of the list
Q: Adding 7 to each entry in a list adds 7 to the standard deviation of the list
R: Doubling each entry in a list doubles the mean of the list
S: Doubling each entry in a list leaves the standard deviation of the list unchanged
(A) P, Q (B) Q, R (C) P, R (D) R, S               Sol:(c)

(* )means marks to all
END OF THE QUESTIONS
(References:  GATE 2013  by GK Publishers)

### Solved Example-Distance of a Point from a Line, whose end Co-ordinates are given.

Hi, Problem: The X and Y co-ordinates(in meters) of station Shore are 246.87 and 659.46 respectively, and those for station Rock are 437...