GATE AG 2012 Question Paper & Solutions

Linear Algebra

The matrix $\begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}$ is:

Calculus

The line y = x - 1 can be expressed in polar coordinates (r, θ) as:

Farm Power

The type of pump used in the forced water cooling system of a tractor engine is:

Tractors and Power Tillers

Which one of the following statements is NOT appropriate regarding cone index:

Farm Machinery

The draft and total power requirement of a rotary cultivator operating in concurrent mode, as compared to a spring-tyne cultivator of equal cutting width under the same operating conditions, respectively are:

Soil and Water Erosion

The soil erodibility factor needs to be determined for use in the Universal Soil Loss Equation. The length (m) and slope (%) of the experimental plot to be used for this purpose, respectively, are:

Surveying and Levelling

The difference between Fore Bearing and Back Bearing of a traverse line is:

Wells and Pumps

A pumping device that combines the advantages of both centrifugal and reciprocating pumps is known as:

Heat and Mass Transfer

If ν is the kinematic viscosity of air–water vapour mixture and D_AB is the mass diffusivity of water vapour in air, the ratio ν/D_AB is known as:

Size Reduction and Material Handling

Work index in size reduction can be obtained by multiplying Bond's energy constant with:

Calculus

The tangent line to y = f(x) at the point (x0, y0), assuming f'(x0) ≠ 0, intersects the x-axis at:

Probability and Statistics

Approximate percentage of scores that fall within ±σ (standard deviation) of the mean in a normal distribution is:

Differential Equations

The integrating factor of the differential equation $(x + 1)\frac{dy}{dx} - y = \sin x$ is:

Sources of Power

The constituent of producer gas which occupies the highest percentage by volume and helps in increasing its overall calorific value is:

Machine Design

During field operation, the shank of a tractor-drawn rigid-tyne sweep-type cultivator is mainly subjected to:

Machine Design

A slider is moving on a straight link at a sliding velocity of 0.5 m/s. The straight link is pivoted at one end and makes angular movement at a rate of 1.0 rad/s. The Coriolis acceleration of the slider, in m/s², is:

Farm Power

The power developed and the exhaust gas temperature of a diesel engine, compared to a spark-ignition engine of the same size running at the same speed, respectively are:

Irrigation Water Conveyance and Application Methods

In a semi-modular outlet, the discharge:

Hydrology

For a reservoir, outflow Q (m³/s) and storage S (m³) are related as Q = S/4000. Inflow, outflow and storage are zero at t = 0. If inflow rises linearly and reaches 300 m³/s at t = 3 h, the outflow rate at t = 3 h, in m³/s, is:

Fluid Mechanics

A trapezoidal grassed waterway on a 4% longitudinal gradient has cross-sectional flow area 1.52 m², wetted perimeter 12.5 m, and Manning's n = 0.04 m⁻¹/³ s. The flow through the waterway, in m³/s, is:

Wells and Pumps

A single-acting reciprocating pump discharges 3.5 litres of water per second at 40 rpm. Piston diameter 150 mm, stroke 300 mm. The percentage slip is:

Heat and Mass Transfer

A pair of parallel glass panes, each 3 mm thick, traps a 2 mm layer of stagnant air. Thermal conductivities: glass 0.5, air 0.02 W/m·K. Film heat-transfer coefficient of air (surface) = 10 W/m²K. The Biot Number is:

Heat and Mass Transfer

Two small parallel plane square surfaces, each 4 mm × 4 mm, are placed 0.5 m apart (centre to centre), with 30° angle between the radial distance and both surface normals. The view factor between the two surfaces is:

Engineering Properties of Agriculture Produce

Tomato catsup with 10 Pa·sⁿ consistency coefficient and 0.8 flow behaviour index is flowing in a pipe. The generalized (apparent) coefficient of viscosity of catsup, in Pa·s, at a representative shear condition is:

Size Reduction and Material Handling

A packed bed of 480 kg solid particles (particle size 0.15 mm, density 800 kg/m³) is fluidized using air at 25°C, 1 atm. Cross-section of the empty bed is 0.45 m², voidage at minimum fluidizing condition is 0.5. The minimum height of the fluidized bed, in m, is:

Numerical Methods

The value of $\int_{0}^{\pi/2} \cos x \, dx$ using the trapezoidal rule with two equal intervals is:

Tractors and Power Tillers

A tractor PTO-driven stationary peg-tooth wheat thresher operating at a cylinder speed of 540 rpm requires a torque of 250 Nm at the PTO. The minimum net engine power required, in kW, is:

Irrigation Water Conveyance and Application Methods

A border strip 8 × 250 m is irrigated by a border stream of 50 L/s. Infiltration capacity of the soil is 25 mm/h (constant). Average depth of the advancing sheet over the land is 70 mm. The time required to irrigate the border strip, in minutes, is:

Preservation of Food

Decimal reduction times for Bacillus subtilis are 37 s and 12 s at 120°C and 125°C respectively. The temperature rise, in °C, necessary to reduce the D-value at 120°C by a factor of 10, is:

Storage Systems

A tall silo (height:diameter = 2) holds 480 tons of wheat, bulk density 960 kg/m³. Angle of internal friction φ=25°, wall friction angle φ'=24°. Using Airy's formula, the maximum lateral pressure, in kPa, at the bottom of the bin section is:

Linear Algebra

The eigenvalues of the matrix $\begin{bmatrix} 6 & 1 \\ 2 & 3 \end{bmatrix}$ are:

Calculus

If f'(x) = eˣ and f(0) = 5, then from the Mean Value Theorem, the value of f(1) lies between:

Differential Equations

The inverse Laplace Transform of $\frac{s^2}{(s-3)^3}$ can be written as $\frac{e^{3t}}{2}[At^2 + Bt + C]$. The values of $A$, $B$ and $C$, respectively are:

Tractors and Power Tillers

A 2WD tractor weighing 15.84 kN, wheelbase 2160 mm, static weight on front:rear axle = 30:70 on level ground. Hitch point is 700 mm high and 120 mm to the rear of the rear-axle centre. Pull acts at 12° below horizontal. The maximum pull in kN, when the front wheels just start to rise, is:

Sources of Power

A horizontal-axis drag-type windmill (square blades) and a horizontal-axis lift-type windmill (airfoil blades) of the same rotor size are installed at 10 m height; average wind speed 25 km/h. Maximum power coefficients: drag type 0.148, lift type 0.593. If the maximum power extracted by the drag type is 5 kW, the corresponding power for the lift type, in kW, is:

Farm Machinery

The thresher of a wheat combine harvester has an optimal throughput capacity of 2400 kg (crop) per hour. The harvester has a forward velocity of 4.5 km/h. Yield of crop in the field is 3000 kg (grain) per ha, grain:straw = 60:40. If the throughput is to be maintained, the width of cut of the harvester in m (neglecting turning losses) is:

Machine Design

In a disc clutch, the inside and outside radii of the clutch plate are 50 and 100 mm respectively. If the axial force exerted on the disc is 4 kN, the maximum pressure experienced by the clutch plate in N/mm² under uniform wear conditions is:

Irrigation Water Conveyance and Application Methods

A regime channel carrying 25 m³/s is designed using Lacey's regime theory. Side slope ½H:1V, Lacey's silt factor f = 1. The bottom width and depth of flow, in m, respectively are:

Soil Mechanics

Flow through a layered soil system (soils M over N, each 1000 mm² cross-section, per the given elevation figure) has head loss in N equal to 20× the head loss in M. If permeability of M = 3×10⁻⁴ mm/s, the permeability of N, in mm/s, is: *Figure (reconstructed): Layered soil column, uniform area 1000 mm², under vertical (series) flow — Soil M (upper layer, 250–400 mm elevation) over Soil N (lower layer, 0–250 mm elevation), total column 0–600/800 mm as marked on the source figure.* For flow in series through layered soils, Darcy velocity v is identical through both layers: v = k_M·i_M = k_N·i_N. Since head loss (and hence gradient, for comparable thickness) in N is 20× that in M, continuity of v requires k_N to be smaller.

Irrigation Water Conveyance and Application Methods

A trapezoidal canal with bottom width 5.0 m and side slope 1H:1V carries a discharge of 20 m³/s. The critical depth, in m, is:

Groundwater Hydrology

A 200 mm well fully penetrates a confined aquifer. After long pumping at 1400 L/min, drawdowns in observation wells at 25 m and 40 m are 2.6 m and 1.9 m respectively. The transmissivity of the aquifer, in m²/day, is:

Agricultural Drainage

Tile drains in soil of permeability 2.3×10⁻³ mm/s. Impermeable stratum at 3.2 m below surface; water table to be kept ≥1.0 m below surface. Average drainage discharge 2.0 mm/day. Drains placed at 1.5 m below surface (equivalent depth = tile depth). The drain spacing in m is: [note: marked "Marks to All" in the official key owing to an ambiguity/error in the original question]

Soil and Water Erosion

Bench terraces are to be constructed on a 10% hill slope with a batter slope of ½ H : 1 V. The percentage area lost for cultivation due to bench terracing is:

Evaporation and Drying

Air at 70°C, 0.015 humidity ratio, is cooled adiabatically by spraying water; final temperature 55°C. c_p(dry air)=1.005, c_p(vapour)=1.88 kJ/kg·K; latent heat at 0°C = 2501.7 kJ/kg. The absolute humidity of the outlet air, in kg water/kg dry air, is:

Evaporation and Drying

Osmotically dehydrated cherries, finish-dried from 18% (dry basis) to 11.5% (wet basis) moisture content, final mass flow rate 5000 kg/h. Dryer efficiency 70%, latent heat 2345 kJ/kg, c_p(air)=1.005 kJ/kg·K, drying temperature 50°C, specific volume of ambient air at 25°C = 0.866 m³/kg. The air flow requirement, in m³/min, is: [note: marked "Marks to All" in the official key owing to an ambiguity/error in the original question]

Heat and Mass Transfer

A single-effect vacuum evaporator has 100 tubes of 25 mm diameter. 1000 kg/h milk feed at 15% TS is concentrated to 20% TS. Film coefficients either side of the tube: 5000 and 800 W/m²K. Thermal conductivity of 1.5 mm thick SS tube wall = 15 W/m·K. Latent heat under vacuum = 2309 kJ/kg. For 10°C ΔT across the tube wall, the height of each tube, in m, is:

Preservation of Food

1000 units of mixed fruit bar, each 100 g (surface area 0.01 m²), frozen from 70°C molten mass to -20°C frozen storage within 3 hours. c_p = 3.6 kJ/kg·K (before freezing, 70→0°C) and 1.97 kJ/kg·K (after freezing, 0→-20°C). Latent heat of crystallization = 250 kJ/kg. The cooling capacity required, in tons of refrigeration (TR), is:

Sources of Power

A diesel engine in dual-fuel mode (diesel pilot + producer gas primary) produces 3.5 kW at rated speed, coupled to a generator. Diesel consumed 460 mL/h, producer gas consumed 12.5 m³/h. Calorific values: diesel 35280 kJ/L (i.e., per litre), producer gas 3.97 MJ/m³. The brake thermal efficiency of the engine, in %, is:

Sources of Power

If generator efficiency is 90%, the maximum electricity produced (from Q.48's engine), in kW, is:

Hydrology

Hourly discharge at a watershed outlet, due to 2 cm excess rainfall (0–1h) and 3 cm excess rainfall (1–2h), constant base flow 1 m³/s: *Table (reproduced from source): Hourly discharge hydrograph*

Hydrology

The peak of the 1-h unit hydrograph, in m³/s, for the watershed of Q.50, and its time of occurrence in h, respectively are:

Farm Machinery

Soybean is planted with a precision planter metering 54 seeds per revolution of the metering disc, powered from a ground wheel of diameter 490 mm. Desired plant population is 44800/ha, row spacing 0.75 m, germination 84%. Planter operated at 2.5 km/h with 10% wheel skid. The angular speed of the ground wheel, in rpm, is:

Farm Machinery

For the planter of Q.52, the angular speed ratio of metering disc to ground wheel for obtaining the desired plant population is:

Preservation of Food

A 1 hp motor runs a dual-cylinder reciprocating compressor (R-134a), cooling capacity 185 kJ/kg, COP 4.2, compressor overall efficiency 80%. Specific volume of refrigerant vapour at suction = 0.15 m³/kg. Bore = 40 mm each, N=1440 rpm. The mass flow rate of refrigerant, in kg/min, is:

Preservation of Food

For the compressor of Q.54, the stroke length, in mm, is:

Verbal Aptitude

Choose the most appropriate alternative to complete the sentence: "I ___ to have bought a diamond ring."

Analytical & Spatial Aptitude

Choose the most appropriate alternative: "Food prices ___ again this month."

Analytical & Spatial Aptitude

"The administrators went on to implement yet another unreasonable measure, arguing that the measures were already ___ and one more would hardly make a difference."

Analytical & Spatial Aptitude

"To those of us who had always thought him timid, his ___ came as a surprise."

Quantitative Aptitude

The arithmetic mean of five different natural numbers is 12. The largest possible value among the numbers is:

Analytical & Spatial Aptitude

Two policemen A and B fire once each simultaneously at an escaping convict. P(A hits) = 3×P(B hits). P(convict not injured) = 0.5. The probability that B hits the convict is:

Quantitative Aptitude

Total runs scored by four cricketers P, Q, R, S in 2009 and 2010 are given. The player with the lowest percentage increase in total runs is:

Quantitative Aptitude

If a prime number on division by 4 gives a remainder of 1, then that number can be expressed as:

Quantitative Aptitude

Two points (4, p) and (0, q) lie on a straight line with slope 3/4. The value of (p - q) is:

Analytical & Spatial Aptitude

Passage on 19th-century theories of social evolution, inspired by belief in growing improvement of social institutions rather than by biology; social scientists took progress for granted and sought its laws and phases. Which inference follows with greatest accuracy? Social scientists: