GATE AG 2011 Question Paper & Solutions

Differential Equations

The differential equation 2(d²z/dx²) + dz/dx + 3y = sin x is considered to be ordinary, as it has

Linear Algebra

A square matrix $A$ will be lower triangular if and only if ($a_{mn}$ represents an element of $m^{\text{th}}$ row and $n^{\text{th}}$ column of the matrix)

Farm Machinery

The metering mechanism of a seed drill is driven by the ground wheels at a velocity ratio of 1:2. When forward speed of the seed drill is increased from 3.0 to 3.45 km h⁻¹, the seed rate would

Farm Machinery

A maize planter drops seeds at 0.20 m interval. The seed weight is 200 g per 1000 seeds. If the row to row spacing is 0.25 m, the seed rate in kg ha⁻¹ is

Tractors and Power Tillers

Which one of the following is NOT a towed wheel?

Fluid Mechanics

If the density of a fluid changes from point to point in a flow region, the flow is called

Soil Mechanics

Capillary water is held in the soil due to

Hydrology

The wedge storage in a river reach during the passage of a flood wave is

Fluid Mechanics

The viscosity of a newtonian fluid depends primarily on X and to a lesser degree on Y. X and Y are

Fluid Mechanics

Milk is agitated in a tank with a rotating impeller. For this system: X = P/(ρN³D⁵), Y = D²Nρ/μ, Z = DN²/g, where P = power imparted by the impeller to the fluid, N = rate of rotation, D = impeller diameter, g = acceleration due to gravity, ρ = fluid density, μ = fluid viscosity. The X, Y, Z are

Numerical Methods

The highest order of polynomial integrand for which Simpson's 1/3 rule of integration is exact is

Calculus

The mean value of a function $f(x)$ from $x = a$ to $x = b$ is given by:

Probability and Statistics

A statistical measure of the variability of a distribution around its mean is referred to as

Farm Power

An engine is to be run in dual fuel mode using diesel and producer gas with diesel as pilot fuel. Operation NOT required while running the engine is

Tractors and Power Tillers

The range of frequency of vertical vibration of tractor most harmful to the operator's body at a root mean square acceleration of 1.0 m s⁻² in Hertz is

Farm Machinery

The Sauter mean diameter of liquid droplets of a hydraulic spray is

Sources of Power

A horizontal axis windmill having 8 blades is used for pumping water. Each blade has a tip radius of 1.0 m and a mean chord of 0.1 m. Assuming blade length equal to tip radius, solidity of the windmill is

Soil Mechanics

A completely saturated clay soil has particle density of 2600 kg m⁻³ and bulk density of 1400 kg m⁻³. It has a moisture fraction of 40% on volume basis. The porosity of the soil is

Farm Power

The magnetic bearing of a line at a station point is found to be 182°. The magnetic declination at the station is 3° E and the local attraction is −1° for correction. The true bearing of the line is

Soil and Water Erosion

In wind erosion process, the rate of soil movement (S) depends on wind speed (v), threshold limit of wind speed required to move the soil (vm) and average soil particle size (d). This interrelationship is expressed as

Hydrology

For a catchment with an area of 400 km², the equivalent discharge of the S-curve obtained by summation of 4-h unit hydrograph in m³ s⁻¹ is

Heat and Mass Transfer

A material having thermal conductivity k insulates a spherical object of diameter

Storage Systems

For a mass of grain stored in a bin, the angle of internal friction of the grain is 30°. The ratio of normal pressure to the applied pressure within the bin is

Heat and Mass Transfer

In a counter current concentric tube heat exchanger, the hot fluid enters at 360 K and leaves at 340 K. The cooling fluid enters at 300 K and leaves at 316 K. The logarithmic mean temperature difference in K is

Heat and Mass Transfer

A constant heat flux of 500 W m⁻² is supplied to one face of a food material having a plate like structure with a thickness of 10 mm. The thermal conductivity of the food material is 1.5 W m⁻¹ °C⁻¹. From the other face of the food material, heat is dissipated by convection into a fluid of 40 °C temperature. The heat transfer coefficient of the fluid is 100 W m⁻² °C⁻¹. The temperature in °C of the surface to which the heat flux is supplied will be

Calculus

The stationary points of $f(x,y) = \frac{1}{3}x^3 - xy^2 - 2y$ are:

Vector Calculus

The divergence of the vector F = sin(xy)i + y cos(z)j + xz cos(z)k (i, j and k represent unit vectors along the three orthogonal axes) at x = y = 0 is

Calculus

A plane contains the following three points: P(2,1,5), Q(−1,3,4) and R(3,0,6). The vector perpendicular to the above plane can be represented as

Differential Equations

The solution of $x\frac{dy}{dx} = x^2 + 3y$ for $x > 0$ and $y(1) = 2$ is:

Farm Power

A two cylinder four stroke diesel engine develops 15 kW power at 2400 rpm. Its brake specific fuel consumption is 0.268 kg kW⁻¹ h⁻¹. If the specific gravity of fuel is 0.85, quantity of fuel injected per cylinder per cycle in m³ is

Farm Machinery

A hydraulic spray nozzle has a discharge of 450 ml min⁻¹ at a pressure of 280 kPa. If the pressure is increased by 10%, the discharge will be

Farm Machinery

In a cutter bar mower, the cutter bar drive consists of an offset slider crank mechanism with a crank of radius 36 mm and a pitman of length 360 mm. If offset of the crank is 72 mm, the stroke length in mm would be

Farm Machinery

The cost of a tractor is Rs. 340000. Its service life is 10 years and the salvage value is 10% of initial cost. Its depreciation in 2nd year, following constant rate method, in rupees would be

Farm Machinery

The resultant soil reaction force acting on a single bottom mould board plough has a component of 1200 N acting at an angle of 31° downwards in a vertical plane. This plane is along the direction of travel. Weight of the plough is 620 N. Neglecting the side forces, the magnitude of pull in N would be

Fluid Mechanics

A concrete-lined non-circular tunnel has a semi-circle at the top and a rectangular section at the bottom. The semi-circle has a diameter of 6 m whereas the rectangular section is 6 m wide and 3 m high. The tunnel carries a discharge of 128 m³ s⁻¹. If the friction factor f = 0.017, then the head loss in 1 km length of the tunnel in m will be

Soil Mechanics

In a falling head permeameter test, the initial head is 0.30 m. The head drops to 0.10 m in 40 min. The permeability of a soil sample, 0.06 m high and 50 × 10⁻⁴ m² in cross-sectional area, is found to be 1.0 × 10⁻⁶ m s⁻¹. The size of the stand pipe in m² is

Hydrology

A 100 ha reservoir receives 2500 mm of rainfall during a period of 2 years. During this period the mean inflow to the stream is 1.0 m³ s⁻¹, the mean outflow from the stream is 0.8 m³ s⁻¹, and the increase in the storage is 500 ha-m. Assuming that there is no seepage loss, the total evaporation during the period in m is

Hydrology

The following figure presents the hyetograph for a 3-hour storm. The surface runoff for the event is estimated to be 38 mm. The phi-index for the event in mm h⁻¹ is Figure (reconstructed): Hyetograph (reconstructed): six 30-min bars at rainfall intensities 10, 30, 40, 50, 25, 8 mm/h over the 180-minute storm, as given in the original question paper.

Agricultural Drainage

A crop grown over an area of 50 ha can tolerate 5 decisiemens m⁻¹ of electrical conductivity in the drainage water. The consumptive use of the crop is 1.0 m, 30% of which is obtained from the rainfall and the remainder is met from the irrigation water having electrical conductivity of 2 decisiemens m⁻¹. For efficient crop production, the leaching requirement and the quantity of water that must be drained from the area are

Groundwater Hydrology

A fully penetrating tubewell of 0.2 m diameter is operational in a confined aquifer of 20 m thickness. The hydraulic conductivity of the formation (K) is 20 m per day and the radius of influence is 1000 m. Water table is 50 m above the bottom of screen and the maximum drawdown is 10 m. Subsequently two more tubewells of the same size are installed in such a way that the three tubewells form an equilateral triangle of side 300 m. All three tubewells have equal discharge when operated independently. If all tubewells are operated simultaneously, the percent loss in the discharge of the first well is

Soil-Water-Plant Relationship

For the eastern Himalayan region having hill slope of 16%, a bench terrace system is to be designed. It is found that the depth of cut for constructing the bench terrace cannot be more than 0.40 m due to the limitation of the soil depth. If a batter slope of 1/2 : 1 is proposed, the maximum possible width of the terrace and the required earthwork per hectare (assuming cut equal to fill) will be

Engineering Properties of Agriculture Produce

The sieve analysis of a finely ground rice powder is represented by a straight line from zero percent mass at 1 μm particle size to 100 percent mass at 101 μm particle size. The volume-surface mean diameter in μm is

Heat and Mass Transfer

A plate shaped frozen food has a thickness of 20 mm, average thermal conductivity of 2.58 W m⁻¹ °C⁻¹, density of 1080 kg m⁻³, specific heat of 3550 J kg⁻¹ °C⁻¹, and a uniform temperature of −20 °C. The food material is suddenly immersed in a well stirred hot water maintained at a constant temperature of 90 °C. The heat transfer coefficient between the food material and hot water is 25 W m⁻² °C⁻¹. The time required for the centre temperature of the food material to reach 30 °C in minutes is

Evaporation and Drying

300 kg of parboiled and wet paddy having an initial moisture content of 28% (dry basis) was dried in a batch type dryer. During drying operation, grain sample was drawn at regular time intervals to measure its moisture content (dry basis). It was found that the moisture content of the grain borne a linear relationship with drying time till 30 minutes of drying operation at which moisture content was measured to be 20%. The drying rate (kg h⁻¹) during this interval was

Preservation of Food

A mixture of microorganisms, P, Q, R, and S are to be inactivated by autoclaving. Number of spores of the above organisms is 6 × 10⁵, 4 × 10⁶, 2 × 10⁴, and 9 × 10⁵ respectively. Further, their decimal reduction time

Size Reduction and Material Handling

The average interstitial velocity in a packed bed is 2.5 m s⁻¹ and the superficial velocity under minimum fluidized condition based on the cross-section of the empty container is 1 m s⁻¹. Solid particles having density of 1200 kg m⁻³ and size of 0.15 mm are to be fluidized using air at 3 atm absolute pressure and 30 °C temperature having a density of 3.5 kg m⁻³. The empty bed cross section is 0.4 m² and the bed contains 400 kg of solid. Pressure drop at minimum fluidized condition in Pa is

Engineering Properties of Agriculture Produce

Sphericity of a particle is defined as: (surface to volume ratio for a sphere having identical volume as that of the particle) / (the surface to volume ratio of the particle). The measure of sphericity of a cube shaped sugar crystal is

Irrigation Water Conveyance and Application Methods

A 5 ha field under wheat crop is irrigated at 40% depletion of available moisture content. The field capacity, wilting point and bulk density of the soil in the field are 32% (on weight basis), 20% (on weight basis) and 1400 kg m⁻³ respectively. To irrigate the field in a day of 10 hours, a pump is used which lifts 270 m³ of water per hour against a total head of 20 m. If the root zone depth is 0.8 m, the volume of water required to irrigate the field in m³ will be

Irrigation Water Conveyance and Application Methods

The pump is driven by an electric motor. If the pump, drive and motor efficiencies are 80%, 82% and 90% respectively, the required size of electric motor in horse-power will be

Machine Design

A vertical hydraulic cylinder having an inside diameter of 100 mm is used in a hoist for lifting load. A constant flow rate pump capable of delivering 12 litre per minute at a maximum pressure of 18 MPa is used for supplying the fluid. Volumetric and mechanical efficiencies of the pump are 91% and 85% respectively. The rate of lifting of load in m s⁻¹ and maximum amount of load that can be lifted in kN are

Machine Design

The size of motor required for operating the pump in kW is

Tractors and Power Tillers

A towed pneumatic wheel (width to diameter ratio of 0.3) is to be rolled on a leveled concrete surface. Total wheel load is 2000 N. The force in N required to roll the wheel on the horizontal concrete surface would be

Tractors and Power Tillers

The minimum slope in degrees of the concrete surface with respect to the horizontal at which the wheel itself will start rolling downward is

Evaporation and Drying

A vegetable with 70% moisture (wet basis) is to be dried in a countercurrent dryer to give a product with 5% moisture (wet basis). The feed rate to the dryer is 0.15 kg s⁻¹. The drying medium consists of air at 100 °C containing water vapour with a partial pressure of 1.0 kPa. The air leaves the dryer at 40 °C and is 70% saturated. The system works under a total pressure of 101.3 kPa. The vapour pressure of water that saturates air at 40 °C is 7.4 kPa. Molecular weight of water and air may be taken as 18 and 29 respectively. The humidity (kg of water vapour / kg of dry air) of the outlet air is

Evaporation and Drying

The flow rate (kg s⁻¹) of inlet air required is

Analytical & Spatial Aptitude

Choose the most appropriate word from the options given below to complete the following sentence: Under ethical guidelines recently adopted by the Indian Medical Association, human genes are to be manipulated only to correct diseases for which ______ treatments are unsatisfactory.

Verbal Aptitude

Choose the word from the options given below that is most nearly opposite in meaning to the given word: Frequency

Analytical & Spatial Aptitude

Choose the most appropriate word from the options given below to complete the following sentence: It was her view that the country's problems had been ______ by foreign technocrats, so that to invite them to come back would be counter-productive.

Quantitative Aptitude

There are two candidates P and Q in an election. During the campaign, 40% of the voters promised to vote for P, and rest for Q. However, on the day of election 15% of the voters went back on their promise to vote for P and instead voted for Q. 25% of the voters went back on their promise to vote for Q and instead voted for P. Suppose, P lost by 2 votes, then what was the total number of voters?

Analytical & Spatial Aptitude

The question below consists of a pair of related words followed by four pairs of words. Select the pair that best expresses the relation in the original pair: Gladiator : Arena

Analytical & Spatial Aptitude

The sum of n terms of the series 4+44+444+.... is

Quantitative Aptitude

Given that f(y) = |y| / y, and q is any non-zero real number, the value of |f(q) − f(−q)| is

Analytical & Spatial Aptitude

Three friends, R, S and T shared toffee from a bowl. R took 1/3rd of the toffees, but returned four to the bowl. S took 1/4th of what was left but returned three toffees to the bowl. T took half of the remainder but returned two back into the bowl. If the bowl had 17 toffees left, how many toffees were originally there in the bowl?

Analytical & Spatial Aptitude

The fuel consumed by a motorcycle during a journey while traveling at various speeds is indicated in the graph below [reproduced as a table below]. The distances covered during four laps of the journey are listed in the table below. From the given data, we can conclude that the fuel consumed per kilometre was least during the lap Figure (reconstructed): Fuel-consumption-vs-speed curve (reconstructed): an inverted-V-shaped curve peaking near 45 km/h at about 90 km/L, rising from about 30 km/L at 10 km/h and falling to about 75 km/L at 75 km/h, as given in the original question paper; plus the lap distance/speed data table.

Verbal Aptitude

The horse has played a little known but very important role in the field of medicine. Horses were injected with toxins of diseases until their blood built up immunities. Then a serum was made from their blood. Serums to fight with diphtheria and tetanus were developed this way. It can be inferred from the passage, that horses were