GATE AG 2010 Question Paper & Solutions

Differential Equations

A particular solution of $\frac{d^2y}{dx^2} + 5\frac{dy}{dx} - 3y = 6$ is:

Differential Equations

The partial differential equation $\frac{\partial^2 u}{\partial x^2} - 7\frac{\partial^2 u}{\partial x \partial y} + 2\frac{\partial^2 u}{\partial y^2} = 0$ is said to be:

Farm Machinery

While carrying out tillage operations, negative slip is sometimes experienced with

Tractors and Power Tillers

A two-wheel drive tractor has a PTO speed of 540 rpm and it produces 35 kW net engine power. Corresponding torque available at PTO in N m will be

Farm Machinery

Raising the hitch on the implement frame of a pull type offset disk harrow without gauge wheel helps in

Fluid Mechanics

While deriving the Chezy formula for uniform flow, it is assumed that there is a balance between

Soil-Water-Plant Relationship

A cross regulator is usually provided

Hydrology

An effective rainfall of 20 mm h⁻¹ occurs for 2 hours in a catchment. The time of concentration of the catchment is 1.5 hour. The peak of the resulting direct runoff hydrograph, in mm h⁻¹, is

Heat and Mass Transfer

The dimensionless number in heat transfer corresponding to Sherwood Number in mass transfer is

Engineering Properties of Agriculture Produce

The interrelationship between thermal conductivity, dynamic viscosity and temperature of gas can be described as

Linear Algebra

A system of equations $\begin{bmatrix} 1 & -1 & 2 \\ 2 & 1 & -4 \\ 1 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix}$ is

Probability and Statistics

There is a significant difference between scores from two groups if

Numerical Methods

The error in using trapezoidal rule for finding the value of $\int_{0}^{1} \frac{dx}{1+x}$ is:

Sources of Power

A farmer constructed a 2 m³ 40 days HRT Deenbandhu biogas plant used solely for cooking, burner efficiency 45%. Gas density 0.94 kg m⁻³, heating value 21 MJ kg⁻¹. Total effective energy available per day in MJ will be

Farm Machinery

A 2×0.3 m tractor drawn mould board plough operating at depth 0.15 m has draft 2.5 kN at forward speed 3 km h⁻¹, field efficiency 75%. Speed increased by 20%, draft increased by 10%. Assuming field efficiency, soil pulverization and soil inversion unchanged, the performance index of the plough increases by

Farm Machinery

While evaluating a stationary power thresher for wheat, grain:straw ratio 45%, moisture content (dry basis) 14%, over 5 min: grain at main outlet = 16 kg; clean grain at bhusa outlet = 0.3 kg; clean grain at sieve underflow+overflow = 0.2 kg; unthreshed grain (all outlets) = 0.5 kg. Percentage of blown and spilled grain are

Hydrology

The Local Apparent Time (LAT) corresponding to 14 h 30' IST at a place in India (19°07'N, 72°51'E) in April with time correction of zero min will be

Irrigation Water Conveyance and Application Methods

Two advance curves M and N for surface irrigation are shown (M slower than N). This could be attributed to: P. lower inflow rate; Q. lower intake rate of soil; R. flatter field slope; S. greater hydraulic roughness for N than M. Figure (reconstruction, not to exact original scale): advance curves M and N vs distance

Soil and Water Erosion

A land survey is conducted on 40 m × 40 m grids; elevations (m, AMSL) of grid points A,B,C (rows 1,2,3) are: Row1: A=102.3, B=103.0, C=103.7 Row2: A=101.5, B=102.4, C=102.3 Row3: A=101.2, B=103.5, C=102.6 Assuming cut = fill, the volume of earthwork required to level the area in m³ is

Soil and Water Erosion

For hydrologic design, the entire runoff hydrograph should be known in case of

Soil and Water Erosion

For a watershed, rainfall erosivity index R=1000 MJ mm ha⁻¹h⁻¹yr⁻¹, soil erodibility index K=0.25 Mg ha h ha⁻¹MJ⁻¹mm⁻¹, crop management factor C=0.75, conservation practice factor P=1.0, slope length factor LS=0.2. If P is reduced to 0.7, the reduction in soil loss, in Mg ha⁻¹yr⁻¹ is

Preservation of Food

Eight log cycle reduction of Clostridium botulinum having z-value of 9°C needs a process time of 1.5 minute at 121°C. The same degree of reduction at 130°C temperature will require a process time of

Preservation of Food

Let m, n and p be the numbers of carbon, hydrogen and fluorine atoms in a refrigerant. The identification number of the refrigerant is

Preservation of Food

A household refrigerator of 1 TR capacity operates half the time during 13-hour long days and 30% time during the nights. COP = 4.7, Rs. 3 per kWh, monthly (30 days) electricity bill in Rupees for the refrigerator is

Engineering Properties of Agriculture Produce

Air at 40°C has partial vapour pressure 2.4 kPa. Universal gas constant 8.314 kJ kg-mole⁻¹K⁻¹, total pressure 101.325 kPa. Humid volume of air in m³ (kg dry air)⁻¹ is

Vector Calculus

The curl of the vector A = xy i + yz j + zx k is

Calculus

The angle of intersection between the planes x − 3y + 2z = 10 and 2x + 4y + 5z = 0 is

Differential Equations

The Laplace transformation of t³e⁴ᵗ is

Calculus

The derivative of $y = \sqrt{x + \sqrt{x + \sqrt{x + \dots}}}$ with respect to $x$ at $y = 0$ is:

Calculus

The value of $\int_{0}^{2} \int_{0}^{y} xy \, dx \, dy$ is:

Machine Design

A thresher requires torque (5000+500 sinα) N m to drive, where α is shaft rotation angle. Directly coupled engine produces torque (5000+600 sin2α) N m. Flywheel and rotary parts have mass 500 kg at radius of gyration 0.4 m. Maximum angular acceleration of the flywheel in rad s⁻² will be

Machine Design

The tractor seat vibrates with a frequency of 1 Hz when there is no damping. When damping is provided, the frequency of damped vibration is reduced by 10%. The damping factor is

Farm Machinery

A disk mower operated by tractor PTO has six discs, swath 0.4 m per disk. Specific energy for cutting is 2.1 kJ m⁻²; specific power losses (air, stubble, gear train friction) are 2 kW m⁻¹ of cutting width. Mower with tractor requires a propelling force of 2 kN. Total power required for mowing in kW at forward speed 3 km h⁻¹ is

Tractors and Power Tillers

In a tractor differential, the pinion on the propeller shaft has 12 teeth and the crown gear has 60 teeth. The propeller shaft rotates at 1000 rpm and the right rear axle rotates at 210 rpm while taking a left turn. The rotation of the left rear axle in rpm will be

Farm Machinery

Match all items in Group I with correct options from Group II: | Group I (Mechanism) | Group II (Application) | |---|---| | i. Slider crank mechanism | a. Tractor steering | | ii. Four bar linkage mechanism | b. Attachment of pitman to knife head | | iii. Ball and socket joint | c. Planting unit of rice transplanter | | iv. Cam and follower mechanism | d. Vertical conveyor reaper |

Soil Mechanics

The mass of a 3.0 mm crumbled soil thread is 17.5×10⁻³ kg. On oven-drying, the mass reduces to 14.9×10⁻³ kg. The liquid limit of the soil sample is 35.4%. The plasticity index of the soil sample is

Fluid Mechanics

A pipeline carrying 500 litres/min branches into two parallel pipes X (L=500 m, D=0.2 m) and Y (L=500 m, D=0.3 m) reconnecting downstream. Friction factor f=0.030 for all pipes. The ratio of flow in pipes X and Y is Figure (reconstruction): pipeline M splitting into parallel pipes X and Y, rejoining at N

Wells and Pumps

A pump in an existing irrigation system delivers 3200 L/min at total head 60.0 m, impeller diameter 0.26 m, rotated at 1800 rpm, output shaft power 54 kW. The system is modified so discharge pressure requirement reduces to 52.0 m at the same flow rate. If the existing pump is utilized (impeller trimmed), the new impeller diameter in m will be

Soil Mechanics

An irrigator estimates the depth of infiltration (mm) around a field on a 5×4 grid: 42,36,32,38 / 40,32,35,34 / 25,28,29,31 / 36,30,32,28 / 40,38,34,44. The distribution uniformity for the irrigation is

Irrigation Water Conveyance and Application Methods

A parabolic shaped grassed waterway has top width 4 m, maximum depth 0.40 m, slope 2.5%, Manning's n = 0.035, no freeboard. The discharge carrying capacity of the waterway in m³ s⁻¹ is

Hydrology

Irrigation interval 20 days, gross application 75 mm, application efficiency 60%, homogeneous soil K=0.9 m/day, impermeable layer 7 m below ground, tile drains at 2 m depth, maximum permissible steady-state mid-way water table height above drain plane is 1.2 m. Using Hooghoudt's steady-state approach with equivalent depth 4.12 m, drain spacing in m will be

Wells and Pumps

A tubewell in a confined aquifer has diameter 0.30 m. For a certain yield, radius of influence is 400 m. All conditions remaining the same, if the diameter of the well is doubled, the percentage increase in yield is

Heat and Mass Transfer

The heating surface of an oven has emissivity 0.7, area 0.1 m², maintained at 280°C. View factor of this surface w.r.t. a piece of bread of 0.01 m² area is 0.05. Bread has emissivity 0.3 and receives 10 W of energy through radiation from the heating surface (σ=5.67×10⁻⁸ W m⁻²K⁻⁴). The steady state bread surface temperature in °C is

Preservation of Food

In parboiling, water:paddy ratio is 1.2. Water (cp=4.2 kJ kg⁻¹K⁻¹) is heated from 25°C to 85°C by condensation of steam (latent heat 2114 kJ kg⁻¹) via a tubular heat exchanger. 1 ton paddy at 30°C is poured into the hot water; mixture stabilizes at 75°C. Assuming no heat loss: P. steam supplied is 431 kg; Q. specific heat of paddy is 1.12 kJ kg⁻¹K⁻¹; R. steam supplied is 143 kg; S. specific heat of paddy is 2.11 kJ kg⁻¹K⁻¹

Heat and Mass Transfer

Density, specific heat capacity and thermal conductivity of air are 0.99 kg m⁻³, 1 kJ kg⁻¹K⁻¹, 0.03 W m⁻¹K⁻¹. Convective heat transfer coefficient of air medium and equimolar counter-diffusion mass transfer coefficient of water vapour into air are 35 W m⁻²K⁻¹ and 0.32 m s⁻¹. The mass diffusivity of water vapour into air in m² s⁻¹ is

Evaporation and Drying

20% sucrose solution is boiled and frozen separately. Latent heat of vapourization at 100°C = 2257 kJ kg⁻¹, latent heat of crystallization at 0°C = 334 kJ kg⁻¹. The ratio of freezing point depression to boiling point elevation is

Evaporation and Drying

Carrot slices of 2 mm thickness are freeze dried from initial free moisture content 80% (wet basis) to final free moisture content 2% (wet basis). Mass density of fresh carrot is 1100 kg m⁻³. Thermal conductivity of dried layer is 0.005 W m⁻¹K⁻¹. Latent heat of sublimation at −35°C is 2840 kJ kg⁻¹, product surface temperature is −5°C. Total drying time in hour is

Farm Machinery

A field sprayer with 16 fan-type nozzles spaced 0.5 m apart moves at 3.5 km h⁻¹, application rate 1 m³ ha⁻¹. The discharge rate per nozzle in m³ h⁻¹ will be

Farm Machinery

At a deposition level 430 mm below the nozzle tip, discharge across a 0.2 m centre strip is constant (15 mL min⁻¹ per 10 mm lateral distance); on each side it decreases uniformly to zero at 0.36 m from the nozzle centreline. The nozzle tip height in mm above the deposition level that would give uniform coverage will be

Evaporation and Drying

In a drying experiment on potato slices of 5 mm thickness, initial moisture 4.2 kg water (kg dry matter)⁻¹ reduced to 0.03 kg water (kg dry matter)⁻¹ using hot air at 65°C with absolute humidity 0.02 kg water vapour (kg dry air)⁻¹, saturation water vapour pressure 6 kPa. Critical moisture content of 2.5 kg water (kg dry matter)⁻¹ was reached after 3 hour of drying time. Dry matter concentration in the drying chamber was 5 kg per m² of surface area. The mass transfer coefficient in kg mole m⁻² s⁻¹ during drying is

Evaporation and Drying

Mass diffusivity of water vapour in m² s⁻¹ during the falling rate phase of drying is

Tractors and Power Tillers

[Linked 52–53] A 37 kW two-wheel drive tractor weighing 20 kN, wheelbase 2.1 m, may be fitted with 12.4–28 12PR or 13.6–28 12PR rear tyres (section height/width ratio 0.75 for both). Weight distribution: front 35%, rear 65%. Cone index of soil is 1200 kPa. The motion resistance ratio of each rear wheel at normal inflation pressure on level ground will be

Tractors and Power Tillers

[Linked 52–53] Net traction developed in kN by the rear wheels when fitted with 13.6–28 12PR tyre at normal inflation pressure on a level ground with 15% wheel slip will be

Hydrology

[Linked 54–55] The peak of a flood hydrograph due to a 1-h duration isolated storm in a catchment of area 13.5 km² is 135 m³ s⁻¹. Total depth of rainfall is 54 mm. Assume constant base flow 10 m³ s⁻¹ and phi-index 4 mm h⁻¹. The peak of the 1-h unit hydrograph for the catchment in m³ s⁻¹ is

Hydrology

[Linked 54–55] Assuming the above 1-h unit hydrograph to be triangular in shape with time to peak 1 hour, the peak of the 2-h unit hydrograph for the catchment in m³ s⁻¹ is

Farm Power

Choose the most appropriate word: His rather casual remarks on politics ______ his lack of seriousness about the subject.

Farm Power

Which of the following options is closest in meaning to: Circuitous

Farm Power

Choose the most appropriate word: If we manage to ______ our natural resources, we would leave a better planet for our children.

Farm Power

25 persons are in a room. 15 of them play hockey, 17 play football and 10 play both. The number of persons playing neither hockey nor football is

Farm Power

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: Unemployed : Worker

Farm Power

If 137 + 276 = 435, how much is 731 + 672?

Farm Power

Hari (H), Gita (G), Irfan (I) and Saira (S) are siblings, all born on 1st January; the age difference between any two successive siblings is less than 3 years. i. Hari's age + Gita's age > Irfan's age + Saira's age. ii. The age difference between Gita and Saira is 1 year; Gita is not the oldest and Saira is not the youngest. iii. There are no twins. In what order were they born (oldest first)?

Farm Power

"Modern warfare has changed from large scale clashes of armies to suppression of civilian populations. Chemical agents that do their work silently appear to be suited to such warfare; and regretfully, there exist people in military establishments who think that chemical agents are useful tools for their cause." Which statement best sums up the meaning of the above passage?

Farm Power

5 skilled workers can build a wall in 20 days; 8 semi-skilled workers can build a wall in 25 days; 10 unskilled workers can build a wall in 30 days. If a team has 2 skilled, 6 semi-skilled and 5 unskilled workers, how long will it take to build the wall?

Farm Power

Given digits 2,2,3,3,3,4,4,4,4 how many distinct 4-digit numbers greater than 3000 can be formed?