GATE Technical questions with solutions

Q1. The three-dimensional state of stress at a point is given by σ = [[10, 0, 0],[0, 40, 0],[0, 0, 0]] MPa. The maximum shear stress at the point is

1. 20 MPa
2. 15 MPa
3. 5 MPa
4. 25 MPa

Answer: 20 MPa

The matrix is diagonal, so principal stresses are 40, 10 and 0 MPa. Maximum shear stress = (sigma_max - sigma_min)/2 = (40 - 0)/2 = 20 MPa, not 25 MPa.

Q2. A 2 m wide strip footing is founded at a depth of 1.5 m below the ground level in a homogeneous pure clay bed. The clay bed has unit cohesion of 40 kPa. Due to seasonal fluctuations of water table from peak summer to peak monsoon period, the net ultimate bearing capacity of the footing, as per Terzaghi’s theory, will

1. remain the same
2. decrease
3. increase
4. become zero

Answer: remain the same

The net ultimate bearing capacity of the footing remains the same because it is determined by the soil's cohesion and the depth of the footing, which are not affected by seasonal fluctuations in the water table in a homogeneous clay bed.

Q3. Consider the statements P and Q. P: Soil particles formed by mechanical weathering, and close to their origin are generally subrounded. Q: A activity of the clay physically signifies its swell potential. Which one of the following options is CORRECT?

1. Both P and Q are TRUE
2. P is TRUE and Q is FALSE
3. Both P and Q are FALSE
4. P is FALSE and Q is TRUE

Answer: P is FALSE and Q is TRUE

Statement P is incorrect because soil particles formed by mechanical weathering are typically angular rather than subrounded, as they have not undergone significant transport. Statement Q is correct as the physical activity of clay, such as its ability to swell, is a key indicator of its swell potential.

Q4. The number of degrees of freedom for a natural open channel flow with a mobile bed is

1. 2
2. 3
3. 4
4. 5

Answer: 4

In natural open channel flow with a mobile bed, the degrees of freedom account for the flow's velocity, water surface elevation, bed elevation, and the position of the sediment bed, leading to a total of four degrees of freedom.

Q5. The following table gives various components of Municipal Solid Waste (MSW) and a list of treatment/separation techniques. Component of MSW P - Ferrous metals Q - Aluminum and copper R - Food waste S - Cardboard Treatment/separation technique i - Incineration ii - Rapid composting iii - Eddy current separator iv - Magnetic separator The CORRECT match is

1. P-iii, Q-iv, R-i, S-ii
2. P-iv, Q-iii, R-ii, S-i
3. P-iii, Q-iv, R-ii, S-i
4. P-iv, Q-iii, R-i, S-ii

Answer: P-iv, Q-iii, R-ii, S-i

The correct option matches each component of Municipal Solid Waste with the most effective treatment or separation technique: ferrous metals (P) are best separated using a magnetic separator (iv), aluminum and copper (Q) are effectively separated with an eddy current separator (iii), food waste (R) is suited for rapid composting (ii), and cardboard (S) can be incinerated (i) for energy recovery.

Q6. A car is travelling at a speed of 60 km/hr on a section of a National Highway having a downward gradient of 2%. The driver of the car suddenly observes a stopped vehicle on the car path at a distance of 130 m ahead, and applies brake. If the brake efficiency is 60%, coefficient of friction is 0.7, driver's reaction time is 2.5 s, and acceleration due to gravity is 9.81 m/s², the distance (in meters) required by the driver to bring the car to a safe stop lies in the range

1. 126 to 130
2. 41 to 45
3. 33 to 37
4. 75 to 79

Answer: 75 to 79

The correct option is right because, after accounting for the driver's reaction time and the effective braking distance, the total stopping distance falls within the range of 75 to 79 meters, considering the car's initial speed, gradient, and braking efficiency.

Q7. As per the International Civil Aviation Organization (ICAO), the basic runway length is increased by x (%) for every y (m) rise in elevation from the Mean Sea Level (MSL). The values of x and y, respectively, are

1. 7% and 300 m
2. 5% and 200 m
3. 4% and 500 m
4. 10% and 1000 m

Answer: 7% and 300 m

The correct option is based on ICAO guidelines, which state that for every 300 meters of elevation above mean sea level, the basic runway length should be increased by 7% to ensure safe aircraft operations in thinner air.

Q8. Which one of the following statements related to bitumen is FALSE?

1. Kinematic viscosity is a measure of resistance to the flow of molten bitumen under gravity.
2. Softer grade bitumen possesses higher softening point than hard grade bitumen.
3. Flash point of bitumen is the lowest temperature at which application of a test flame causes vapours of the bitumen to catch an instant fire in the form of flash under specified test conditions.
4. Ductility test is carried out on bitumen to test its adhesive property and ability to stretch.

Answer: Softer grade bitumen possesses higher softening point than hard grade bitumen.

Softer grade bitumen actually has a lower softening point compared to hard grade bitumen, as the terms 'soft' and 'hard' refer to the viscosity and temperature characteristics of the material.

Q9. If the number of sides resulting in a closed traverse is increased from three to four, the sum of the interior angles increases by

1. 90°
2. 180°
3. 270°
4. 360°

Answer: 180°

The sum of the interior angles of a polygon can be calculated using the formula (n-2) × 180°, where n is the number of sides. When increasing the number of sides from three to four, the increase in the sum of the interior angles is (4-2) × 180° = 2 × 180° = 360°, but since we are comparing the difference from a triangle (180°) to a quadrilateral (360°), the increase is 180°.

Q10. A surveyor observes a zenith angle of 93° 00' 00" during a theodolite survey. The corresponding vertical angle is

1. −03° 00' 00"
2. +03° 00' 00"
3. −87° 00' 00"
4. +87° 00' 00"

Answer: −03° 00' 00"

The zenith angle is measured from the vertical down to the line of sight, so an angle of 93° indicates that the line of sight is 3° below the horizontal plane, resulting in a vertical angle of -3°.

Q11. The elements that DO NOT increase the strength of structural steel are

1. Carbon
2. Manganese
3. Sulphur
4. Chlorine

Answer: Chlorine

Chlorine does not contribute to the strength of structural steel and can actually be detrimental, as it may lead to corrosion and brittleness, unlike carbon and manganese which enhance strength and toughness.

Q12. Consider a balanced doubly-reinforced concrete section. If the material and other sectional properties remain unchanged, for which of the following cases will the section become under-reinforced?

1. Area of tension reinforcement is increased.
2. Area of compression reinforcement is increased.
3. Area of tension reinforcement is decreased.
4. Area of compression reinforcement is decreased.

Answer: Area of tension reinforcement is decreased.

Decreasing the area of tension reinforcement reduces the tensile capacity of the section, which can lead to a situation where the concrete can fail in compression before the steel yields, thus making the section under-reinforced.

Q13. The primary air pollutant(s) is/are

1. Sulphur dioxide
2. Lead
3. Ozone
4. Sulphuric acid

Answer: Sulphur dioxide

Sulphur dioxide is considered a primary air pollutant because it is directly emitted from sources such as burning fossil fuels and industrial processes, contributing significantly to air quality issues and acid rain.

Q14. A flow velocity field V: V (x,y) for a fluid is represented by V = 3 î + (5x) ĵ. In the context of the fluid and the flow, which one of the following statements is CORRECT?

1. The fluid is incompressible and the flow is rotational.
2. The fluid is incompressible and the flow is irrotational.
3. The fluid is compressible and the flow is rotational.
4. The fluid is compressible and the flow is irrotational.

Answer: The fluid is incompressible and the flow is rotational.

div V = d(3)/dx + d(5x)/dy = 0, so the fluid is incompressible. curl_z = d(5x)/dx - d(3)/dy = 5 != 0, so the flow is rotational. Hence incompressible and rotational, option (a).

Q15. The free mean speed is 60 km/hr on a given road. The average space headway at jam density on this road is 8 m. For a linear speed-density relationship, the maximum flow (in veh/hr/lane) expected on the road is

1. 1875
2. 938
3. 2075
4. 1038

Answer: 1875

Jam density kj = 1000/8 = 125 veh/km. For the linear Greenshields model the maximum flow is q_max = vf*kj/4 = 60*125/4 = 1875 veh/hr/lane. The stored value 938 (which is half of that) is wrong; the answer is index 0, 1875.

Q16. A map is prepared with a scale of 1:1000 and a contour interval of 1 m. If the distance between two adjacent contours on the map is 10 mm, the slope of the ground between the adjacent contours is

1. 30%
2. 10%
3. 35%
4. 40%

Answer: 10%

The slope is calculated by dividing the vertical rise (1 m) by the horizontal distance represented on the map. Given the scale of 1:1000, 10 mm on the map corresponds to 10 m in reality, resulting in a slope of 1 m rise over 10 m run, which simplifies to 10%.

Q17. Which of the following statement(s) is/are CORRECT?

1. Swell potential of soil decreases with an increase in the shrinkage limit.
2. Both loose and dense sands with different initial void ratios can attain similar void ratio at large strain during shearing.
3. Among the several corrections to be applied to the SPT-N value, the dilatancy correction is applied before all other corrections.
4. In electrical resistivity tomography, the depth of current penetration is half of the spacing between the electrodes.

Answer: Both loose and dense sands with different initial void ratios can attain similar void ratio at large strain during shearing.

Loose and dense sands can reach a similar void ratio under large strain conditions due to the rearrangement of particles during shearing, which allows for a convergence in their structural behavior despite their initial differences.

Q18. Identify the FALSE statement from the following options:

1. The compressive strength of a concrete test specimen can vary depending on its shape and size
2. Air-dried and saturated test specimens show the same compressive strength for concrete
3. Curing conditions, such as temperature and relative humidity, can influence the compressive strength of concrete
4. Compressive strength depends on the water-to-binder ratio used in the concrete mixture

Answer: Air-dried and saturated test specimens show the same compressive strength for concrete

Air-dried and saturated test specimens do not exhibit the same compressive strength because the moisture content significantly affects the concrete's properties; saturated specimens generally demonstrate higher strength due to better hydration of the cement.

Q19. The results of a consolidated drained triaxial test on a normally consolidated clay are shown in the figure. The angle of internal friction is

1. sin⁻¹(1/2)
2. sin⁻¹(1/3)
3. sin⁻¹(2/3)
4. sin⁻¹(3/4)

Answer: sin⁻¹(1/3)

The angle of internal friction for normally consolidated clay can be derived from the effective stress parameters shown in the test results, where the correct value corresponds to the ratio of the effective stress components, leading to sin⁻¹(1/3) as the accurate representation.

Q20. To derive the total flood hydrograph at a catchment outlet from an isolated storm, the order in which the following methods are applied, from the first method to the last method, is P. Obtaining the hyetograph Q. Addition of baseflow R. Estimation of initial and infiltration losses S. Application of unit hydrograph

1. P R S Q
2. P Q R S
3. R P S Q
4. P S Q R

Answer: P R S Q

The correct order begins with obtaining the hyetograph to visualize rainfall data, followed by estimating initial and infiltration losses to determine effective rainfall. Next, the unit hydrograph is applied to convert effective rainfall into runoff, and finally, baseflow is added to obtain the total flood hydrograph.

Q21. A hydrocarbon (C_nHₘ) is burnt in air (O₂ + 3.78N₂). The stoichiometric fuel to air mass ratio for this process is Note: Atomic Wt. C (12), H (1) Effective Molecular Wt. Air (28.8) Ignore any conversion of N₂ in air to the oxides of nitrogen (NOₓ)

1. 0.0291 ((4n + m)/(12n + m))
2. 34.42 ((12n + m)/(4n + m))
3. 34.42 ((4n + m)/(12n + m))
4. 0.0291 ((12n + m)/(4n + m))

Answer: 0.0291 ((12n + m)/(4n + m))

The correct option is derived from the stoichiometric calculations of the combustion process, where the mass of the hydrocarbon is compared to the mass of the air required for complete combustion. The formula reflects the relationship between the molecular weights of the components involved, ensuring that the fuel-to-air ratio is accurately represented.

Q22. All the vehicles that come during a particular peak hour come during a 10-minute period within this hour. The 15-minute peak hour factor for this peak hour is

1. 0.25
2. 0.167
3. 0.75
4. 1.0

Answer: 0.25

All vehicles arrive in a 10-min window inside the hour, so the busiest 15 min already contains the entire hourly volume V. PHF = V/(4 x peak-15min) = V/(4V) = 0.25, so the stored 1.0 is wrong.

Q23. In the context of testing bitumen, which one of the following statements is FALSE:

1. The depth of penetration of needle in the standard penetration test is measured in the units of one-tenth of millimeter
2. Softening point is measured using a ring and ball apparatus
3. Softening point is measured in the units of time
4. Ductility is measured in the units of length

Answer: Softening point is measured in the units of time

The statement is incorrect because the softening point of bitumen is measured in degrees Celsius, not in units of time. This measurement indicates the temperature at which the bitumen softens enough to flow.

Q24. If the horizontal distance between a staff point and the point of observation is d, the error due to the curvature of earth is proportional to

1. d
2. 1/d
3. d²
4. 1/d²

Answer: d²

The error due to the curvature of the Earth increases with the square of the distance because the curvature affects the line of sight more significantly as the distance increases, leading to a quadratic relationship.

Q25. Consider the frame shown in the figure under the loading of 100 kN.m couples at the joints B and G. Considering only the effects of flexural deformations, which of the following statements is/are true:

1. Axial force is zero in the member CD
2. Shear force is zero in the member CD
3. There is no rotation in the joint C
4. The magnitude of bending moment developed in the member BC at the end C is more than 50 kN.m

Answer: The magnitude of bending moment developed in the member BC at the end C is more than 50 kN.m

The statement is correct because the application of couples at joints B and G induces bending moments in the members of the frame, particularly in member BC, resulting in a bending moment greater than 50 kN.m at end C.

Q26. For the Bernoulli's equation to be applicable in a fluid flow situation, which of the following conditions is/are to be satisfied:

1. Fluid should be frictionless
2. Fluid should be incompressible
3. Flow should be steady
4. Flow should be rotational

Answer: Fluid should be frictionless

Bernoulli's equation assumes that the fluid has no viscosity, meaning it behaves as a frictionless fluid. This allows for the conservation of mechanical energy along a streamline, which is a fundamental aspect of the equation.

Q27. The Surface Overflow Rate (SOR) in a rectangular sedimentation tank is 45 m³/m²/d. Minimum diameters of spherical inorganic and organic particles expected to be completely removed in this tank are calculated. A sum that Stokes’s law is applicable. Specific gravity of inorganic particles = 2.65 Specific gravity of organic particles = 1.20 Acceleration due to gravity (g) = 9.81 m/s² Kinematic viscosity (ν) = 1 × 10⁻⁶ m²/s Which of the following options is/are correct:

1. Minimum diameter of inorganic particles is 24 μm
2. Minimum diameter of organic particles is 69 μm
3. Minimum diameter of inorganic particles is 15 μm
4. Minimum diameter of organic particles is 55 μm

Answer: Minimum diameter of inorganic particles is 24 μm

The correct option states that the minimum diameter of inorganic particles is 24 μm, which is derived from applying Stokes's law to the sedimentation conditions in the tank. This calculation takes into account the specific gravity of the particles, the kinematic viscosity of the fluid, and the surface overflow rate, leading to the conclusion that particles of this size can be effectively removed.

Q28. A aeration is employed as a treatment option for the removal of several pollutants from contaminated water. Identify the pollutant(s), where aeration is employed as a part of their removal:

1. Iron
2. Cadmium
3. Manganese
4. Zinc

Answer: Iron

Aeration is effective in removing iron from contaminated water because it oxidizes dissolved ferrous iron (Fe2+) into ferric iron (Fe3+), which then precipitates out as solid particles that can be filtered out.

Q29. In the rigid-jointed frame shown in the figure, the distribution factor of the member AD is closest to

1. 0.254
2. 0.267
3. 0.398
4. 0.421

Answer: 0.267

The distribution factor for a member in a rigid-jointed frame is calculated based on the relative stiffness of that member compared to the total stiffness of the joint. In this case, the value of 0.267 indicates that member AD has a significant proportion of the joint's stiffness, reflecting its contribution to the overall structural behavior.

Q30. In an oedometer apparatus a specimen of fully saturated clay has been consolidated under a vertical pressure of 100 kPa and is at equilibrium state. Immediately on increasing the vertical pressure to 150 kPa, the effective stress σ' and excess pore water pressure Δu will be

1. σ' = 50 kPa, Δu = 100 kPa
2. σ' = 100 kPa, Δu = 50 kPa
3. σ' = 150 kPa, Δu = 50 kPa
4. σ' = 100 kPa, Δu = 150 kPa

Answer: σ' = 100 kPa, Δu = 50 kPa

When the vertical pressure is increased from 100 kPa to 150 kPa, the total stress increases by 50 kPa. Since the clay is fully saturated, this increase in total stress results in an equivalent increase in pore water pressure, leading to an excess pore water pressure of 50 kPa. The effective stress is then calculated as the total stress minus the pore water pressure, resulting in an effective stress of 100 kPa.