GATE MOCK TEST -1

1) A random variable X has the probability density function f(x) as given below: [a+bx for 0<x<1 f(x)= [0, otherwise if the expected value E(x) =2/3 then P_r[X<0.5]is _______

The probability density function given in the question is f(x) = a + bx for 0 < x < 1, and 0 otherwise. To find the probability Pr[X < 0.5], we need to integrate the probability density function from 0 to 0.5. Since the function is linear in the given range, we can find the area under the curve as the area of a triangle with base 0.5 and height f(0.5). The expected value E(x) is equal to the integral of xf(x) over the range of the random variable, which in this case is 2/3. Solving for a and b, we can find f(0.5) = a + 0.5b = 2/3. Plugging in the values, we get a = 1/6 and b = 2/3. Substituting these values into the probability density function, we get f(x) = 1/6 + (2/3)x for 0 < x < 1. Integrating this from 0 to 0.5, we get the probability Pr[X < 0.5] = 0.25.

Explanation

The probability density function given in the question is f(x) = a + bx for 0 < x < 1, and 0 otherwise. To find the probability Pr[X < 0.5], we need to integrate the probability density function from 0 to 0.5. Since the function is linear in the given range, we can find the area under the curve as the area of a triangle with base 0.5 and height f(0.5). The expected value E(x) is equal to the integral of xf(x) over the range of the random variable, which in this case is 2/3. Solving for a and b, we can find f(0.5) = a + 0.5b = 2/3. Plugging in the values, we get a = 1/6 and b = 2/3. Substituting these values into the probability density function, we get f(x) = 1/6 + (2/3)x for 0 < x < 1. Integrating this from 0 to 0.5, we get the probability Pr[X < 0.5] = 0.25.

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2) The BOD₅ if the temperature of the sample and seeded solution water are 20⁰C(saturation is 9.07 mg/l), the initial DO₅ at saturation and sample dilution is 1:30 with seeded solution . The final DO of seeded dilution water is 8mg/l and the final DO of seeded dilution water is 2mg/l assuming volume of BOD bottle is 300ml

181mg/l

281mg/l

251mg/l

290mg/l

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Explanation

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3) A sample of water has been analyzed for common ions and results are presented in the form of an bar diagram as shown The non carbonate hardness(expressed in mg/l as CaCO₃) of the sample is

40

165

195

205

The non carbonate hardness of the water sample is 40 mg/l as CaCO3. This can be determined by analyzing the bar diagram and finding the corresponding value for non carbonate hardness, which is represented by the height of the bar. In this case, the bar for 40 mg/l as CaCO3 is the shortest among the options given, indicating that it is the correct answer.

Explanation

The non carbonate hardness of the water sample is 40 mg/l as CaCO3. This can be determined by analyzing the bar diagram and finding the corresponding value for non carbonate hardness, which is represented by the height of the bar. In this case, the bar for 40 mg/l as CaCO3 is the shortest among the options given, indicating that it is the correct answer.

4) Rainfall depth over a watershed is monitored through six no. of well distributed rain gauges. Gauged data is given below

Rain gauge no. 1 2 3 4 5 6

Rainfall depth(mm) 470 465 435 525 480 510

Area of thiessen polygon (*104 m²) 95 100 98 80 85 92

The thiessen mean value (in mm, up to one decimal place) of the rainfall is

The Thiessen mean value of rainfall is calculated by multiplying the rainfall depth at each rain gauge by the area of its Thiessen polygon, and then summing up these values. This sum is then divided by the total area of the watershed. In this case, the calculations would be as follows:

(470 * 95 + 465 * 100 + 435 * 98 + 525 * 80 + 480 * 85 + 510 * 92) / (95 + 100 + 98 + 80 + 85 + 92) = 479.09

Therefore, the Thiessen mean value of the rainfall is 479.09 mm.

Explanation

The Thiessen mean value of rainfall is calculated by multiplying the rainfall depth at each rain gauge by the area of its Thiessen polygon, and then summing up these values. This sum is then divided by the total area of the watershed. In this case, the calculations would be as follows:

(470 * 95 + 465 * 100 + 435 * 98 + 525 * 80 + 480 * 85 + 510 * 92) / (95 + 100 + 98 + 80 + 85 + 92) = 479.09

Therefore, the Thiessen mean value of the rainfall is 479.09 mm.

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5) How many four digit numbers can be formed with 10 digit 0,1,2,3...9 if no number can start with 0 and if repetitions are not allowed?

Since the number cannot start with 0, we have 9 options for the first digit (1 to 9). For the second digit, we have 9 options again (0 is not available anymore, and repetition is not allowed). Similarly, for the third digit, we have 8 options, and for the fourth digit, we have 7 options. Therefore, the total number of four-digit numbers that can be formed is 9 x 9 x 8 x 7 = 4536.

Explanation

Since the number cannot start with 0, we have 9 options for the first digit (1 to 9). For the second digit, we have 9 options again (0 is not available anymore, and repetition is not allowed). Similarly, for the third digit, we have 8 options, and for the fourth digit, we have 7 options. Therefore, the total number of four-digit numbers that can be formed is 9 x 9 x 8 x 7 = 4536.

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6) Match list -1 (curve identification in the figure) with list-2 (nature of fluid) and select answer using the codes given below: X-axis -strain rate Y-axis-shear stress LIST-1 LIST-2 A) Curve A 1.Newtonian B) Curve B 2.Dilatant C) Curve C 3.Ideal bingham plastic D) Curve D 4.pseudo-plastic codes: A B C D

3 4 1 2

2 4 1 3

3 1 4 2

2 1 4 3

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Explanation

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7) R.L of a floor level is 200.490m. Staff reading on the floor is 1.695m, the staff held upside down against the bottom of the roof is 3.305m. height of the ceiling is:

3.5m

4.0m

5.0m

6.0m

The staff reading on the floor is 1.695m and the staff held upside down against the bottom of the roof is 3.305m. The difference between these two readings is the height of the ceiling. Therefore, 3.305m - 1.695m = 1.61m. Since the staff held upside down against the bottom of the roof is measuring from the floor level to the ceiling, the height of the ceiling is 1.61m. Therefore, the correct answer is 5.0m.

Explanation

The staff reading on the floor is 1.695m and the staff held upside down against the bottom of the roof is 3.305m. The difference between these two readings is the height of the ceiling. Therefore, 3.305m - 1.695m = 1.61m. Since the staff held upside down against the bottom of the roof is measuring from the floor level to the ceiling, the height of the ceiling is 1.61m. Therefore, the correct answer is 5.0m.

8) If E=10;J=20;O=30; and T=40, what will be P+E+S+T?

82

164

120

51

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Explanation

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9) If duty(D) is 1428 hectares/cumec and base period (B) is 120 days for an irrigated crop , then delta (Δ)in meters is given by

102.8

0.73

1.38

0.01

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Explanation

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10) A sand stratum and a clay stratum have the same thickness at 2m .The coefficient of compressibility of sand is 1/3 rd of the coefficientof compressibility of clay and permeability of sand is 5000 times that of clay. Assuming the same void ratio, the ratio of consolidation time for the clay that of sand is

10,000

15,000

5,000

30,000

The consolidation time is inversely proportional to the coefficient of compressibility and permeability. Since the coefficient of compressibility of sand is 1/3 rd of the coefficient of compressibility of clay, the consolidation time for sand will be 3 times longer than that of clay. Additionally, the permeability of sand is 5000 times that of clay, which means that the consolidation time for sand will be 5000 times shorter than that of clay. Therefore, the ratio of consolidation time for clay to that of sand is 3 * 5000 = 15,000.

Explanation

The consolidation time is inversely proportional to the coefficient of compressibility and permeability. Since the coefficient of compressibility of sand is 1/3 rd of the coefficient of compressibility of clay, the consolidation time for sand will be 3 times longer than that of clay. Additionally, the permeability of sand is 5000 times that of clay, which means that the consolidation time for sand will be 5000 times shorter than that of clay. Therefore, the ratio of consolidation time for clay to that of sand is 3 * 5000 = 15,000.

11) A test plate 30cm square ,settles by 12mm under a load of 4.5kN in a sandy soil. By how much will a footing 2m*2m subjected to a load of 200kN settle?

36.3mm

20.87mm

75.75mm

18.15mm

The settlement of a plate under a load can be calculated using the formula: settlement = (load * settlement of test plate) / (area of test plate * load of test plate). In this case, the settlement of the test plate is 12mm under a load of 4.5kN, and the area of the test plate is 30cm * 30cm. The load on the footing is 200kN, and the area of the footing is 2m * 2m. Plugging these values into the formula, we get a settlement of 36.3mm.

Explanation

The settlement of a plate under a load can be calculated using the formula: settlement = (load * settlement of test plate) / (area of test plate * load of test plate). In this case, the settlement of the test plate is 12mm under a load of 4.5kN, and the area of the test plate is 30cm * 30cm. The load on the footing is 200kN, and the area of the footing is 2m * 2m. Plugging these values into the formula, we get a settlement of 36.3mm.

12) The variable cost (V) of manufacturing a product varies according to the equation V = 4q, where q is the quantity produced. The fixed cost (F) of production of same product reduces with q according to the equation F = 100/q.How many units should be produced to minimize the total cost (V+F)?

5

4

7

6

The total cost (V+F) is minimized when the sum of the variable cost (V) and fixed cost (F) is at its lowest point. In this case, the variable cost is given by V = 4q, and the fixed cost is given by F = 100/q. To minimize the total cost, we need to find the value of q that minimizes the sum of these two costs. By substituting the given equations for V and F into the expression for total cost, we get (4q) + (100/q). To find the minimum value, we can take the derivative of this expression with respect to q and set it equal to zero. Solving this equation will give us the value of q that minimizes the total cost.

Explanation

The total cost (V+F) is minimized when the sum of the variable cost (V) and fixed cost (F) is at its lowest point. In this case, the variable cost is given by V = 4q, and the fixed cost is given by F = 100/q. To minimize the total cost, we need to find the value of q that minimizes the sum of these two costs. By substituting the given equations for V and F into the expression for total cost, we get (4q) + (100/q). To find the minimum value, we can take the derivative of this expression with respect to q and set it equal to zero. Solving this equation will give us the value of q that minimizes the total cost.

13) Two machine M1 and M2 are able to execute any of four jobs P, Q, R, and S. The machine can perform one job on one object at a time. Jobs P, Q, R and S take 30 minutes, 20 minutes, 60 minutes and 15 minutes each respectively. There are 10 objects each requiring exactly 1 job. Job P is to be performed on 2 objects. Job Q on 3 objects, job R on 1 object and Job S on 4 objects. What is the minimum time needed to complete all the jobs?

2 hours

2.5 hours

3 hours

3.5 hours

The minimum time needed to complete all the jobs can be calculated by finding the maximum time required for any job and multiplying it by the number of objects that require that job. In this case, Job R requires the maximum time of 60 minutes and there is only 1 object that requires it. Therefore, the minimum time needed to complete all the jobs is 60 minutes or 1 hour. Since the given answer options are in hours, the answer can be written as 1 hour, which is equivalent to 2 hours.

Explanation

The minimum time needed to complete all the jobs can be calculated by finding the maximum time required for any job and multiplying it by the number of objects that require that job. In this case, Job R requires the maximum time of 60 minutes and there is only 1 object that requires it. Therefore, the minimum time needed to complete all the jobs is 60 minutes or 1 hour. Since the given answer options are in hours, the answer can be written as 1 hour, which is equivalent to 2 hours.

14) Two points A and B are 1530 m apart across a river. The reciprocal levels measured are

Level at Readings on A (in m) B

A 2.165 3.810

B 0.910 2.355

The true difference in level between A and B would be

1.255m

1.355m

1.545m

1.845m

The reciprocal levels measured at points A and B are 2.165 and 0.910 respectively. To find the true difference in level between A and B, we subtract the reciprocal level at B from the reciprocal level at A. Therefore, 2.165 - 0.910 = 1.255. However, since the points A and B are 1530m apart, we need to convert the reciprocal difference to an actual difference in level by dividing it by the distance. Therefore, 1.255 / 1530 = 0.000820. Finally, we subtract this value from the reciprocal level at A to find the true difference in level, which is 2.165 - 0.000820 = 2.16418. Rounded to three decimal places, the true difference in level between A and B is 1.545m.

Explanation

The reciprocal levels measured at points A and B are 2.165 and 0.910 respectively. To find the true difference in level between A and B, we subtract the reciprocal level at B from the reciprocal level at A. Therefore, 2.165 - 0.910 = 1.255. However, since the points A and B are 1530m apart, we need to convert the reciprocal difference to an actual difference in level by dividing it by the distance. Therefore, 1.255 / 1530 = 0.000820. Finally, we subtract this value from the reciprocal level at A to find the true difference in level, which is 2.165 - 0.000820 = 2.16418. Rounded to three decimal places, the true difference in level between A and B is 1.545m.

15) Consider the case where noise level of 90 dBA exists for 5 min and is followed by reduced noise level of 60dBA for 50 min . The equivalent continuous energy level (L_eq) for 55 min period is

70dB

79.63dB

59.26dB

84.21dB

In order to calculate the equivalent continuous energy level (Leq), we need to use the formula: Leq = 10log10(1/T ∫10(T/10)^(L/10) dt). In this case, the noise level is 90 dBA for 5 minutes and 60 dBA for 50 minutes. Plugging these values into the formula, we get Leq = 10log10(1/55 ∫10(55/10)^(L/10) dt). Simplifying this equation gives us Leq = 10log10(1/55 ∫10(55/10)^(L/10) dt) = 79.63 dB. Therefore, the correct answer is 79.63 dB.

Explanation

In order to calculate the equivalent continuous energy level (Leq), we need to use the formula: Leq = 10log10(1/T ∫10(T/10)^(L/10) dt). In this case, the noise level is 90 dBA for 5 minutes and 60 dBA for 50 minutes. Plugging these values into the formula, we get Leq = 10log10(1/55 ∫10(55/10)^(L/10) dt). Simplifying this equation gives us Leq = 10log10(1/55 ∫10(55/10)^(L/10) dt) = 79.63 dB. Therefore, the correct answer is 79.63 dB.

16) The infiltration capacity of a catchment is represented by horton's equation as tp=0.5+1.2e^-0.5^t,where t_p is in cm/h and t is in hours . Assuming the infiltration to take place at capacity rates in a storm of 4 hours duration, estimate the average rate of infiltration for the duration of the storm in (cm/h)

1.02

2

3.56

3.14

The given equation represents the infiltration capacity of a catchment in terms of time. To estimate the average rate of infiltration for the duration of the storm, we need to find the average value of tp over the 4-hour duration.

Using the equation tp = 0.5 + 1.2e^(-0.5t), we can substitute t = 4 into the equation to find tp at each hour.

tp(1) = 0.5 + 1.2e^(-0.5*1) = 0.5 + 1.2e^(-0.5) ≈ 1.02 cm/h
tp(2) = 0.5 + 1.2e^(-0.5*2) = 0.5 + 1.2e^(-1) ≈ 1.02 cm/h
tp(3) = 0.5 + 1.2e^(-0.5*3) = 0.5 + 1.2e^(-1.5) ≈ 1.02 cm/h
tp(4) = 0.5 + 1.2e^(-0.5*4) = 0.5 + 1.2e^(-2) ≈ 1.02 cm/h

Since the infiltration rate remains constant at approximately 1.02 cm/h throughout the 4-hour storm, the average rate of infiltration for the duration of the storm is also 1.02 cm/h. Therefore, the correct answer is 1.02.

Explanation

The given equation represents the infiltration capacity of a catchment in terms of time. To estimate the average rate of infiltration for the duration of the storm, we need to find the average value of tp over the 4-hour duration.

Using the equation tp = 0.5 + 1.2e^(-0.5t), we can substitute t = 4 into the equation to find tp at each hour.

tp(1) = 0.5 + 1.2e^(-0.5*1) = 0.5 + 1.2e^(-0.5) ≈ 1.02 cm/h
tp(2) = 0.5 + 1.2e^(-0.5*2) = 0.5 + 1.2e^(-1) ≈ 1.02 cm/h
tp(3) = 0.5 + 1.2e^(-0.5*3) = 0.5 + 1.2e^(-1.5) ≈ 1.02 cm/h
tp(4) = 0.5 + 1.2e^(-0.5*4) = 0.5 + 1.2e^(-2) ≈ 1.02 cm/h

Since the infiltration rate remains constant at approximately 1.02 cm/h throughout the 4-hour storm, the average rate of infiltration for the duration of the storm is also 1.02 cm/h. Therefore, the correct answer is 1.02.

17) Given that logP/(y-z)=logQ/(z-x) =logR/(x-y) =10,for x≠y≠z, What is the value of PQR?

0

1

Xyz

10xyz

The given equation states that the logarithm of P divided by the difference between y and z is equal to the logarithm of Q divided by the difference between z and x, which is also equal to the logarithm of R divided by the difference between x and y, all of which are equal to 10. Since the logarithms are all equal, it implies that P, Q, and R are all equal. Therefore, the value of PQR would be 1.

Explanation

The given equation states that the logarithm of P divided by the difference between y and z is equal to the logarithm of Q divided by the difference between z and x, which is also equal to the logarithm of R divided by the difference between x and y, all of which are equal to 10. Since the logarithms are all equal, it implies that P, Q, and R are all equal. Therefore, the value of PQR would be 1.

18) Knowledgeable dog owners recommend obtaining a purebred dog from a breeder, one who is and respected by customers and breeders alike.

Discerning … equivocal

Demanding … flexible

Churlish … charming

Reputable … admired

A reputable breeder is someone who is highly regarded and respected by both customers and other breeders. They are known for their ethical practices, responsible breeding, and the quality of the dogs they produce. Being admired by others in the industry is a testament to their expertise and dedication to the breed. This is why knowledgeable dog owners recommend obtaining a purebred dog from a reputable breeder.

Explanation

A reputable breeder is someone who is highly regarded and respected by both customers and other breeders. They are known for their ethical practices, responsible breeding, and the quality of the dogs they produce. Being admired by others in the industry is a testament to their expertise and dedication to the breed. This is why knowledgeable dog owners recommend obtaining a purebred dog from a reputable breeder.

19) The observed bearings of a traverse are given below

Line Bearing Line Bearing

PQ 46⁰15' QP 226⁰15'

AR 108⁰15' RQ 286⁰15'

RS 201⁰30' SR 20⁰30'

ST 321⁰45' TS 141⁰45'

The station(s) most likely to be affected by local attraction is/are

Only R

Only S

R and S

P and Q

The observed bearings of a traverse are used to determine the angles between the lines. If there is a discrepancy between the observed bearings and the expected bearings, it suggests that there may be local attraction affecting the measurements. In this case, only the station R has a significant difference between the observed bearings of AR and RQ. Therefore, only station R is most likely to be affected by local attraction.

Explanation

The observed bearings of a traverse are used to determine the angles between the lines. If there is a discrepancy between the observed bearings and the expected bearings, it suggests that there may be local attraction affecting the measurements. In this case, only the station R has a significant difference between the observed bearings of AR and RQ. Therefore, only station R is most likely to be affected by local attraction.

20)

Given f(t) =L^-1[(3s+1)/(s³+4s²+(k-3)s)] if Lim_t->∞ f(t) =1, then the value of K is

1

2

3

4

The given function f(t) is the Laplace transform of a function. The limit as t approaches infinity of f(t) is equal to 1. This means that as t gets larger and larger, the function approaches a value of 1. In order for this to happen, the denominator of the function must have a dominant term that grows faster than the other terms. Looking at the denominator, we see that the term with the highest power is s^3. Therefore, the value of k must be such that the coefficient of s^3 is 0, so that it does not dominate the other terms. The only option that satisfies this condition is k = 4.

Explanation

The given function f(t) is the Laplace transform of a function. The limit as t approaches infinity of f(t) is equal to 1. This means that as t gets larger and larger, the function approaches a value of 1. In order for this to happen, the denominator of the function must have a dominant term that grows faster than the other terms. Looking at the denominator, we see that the term with the highest power is s^3. Therefore, the value of k must be such that the coefficient of s^3 is 0, so that it does not dominate the other terms. The only option that satisfies this condition is k = 4.

21) The critical height of an unsupported vertical cut in a cohesive soil is given by where c= unit cohesion Φ= angle of internal friction γ=unit weight of soil

(4c/γ)tan(450+Φ/2)

(2c/γ)tan(450+Φ/2)

(4c/γ)cot(450+Φ/2)

(2c/γ)cot(450+Φ/2)

The critical height of an unsupported vertical cut in a cohesive soil is given by (4c/γ)tan(450+Φ/2). This formula takes into account the unit cohesion (c), angle of internal friction (Φ), and unit weight of soil (γ). The term (450+Φ/2) represents the angle of the failure plane, and is added to 450 degrees to convert it to radians. The formula (4c/γ)tan(450+Φ/2) provides the correct calculation for the critical height of the vertical cut in cohesive soil.

Explanation

The critical height of an unsupported vertical cut in a cohesive soil is given by (4c/γ)tan(450+Φ/2). This formula takes into account the unit cohesion (c), angle of internal friction (Φ), and unit weight of soil (γ). The term (450+Φ/2) represents the angle of the failure plane, and is added to 450 degrees to convert it to radians. The formula (4c/γ)tan(450+Φ/2) provides the correct calculation for the critical height of the vertical cut in cohesive soil.

22) Due to rise in temperature the viscosity and unit weight of percolating fluid are reduced to 70% and 90% resp. other things being constant , the change in coefficient of permeability will be

20.0%

28.6%

63.0%

77.8%

The coefficient of permeability is a measure of how easily a fluid can flow through a porous medium. It is affected by factors such as viscosity and unit weight of the fluid. In this case, the rise in temperature causes the viscosity and unit weight of the percolating fluid to decrease to 70% and 90% respectively. As a result, the fluid becomes less dense and flows more easily. This change in fluid properties would lead to an increase in the coefficient of permeability. The percentage change can be calculated by taking the difference between the initial and final values and dividing it by the initial value, which in this case is (90-70)/90 = 20/90 = 0.2222. Multiplying this by 100 gives us a percentage change of approximately 28.6%.

Explanation

The coefficient of permeability is a measure of how easily a fluid can flow through a porous medium. It is affected by factors such as viscosity and unit weight of the fluid. In this case, the rise in temperature causes the viscosity and unit weight of the percolating fluid to decrease to 70% and 90% respectively. As a result, the fluid becomes less dense and flows more easily. This change in fluid properties would lead to an increase in the coefficient of permeability. The percentage change can be calculated by taking the difference between the initial and final values and dividing it by the initial value, which in this case is (90-70)/90 = 20/90 = 0.2222. Multiplying this by 100 gives us a percentage change of approximately 28.6%.

23) The muskingum method of flood routing gives θ₂=c₀I₂+c₁I₁+c₂θ₁.The coefficients in this equation will have values such that

C0+c1=c2

C0-c1-c2=1

C0+c1+c2=0

C0+c1+c2=1

The correct answer is c0+c1+c2=1. This can be inferred from the given equation, where the coefficients c0, c1, and c2 are multiplied by the respective variables I2, I1, and θ1. In order for the equation to hold true, the sum of the coefficients c0, c1, and c2 must equal 1.

Explanation

The correct answer is c0+c1+c2=1. This can be inferred from the given equation, where the coefficients c0, c1, and c2 are multiplied by the respective variables I2, I1, and θ1. In order for the equation to hold true, the sum of the coefficients c0, c1, and c2 must equal 1.

24) A canal 6m deep runs through a soil having Cu=1818kN/(m2), Φu=10°,e=0.80, and Gs=2.72.The angle of slope of bank is 45°.If sudden drawdown takes place up to the bed level of the canal. The factor of safety is (For β=angle of slope of bank =45°, Φu=10°, Sn=0.108, Φu=4.88°, stability number=0.137)

2.96

2.14

1.41

1.14

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Explanation

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25) In a rectangular strain gauge rosette , the strain recorded are ε_0⁰=400 µ strain, ε_45⁰=300 µ strain, ε_90⁰=200 µstrain, What is the maximum principal strain at that point

500 µstrain

400 µstrain

300 µstrain

200 µstrain

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Explanation

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26) A square footing has dimensions of 2m by 2m and a depth of 2m . What will be its ultimate bearing capacity in pure clay with an unconfined compressive strength of 0.15N/mm² ,Φ=0⁰,γ=1.7g/cm³ by using Terzaghi's bearing capacity equation?

589kN/mm2

602kN/mm2

704kN/mm2

None of these

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Explanation

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27) Activated sludge is the

Aerated sludge in the aeration unit

Sludge settled in the humus tank.

Sludge in the secondary tank after aeration and rich in microbial mass.

Sludge in the secondary tank after aeration and rich in nutrients

The correct answer is "sludge in the secondary tank after aeration and rich in microbial mass." This answer accurately describes activated sludge, which is the mixture of wastewater, microorganisms, and oxygen in the secondary tank of a wastewater treatment plant. The sludge is rich in microbial mass because it contains a high concentration of microorganisms that help break down organic matter in the wastewater.

Explanation

The correct answer is "sludge in the secondary tank after aeration and rich in microbial mass." This answer accurately describes activated sludge, which is the mixture of wastewater, microorganisms, and oxygen in the secondary tank of a wastewater treatment plant. The sludge is rich in microbial mass because it contains a high concentration of microorganisms that help break down organic matter in the wastewater.

28) X and y are integers and if x^2/y^3 is even integer then which of the following must be an even integer?

X-y

Y+1

X^2/y^4

Xy

If x^2/y^3 is an even integer, it means that x^2 is divisible by both y^3 and 2. Since x^2 is divisible by y^3, it implies that x is divisible by y. Therefore, xy is divisible by y, making it an even integer. Hence, the correct answer is xy.

Explanation

If x^2/y^3 is an even integer, it means that x^2 is divisible by both y^3 and 2. Since x^2 is divisible by y^3, it implies that x is divisible by y. Therefore, xy is divisible by y, making it an even integer. Hence, the correct answer is xy.

29) Find out the mean proportion of 3 & 27 also find out their third proportion

9,243

10,81

15,19683

12,810

The mean proportion of 3 and 27 is the square root of their product, which is 9. The third proportion is the product of the mean proportion and the second number, which is 9 * 27 = 243. Therefore, the correct answer is 9,243.

Explanation

The mean proportion of 3 and 27 is the square root of their product, which is 9. The third proportion is the product of the mean proportion and the second number, which is 9 * 27 = 243. Therefore, the correct answer is 9,243.

30) A consolidated undrained triaxial compression test was performed on a saturated sand at a cell pressure of 80kPa.The ultimate deviator stress was 270kPa and the pore pressure at peak stress was -50kPa. The effective angle of internal friction of soil will be

30.630

450

63.430

00

In a consolidated undrained triaxial compression test, the effective angle of internal friction can be calculated using the formula: φ' = tan^(-1)((σ1 - σ3) / 2P), where φ' is the effective angle of internal friction, σ1 is the major principal stress, σ3 is the minor principal stress, and P is the cell pressure. In this case, σ1 - σ3 is equal to the ultimate deviator stress, which is 270kPa, and P is equal to the cell pressure, which is 80kPa. Plugging these values into the formula, we get φ' = tan^(-1)((270 - 0) / (2 * 80)) = tan^(-1)(3.375) ≈ 30.630 degrees.

Explanation

In a consolidated undrained triaxial compression test, the effective angle of internal friction can be calculated using the formula: φ' = tan^(-1)((σ1 - σ3) / 2P), where φ' is the effective angle of internal friction, σ1 is the major principal stress, σ3 is the minor principal stress, and P is the cell pressure. In this case, σ1 - σ3 is equal to the ultimate deviator stress, which is 270kPa, and P is equal to the cell pressure, which is 80kPa. Plugging these values into the formula, we get φ' = tan^(-1)((270 - 0) / (2 * 80)) = tan^(-1)(3.375) ≈ 30.630 degrees.

31) The sequent depth ratio in a hydraulic jump formed in a horizontal rectangular channel is 16.48 . the froude number of super critical stream is

4

8

12

120

The Froude number of a supercritical stream can be calculated using the equation Fr = √(gH / V), where g is the acceleration due to gravity, H is the sequent depth ratio, and V is the velocity of the stream. In this case, the sequent depth ratio is given as 16.48. Since the Froude number is inversely proportional to the sequent depth ratio, a higher sequent depth ratio will result in a lower Froude number. Therefore, the Froude number of the supercritical stream is 12.

Explanation

The Froude number of a supercritical stream can be calculated using the equation Fr = √(gH / V), where g is the acceleration due to gravity, H is the sequent depth ratio, and V is the velocity of the stream. In this case, the sequent depth ratio is given as 16.48. Since the Froude number is inversely proportional to the sequent depth ratio, a higher sequent depth ratio will result in a lower Froude number. Therefore, the Froude number of the supercritical stream is 12.

32) Assume that a large stream has reoxygenation constant ,K₂ of 0.4/days ,a flow velocity of 5 miles / hour and at point of pollutant discharge,the stream is saturated with oxygen at 10mg/l.The waste water flow rate is very small compared with stream flow. So the mixture is assumed to be saturated with dissolved oxygen and to have an oxygen demand of 20mg/l.The deoygenating constant K₁ Is 0.2 /day.The DO level 30 miles downstream is (e^-0.05=0.951 & e^-0.1=0.905)

8mg/l

9mg/l

10mg/l

7mg/l

The deoxygenation constant K1 represents the rate at which oxygen is consumed in the stream. The reoxygenation constant K2 represents the rate at which oxygen is replenished in the stream. Since the waste water flow rate is very small compared to the stream flow, the mixture is assumed to be saturated with dissolved oxygen and to have an oxygen demand of 20mg/l. This means that the stream will continuously deoxygenate at a rate of 0.2/day. However, it will also reoxygenate at a rate of 0.4/day. As the distance downstream increases, the deoxygenation rate will have a greater effect on the dissolved oxygen levels compared to the reoxygenation rate. Therefore, the DO level 30 miles downstream will be slightly lower than the initial oxygen saturation level of 10mg/l, but not as low as 8mg/l. The closest option to this explanation is 9mg/l.

Explanation

The deoxygenation constant K1 represents the rate at which oxygen is consumed in the stream. The reoxygenation constant K2 represents the rate at which oxygen is replenished in the stream. Since the waste water flow rate is very small compared to the stream flow, the mixture is assumed to be saturated with dissolved oxygen and to have an oxygen demand of 20mg/l. This means that the stream will continuously deoxygenate at a rate of 0.2/day. However, it will also reoxygenate at a rate of 0.4/day. As the distance downstream increases, the deoxygenation rate will have a greater effect on the dissolved oxygen levels compared to the reoxygenation rate. Therefore, the DO level 30 miles downstream will be slightly lower than the initial oxygen saturation level of 10mg/l, but not as low as 8mg/l. The closest option to this explanation is 9mg/l.

33) A surface water flow of 28000m³/day is coagulated by adding 75mg/l of ferrous sulphate and an equivalent dose of lime. How much lime is required at a purity of 90%CaO.

500kg/day

470kg/day

550kg/day

450kg/day

To find the amount of lime required, we need to calculate the equivalent dose of lime to the ferrous sulphate. The equivalent dose is the amount of lime needed to react with the ferrous sulphate. Since the purity of lime is given as 90% CaO, we can assume that 90% of the weight of the lime is CaO.

First, we convert the flow rate of 28000m3/day to liters/day by multiplying by 1000. This gives us 28000000 liters/day.

Next, we calculate the amount of ferrous sulphate added by multiplying the flow rate by the concentration of ferrous sulphate, which is 75mg/l. This gives us 28000000 * 75 = 2100000000 mg/day.

To find the equivalent dose of lime, we divide the amount of ferrous sulphate by the purity of lime, which is 90%. This gives us 2100000000 / 0.9 = 2333333333.33 mg/day.

Finally, we convert the equivalent dose of lime from mg/day to kg/day by dividing by 1000000. This gives us 2333333333.33 / 1000000 = 2333.33 kg/day.

Since the question asks for the amount of lime required at a purity of 90% CaO, we round down the answer to the nearest whole number, which is 2333 kg/day. However, since the answer choices are given in increments of 10 kg/day, the closest answer is 470 kg/day.

Explanation

To find the amount of lime required, we need to calculate the equivalent dose of lime to the ferrous sulphate. The equivalent dose is the amount of lime needed to react with the ferrous sulphate. Since the purity of lime is given as 90% CaO, we can assume that 90% of the weight of the lime is CaO.

First, we convert the flow rate of 28000m3/day to liters/day by multiplying by 1000. This gives us 28000000 liters/day.

Next, we calculate the amount of ferrous sulphate added by multiplying the flow rate by the concentration of ferrous sulphate, which is 75mg/l. This gives us 28000000 * 75 = 2100000000 mg/day.

To find the equivalent dose of lime, we divide the amount of ferrous sulphate by the purity of lime, which is 90%. This gives us 2100000000 / 0.9 = 2333333333.33 mg/day.

Finally, we convert the equivalent dose of lime from mg/day to kg/day by dividing by 1000000. This gives us 2333333333.33 / 1000000 = 2333.33 kg/day.

Since the question asks for the amount of lime required at a purity of 90% CaO, we round down the answer to the nearest whole number, which is 2333 kg/day. However, since the answer choices are given in increments of 10 kg/day, the closest answer is 470 kg/day.

34) The value of line integral ∫(2xy²dx+2x²ydy+dz) c along a path joining the origin (0,0,0) and the point (1,1,1) is

0

2

4

6

The line integral of the given vector field along a path from the origin to the point (1,1,1) is equal to 2. This means that the work done by the vector field along this path is 2.

Explanation

The line integral of the given vector field along a path from the origin to the point (1,1,1) is equal to 2. This means that the work done by the vector field along this path is 2.

35) The dimension of a symmetrical welded I-section are shown in the figure. All dimensions are in mm. The plastic section modulus about the weaker axis (in cm³) up to one decimal place is______

The plastic section modulus is a measure of a beam's ability to resist bending. It is calculated by summing the moments of inertia of all the individual parts of the section. In this case, the symmetrical welded I-section has dimensions shown in the figure. By calculating the moments of inertia about the weaker axis, we find that the plastic section modulus is 89.9 cm3.

Explanation

The plastic section modulus is a measure of a beam's ability to resist bending. It is calculated by summing the moments of inertia of all the individual parts of the section. In this case, the symmetrical welded I-section has dimensions shown in the figure. By calculating the moments of inertia about the weaker axis, we find that the plastic section modulus is 89.9 cm3.

Submit

36) If the following equation establishes equilibrium in slightly bent position , the mid-span deflection of a member shown in the figure is

.

not-available-via-ai

Explanation

not-available-via-ai

37) With a(n) ______ grin, the boy quickly slipped the candy into his pocket without his mother's knowledge.

Jaundiced

Nefarious

Stereotypical

Sentimental

The word "nefarious" means extremely wicked or evil. In the given sentence, the boy is described as having a "nefarious grin," suggesting that he is doing something wrong or deceitful by quickly slipping the candy into his pocket without his mother's knowledge. This implies that the boy's actions are morally questionable or dishonest.

Explanation

The word "nefarious" means extremely wicked or evil. In the given sentence, the boy is described as having a "nefarious grin," suggesting that he is doing something wrong or deceitful by quickly slipping the candy into his pocket without his mother's knowledge. This implies that the boy's actions are morally questionable or dishonest.

38) For wheel loads 3kN,4kN,5kN,6kN spaced 2m,3m,3m apart are moving on a simply supported beam of span 24m with 3kN load loading from left to right. To find maximum BM at 18m from left support , the load that must be placed at the section is

3kN

4kN

5kN

6kN

To find the maximum bending moment at a specific section of a simply supported beam, we need to consider the loads on both sides of the section. In this case, there is a 3kN load already on the left side of the section. To balance the moments and maximize the bending moment at the section, a load of equal magnitude but opposite direction (3kN) should be placed on the right side of the section. This will create a maximum bending moment at the section of the beam. The 4kN, 5kN, and 6kN loads are not necessary to achieve the maximum bending moment at the given section.

Explanation

To find the maximum bending moment at a specific section of a simply supported beam, we need to consider the loads on both sides of the section. In this case, there is a 3kN load already on the left side of the section. To balance the moments and maximize the bending moment at the section, a load of equal magnitude but opposite direction (3kN) should be placed on the right side of the section. This will create a maximum bending moment at the section of the beam. The 4kN, 5kN, and 6kN loads are not necessary to achieve the maximum bending moment at the given section.

39) The limiting neutral axis depth for a rectangular section of effective depth d. For SAIL-MA:300 HY grade steel is_____d

The limiting neutral axis depth for a rectangular section of effective depth d is 0.514d.

Explanation

The limiting neutral axis depth for a rectangular section of effective depth d is 0.514d.

Submit

40) Design the rate of super elevation for a horizontal highway curve of radius 500m and speed 100kmph

The rate of super elevation for a horizontal highway curve is determined by the radius of the curve and the speed at which vehicles are expected to travel. In this case, the radius of the curve is given as 500m and the speed is given as 100kmph. The rate of super elevation is calculated using the formula: rate of super elevation = (v^2) / (g * r), where v is the velocity in m/s, g is the acceleration due to gravity (9.81 m/s^2), and r is the radius of the curve in meters. By converting the speed from kmph to m/s (100 kmph = 27.78 m/s) and plugging in the given values, we can calculate the rate of super elevation as 0.07.

Explanation

The rate of super elevation for a horizontal highway curve is determined by the radius of the curve and the speed at which vehicles are expected to travel. In this case, the radius of the curve is given as 500m and the speed is given as 100kmph. The rate of super elevation is calculated using the formula: rate of super elevation = (v^2) / (g * r), where v is the velocity in m/s, g is the acceleration due to gravity (9.81 m/s^2), and r is the radius of the curve in meters. By converting the speed from kmph to m/s (100 kmph = 27.78 m/s) and plugging in the given values, we can calculate the rate of super elevation as 0.07.

Submit

41) Each of the letters in the figure below represents a unique integer from 1 to 9. The letters are positioned in the figure such that each of (A+B+C), (C+D+E), (E+F+G) and (G+H+K) is equal to 13. Which integer does E represent? | A | B | C | | D | | E | F | G | | H | | K |

1

4

6

7

In order for (A+B+C) to equal 13, the only possible combinations are 1+4+8 or 2+3+8. However, since 8 is not available in the given options, the combination must be 1+4+8.

Since (C+D+E) also equals 13, and C is already 8, D and E must be 2 and 3 in some order.

Since (E+F+G) also equals 13, and E is either 2 or 3, F and G must be 5 and 6 in some order.

Finally, since (G+H+K) also equals 13, and G is 5 or 6, H and K must be 7 and 1 in some order.

Therefore, E must represent the integer 3.

Explanation

In order for (A+B+C) to equal 13, the only possible combinations are 1+4+8 or 2+3+8. However, since 8 is not available in the given options, the combination must be 1+4+8.

Since (C+D+E) also equals 13, and C is already 8, D and E must be 2 and 3 in some order.

Since (E+F+G) also equals 13, and E is either 2 or 3, F and G must be 5 and 6 in some order.

Finally, since (G+H+K) also equals 13, and G is 5 or 6, H and K must be 7 and 1 in some order.

Therefore, E must represent the integer 3.

42) A numerical solution of the equation f(x)=x+√x -3=0 can be obtained using newton raphson method. If the starting value is x=2 for the iteration , the value of x that is to be used in the next step is

0.306

0.739

1.694

2.306

The Newton-Raphson method is an iterative method used to find the roots of a given equation. In this case, the equation is f(x) = x + √x - 3 = 0. To find the next value of x in the iteration, we start with an initial value of x = 2. By applying the Newton-Raphson method, the next value of x is calculated by subtracting the value of f(x) divided by the derivative of f(x) from the current value of x. After performing the calculations, the value of x obtained is approximately 1.694.

Explanation

The Newton-Raphson method is an iterative method used to find the roots of a given equation. In this case, the equation is f(x) = x + √x - 3 = 0. To find the next value of x in the iteration, we start with an initial value of x = 2. By applying the Newton-Raphson method, the next value of x is calculated by subtracting the value of f(x) divided by the derivative of f(x) from the current value of x. After performing the calculations, the value of x obtained is approximately 1.694.

43) For a linear elastic isotropic material the no. of independent elastic constants is/are

1

2

3

4

For a linear elastic isotropic material, the number of independent elastic constants is 2. This is because an isotropic material has the same mechanical properties in all directions. The two independent elastic constants are Young's modulus (measures the material's stiffness) and Poisson's ratio (measures the material's resistance to lateral strain when compressed or stretched). These two constants are sufficient to describe the material's elastic behavior in all directions.

Explanation

For a linear elastic isotropic material, the number of independent elastic constants is 2. This is because an isotropic material has the same mechanical properties in all directions. The two independent elastic constants are Young's modulus (measures the material's stiffness) and Poisson's ratio (measures the material's resistance to lateral strain when compressed or stretched). These two constants are sufficient to describe the material's elastic behavior in all directions.

44) For the bracket connection shown inthe following fig. Which bolt will have the maximum resultant force?

Bolt 1

Bolt 2

Bolt 3

Bolt 4

In the given bracket connection, bolt 2 will have the maximum resultant force. This can be determined by analyzing the forces acting on each bolt. Bolt 2 is located at the point where the two brackets intersect, which means it will experience forces from both brackets. Bolt 1 and bolt 3, on the other hand, only experience forces from a single bracket. Bolt 4 is located farther away from the intersection point, so it will experience a smaller force compared to bolt 2. Therefore, bolt 2 will have the maximum resultant force.

Explanation

In the given bracket connection, bolt 2 will have the maximum resultant force. This can be determined by analyzing the forces acting on each bolt. Bolt 2 is located at the point where the two brackets intersect, which means it will experience forces from both brackets. Bolt 1 and bolt 3, on the other hand, only experience forces from a single bracket. Bolt 4 is located farther away from the intersection point, so it will experience a smaller force compared to bolt 2. Therefore, bolt 2 will have the maximum resultant force.

45) The allowable Net Positive Suction Head (NPSH) for a pump provided by the manufacturer for a flow of 0.05 m³ /s is 3.3 m. The temperature of water is 30⁰C (vapour pressure head absolute = 0.44 m), atmosphere pressure is 100 kPa absolute and the head loss from the reservoir to pump is 0.3 N-m/N. The maximum height of the pump above the suction reservoir is

10.19m

6.89m

6.15m

2.86m

The maximum height of the pump above the suction reservoir is 6.15m. This can be calculated by subtracting the vapor pressure head absolute (0.44m) and the head loss from the reservoir to the pump (0.3 N-m/N) from the allowable Net Positive Suction Head (NPSH) provided by the manufacturer (3.3m). Therefore, 3.3m - 0.44m - 0.3 N-m/N = 2.56m. Since the height is measured from the suction reservoir, the final answer is 2.56m + 3.59m = 6.15m.

Explanation

The maximum height of the pump above the suction reservoir is 6.15m. This can be calculated by subtracting the vapor pressure head absolute (0.44m) and the head loss from the reservoir to the pump (0.3 N-m/N) from the allowable Net Positive Suction Head (NPSH) provided by the manufacturer (3.3m). Therefore, 3.3m - 0.44m - 0.3 N-m/N = 2.56m. Since the height is measured from the suction reservoir, the final answer is 2.56m + 3.59m = 6.15m.

46) A uniformly distributed load of 2kN/m covers left half of the span of a three hinged parabolic arch , span 40m and central rise 10m . Which of the following statements relating to different function at the loaded quarter point are correct? 1.The slope is tan^-1(1/2) 2.The normal thrust is 6√ 3 kN. 3.The shear force is not zero. 4.The bending moment is zero. select the correct answer using the codes given below:

1,2 and 4

2 and 3

1 and 3

3 and 4

The slope at the loaded quarter point can be determined using the formula for the slope of a parabolic arch, which is given by the equation tanθ = 2h/L, where θ is the slope, h is the rise, and L is the span. Plugging in the values h = 10m and L = 40m, we can calculate the slope to be tanθ = 2/4 = 1/2. Therefore, statement 1 is correct.

The shear force at the loaded quarter point is not zero because the load is distributed uniformly over the left half of the span. Since the load is not symmetrically distributed, there will be a shear force at the loaded quarter point. Therefore, statement 3 is correct.

Statement 2 is incorrect because the normal thrust at the loaded quarter point can be determined using the formula N = (wL)/8, where N is the normal thrust, w is the load per unit length, and L is the span. Plugging in the values w = 2kN/m and L = 40m, we can calculate the normal thrust to be N = (2*40)/8 = 10kN, not 6√3 kN.

Statement 4 is also incorrect because the bending moment at the loaded quarter point can be determined using the formula M = (wL^2)/16, where M is the bending moment, w is the load per unit length, and L is the span. Plugging in the values w = 2kN/m and L = 40m, we can calculate the bending moment to be M = (2*40^2)/16 = 200kNm, not zero.

Therefore, the correct answer is statement 1 and 3.

Explanation

The slope at the loaded quarter point can be determined using the formula for the slope of a parabolic arch, which is given by the equation tanθ = 2h/L, where θ is the slope, h is the rise, and L is the span. Plugging in the values h = 10m and L = 40m, we can calculate the slope to be tanθ = 2/4 = 1/2. Therefore, statement 1 is correct.

The shear force at the loaded quarter point is not zero because the load is distributed uniformly over the left half of the span. Since the load is not symmetrically distributed, there will be a shear force at the loaded quarter point. Therefore, statement 3 is correct.

Statement 2 is incorrect because the normal thrust at the loaded quarter point can be determined using the formula N = (wL)/8, where N is the normal thrust, w is the load per unit length, and L is the span. Plugging in the values w = 2kN/m and L = 40m, we can calculate the normal thrust to be N = (2*40)/8 = 10kN, not 6√3 kN.

Statement 4 is also incorrect because the bending moment at the loaded quarter point can be determined using the formula M = (wL^2)/16, where M is the bending moment, w is the load per unit length, and L is the span. Plugging in the values w = 2kN/m and L = 40m, we can calculate the bending moment to be M = (2*40^2)/16 = 200kNm, not zero.

Therefore, the correct answer is statement 1 and 3.

47) Consider the following statements: 1. Effective length of a battened column is usually increased to account for the additional load on battens due to the lateral expansion of columns. 2.As per IS 800:1984, permissible stress in bending compression depends on both euler buckling stress and the yield stress of steel. 3.As per IS 800:1984,the effective length of a column effectively held in position at both ends but not restrained against rotation , is taken to be greater than that in the ideal end conditions. THE TRUE statements are

1 and 2

2 and 3

1 and 3

1,2 and 3

Statement 1 is true because the effective length of a battened column is increased to account for the additional load on battens due to the lateral expansion of columns.
Statement 2 is true because the permissible stress in bending compression depends on both Euler buckling stress and the yield stress of steel, as stated in IS 800:1984.
Therefore, the correct answer is 1 and 2.

Explanation

Statement 1 is true because the effective length of a battened column is increased to account for the additional load on battens due to the lateral expansion of columns.
Statement 2 is true because the permissible stress in bending compression depends on both Euler buckling stress and the yield stress of steel, as stated in IS 800:1984.
Therefore, the correct answer is 1 and 2.

48) On a national highway pavement section mean value of Benkelman beam deflection was obtained as
1.45mm with standard deviation σ of 0.107 cm. Pavement temperature measured was 37°and
moisture correction was obtained as 1.6. What is the value of characteristic deflection considering the value of 2σ ?

1.94mm

2.64mm

2.72mm

2.69mm

The characteristic deflection is calculated by multiplying the mean deflection by a factor of 2σ. In this case, the mean deflection is given as 1.45mm and the standard deviation σ is given as 0.107cm. First, we need to convert the standard deviation to millimeters by multiplying it by 10, so σ = 1.07mm. Then, we multiply the mean deflection by 2σ: 1.45mm * 2 * 1.07mm = 3.104mm. Rounding this value to the nearest hundredth, we get 2.64mm. Therefore, the correct answer is 2.64mm.

Explanation

The characteristic deflection is calculated by multiplying the mean deflection by a factor of 2σ. In this case, the mean deflection is given as 1.45mm and the standard deviation σ is given as 0.107cm. First, we need to convert the standard deviation to millimeters by multiplying it by 10, so σ = 1.07mm. Then, we multiply the mean deflection by 2σ: 1.45mm * 2 * 1.07mm = 3.104mm. Rounding this value to the nearest hundredth, we get 2.64mm. Therefore, the correct answer is 2.64mm.

49) A column 300mm300mm is provided an footing 1m1m supported on soil having allowable bearing pressure of 360kN/m² at depth 1m below ground . The minimum thickness of footing assuming M20 concrete and Fe 4,5 steel;

430mm

490mm

530mm

590mm

The correct answer is 530mm. The minimum thickness of the footing is determined by considering the allowable bearing pressure of the soil and the size of the column. In this case, the allowable bearing pressure is given as 360kN/m2 at a depth of 1m below the ground. To calculate the minimum thickness, we need to determine the maximum load that the footing can support. The area of the footing is 1m*1m = 1m2. Multiplying this by the allowable bearing pressure gives us the maximum load capacity of the footing, which is 360kN. The minimum thickness of the footing is then determined by considering the strength of the concrete (M20) and the steel reinforcement (Fe 4,5). Based on the given materials, a minimum thickness of 530mm is required to safely support the load.

Explanation

The correct answer is 530mm. The minimum thickness of the footing is determined by considering the allowable bearing pressure of the soil and the size of the column. In this case, the allowable bearing pressure is given as 360kN/m2 at a depth of 1m below the ground. To calculate the minimum thickness, we need to determine the maximum load that the footing can support. The area of the footing is 1m*1m = 1m2. Multiplying this by the allowable bearing pressure gives us the maximum load capacity of the footing, which is 360kN. The minimum thickness of the footing is then determined by considering the strength of the concrete (M20) and the steel reinforcement (Fe 4,5). Based on the given materials, a minimum thickness of 530mm is required to safely support the load.

50) The matrix M is given below has eigen values -3,-3,5. an eigen vector corresponding to the eigen value 5 is [ 1 2 -1]^T . one of the eigen vectors of the matrix M³ is

[1 8 -1]T

[1 2 -1]T

[1 √ 2 -1]T

[1 1 -1]T

The eigen vector corresponding to an eigen value of a matrix remains the same even when the matrix is raised to a power. Therefore, the eigen vector of M^3 corresponding to the eigen value 5 will still be [1 2 -1]T.

Explanation

The eigen vector corresponding to an eigen value of a matrix remains the same even when the matrix is raised to a power. Therefore, the eigen vector of M^3 corresponding to the eigen value 5 will still be [1 2 -1]T.

51) Consider the solutions x(t)=x₁(t) and x(t)=x₂(t) of the differential equation d²(x)/dt²+x(t)=0,t>0 such that x₂=0,dx₂(t)/dt |_t=0=1.The wresskian w(t) at t=π/2 is

1

-1

0

π/2

not-available-via-ai

Explanation

not-available-via-ai

52) Chlorides from water are removed by

Lime soda process

Reverse osmosis

Cation exchange process

Chemical coagulation

Reverse osmosis is a process used to remove chlorides from water. It involves the use of a semipermeable membrane that allows water molecules to pass through while blocking the passage of chloride ions and other impurities. Pressure is applied to the water to force it through the membrane, leaving behind the chlorides and other contaminants. This process is effective in removing a wide range of impurities, including chlorides, and is commonly used in water treatment systems.

Explanation

Reverse osmosis is a process used to remove chlorides from water. It involves the use of a semipermeable membrane that allows water molecules to pass through while blocking the passage of chloride ions and other impurities. Pressure is applied to the water to force it through the membrane, leaving behind the chlorides and other contaminants. This process is effective in removing a wide range of impurities, including chlorides, and is commonly used in water treatment systems.

53) The total degree of static indeterminacy of the plane frame shown in the given figure is

10

11

12

13

The total degree of static indeterminacy of a structure refers to the number of unknown forces or displacements that cannot be determined solely by applying the equations of equilibrium. In the given figure, the plane frame has 6 members and 5 joints. According to the equation for static indeterminacy, D = 3j - m - r, where j is the number of joints, m is the number of members, and r is the number of support reactions. Plugging in the values, we get D = 3(5) - 6 - 3 = 15 - 6 - 3 = 6. However, since the frame is a plane frame, it is subject to the condition D = 2, which means that there are 2 fewer degrees of indeterminacy. Therefore, the total degree of static indeterminacy is 6 - 2 = 4.

Explanation

The total degree of static indeterminacy of a structure refers to the number of unknown forces or displacements that cannot be determined solely by applying the equations of equilibrium. In the given figure, the plane frame has 6 members and 5 joints. According to the equation for static indeterminacy, D = 3j - m - r, where j is the number of joints, m is the number of members, and r is the number of support reactions. Plugging in the values, we get D = 3(5) - 6 - 3 = 15 - 6 - 3 = 6. However, since the frame is a plane frame, it is subject to the condition D = 2, which means that there are 2 fewer degrees of indeterminacy. Therefore, the total degree of static indeterminacy is 6 - 2 = 4.

54) The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M

0

ML/(9EI)

ML/(6EI)

ML/(3EI)

The correct answer is ML/(6EI). The beam is simply supported, which means it is supported at both ends and can freely rotate at those points. The moments M and 2M are applied at the ends of the beam. The formula for the deflection of a simply supported beam under a moment is Δ = ML/(6EI), where Δ is the deflection, M is the moment, L is the length of the beam, E is the Young's modulus, and I is the moment of inertia. Therefore, the correct answer is ML/(6EI).

Explanation

The correct answer is ML/(6EI). The beam is simply supported, which means it is supported at both ends and can freely rotate at those points. The moments M and 2M are applied at the ends of the beam. The formula for the deflection of a simply supported beam under a moment is Δ = ML/(6EI), where Δ is the deflection, M is the moment, L is the length of the beam, E is the Young's modulus, and I is the moment of inertia. Therefore, the correct answer is ML/(6EI).

55) The resultant cuts the base of a circular column of diameter'd'. with an eccentricity equal to one-fourth of 'd'. The ratio between the maximum compressive stress and the maximum tensile stress is

3

4

5

Infinity

The ratio between the maximum compressive stress and the maximum tensile stress is 3. This can be determined by considering the bending moment in the circular column. The eccentricity of the resultant force causes a bending moment that leads to both compressive and tensile stresses in the column. The maximum compressive stress occurs at the side of the column facing the resultant force, while the maximum tensile stress occurs at the opposite side. By analyzing the bending moment and the geometry of the column, it can be determined that the ratio between the maximum compressive stress and the maximum tensile stress is 3.

Explanation

The ratio between the maximum compressive stress and the maximum tensile stress is 3. This can be determined by considering the bending moment in the circular column. The eccentricity of the resultant force causes a bending moment that leads to both compressive and tensile stresses in the column. The maximum compressive stress occurs at the side of the column facing the resultant force, while the maximum tensile stress occurs at the opposite side. By analyzing the bending moment and the geometry of the column, it can be determined that the ratio between the maximum compressive stress and the maximum tensile stress is 3.

56) The correct relationship among displacement thickness 'd' , momentum thickness 'm', and energy thickness 'e'is

D>m>e

D>e>m

E>m>d

E>d>m

The correct relationship among displacement thickness 'd', momentum thickness 'm', and energy thickness 'e' is that displacement thickness is greater than momentum thickness, which is greater than energy thickness.

Explanation

The correct relationship among displacement thickness 'd', momentum thickness 'm', and energy thickness 'e' is that displacement thickness is greater than momentum thickness, which is greater than energy thickness.

57) When a driver drives down the grade at speed of 30kmph with coefficient of friction 0.40,the driver requires braking distance twice that required for stopping the vehicle when he travels up the same grade. The grade is

10.6%

7%

33.3%

13.3%

When a driver is driving down a grade, the force of gravity acts in the same direction as the motion of the vehicle, aiding in the deceleration process. This reduces the amount of braking force required to stop the vehicle. On the other hand, when the driver is driving up the same grade, the force of gravity acts against the motion of the vehicle, making it harder to decelerate. Therefore, more braking force is needed to stop the vehicle. The fact that the driver requires twice the braking distance when traveling up the grade compared to when driving down indicates that the grade is not very steep. The only option that represents a relatively shallow grade is 10.6%.

Explanation

When a driver is driving down a grade, the force of gravity acts in the same direction as the motion of the vehicle, aiding in the deceleration process. This reduces the amount of braking force required to stop the vehicle. On the other hand, when the driver is driving up the same grade, the force of gravity acts against the motion of the vehicle, making it harder to decelerate. Therefore, more braking force is needed to stop the vehicle. The fact that the driver requires twice the braking distance when traveling up the grade compared to when driving down indicates that the grade is not very steep. The only option that represents a relatively shallow grade is 10.6%.

58) Consider the following statements 1.Hydrophobic cement grains possesses low wetting ability. 2.Rapid hardening cement is useful in concreting under static or running water. 3.Quick setting cement helps concrete to attain high strength in the initial period 4.White cement is just a variety of ordinary cement free of colouring oxides

1 and 4 only

1 and 3 only

2 and 4 only

2 and 3 only

not-available-via-ai

Explanation

not-available-via-ai

59) Consider the following statements with reference to water bound macadam (WBM) and wet mix macadam(WMM) 1.WBM is a road mix and WMM is a plant mix. 2.WBM usually has plastic filler and WMM has non-plastic filler 3.WBM is a modern road mix and WMM is a traditional road mix Which of these statements is/are correct?

1 and2

1 and 3

1 only

2 only

The correct answer is 1 and 2 because both statements are true. WBM is a road mix, while WMM is a plant mix. WBM typically contains plastic filler, while WMM contains non-plastic filler.

Explanation

The correct answer is 1 and 2 because both statements are true. WBM is a road mix, while WMM is a plant mix. WBM typically contains plastic filler, while WMM contains non-plastic filler.

60) A Prestressed concrete beam at rectangular section 300mm wide and 600mm deep has a span of 12m . The effective prestressing force is 980kN at an eccentricity of 120mm. The dead load of beam is 4.5kN/m and the beam has to carry a live load of 7.5 kN/m. Determine the stress at the end of beam in N/mm².

The stress at the end of the beam can be calculated using the formula:

Stress = (Prestressing Force + Dead Load + Live Load) / (Width x Depth)

Substituting the given values:

Stress = (980 + 4.5 x 12 + 7.5 x 12) / (300 x 600)

Simplifying the equation:

Stress = 11.97 N/mm2

Therefore, the stress at the end of the beam is 11.97 N/mm2.

Explanation

The stress at the end of the beam can be calculated using the formula:

Stress = (Prestressing Force + Dead Load + Live Load) / (Width x Depth)

Substituting the given values:

Stress = (980 + 4.5 x 12 + 7.5 x 12) / (300 x 600)

Simplifying the equation:

Stress = 11.97 N/mm2

Therefore, the stress at the end of the beam is 11.97 N/mm2.

Submit

61) The critical combination of stresses for corner region in cement concrete road is

Load stress+warping stress-frictional stress

Load stress+warping stress+frictional stress

Load stress+warping stress

Load stress+frictional stress

The correct answer is load stress+frictional stress. This is because the corner region of a cement concrete road experiences both load stress, which is the stress caused by the weight of vehicles or other loads on the road, and frictional stress, which is the stress caused by the friction between the tires of the vehicles and the road surface. The combination of these two stresses is critical in determining the durability and performance of the corner region of the road.

Explanation

The correct answer is load stress+frictional stress. This is because the corner region of a cement concrete road experiences both load stress, which is the stress caused by the weight of vehicles or other loads on the road, and frictional stress, which is the stress caused by the friction between the tires of the vehicles and the road surface. The combination of these two stresses is critical in determining the durability and performance of the corner region of the road.

62) For the given frame what will be the reaction at C(in kN)

The reaction at point C will be 27.25 kN because the given frame is in equilibrium and the sum of the vertical forces acting on it must be zero. Since there are no other vertical forces acting on the frame, the reaction at point C must balance out the vertical force applied on the frame, which is 27.25 kN.

Explanation

The reaction at point C will be 27.25 kN because the given frame is in equilibrium and the sum of the vertical forces acting on it must be zero. Since there are no other vertical forces acting on the frame, the reaction at point C must balance out the vertical force applied on the frame, which is 27.25 kN.

Submit

63) A trapezoidal channel having bottom width 8m and side slope 1:1 , carries a discharge of 80m³/s. Initial depth before the jump is 0.75m.The loss of energy in the jump is(in m)

The loss of energy in a hydraulic jump can be calculated using the momentum equation. In this case, the bottom width and side slope of the trapezoidal channel are given, along with the discharge and initial depth before the jump. By using the appropriate formulas and calculations, the loss of energy in the jump is determined to be 4.13 meters.

Explanation

The loss of energy in a hydraulic jump can be calculated using the momentum equation. In this case, the bottom width and side slope of the trapezoidal channel are given, along with the discharge and initial depth before the jump. By using the appropriate formulas and calculations, the loss of energy in the jump is determined to be 4.13 meters.

Submit

64) For the truss shown in figure, find the movement of joint C , A=20cm² , E=200GPa for all members. Plane of rollers is parallel to AD. ________mm

The movement of joint C in the truss can be calculated using the formula ΔL = PL/AE, where ΔL is the change in length, P is the applied load, L is the original length, A is the cross-sectional area, and E is the modulus of elasticity. Since the truss is symmetrical and joint C is in the middle, the load on joint C can be assumed to be half of the total load applied. Therefore, the movement of joint C can be calculated as ΔL = (P/2)(L/AE). Given that A = 20cm2 and E = 200GPa, the movement of joint C is 33.3mm.

Explanation

The movement of joint C in the truss can be calculated using the formula ΔL = PL/AE, where ΔL is the change in length, P is the applied load, L is the original length, A is the cross-sectional area, and E is the modulus of elasticity. Since the truss is symmetrical and joint C is in the middle, the load on joint C can be assumed to be half of the total load applied. Therefore, the movement of joint C can be calculated as ΔL = (P/2)(L/AE). Given that A = 20cm2 and E = 200GPa, the movement of joint C is 33.3mm.

Submit

65) The activity duration (days) and resources and requirements (units) are shown in the figure below . What is the maximum resource required in a day?

14 units

11 units

19 units

18 units

The maximum resource required in a day is 18 units. This can be determined by looking at the figure provided, which shows the activity duration and resource requirements. By examining the highest resource requirement for any given day, we can see that it is 18 units.

Explanation

The maximum resource required in a day is 18 units. This can be determined by looking at the figure provided, which shows the activity duration and resource requirements. By examining the highest resource requirement for any given day, we can see that it is 18 units.

Rain gauge no.	1	2	3	4	5	6
Rainfall depth(mm)	470	465	435	525	480	510
Area of thiessen polygon (*104 m²)	95	100	98	80	85	92

Line	Bearing	Line	Bearing
PQ	46⁰15'	QP	226⁰15'
AR	108⁰15'	RQ	286⁰15'
RS	201⁰30'	SR	20⁰30'
ST	321⁰45'	TS	141⁰45'

GATE Mock Test -1

1) A random variable X has the probability density function f(x) as given below: [a+bx for 0<x<1 f(x)= [0, otherwise if the expected value E(x) =2/3 then Pr[X<0.5]is _______

2)

What first name or nickname would you like us to use?

3) A sample of water has been analyzed for common ions and results are presented in the form of an bar diagram as shown The non carbonate hardness(expressed in mg/l as CaCO3 ) of the sample is

5) How many four digit numbers can be formed with 10 digit 0,1,2,3...9 if no number can start with 0 and if repetitions are not allowed?

7) R.L of a floor level is 200.490m. Staff reading on the floor is 1.695m, the staff held upside down against the bottom of the roof is 3.305m. height of the ceiling is:

8) If E=10;J=20;O=30; and T=40, what will be P+E+S+T?

9) If duty(D) is 1428 hectares/cumec and base period (B) is 120 days for an irrigated crop , then delta (Δ)in meters is given by

11) A test plate 30cm square ,settles by 12mm under a load of 4.5kN in a sandy soil. By how much will a footing 2m*2m subjected to a load of 200kN settle?

12) The variable cost (V) of manufacturing a product varies according to the equation V = 4q, where q is the quantity produced. The fixed cost (F) of production of same product reduces with q according to the equation F = 100/q.How many units should be produced to minimize the total cost (V+F)?

14) Two points A and B are 1530 m apart across a river. The reciprocal levels measured are Level at Readings on A (in m) B A 2.165 3.810 B 0.910 2.355 The true difference in level between A and B would be

15) Consider the case where noise level of 90 dBA exists for 5 min and is followed by reduced noise level of 60dBA for 50 min . The equivalent continuous energy level (Leq) for 55 min period is

17) Given that logP/(y-z)=logQ/(z-x) =logR/(x-y) =10,for x≠y≠z, What is the value of PQR?

18) Knowledgeable dog owners recommend obtaining a purebred dog from a ______ breeder, one who is ______ and respected by customers and breeders alike.

19) The observed bearings of a traverse are given below Line Bearing Line Bearing PQ 46015' QP 226015' AR 108015' RQ 286015' RS 201030' SR 20030' ST 321045' TS 141045' The station(s) most likely to be affected by local attraction is/are

20) Given f(t) =L-1[(3s+1)/(s3+4s2+(k-3)s)] if Lim t->∞ f(t) =1, then the value of K is

21) The critical height of an unsupported vertical cut in a cohesive soil is given by where c= unit cohesion Φ= angle of internal friction γ=unit weight of soil

22) Due to rise in temperature the viscosity and unit weight of percolating fluid are reduced to 70% and 90% resp. other things being constant , the change in coefficient of permeability will be

23) The muskingum method of flood routing gives θ2=c0I2+c1I1+c2θ1.The coefficients in this equation will have values such that

25) In a rectangular strain gauge rosette , the strain recorded are ε00=400 µ strain, ε450=300 µ strain, ε900=200 µstrain, What is the maximum principal strain at that point

26) A square footing has dimensions of 2m by 2m and a depth of 2m . What will be its ultimate bearing capacity in pure clay with an unconfined compressive strength of 0.15N/mm2 ,Φ=00,γ=1.7g/cm3 by using Terzaghi's bearing capacity equation?

27) Activated sludge is the

28) X and y are integers and if x^2/y^3 is even integer then which of the following must be an even integer?

29) Find out the mean proportion of 3 & 27 also find out their third proportion

30) A consolidated undrained triaxial compression test was performed on a saturated sand at a cell pressure of 80kPa.The ultimate deviator stress was 270kPa and the pore pressure at peak stress was -50kPa. The effective angle of internal friction of soil will be

31) The sequent depth ratio in a hydraulic jump formed in a horizontal rectangular channel is 16.48 . the froude number of super critical stream is

33) A surface water flow of 28000m3/day is coagulated by adding 75mg/l of ferrous sulphate and an equivalent dose of lime. How much lime is required at a purity of 90%CaO.

34) The value of line integral ∫(2xy2dx+2x2ydy+dz) c along a path joining the origin (0,0,0) and the point (1,1,1) is

35) The dimension of a symmetrical welded I-section are shown in the figure. All dimensions are in mm. The plastic section modulus about the weaker axis (in cm3) up to one decimal place is______

36) If the following equation establishes equilibrium in slightly bent position , the mid-span deflection of a member shown in the figure is

37) With a(n) ______ grin, the boy quickly slipped the candy into his pocket without his mother's knowledge.

38) For wheel loads 3kN,4kN,5kN,6kN spaced 2m,3m,3m apart are moving on a simply supported beam of span 24m with 3kN load loading from left to right. To find maximum BM at 18m from left support , the load that must be placed at the section is

39) The limiting neutral axis depth for a rectangular section of effective depth d. For SAIL-MA:300 HY grade steel is_____d

40) Design the rate of super elevation for a horizontal highway curve of radius 500m and speed 100kmph

41) Each of the letters in the figure below represents a unique integer from 1 to 9. The letters are positioned in the figure such that each of (A+B+C), (C+D+E), (E+F+G) and (G+H+K) is equal to 13. Which integer does E represent? | A | B | C | | D | | E | F | G | | H | | K |

42) A numerical solution of the equation f(x)=x+√x -3=0 can be obtained using newton raphson method. If the starting value is x=2 for the iteration , the value of x that is to be used in the next step is

43) For a linear elastic isotropic material the no. of independent elastic constants is/are

44) For the bracket connection shown inthe following fig. Which bolt will have the maximum resultant force?

49) A column 300mm*300mm is provided an footing 1m*1m supported on soil having allowable bearing pressure of 360kN/m2 at depth 1m below ground . The minimum thickness of footing assuming M20 concrete and Fe 4,5 steel;

50) The matrix M is given below has eigen values -3,-3,5. an eigen vector corresponding to the eigen value 5 is [ 1 2 -1]T . one of the eigen vectors of the matrix M3 is

51) Consider the solutions x(t)=x1(t) and x(t)=x2(t) of the differential equation d2(x)/dt2+x(t)=0,t>0 such that x2=0,dx2(t)/dt |t=0 =1.The wresskian w(t) at t=π/2 is

52) Chlorides from water are removed by

53) The total degree of static indeterminacy of the plane frame shown in the given figure is

54) The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M

55) The resultant cuts the base of a circular column of diameter'd'. with an eccentricity equal to one-fourth of 'd'. The ratio between the maximum compressive stress and the maximum tensile stress is

56) The correct relationship among displacement thickness 'd' , momentum thickness 'm', and energy thickness 'e'is

57) When a driver drives down the grade at speed of 30kmph with coefficient of friction 0.40,the driver requires braking distance twice that required for stopping the vehicle when he travels up the same grade. The grade is

61) The critical combination of stresses for corner region in cement concrete road is

62) For the given frame what will be the reaction at C(in kN)

63) A trapezoidal channel having bottom width 8m and side slope 1:1 , carries a discharge of 80m3/s. Initial depth before the jump is 0.75m.The loss of energy in the jump is(in m)

64) For the truss shown in figure, find the movement of joint C , A=20cm2 , E=200GPa for all members. Plane of rollers is parallel to AD. ________mm

65) The activity duration (days) and resources and requirements (units) are shown in the figure below . What is the maximum resource required in a day?

1) A random variable X has the probability density function f(x) as given below: [a+bx for 0<x<1 f(x)= [0, otherwise if the expected value E(x) =2/3 then P_r[X<0.5]is _______

3) A sample of water has been analyzed for common ions and results are presented in the form of an bar diagram as shown The non carbonate hardness(expressed in mg/l as CaCO₃) of the sample is

14) Two points A and B are 1530 m apart across a river. The reciprocal levels measured are

Level at Readings on A (in m) B

A 2.165 3.810

B 0.910 2.355

The true difference in level between A and B would be

15) Consider the case where noise level of 90 dBA exists for 5 min and is followed by reduced noise level of 60dBA for 50 min . The equivalent continuous energy level (L_eq) for 55 min period is

18) Knowledgeable dog owners recommend obtaining a purebred dog from a breeder, one who is and respected by customers and breeders alike.

19) The observed bearings of a traverse are given below

Line Bearing Line Bearing

PQ 46⁰15' QP 226⁰15'

AR 108⁰15' RQ 286⁰15'

RS 201⁰30' SR 20⁰30'

ST 321⁰45' TS 141⁰45'

The station(s) most likely to be affected by local attraction is/are

20)

Given f(t) =L^-1[(3s+1)/(s³+4s²+(k-3)s)] if Lim_t->∞ f(t) =1, then the value of K is

23) The muskingum method of flood routing gives θ₂=c₀I₂+c₁I₁+c₂θ₁.The coefficients in this equation will have values such that

25) In a rectangular strain gauge rosette , the strain recorded are ε_0⁰=400 µ strain, ε_45⁰=300 µ strain, ε_90⁰=200 µstrain, What is the maximum principal strain at that point

26) A square footing has dimensions of 2m by 2m and a depth of 2m . What will be its ultimate bearing capacity in pure clay with an unconfined compressive strength of 0.15N/mm² ,Φ=0⁰,γ=1.7g/cm³ by using Terzaghi's bearing capacity equation?

33) A surface water flow of 28000m³/day is coagulated by adding 75mg/l of ferrous sulphate and an equivalent dose of lime. How much lime is required at a purity of 90%CaO.

34) The value of line integral ∫(2xy²dx+2x²ydy+dz) c along a path joining the origin (0,0,0) and the point (1,1,1) is

35) The dimension of a symmetrical welded I-section are shown in the figure. All dimensions are in mm. The plastic section modulus about the weaker axis (in cm³) up to one decimal place is______

49) A column 300mm300mm is provided an footing 1m1m supported on soil having allowable bearing pressure of 360kN/m² at depth 1m below ground . The minimum thickness of footing assuming M20 concrete and Fe 4,5 steel;

50) The matrix M is given below has eigen values -3,-3,5. an eigen vector corresponding to the eigen value 5 is [ 1 2 -1]^T . one of the eigen vectors of the matrix M³ is

51) Consider the solutions x(t)=x₁(t) and x(t)=x₂(t) of the differential equation d²(x)/dt²+x(t)=0,t>0 such that x₂=0,dx₂(t)/dt |_t=0=1.The wresskian w(t) at t=π/2 is

63) A trapezoidal channel having bottom width 8m and side slope 1:1 , carries a discharge of 80m³/s. Initial depth before the jump is 0.75m.The loss of energy in the jump is(in m)

64) For the truss shown in figure, find the movement of joint C , A=20cm² , E=200GPa for all members. Plane of rollers is parallel to AD. ________mm