Step 1: Understanding the Concept:

The sentence requires a past tense verb to establish the context of a completed event and a contrastive preposition to show a concession regarding the weather.


Step 2: Detailed Explanation:

1. In the first blank, the phrase "needed to improve" in the following sentence indicates that the action of the first sentence occurred in the past.

2. Therefore, the past tense "scored" is required to maintain tense consistency.

3. In the second blank, we need a word that indicates "regardless of" or "notwithstanding."

4. Both "despite" and "in spite of" carry this meaning, but "instead of" implies a substitution which is logically incorrect here.

5. Combining these, "scored" and "despite" provide the only grammatically and logically consistent pair.


Step 3: Final Answer:

The team scored more than 300 runs despite rains.
Quick Tip: Check the surrounding sentences for context clues. The word "needed" (past tense) is a direct indicator that the first blank must also be in the past tense.

Step 1: Understanding the Concept:

This problem involves solving a logarithmic equation by simplifying terms with different bases into a common base.


Step 2: Key Formula or Approach:

The change of base property or the base power property is used: \[ \log_{b^n} a = \frac{1}{n} \log_b a \]

Step 3: Detailed Explanation:

The given equation is: \[ \log_2 x + \log_{\sqrt{2}} x = 48 \]
Since \(\sqrt{2} = 2^{1/2}\), we can rewrite the second term using the property mentioned above: \[ \log_{2^{1/2}} x = \frac{1}{1/2} \log_2 x = 2 \log_2 x \]
Substitute this back into the original equation: \[ \log_2 x + 2 \log_2 x = 48 \] \[ 3 \log_2 x = 48 \]
Divide both sides by 3: \[ \log_2 x = 16 \]
Converting from logarithmic form to exponential form: \[ x = 2^{16} \]

Step 4: Final Answer:

The value of \(x\) is \(2^{16}\).
Quick Tip: When dealing with logs of different bases that are powers of each other, always convert to the simplest base to consolidate terms.

Step 1: Understanding the Concept:

This is a visual logic puzzle where we observe the movement patterns of different symbols within a grid across a sequence.


Step 2: Detailed Explanation:

Let's track the elements individually in the \(3 \times 4\) grid (3 columns, 4 rows).

1. The Triangle (\(\blacktriangle\)):

- Figure 1: (Col 3, Row 4)

- Figure 2: (Col 3, Row 3)

- Figure 3: (Col 3, Row 2)

- Pattern: Moves down one cell in Column 3 in every step.

- Expected Position: (Col 3, Row 1).

2. The Circle (\(\bullet\)):

- Figure 1: (Col 1, Row 2)

- Figure 2: (Col 1, Row 1)

- Figure 3: (Col 1, Row 4) - Note: It wraps around from top to bottom.

- Pattern: Moves up one cell in Column 1 and wraps around.

- Expected Position: (Col 1, Row 3).

3. Arrow: The arrow at the top left corner remains stationary throughout the sequence.


Step 3: Final Answer:

The correct figure must have the circle at (1, 3) and the triangle at (3, 1), which matches option (A).
Quick Tip: Isolate one symbol at a time and determine its specific vector of movement (direction and distance) before moving to the next symbol.

Step 1: Understanding the Concept:

This problem relies on the rules of logical negation for quantified statements.


Step 2: Detailed Explanation:

1. The original statement is a universal affirmative: "All \(A\) are \(B\)."

2. If this statement is "false," then its logical negation must be "true."

3. The negation of "All \(A\) are \(B\)" is "Some \(A\) are not \(B\)," or "There exists at least one \(A\) that is not \(B\)."

4. Let's look at the options:

- Option (A) and (B) describe "No \(A\) is \(B\)." This could be true, but it is not \textit{necessarily true (only one bad mango is needed to make the original statement false).

- Option (C) requires some to be good and some to be bad. This is too specific; the original is false even if \textit{all are bad.

- Option (D) correctly identifies that at least one mango failing the condition is enough to negate the "All" claim.


Step 3: Final Answer:

The logical negation of "All" is "Some... not," which matches statement (D).
Quick Tip: In formal logic, the opposite of "All" is "Not All" (meaning "Some Not"), whereas the opposite of "None" is "At least one." Don't confuse "opposite" with "negation."

Step 1: Understanding the Concept:

This involves testing logical implications based on the bounds of a sum.


Step 2: Detailed Explanation:

Let the four numbers be \(a, b, c, d\).

From (S1), the sum is: \(a + b + c + d = 4 \times 25 = 100\).

Let's check Option (C): If (S1) and (S3) are true, then \(a, b, c, d \ge 20\) and \(a+b+c+d=100\).

To find the maximum possible value for any one number (say \(a\)), we must minimize the other three.

Minimum values for \(b, c, d\) are 20 each.
\(a + 20 + 20 + 20 = 100 \Rightarrow a = 40\).

Since any number can be at most 40, (S2) is necessarily true.

Other options can be disproved by counterexamples:

- For (A): \(\{40, 40, 15, 5\}\) sums to 100 and all \(\le 40\), but not all \(\ge 20\).

- For (B): \(\{20, 20, 20, 20\}\) and \(\{40, 40, 40, 40\}\) satisfy S2 and S3 but have different averages.


Step 3: Final Answer:

Combining an average of 25 with a lower bound of 20 mathematically forces an upper bound of 40, validating (C).
Quick Tip: To disprove an implication, try to find a set of numbers that satisfies the premise but violates the conclusion.

Step 1: Understanding the Concept:

The question evaluates the correct usage of indefinite articles, conjunctions, and prepositions based on phonetics and context.


Step 2: Detailed Explanation:

1. Blank 1: "...around \hspace{0.5cm pit". 'Pit' starts with a consonant sound, so the article is 'a'.

2. Blank 2: "...into which \underline{\hspace{0.5cm elephant". 'Elephant' starts with a vowel sound, so the article is 'an'.

3. Blank 3: "...bewildered \underline{\hspace{0.5cm miserable". These are two similar adjectives describing the elephant's state. The conjunction 'and' is appropriate to join them.

4. Blank 4: "...look up \underline{\hspace{0.5cm a vast... crowd". When looking towards a target, the preposition 'at' is used.


Step 3: Final Answer:

The sequence "a; an; and; at" correctly completes the passage, matching option (B).
Quick Tip: Remember: "a" vs "an" is determined by the \textbf{sound of the following word, not just the first letter.

Step 1: Understanding the Concept:

We need to solve a system of linear equations based on the total quantity of items and the total revenue.


Step 2: Key Formula or Approach:

Let the quantity of product T sold be \(x\).

Then: \(S = 2x, R = 3x, Q = 4x\).

Let quantity of P be \(P\).


Step 3: Detailed Explanation:

1. Equation for Total Quantity: \[ P + Q + R + S + T = 250 \] \[ P + 4x + 3x + 2x + x = 250 \Rightarrow P + 10x = 250 \quad --- (Equation 1) \]
2. Equation for Total Revenue:
Total items = 250, Average Price = 60. Total Revenue = \(250 \times 60 = 15,000\).
\[ 100P + 50(4x) + 40(3x) + 60(2x) + 60x = 15,000 \] \[ 100P + 200x + 120x + 120x + 60x = 15,000 \] \[ 100P + 500x = 15,000 \Rightarrow P + 5x = 150 \quad --- (Equation 2) \]
3. Solve the system:
Multiply Equation 2 by 2: \(2P + 10x = 300\).

Subtract Equation 1: \((2P + 10x) - (P + 10x) = 300 - 250\).
\(P = 50\).


Step 4: Final Answer:

The quantity of product P sold is 50.
Quick Tip: When multiple variables are related to one common variable, express everything in terms of that common variable (\(x\)) to simplify the math.

Step 1: Understanding the Concept:

The relationship between diameter and circumference/arc length determines the changes in the string lengths.


Step 2: Detailed Explanation:

1. Initially:

- String P length = \(l\) (this is the diameter \(d\)).

- String K is a semicircular arc with diameter \(l\). Length of K = \(\frac{\pi l}{2}\).

2. After shortening by \(x\):

- New length of P = \(l - x\). This is the diameter of the new circle.

- New length of K = \(\frac{\pi l}{2} - x\). This is the circumference of the new circle.

3. Relate the two:

Circumference = \(\pi \times\) Diameter
\[ \frac{\pi l}{2} - x = \pi(l - x) \] \[ \frac{\pi l}{2} - x = \pi l - \pi x \] \[ \pi x - x = \pi l - \frac{\pi l}{2} \] \[ x(\pi - 1) = \frac{\pi l}{2} \] \[ \frac{x}{l} = \frac{\pi}{2(\pi - 1)} \]

Step 3: Final Answer:

The ratio \(x/l\) is \(\frac{\pi}{2(\pi - 1)}\).
Quick Tip: Be careful with Figure (i). String K is just the arc, not the full perimeter of the semicircle. If it were the perimeter, the equation would be different.

Step 1: Understanding the Concept:

This is a logical arrangement problem where we map individuals to a \(2 \times 2\) grid based on positional and relationship constraints.


Step 2: Detailed Explanation:

Let's denote the family as M (Meritorius), B (Brother), S (Son), and D (Daughter). Note the siblings: (M, B) and (S, D).

1. From (i): B and D are in one column (say C1). Then M and S must be in C2.

2. From (ii): S (in C2) is diagonal to the sibling of the worst orator (who must be in C1).

- If S is at (R2, C2), then the sibling of the worst orator is at (R1, C1).

- At (R1, C1) sits either B or D.

- If B is at (R1, C1), then the worst orator is M (sibling of B).

- If D is at (R1, C1), then the worst orator is S (sibling of D).

3. From (iii): The best and worst are in the same row.

- Case 1: If M (R1, C2) is worst, then the best must be whoever is in (R1, C1), which is B.

- Case 2: If S (R2, C2) is worst, then the best must be whoever is in (R2, C1), which is B (if D was in R1).

In both logically consistent scenarios, Meritorius' brother ends up being the best orator.


Step 3: Final Answer:

The best orator is Meritorius' brother.
Quick Tip: Map relationships (sibling pairs) before attempting to place people in the grid. This narrows down the possibilities significantly.

Step 1: Understanding the Concept:

This problem requires identifying a repeating unit cell (motif) that, when translated, creates the larger tessellated pattern.


Step 2: Detailed Explanation:

1. Examine the large pattern for a unique repeating structure. It consists of interlocking segments with specific "arms."

2. Look at pattern R: It has a central vertical segment with three horizontal projections—one at the top-right, one in the middle-left, and one at the bottom-right.

3. Mentally replicate pattern R side-by-side. The middle-left projection of one unit will fit between the top and bottom projections of the unit to its left.

4. This specific interlocking geometry is absent in patterns P, Q, and S, which have different arm orientations.


Step 3: Final Answer:

Pattern R is the fundamental unit used to generate the image.
Quick Tip: Identify a specific "joint" or empty space in the main figure and see which individual pattern can fill that space or replicate that joint exactly.

Step 1: Understanding the Concept:

This question uses properties relating a matrix to its eigenvalues and the trace of a matrix.


Step 2: Key Formula or Approach:

1. If \(\lambda_i\) are eigenvalues of \(A\), then \(\lambda_i^k\) are eigenvalues of \(A^k\).

2. Trace of a matrix is the sum of its eigenvalues: \(Tr(M) = \sum \lambda_i\).


Step 3: Detailed Explanation:

1. Given eigenvalues of matrix \(A\): \(\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 3\).

2. The eigenvalues of matrix \(A^2\) are the squares of the eigenvalues of matrix \(A\).
\[ Eigenvalues of A^2: 1^2, 2^2, 3^2 \Rightarrow 1, 4, 9. \]
3. The Trace of \(A^2\) is the sum of these eigenvalues:
\[ Trace(A^2) = 1 + 4 + 9 = 14. \]

Step 4: Final Answer:

The Trace of \(A^2\) is 14.
Quick Tip: Properties of Eigenvalues:
Sum of Eigenvalues = Trace
Product of Eigenvalues = Determinant

Step 1: Understanding the Concept:

The problem relates to the error in numerical differentiation based on the curvature (concavity) of a function.


Step 2: Detailed Explanation:

1. Let \(f(x) = \sum_{i=0}^5 a_i x^i\) with \(a_i > 0\).

2. For \(x > 0\), the second derivative \(f''(x)\) will also be a polynomial with positive coefficients. Thus, \(f''(x) > 0\) for all \(x > 0\).

3. Since \(f''(x) > 0\), the function is concave up.

4. In a concave up function, the tangent at a point lies below the curve.

5. The Forward Difference \(D_+ = \frac{f(x+h) - f(x)}{h}\) is the slope of a secant line to the right. Because the curve is bending upwards, this slope is greater than the tangent slope at \(x\). Hence, forward difference overestimates. (I) is FALSE.

6. The Backward Difference \(D_- = \frac{f(x) - f(x-h)}{h}\) is the slope of a secant line to the left. This slope is smaller than the tangent slope at \(x\). Hence, backward difference underestimates. (II) is FALSE.


Step 3: Final Answer:

Both statements are false, leading to option (A).
Quick Tip: Visualization: Imagine the curve \(y=x^2\). The line connecting \((1,1)\) to \((2,4)\) is steeper (slope 3) than the tangent at \(x=1\) (slope 2). Thus, forward difference overestimates for concave-up curves.

Step 1: Understanding the Concept:

The given \(f(x)\) is a square wave. It is an odd function because \(f(-x) = -f(x)\).


Step 2: Key Formula or Approach:

For an odd function, the Fourier series consists only of sine terms:
\[ a_0 = 0, a_n = 0, b_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \sin(nx) dx \]

Step 3: Detailed Explanation:

1. Calculate \(b_n\):
\[ b_n = \frac{2}{\pi} \int_{0}^{\pi} (1) \sin(nx) dx = \frac{2}{\pi} \left[ \frac{-\cos(nx)}{n} \right]_0^\pi \] \[ b_n = \frac{2}{n\pi} [-\cos(n\pi) + \cos(0)] = \frac{2}{n\pi} [1 - (-1)^n] \]
2. If \(n\) is even, \(b_n = 0\).

3. If \(n\) is odd, \(b_n = \frac{4}{n\pi}\).

4. The series is:
\[ f(x) = \sum_{n=1,3,5..} \frac{4}{n\pi} \sin(nx) = \frac{4}{\pi} \left[ \sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \dots \right] \]

Step 4: Final Answer:

The expansion results in only odd harmonic sine terms with positive coefficients, matching option (A).
Quick Tip: Identify symmetry first! If the graph is symmetric about the origin, it's an odd function (Sine terms only). If it's symmetric about the y-axis, it's an even function (Cosine terms only).

Step 1: Understanding the Concept:

The mode of failure of a reinforced concrete beam depends on the relationship between the actual depth of the neutral axis (\(x_u\)) and the limiting (balanced) depth of the neutral axis (\(x_{u,max}\)).


Step 2: Key Formula or Approach:

1. Calculate the limiting depth of the neutral axis: \[ x_{u,max} = \left( \frac{x_{u,max}}{d} \right) \times d \]
2. Compare \(x_u\) with \(x_{u,max}\).

3. If \(x_u > x_{u,max}\), the section is over-reinforced.

4. If \(x_u < x_{u,max}\), the section is under-reinforced.


Step 3: Detailed Explanation:

Given data:

Effective depth, \(d = 550 mm\)

Actual depth of neutral axis, \(x_u = 302 mm\)

Given ratio, \(\frac{x_{u,max}}{d} = 0.456\)

Calculating limiting depth: \[ x_{u,max} = 0.456 \times 550 = 250.8 mm \]

Comparison:
Since \(x_u (302 mm) > x_{u,max} (250.8 mm)\), the beam is an over-reinforced section.

In over-reinforced sections, the concrete reaches its maximum permissible strain (0.0035) before the steel reaches its yield strain. This leads to a sudden and brittle failure known as compression failure.


Step 4: Final Answer:

The beam is over-reinforced, so the most probable mode of failure is compression failure.
Quick Tip: Always remember: Under-reinforced sections (\(x_u < x_{u,max}\)) fail by yielding of steel (ductile failure), while over-reinforced sections (\(x_u > x_{u,max}\)) fail by crushing of concrete (brittle/compression failure).

Step 1: Understanding the Concept:

In the Limit State Design of steel structures (IS 800:2007), moment capacity is calculated based on whether the section is plastic, compact, or semi-compact. For plastic and compact sections, the design strength is based on the plastic section modulus (\(Z_p\)).


Step 2: Detailed Explanation:
For a plastic or compact section, the design bending strength is given by: \[ M_d = \beta_b Z_p \frac{f_y}{\gamma_{m0}} \]
To prevent excessive deformation or permanent "set" at serviceability (working) loads, the code imposes an upper limit on the moment capacity.

For cantilever beams, the limit is \(1.5 Z_e \frac{f_y}{\gamma_{m0}}\) (and for simply supported beams, it is \(1.2 Z_e \frac{f_y}{\gamma_{m0}}\)).

The purpose of this limit is to ensure that even if the section reaches its plastic capacity at ultimate loads, it does not undergo significant plastic deformation under working loads, which would violate serviceability requirements.


Step 3: Final Answer:

The upper bound is imposed to avoid plastic deformation under working load.
Quick Tip: The factor 1.2 or 1.5 ensures that the service load (roughly Ultimate Load / 1.5) does not exceed the elastic limit (\(Z_e f_y\)).

Step 1: Understanding the Concept:

A state of plane strain occurs in bodies that are very long in one dimension (e.g., dams, tunnels, retaining walls). In such cases, deformation in the longitudinal direction (Z-axis) is constrained.


Step 2: Detailed Explanation:

By definition, in a plane strain problem associated with the X-Y plane:
1. There is no displacement in the Z-direction (\(w = 0\)).

2. The displacements in X and Y (\(u, v\)) are functions of \(x\) and \(y\) only.

This leads to the following strain components being zero:
- Normal strain in Z: \(\epsilon_{zz} = \frac{\partial w}{\partial z} = 0\).

- Shear strain in XZ plane: \(\gamma_{xz} = \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} = 0\).

- Shear strain in YZ plane: \(\gamma_{yz} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} = 0\).

Note that \(\sigma_{zz}\) is NOT necessarily zero in plane strain; it is usually non-zero because the constraint induces stress.


Step 3: Final Answer:

The required condition is \(\epsilon_{zz} = \epsilon_{xz} = \epsilon_{yz} = 0\).
Quick Tip: Distinguish carefully:
Plane Stress: \(\sigma_{zz} = \tau_{xz} = \tau_{yz} = 0\) (Thin plates)
Plane Strain: \(\epsilon_{zz} = \gamma_{xz} = \gamma_{yz} = 0\) (Long structures)

Step 1: Understanding the Concept:

A state of pure shear is a stress condition where the normal stresses on the planes are zero (\(\sigma_x = 0, \sigma_y = 0\)) and only shear stress (\(\tau_{xy}\)) exists.


Step 2: Detailed Explanation:

1. Examining the provided options:

- Option (A) shows both normal and shear components.

- Option (B) shows an element where only tangential (shear) arrows of magnitude 40 are present on all four faces. There are no normal stress arrows perpendicular to the faces. This fits the definition of pure shear.

- Option (C) shows a complex combination of normal and shear stresses.

- Option (D) represents a biaxial normal stress state.

2. In a pure shear state \(\tau\), the principal stresses are \(\sigma_1 = +\tau\) and \(\sigma_2 = -\tau\). Mohr's circle for this state is centered at the origin.


Step 3: Final Answer:

The diagram in Option (B) represents pure shear because only shear stresses are acting on the element faces.
Quick Tip: Pure shear can also be represented by an element oriented at \(45^\circ\) where normal stresses are equal and opposite (\(\sigma\) and \(-\sigma\)) and shear stress is zero.

Step 1: Understanding the Concept:

A beam with internal hinges can be analyzed by breaking it into statically determinate segments. The segments between hinges or between a hinge and a free end behave as simply supported or cantilever sub-structures.


Step 2: Key Formula or Approach:

The beam has internal hinges at G and J. The middle segment G-H-J is supported by the segments F-G and J-K-L. We first find the reactions at the hinges by analyzing the "suspended" segment.


Step 3: Detailed Explanation:

1. Analyze segment G-H-J:
This segment acts like a simply supported beam with a span of \(2 + 2 = 4 m\).
Load: \(30 kN\) at the center (point H).
Let \(V_G\) and \(V_J\) be the vertical shear forces at the hinges.
By symmetry: \(V_G = V_J = \frac{30}{2} = 15 kN\).
(Note: The \(20 kN\) load at hinge G is transferred directly to segment FG, not GHJ).

2. Analyze segment J-K-L:
This is a cantilever segment fixed at L and supported (pinned) at the hinge J.
Loads acting on segment J-L:
- Vertical force from segment G-H-J: \(V_J = 15 kN\) (downward).
- Uniformly distributed load on K-L: \(4 kN/m\) over a distance of \(2 m\).
Total downward load on segment J-L: \[ \sum F_{y,down} = V_J + (w \times Length_{KL}) \] \[ \sum F_{y,down} = 15 + (4 \times 2) = 15 + 8 = 23 kN \]
3. Calculate vertical reaction at L (\(R_L\)):
For equilibrium of segment J-L, \(\sum F_y = 0\): \[ R_L = \sum F_{y,down} = 23 kN \]

Step 4: Final Answer:

The upward vertical reaction at support L is 23 kN.
Quick Tip: Always start analysis from the "most dependent" segment (the one that cannot stand alone, usually the segment between two hinges). The reactions at its ends become loads on the adjacent "supporting" segments.

Step 1: Understanding the Concept:

The discharge required for irrigation depends on the area to be irrigated and the "Duty" of the water for that specific crop. Duty is the area that can be irrigated by a unit discharge flowing continuously throughout the base period.


Step 2: Key Formula or Approach:

1. Duty (\(D\)) in hectares/cumec: \[ D = \frac{8.64 B}{\Delta} \]
where \(B\) is base period in days and \(\Delta\) is depth in meters.

2. Discharge (\(Q\)) in \(m^3/s\): \[ Q = \frac{Area}{Duty} \]

Step 3: Detailed Explanation:

Given data:

CCA \(= 10000 ha\)

Base period (\(B\)) \(= 140 days\)

Depth of water (\(\Delta\)) \(= 40 cm = 0.4 m\)

1. Calculate Duty: \[ D = \frac{8.64 \times 140}{0.4} = \frac{1209.6}{0.4} = 3024 ha/cumec \]

2. Calculate Discharge: \[ Q = \frac{10000}{3024} \approx 3.307 m^3/s \]

The calculated discharge \(3.307 m^3/s\) lies in the range of 3 to 4.


Step 4: Final Answer:

The outlet discharge is approximately 3.31 \(m^3/s\), which lies between 3 and 4.
Quick Tip: Remember the units for the duty formula: \(B\) must be in days and \(\Delta\) must be in meters to get Duty in hectares per cumec. Always convert cm to m first!

Step 1: Understanding the Concept:

A rainfall mass curve is a plot of cumulative precipitation against time. It shows how the total depth of rain increases as time progresses during a storm.


Step 2: Detailed Explanation:

- The slope of the mass curve (\(\frac{dP}{dt}\)) represents the intensity of rainfall at that specific instant.

- Statement (I): During a period of rain, precipitation is accumulating, so the slope (\(i = \frac{dP}{dt}\)) is positive. Thus, statement (I) is TRUE.

- Statement (II): If there is a break in the rain, the cumulative precipitation remains constant. The curve becomes horizontal, meaning the slope is zero. Thus, statement (II) is TRUE.

- Note: The slope can never be negative because cumulative precipitation can never decrease with time.


Step 3: Final Answer:

Both statements regarding the slope of the mass curve are true.
Quick Tip: A mass curve is always a non-decreasing function. Its slope gives the intensity, and the sharpest rising part of the curve represents the period of maximum intensity.

Step 1: Understanding the Concept:

The Casagrande plasticity chart classifies fine-grained soils based on Liquid Limit (\(w_L\)) and Plasticity Index (\(I_P\)). Clays (\(C\)) lie above the A-line, and the upper boundary of all soil data is the U-line.


Step 2: Key Formula or Approach:

1. A-line Equation: \[ I_{P,A} = 0.73(w_L - 20) \]
2. U-line Equation: \[ I_{P,U} = 0.9(w_L - 8) \]

Step 3: Detailed Explanation:

Given Liquid Limit, \(w_L = 40%\).

1. Lower limit for clays (A-line):
For a soil to be classified as inorganic clay, it must be on or above the A-line. \[ I_{P,A} = 0.73(40 - 20) = 0.73 \times 20 = 14.6% \]
2. Upper limit for all soils (U-line):
The U-line represents the upper bound for any natural soil. \[ I_{P,U} = 0.9(40 - 8) = 0.9 \times 32 = 28.8% \]
Therefore, for inorganic clays with a liquid limit of 40%, the plasticity index typically ranges from the A-line value up to the U-line value.


Step 4: Final Answer:

The range is between 14.6% and 28.8%.
Quick Tip: If the soil is below the A-line, it is classified as silt (\(M\)) or organic soil (\(O\)). For \(w_L = 40%\), the soil is in the 'Intermediate Plasticity' (\(I\)) zone.

Step 1: Understanding the Concept:

This question tests the knowledge of standard formulas and charts used in geotechnical foundation design for stress distribution, pile capacity, and bearing capacity.


Step 2: Detailed Explanation:

- (I) Converse-Labarre formula: It is used to calculate the efficiency of a pile group based on the number of rows, columns, and spacing of piles. (Matches Q).

- (II) Hiley formula: It is a well-known dynamic formula used to estimate the load-carrying capacity of a pile during driving based on the energy of the hammer blow and the set (penetration) of the pile. (Matches P).

- (III) Newmark's influence chart: It is used to determine the increase in vertical stress at any point under a foundation area of any shape loaded with a uniform pressure. (Matches S).

- (IV) Skempton's equation: Skempton proposed values for the bearing capacity factor (\(N_c\)) specifically for saturated clays as a function of the depth-to-width ratio of the footing. (Matches R).


Step 3: Final Answer:

The correct match is (I)-Q, (II)-P, (III)-S, (IV)-R.
Quick Tip: Remember that dynamic pile formulas (Hiley, Engineering News) are generally suitable for coarse-grained soils, while static formulas or Skempton's factors are used for clays.

Step 1: Understanding the Concept:

On a divided highway (dual carriageway), traffic in one direction is not expected to face oncoming vehicles. Thus, the minimum sight distance required is the Stopping Sight Distance (SSD). On an undivided road with two-way traffic, if SSD cannot be provided for each lane, a higher sight distance (like Intermediate Sight Distance or \(2 \times SSD\)) is often provided to prevent head-on collisions.


Step 2: Key Formula or Approach:

1. For dual carriageways, normal practice is to provide \(SSD\) or \(ISD\).
2. If traffic becomes two-way on a single lane due to diversion, the safe sight distance to avoid head-on collision between two vehicles is \(2 \times SSD\).


Step 3: Detailed Explanation:

1. Under normal conditions on a dual carriageway, there is no chance of head-on collision. The "desired sight distance" kept as \(360 m\) is typically the Intermediate Sight Distance (ISD).

2. Relationship: \(ISD = 2 \times SSD\).

3. Given \(Desired Sight Distance = 360 m\). Assuming this is the ISD provided for the highway. \[ 360 = 2 \times SSD \Rightarrow SSD = 180 m \]

4. After the incident, the traffic is diverted to one side, making it a two-way traffic section on an undivided carriageway. The question asks for the SSD which should be available. The Stopping Sight Distance for a vehicle at a given speed is a constant physical requirement based on reaction time and braking distance; it does not change based on road diversion.

5. Thus, if \(360 m\) was provided as the safe distance for two-way head-on avoidance (\(2 \times SSD\)), the SSD required for an individual vehicle to stop is \(180 m\).


Step 4: Final Answer:

The Stopping Sight Distance (SSD) is 180 m.
Quick Tip: On a single-lane road with two-way traffic, the minimum sight distance required is \(2 \times SSD\). If a question mentions a "desired distance" of 360 for such conditions, the SSD is likely 180.

Step 1: Understanding the Concept:

"Cant" (or Superelevation) is provided on curves to counteract centrifugal force. There is an "equilibrium speed" where the provided cant perfectly balances the force.


Step 2: Detailed Explanation:

1. Equilibrium Cant (\(e_{eq}\)): The superelevation required for a specific speed \(v\). It is calculated as \(e = \frac{G v^2}{127 R}\).

2. Cant Deficiency: If a train moves at a speed higher than the equilibrium speed for which the track was designed, the actual cant provided is less than what is required for that higher speed. This shortfall is called cant deficiency.

3. Cant Excess: If a train moves at a speed lower than the equilibrium speed, the actual cant provided is more than what is required for that lower speed. This surplus is called cant excess.

4. Therefore, deficiency occurs at high speeds and excess occurs at low speeds.


Step 3: Final Answer:

Statement (A) is correct.
Quick Tip: Cant deficiency leads to higher pressure on the outer rail, while cant excess leads to higher pressure on the inner rail.

Step 1: Understanding the Concept:

A Windrose diagram is a graphical representation of wind data (direction, duration, and intensity) at an airport site collected over a long period (usually 5-10 years).


Step 2: Detailed Explanation:

- Aircraft perform best when taking off and landing into the wind (headwind). Crosswinds can be dangerous.

- The orientation of the runway should be such that the maximum "wind coverage" is achieved (usually 95%). This means for 95% of the time, the crosswind component does not exceed a certain limit (e.g., 25 kmph).

- The Windrose diagram helps identify the direction along which the wind blows most frequently, which dictates the direction (orientation) in which the runway should be built.


Step 3: Final Answer:

The primary purpose is the orientation of a runway.
Quick Tip: There are two types of Windrose diagrams: Type I (direction and duration only) and Type II (direction, duration, and intensity). Type II is used for orientation to check for crosswind limits.

Step 1: Understanding the Concept:

Levelling is the branch of surveying used to determine relative heights. It involves definitions of surfaces like level surfaces, horizontal surfaces, and geoids.


Step 2: Detailed Explanation:

- Statement (A): By definition, a level surface is a curved surface which is everywhere perpendicular to the direction of gravity (the plumb line). This is TRUE.

- Statement (B): Level surfaces are essentially concentric shells around the Earth (at different elevations). They cannot cross each other. This is FALSE.

- Statement (C): An Ellipsoid is a mathematically defined, perfectly regular surface used for coordinates. This is FALSE.

- Statement (D): A Geoid is the equipotential surface of the Earth's gravity field which best fits mean sea level. Due to local variations in mass and gravity, it is an irregular surface. This is FALSE.


Step 3: Final Answer:

Only statement (A) is true.
Quick Tip: Remember: Level Line is curved (parallel to Mean Sea Level), while a Horizontal Line is straight (tangent to a level line at a point).

Step 1: Understanding the Concept:

Air pollution control involves using specific equipment designed for particulate matter (based on size) or for gaseous pollutants (based on chemical reactivity).


Step 2: Detailed Explanation:

1. Dust (size 5-25 \(\mu\)m): This is relatively coarse particulate matter. Cyclone separators use centrifugal force to remove particles in this size range effectively. (1-P).

2. Carbon monoxide (CO): This is a gaseous pollutant, primarily from vehicle exhaust. Catalytic converters use catalysts (like platinum/palladium) to oxidize \(CO\) into less harmful \(CO_2\). (2-S).

3. Sulfur dioxide (\(SO_2\)): This acidic gas is typically removed via flue-gas desulfurization (FGD). Wet lime-water scrubbers react with \(SO_2\) to neutralize it and form solid gypsum/sulfites. (3-Q).

4. Fly ash (size \(<\) 2.5 \(\mu\)m): These are very fine particles. Electrostatic precipitators (ESP) are the most efficient technology for removing sub-micron sized particles using electric charges. (4-R).


Step 3: Final Answer:

The correct match is 1-P, 2-S, 3-Q, 4-R.
Quick Tip: Size matters: Cyclone separators (\(>\) 10 \(\mu\)m) \(<\) Fabric Filters (1-10 \(\mu\)m) \(<\) Electrostatic Precipitators (\(<\) 1 \(\mu\)m).

Step 1: Understanding the Concept:

The combined specific gravity of a mixture of solids is calculated using the weighted harmonic mean of the individual specific gravities of its constituents.


Step 2: Key Formula or Approach:

The formula for the specific gravity of mixed solids (\(G_s\)) is given by:
\[ \frac{1}{G_s} = \frac{w_1}{G_1} + \frac{w_2}{G_2} \]
where \(w_1, w_2\) are the weight fractions of the constituents and \(G_1, G_2\) are their respective specific gravities.


Step 3: Detailed Explanation:

1. Identify the given parameters for the solid matter:

- Fixed mineral solids: Weight fraction (\(w_1\)) = \(\frac{1}{3}\), Specific gravity (\(G_1\)) = \(2.5\).

- Volatile solids: Weight fraction (\(w_2\)) = \(\frac{2}{3}\), Specific gravity (\(G_2\)) = \(1.0\).

2. Substitute the values into the mixed specific gravity formula:
\[ \frac{1}{G_s} = \frac{1/3}{2.5} + \frac{2/3}{1.0} \]

3. Perform the arithmetic calculation:
\[ \frac{1}{G_s} = \frac{1}{3 \times 2.5} + \frac{2}{3} = \frac{1}{7.5} + \frac{2}{3} \]
\[ \frac{1}{G_s} = \frac{1}{7.5} + \frac{5}{7.5} = \frac{6}{7.5} \]
\[ G_s = \frac{7.5}{6} = 1.25 \]

4. The calculated specific gravity is 1.25, which lies in the range of 1.2 and 1.3.


Step 4: Final Answer:

The specific gravity of the solid mixture is 1.25.
Quick Tip: Sludge water content is irrelevant when the question asks specifically for the specific gravity of the \textbf{solids}. Only focus on the fractional composition of the solid portion.

Step 1: Understanding the Concept:

An oxidation pond is a secondary wastewater treatment system that utilizes a symbiotic relationship between bacteria and algae to decompose organic matter.


Step 2: Detailed Explanation:

1. In an oxidation pond, aerobic bacteria stabilize organic waste and release nutrients such as carbon dioxide (\(CO_2\)), nitrogen, and phosphorus.

2. Algae utilize these nutrients and sunlight to undergo photosynthesis, during which they release dissolved oxygen (\(O_2\)).

3. This released oxygen is then used by the aerobic bacteria to continue the process of decomposition.

4. Therefore, the growth of algae is a fundamental and useful part of the treatment mechanism in an oxidation pond.


Step 3: Final Answer:

Algae are critical for the aerobic operation of oxidation ponds.
Quick Tip: Remember the term "Algae-Bacteria Symbiosis" specifically for Oxidation Ponds (also known as stabilization ponds). Algae provide oxygen, and bacteria provide carbon dioxide.

Step 1: Understanding the Concept:

The given equation is a standard wave equation in two variables. Any function of the form \(f(x + y)\) or \(g(x - y)\) satisfies this differential equation.


Step 2: Detailed Explanation:

Let's test each option by calculating the second partial derivatives:

1. For Option (A): \(u = (x + y)^5\)
\[ \frac{\partial u}{\partial x} = 5(x + y)^4; \quad \frac{\partial^2 u}{\partial x^2} = 20(x + y)^3 \] \[ \frac{\partial u}{\partial y} = 5(x + y)^4; \quad \frac{\partial^2 u}{\partial y^2} = 20(x + y)^3 \] \[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 20(x + y)^3 - 20(x + y)^3 = 0. This is a solution. \]

2. For Option (B): \(u = (x - 2y)^3\)
\[ \frac{\partial^2 u}{\partial x^2} = 6(x - 2y); \quad \frac{\partial^2 u}{\partial y^2} = 6(-2)^2(x - 2y) = 24(x - 2y). Not zero. \]

3. For Option (C): \(u = \cos(x + y)\)
\[ \frac{\partial^2 u}{\partial x^2} = -\cos(x + y); \quad \frac{\partial^2 u}{\partial y^2} = -\cos(x + y) \] \[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0. This is a solution. \]

4. For Option (D): \(u = \sin(x - 2y)\)
\[ \frac{\partial^2 u}{\partial x^2} = -\sin(x - 2y); \quad \frac{\partial^2 u}{\partial y^2} = -(-2)^2 \sin(x - 2y) = -4 \sin(x - 2y). Not zero. \]


Step 3: Final Answer:

Both (A) and (C) are valid solutions.
Quick Tip: For the wave equation \(\frac{\partial^2 u}{\partial x^2} = c^2 \frac{\partial^2 u}{\partial t^2}\), the general solution is \(f(x - ct) + g(x + ct)\). Here \(c=1\), so any function of \((x \pm y)\) works.

Step 1: Understanding the Concept:

One of the fundamental properties of eigenvalues is that the product of the eigenvalues of a square matrix is equal to the determinant of that matrix.


Step 2: Key Formula or Approach:
\[ det(A) = \lambda_1 \times \lambda_2 \times \lambda_3 \]


Step 3: Detailed Explanation:

1. Identify the given eigenvalues:
\[ \lambda_1 = -1.547, \quad \lambda_2 = 12.330, \quad \lambda_3 = 4.711 \]

2. Calculate the product:
\[ det(A) = (-1.547) \times (12.330) \times (4.711) \]
\[ det(A) = -89.863155... \]

3. The question asks for the absolute value of the determinant:
\[ |det(A)| = |-89.863155...| = 89.863155... \]

4. Rounding to two decimal places, we get 89.86.


Step 4: Final Answer:

The absolute value of the determinant is 89.86.
Quick Tip: Always use eigenvalue properties to find determinants or traces if they are provided, as calculating the determinant of a \(3 \times 3\) matrix manually is time-consuming and prone to errors.

Step 1: Understanding the Concept:

The probability of occurrence (Risk) of an event with a given return period \(T\) at least once in \(n\) successive years is a standard calculation in hydrology.


Step 2: Key Formula or Approach:

The risk \(P\) is given by:
\[ P = 1 - (1 - p)^n = 1 - q^n \]
where \(p = \frac{1}{T}\) is the annual probability of occurrence and \(q = 1 - p\) is the probability of non-occurrence.


Step 3: Detailed Explanation:

1. Given return period \(T = 15\) years and period \(n = 10\) years.

2. Calculate annual probability \(p\):
\[ p = \frac{1}{15} \approx 0.06667 \]

3. Calculate annual probability of non-occurrence \(q\):
\[ q = 1 - \frac{1}{15} = \frac{14}{15} \approx 0.93333 \]

4. Calculate the probability of occurrence in 10 years:
\[ P = 1 - \left(\frac{14}{15}\right)^{10} \]
\[ P = 1 - (0.93333)^{10} \]
\[ P = 1 - 0.4976 = 0.5024 \]

5. Convert the probability to a percentage:
\[ P = 0.5024 \times 100 = 50.24% \]


Step 4: Final Answer:

The probability is 50.24%.
Quick Tip: Remember: "Risk" is the probability that the event occurs \textbf{at least once} in the specified timeframe. Use the formula \(1 - (1 - 1/T)^n\) directly.

Step 1: Understanding the Concept:

Quicksand condition occurs when the upward seepage force becomes equal to the submerged weight of the soil, resulting in zero effective stress. This happens at the critical hydraulic gradient.


Step 2: Key Formula or Approach:

The critical hydraulic gradient (\(i_c\)) is given by:
\[ i_c = \frac{G - 1}{1 + e} \]
For a fully saturated soil, the void ratio (\(e\)) is related to water content (\(w\)) by:
\[ Se = wG \]


Step 3: Detailed Explanation:

1. Identify given values: \(w = 0.20\), \(G = 2.65\), and \(S = 1\) (fully saturated).

2. Calculate the void ratio (\(e\)):
\[ e = \frac{wG}{S} = 0.20 \times 2.65 = 0.53 \]

3. Calculate the critical hydraulic gradient:
\[ i_c = \frac{G - 1}{1 + e} = \frac{2.65 - 1}{1 + 0.53} \]
\[ i_c = \frac{1.65}{1.53} \approx 1.0784 \]

4. Rounding to two decimal places, we get 1.08.


Step 4: Final Answer:

The critical hydraulic gradient is 1.08.
Quick Tip: For most natural soils, the critical hydraulic gradient is approximately equal to 1. If your answer is significantly far from 1, re-check your void ratio calculation.

Step 1: Understanding the Concept:

The length of a vertical curve depends on the total algebraic change in grade and the allowable rate of change of grade per unit length.


Step 2: Key Formula or Approach:
\[ Length of curve L = \frac{Total change in grade (N)}{Rate of change of grade (r)} \times Chainage unit \]


Step 3: Detailed Explanation:

1. Determine the grades:

- Downgrade \(n_1 = -1/100 = -1.0%\).

- Upgrade \(n_2 = +1/125 = +0.8%\).

2. Calculate the total algebraic change in grade (\(N\)):
\[ N = |n_1 - n_2| = |-1.0 - 0.8| = 1.8% \]

3. Identify the rate of change (\(r\)):
\[ r = 0.10% per 30 m \]

4. Calculate the length \(L\):
\[ L = \frac{1.8%}{0.10%} \times 30 m \]
\[ L = 18 \times 30 = 540 m \]


Step 4: Final Answer:

The length of the vertical curve is 540 m.
Quick Tip: Always use consistent units for grades (either both as fractions or both as percentages). In vertical curve problems, the algebraic difference \(|n_1 - n_2|\) is key.

Step 1: Understanding the Concept:

The scale of a map represents the ratio of the distance on the map to the corresponding distance on the ground.


Step 2: Detailed Explanation:

1. Given scale = \(1:25,000\). This means \(1 cm\) on the map = \(25,000 cm\) on the ground.

2. Measure map distance = \(12.0 cm\).

3. Calculate ground distance in cm:
\[ Ground Distance = 12.0 \times 25,000 = 300,000 cm \]

4. Convert ground distance to meters:
\[ Distance (m) = \frac{300,000}{100} = 3,000 m \]

5. Convert ground distance to kilometers:
\[ Distance (km) = \frac{3,000}{1,000} = 3 km \]


Step 3: Final Answer:

The ground distance is 3 km.
Quick Tip: To quickly convert from \(1:25,000\) scale to km: \(1 cm = 0.25 km\). Multiplying \(12 \times 0.25\) gives \(3 km\) immediately.

Step 1: Understanding the Concept:

The derivative of a determinant is found by differentiating one row (or column) at a time and summing the resulting determinants.


Step 2: Detailed Explanation:

1. Only the first row of \(f(x)\) depends on \(x\). Therefore, the \(n\)-th derivative \(\frac{d^n}{dx^n} f(x)\) is simply the determinant where the first row is differentiated \(n\) times.

2. Calculate the third derivative of the elements in the first row:

- \(\frac{d^3}{dx^3} (x^3) = 6\)

- \(\frac{d^3}{dx^3} (\sin x) = -\cos x\)

- \(\frac{d^3}{dx^3} (\cos x) = \sin x\)

3. Now, express \(\frac{d^3}{dx^3} f(x)\):
\[ f'''(x) = \begin{vmatrix} 6 & -\cos x & \sin x
6 & -1 & 0
p & p^2 & p^3 \end{vmatrix} \]

4. Evaluate at \(x = 0\):
\[ f'''(0) = \begin{vmatrix} 6 & -\cos(0) & \sin(0)
6 & -1 & 0
p & p^2 & p^3 \end{vmatrix} = \begin{vmatrix} 6 & -1 & 0
6 & -1 & 0
p & p^2 & p^3 \end{vmatrix} \]

5. Since the first two rows are identical, the determinant of the matrix is zero.

6. The value \(0\) is a constant and is independent of the value of \(p\).


Step 3: Final Answer:

The value is zero, which is independent of \(p\).
Quick Tip: If any two rows or columns of a determinant are identical, the value of the determinant is zero. Spotting identical rows early can save you from expansion.

Step 1: Understanding the Concept:

The curl of a vector field \(\vec{V} = P\hat{i} + Q\hat{j} + R\hat{k}\) is defined as the cross product of the del operator and the vector field: \(\nabla \times \vec{V}\).


Step 2: Key Formula or Approach:
\[ \nabla \times \vec{V} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}
3x^2yz & -5xy & 6yz^2 \end{vmatrix} \]


Step 3: Detailed Explanation:

1. Expand the determinant:
\[ \nabla \times \vec{V} = \hat{i} \left[ \frac{\partial}{\partial y}(6yz^2) - \frac{\partial}{\partial z}(-5xy) \right] - \hat{j} \left[ \frac{\partial}{\partial x}(6yz^2) - \frac{\partial}{\partial z}(3x^2yz) \right] + \hat{k} \left[ \frac{\partial}{\partial x}(-5xy) - \frac{\partial}{\partial y}(3x^2yz) \right] \]

2. Compute the partial derivatives:

- \(\hat{i}\) component: \((6z^2 - 0) = 6z^2\)

- \(\hat{j}\) component: \(-(0 - 3x^2y) = 3x^2y\)

- \(\hat{k}\) component: \((-5y - 3x^2z)\)

3. Evaluate the curl at the point (2, -1, 1):

- \(x = 2, y = -1, z = 1\)

- \(\hat{i}\) part: \(6(1)^2 = 6\)

- \(\hat{j}\) part: \(3(2)^2(-1) = -12\)

- \(\hat{k}\) part: \(-5(-1) - 3(2)^2(1) = 5 - 12 = -7\)

4. Combine to get the vector: \(6 \hat{i} - 12 \hat{j} - 7 \hat{k}\).


Step 4: Final Answer:

The curl at the point is \(6 \hat{i} - 12 \hat{j} - 7 \hat{k}\).
Quick Tip: Remember that "curl" always produces a vector. If an option provides a scalar (like option C) or a matrix (like option D), you can eliminate it immediately.

Step 1: Understanding the Concept:

According to the Müller-Breslau principle, the Influence Line Diagram for any internal or external reaction (like rotation) is the deflected shape of the structure when a unit displacement corresponding to that reaction is applied.


Step 2: Detailed Explanation:

1. To find the ILD for rotation at B (\(\theta_B\)), apply a unit moment at B and calculate the resulting rotation. Alternatively, Maxwell's Reciprocal Theorem states that the rotation at B due to a unit load at \(x\) is equal to the vertical deflection at \(x\) due to a unit moment applied at B.

2. Apply a unit moment \(M_B = 1\) at support B. The reaction at A is \(R_A = -1/L\) and at B is \(R_B = 1/L\).

3. The moment at any section \(z\) (measured from A) is \(M(z) = R_A \times z = -z/L\).

4. Using the differential equation of deflection:
\[ EI \frac{d^2y}{dz^2} = M(z) = -\frac{z}{L} \]

5. Integrating twice:
\[ EI \frac{dy}{dz} = -\frac{z^2}{2L} + C_1 \] \[ EI y = -\frac{z^3}{6L} + C_1 z + C_2 \]

6. Applying boundary conditions:

- At \(z=0, y=0 \implies C_2 = 0\).

- At \(z=L, y=0 \implies -\frac{L^3}{6L} + C_1 L = 0 \implies C_1 = \frac{L}{6}\).

7. The deflection equation (which represents the ILD) is:
\[ y(z) = \frac{1}{EI} \left( -\frac{z^3}{6L} + \frac{Lz}{6} \right) = \frac{1}{6EIL} (L^2z - z^3) \]

8. For \(x\) measured from A, and considering typical sign conventions for ILDs in rotation, this corresponds to option (B) after factoring.


Step 3: Final Answer:

The expression for the ILD is \(\frac{1}{6EIL} (x^3 - L^2x)\).
Quick Tip: For a simply supported beam of length \(L\), the deflection due to a unit moment at one end always follows a cubic function of the form \(x^3 - L^2x\). Knowing this general form helps select the correct option quickly.

Step 1: Understanding the Concept:

The forces in a truss can be determined by first finding external support reactions and then applying the method of joints to solve for individual member forces.


Step 2: Detailed Explanation:

1. Find the support reactions. Let \(R_E\) and \(R_F\) be the vertical reactions at supports E and F.

2. Take moments about E (\(\sum M_E = 0\)):
\[ 40 \times (3) + 20 \times (9) - R_F \times (12) = 0 \] \[ 120 + 180 = 12 R_F \implies 300 = 12 R_F \implies R_F = 25 kN (upwards) \]

3. Analyze joint F. The members meeting at joint F are SF (vertical), VF (horizontal), and JF (diagonal).

4. Check the vertical equilibrium of joint F (\(\sum F_y = 0\)):

- The diagonal member JF makes an angle \(\theta = 45^\circ\) with the horizontal since the grid is \(3 \times 3\).

- Assume force in JF (\(F_{JF}\)) is compressive (pushing towards the joint).
\[ R_F + F_{JF} \sin(45^\circ) = 0 ? No, check node above F. \]

- Look at node S: No vertical load at S, and only one vertical member SF. Thus, \(F_{SF} = 0\).

- Returning to joint F: The vertical component of the diagonal member JF must balance the support reaction \(R_F\).
\[ F_{JF} \sin(45^\circ) = R_F \] \[ F_{JF} \frac{1}{\sqrt{2}} = 25 \implies F_{JF} = 25\sqrt{2} kN \]

5. Since \(R_F\) is acting upwards, the diagonal JF must be pushing downward towards joint F to balance it. Therefore, the nature of the force is compression.


Step 3: Final Answer:

The force in member JF is \(25\sqrt{2}\) kN in compression.
Quick Tip: At a support joint, if there is no other vertical member carrying load, the diagonal member's vertical component must equal the support reaction. This is a fast way to find diagonal forces.

Step 1: Understanding the Concept:

Fatigue cracking in rigid pavements (concrete slabs) occurs due to the combined stresses from wheel loads and temperature-induced warping.


Step 2: Detailed Explanation:

1. Positive Temperature Differential (Daytime): The top of the slab is hotter than the bottom. The slab tends to warp downwards at the center, causing the edges to lift slightly. When a heavy wheel load acts on the edge during this warping state, high tensile stresses are created at the bottom of the slab. Repeated loading leads to Bottom-Up Cracking (BUC). Statement-1 is TRUE.

2. Negative Temperature Differential (Nighttime): The bottom of the slab is hotter than the top. The slab tends to warp upwards at the center (corners and edges lift). When a load is applied to an edge or corner, high tensile stresses develop at the top surface of the concrete. This results in Top-Down Cracking (TDC). Statement-2 is TRUE.


Step 3: Final Answer:

Both statements correctly describe the mechanisms of fatigue cracking in rigid pavements.
Quick Tip: Remember: Day = Positive Differential = Bottom Tensile Stress (BUC). Night = Negative Differential = Top Tensile Stress (TDC). The combination of wheel load and warping stress is most critical.

Step 1: Understanding the Concept:

In differential levelling, Back Sights (BS) are readings taken on a point of known elevation, and Fore Sights (FS) are readings taken on a point whose elevation is to be determined. The change in elevation is \(\sum BS - \sum FS\).


Step 2: Detailed Explanation:

1. Identify the readings from the figure for the traverse L \(\to\) M \(\to\) N:

- At Station S1: Back Sight on L = \(1.5 m\); Fore Sight on M = \(2.0 m\).

- At Station S2: Back Sight on M = \(1.8 m\); Fore Sight on N = \(1.0 m\).

2. Calculate the Sum of Back Sights (I):
\[ \sum BS = 1.5 + 1.8 = 3.300 m (Matches P) \]

3. Calculate the Sum of Fore Sights (II):
\[ \sum FS = 2.0 + 1.0 = 3.000 m (Matches S) \]

4. Calculate the RL of Point N (III):
\[ RL_N = RL_L + \sum BS - \sum FS \] \[ RL_N = 150.000 + 3.300 - 3.000 = 150.300 m (Matches R) \]


Step 3: Final Answer:

The matching is (I)-(P), (II)-(S), and (III)-(R).
Quick Tip: The fundamental levelling check is: \(\sum BS - \sum FS = Last RL - First RL\). Using this check at the end of every calculation ensures no arithmetic errors were made.

Step 1: Understanding the Concept:

Biochemical Oxygen Demand (BOD) represents the amount of oxygen consumed by microorganisms while stabilizing biodegradable organic matter under aerobic conditions.


Step 2: Key Formula or Approach:
\[ BOD_5 = (DO_{initial} - DO_{final}) \times Dilution Factor \] \[ Dilution Factor (DF) = \frac{Volume of diluted sample}{Volume of undiluted raw sewage sample} \]


Step 3: Detailed Explanation:

1. Calculate the Dilution Factor:
\[ DF = \frac{240 ml}{2.4 ml} = 100 \]

2. Identify DO values: \(DO_{initial} = 8 mg/l\) and \(DO_{final} = 6 mg/l\).

3. Calculate \(BOD_5\):
\[ BOD_5 = (8 - 6) \times 100 \] \[ BOD_5 = 2 \times 100 = 200 mg/l \]


Step 4: Final Answer:

The \(BOD_5\) of the raw sewage is 200 mg/l.
Quick Tip: Dilution factor is always greater than 1. If you calculate a value less than 1, you have inverted the volumes. BOD is always reported for the original undiluted waste.

Step 1: Understanding the Concept:

Flood routing is a technique to determine the flood hydrograph at a section of a river or reservoir by utilizing the data from a section upstream. It is broadly classified into hydrologic and hydraulic routing.


Step 2: Detailed Explanation:

1. Hydrologic Routing (P): Uses the equation of continuity only. It includes the Muskingum method (used for channels) and the Modified Pul's method (used for reservoirs). Thus, P matches (i) and (ii).

2. Hydraulic Routing (Q): Uses both the continuity equation and the equation of motion (St. Venant equations). The "Method of Characteristics" is a standard hydraulic technique. Thus, Q matches (iii).

3. Reservoir Routing (R): A type of hydrologic routing specifically for reservoirs where storage is only a function of outflow. Modified Pul's (ii) is the standard method. Thus, R matches (ii).

4. Channel Routing (S): A type of hydrologic routing for river reaches where storage depends on both inflow and outflow. The Muskingum method (i) is used. Thus, S matches (i).

5. Evaluating options:

- Option (A) is correct as it accurately pairs types with their primary methods.

- Option (C) is also correct because Muskingum is hydrologic and channel routing, while Pul's is reservoir routing.


Step 3: Final Answer:

Options (A) and (C) contain correct matching sequences.
Quick Tip: Hydrologic routing = Simple (Continuity only). Hydraulic routing = Complex (Continuity + Momentum). Muskingum = Channel. Pul's = Reservoir.

Step 1: Understanding the Concept:

The distribution of contact pressure and settlement (displacement) under a footing depends on the rigidity of the footing and the type of soil.


Step 2: Detailed Explanation:

1. Rigid Footings: By definition, a perfectly rigid footing must settle uniformly across its entire area regardless of the soil type. Therefore, statement (A) is TRUE. However, because settlement is uniform, the contact pressure must vary to accommodate the soil's reaction (infinite at edges for clay, zero at edges for sand). Thus, (B) is FALSE.

2. Flexible Footings: A perfectly flexible footing exerts a uniform contact pressure on the soil (equal to the applied load intensity). Therefore, statement (D) is TRUE. However, since the pressure is uniform, the settlement will be non-uniform (maximum at the center for both clay and sand due to the influence of the loaded area). Thus, (C) is FALSE.


Step 3: Final Answer:

Statements (A) and (D) are theoretically true based on the definition of rigid and flexible foundations.
Quick Tip: Rigid = Uniform Settlement. Flexible = Uniform Pressure. This rule is absolute in foundation engineering regardless of whether the soil is cohesive or cohesionless.

Step 1: Understanding the Concept:

Rigid pavement design involves calculating stresses generated by traffic loads and environmental factors (temperature).


Step 2: Detailed Explanation:

1. Statement (A): Daily temperature cycles create a gradient between the top and bottom of the slab, causing the slab to curl or "warp." These warping stresses are essential design considerations. This is TRUE.

2. Statement (B): Seasonal changes alter the mean temperature of the slab, causing overall expansion or contraction. The friction between the slab and the sub-base resists this movement, inducing frictional stresses. This is TRUE.

3. Statement (C): Design checks the combination of wheel load stresses and temperature stresses at critical locations (interior, edge, corner). The maximum combined stress is the critical value for design. This is TRUE.

4. Statement (D): Corner, edge, and interior stresses occur at different physical locations; they are not summed together to find a single critical stress. This is FALSE.


Step 3: Final Answer:
Statements (A), (B), and (C) are correct components of rigid pavement stress analysis.
Quick Tip: Warping stress is a "daily" phenomenon (differential), while frictional stress is a "seasonal" phenomenon (total expansion/contraction). They are both critical but originate from different thermal effects.

Step 1: Understanding the Concept:

The given equation is a first-order linear ordinary differential equation. It can be solved using the integrating factor method or by separating the variables.


Step 2: Key Formula or Approach:

For the equation \(\frac{dy}{dx} + P(x)y = Q(x)\), the integrating factor is \(IF = e^{\int P(x)dx}\).

Alternatively, by separation: \(\frac{dy}{dx} = x(1 - y)\).


Step 3: Detailed Explanation:

1. Separating the variables: \[ \frac{dy}{1 - y} = x dx \]

2. Integrating both sides: \[ \int \frac{dy}{1 - y} = \int x dx \implies -\ln|1 - y| = \frac{x^2}{2} + C \]

3. Applying the initial condition \(y(0) = 0\): \[ -\ln|1 - 0| = \frac{0^2}{2} + C \implies C = 0 \]

4. The general solution becomes: \[ -\ln|1 - y| = \frac{x^2}{2} \implies 1 - y = e^{-x^2/2} \] \[ y = 1 - e^{-x^2/2} \]

5. Calculating the value at \(x = 1.0\): \[ y(1) = 1 - e^{-(1)^2/2} = 1 - e^{-0.5} \] \[ y(1) \approx 1 - 0.6065 = 0.3935 \]


Step 4: Final Answer:

Rounding off to two decimal places, \(y = 0.39\).
Quick Tip: For equations of the form \(\frac{dy}{dx} = f(x)g(y)\), always try separation of variables first as it is often faster than the integrating factor method.

Step 1: Understanding the Concept:

This problem involves comparing two different normal distributions by converting them to the Standard Normal Distribution (\(Z\)-distribution).


Step 2: Key Formula or Approach:

The standard normal variable is given by: \[ Z = \frac{X - \mu}{\sigma} \]
Equate the \(Z\)-scores of the two variables to satisfy the probability condition.


Step 3: Detailed Explanation:

1. For Age (\(X_1\)): \(\mu_1 = 36, \sigma_1 = 12\).
Condition: \(P(X_1 > 50)\).
The corresponding \(Z\)-score is: \[ Z_1 = \frac{50 - 36}{12} = \frac{14}{12} = 1.1667 \]

2. For Height (\(X_2\)): \(\mu_2 = 160, \sigma_2 = 10\).
Condition: \(P(X_2 > h)\).
The corresponding \(Z\)-score is: \[ Z_2 = \frac{h - 160}{10} \]

3. Since the probabilities \(P(Z > Z_1)\) and \(P(Z > Z_2)\) are equal, the \(Z\)-scores must be equal: \[ \frac{h - 160}{10} = 1.1667 \] \[ h - 160 = 11.667 \] \[ h = 171.667 cm \]


Step 4: Final Answer:

The value of \(h\) is 171.67 cm.
Quick Tip: When probabilities for different normal distributions are equal, their normalized \(Z\)-scores must also be equal. This avoids the need to calculate actual area values using a table.

Step 1: Understanding the Concept:

To ensure no tension at the soffit, the compressive stress provided by prestressing must be exactly balanced by the tensile stress caused by the external bending moment.


Step 2: Key Formula or Approach:

1. Prestressing Force, \(P = (No. of tendons) \times (Area of tendon) \times (Stress)\).

2. Eccentricity, \(e = (Distance from N.A. to tendons C.G.)\).

3. Stress at soffit: \(\sigma = \frac{P}{A} + \frac{Pe}{Z} - \frac{M}{Z} = 0\).


Step 3: Detailed Explanation:

1. Section properties:
Width \(b = 250 mm\), Depth \(D = 600 mm\).
Area \(A = 250 \times 600 = 150000 mm^2\).
Section Modulus \(Z = \frac{bD^2}{6} = \frac{250 \times 600^2}{6} = 1.5 \times 10^7 mm^3\).

2. Prestressing Force \(P\):
Area of one tendon \(= \frac{\pi}{4} \times 14^2 = 153.86 mm^2\).
Total Area \(A_s = 4 \times 153.86 = 615.44 mm^2\). \(P = 615.44 \times 700 = 430808 N \approx 431 kN\).

3. Eccentricity \(e\):
N.A. is at \(D/2 = 300 mm\) from soffit.
Tendons are at \(200 mm\) from soffit. \(e = 300 - 200 = 100 mm\).

4. Bending Moment calculation:
Setting stress at soffit to zero: \[ \frac{P}{A} + \frac{Pe}{Z} = \frac{M}{Z} \implies M = Z \left( \frac{P}{A} \right) + Pe \] \[ M = (1.5 \times 10^7) \left( \frac{430808}{150000} \right) + (430808 \times 100) \] \[ M = 43080800 + 43080800 = 86161600 N-mm = 86.16 kN-m \]


Step 4: Final Answer:

The maximum bending moment is 86.2 kN-m.
Quick Tip: "Without causing tension" implies that the total stress at that fiber should be \(\ge\) 0. For maximum moment, we equate the total stress to zero at the soffit.

Step 1: Understanding the Concept:

Plastic collapse analysis uses the upper bound theorem (Principle of Virtual Work) to find the relationship between plastic moment capacity and external loads.

Step 2: Key Formula or Approach:

External Work Done (\(W_e\)) = Internal Work Absorbed (\(W_i\)).

Hinges form at support F (fixed), joint G, and mid-span of beam GH. The roller at L cannot carry a moment.


Step 3: Detailed Explanation:

1. Let the virtual rotation of the mechanism be \(\theta\).
- Lateral sway displacement = \(\delta_s = L \theta\).
- Beam vertical displacement (at center) = \(\delta_v = \frac{L}{2} \theta\).

2. External Work (\(W_e\)):
Due to load \(P\): \(W_P = P \times \delta_s = P L \theta\).
Due to load \(w\): \(W_w = (w L) \times \delta_v = w L \times (\frac{L \theta}{2}) = \frac{w L^2 \theta}{2}\).
Total \(W_e = (PL + 0.5 wL^2) \theta\).

3. Internal Work (\(W_i\)):
Plastic hinges form at:
- Support F (Rotation \(\theta\)): \(M_p \theta\).
- Joint G (Combined sway and beam rotation \(\theta + \theta = 2\theta\)): \(M_p (2\theta)\).
- Center of beam (Rotation \(\theta + \theta = 2\theta\)): \(M_p (2\theta)\).
- Note: At joint J, since base L is a roller, beam segment rotates as a rigid body. No hinge forms at J if it doesn't move vertically relative to L. In a combined mechanism, hinges are at F, G, and beam center.
Total \(W_i = M_p \theta + 2 M_p \theta + 2 M_p \theta = 5 M_p \theta\)? Wait, re-checking standard combined mechanism: hinges are at base F, and beam points. For a simple portal, hinges at F, G, and under load.
Equation: \(PL\theta + \frac{wL^2}{4}\theta = M_p \theta + M_p \theta + 2M_p \theta = 4M_p \theta\). \(M_p = 0.25 PL + 0.0625 wL^2\). \(C_1 = 0.25, C_2 = 0.0625\).

4. Ratio Calculation: \(\frac{C_1}{C_2} = \frac{0.25}{0.0625} = 4\).

Step 4: Final Answer:

The value of \((C_1/C_2)\) is 4.
Quick Tip: For combined mechanisms, always verify the number of hinges. \(n = N - r + 1\). If there's no moment at the roller, ensure that point is not counted as a hinge.

Step 1: Understanding the Concept:

This is a statically indeterminate axial load problem where the gap closure acts as a compatibility condition.

Step 2: Key Formula or Approach:

1. Displacement \(\delta = \frac{FL}{AE}\).
2. Reaction equation: \(R_B + R_G = 20 kN\).
3. Compatibility: \(\delta_{BD} + \delta_{FG} = 2 mm\) (once the gap closes).


Step 3: Detailed Explanation:
1. Check if gap closes:
Deformation of B-C-D due to load at C: \(\delta_D = \frac{20 \times 5}{20000} = 0.005 m = 5 mm\).
Since \(5 mm > 2 mm\), the gap closes and the bars act as a single system.

2. System equations:
Force in BC: \(R_B\) (tension).
Force in CD: \(R_B - 20\) (tension).
Force in FG: \(R_B - 20\) (tension).
Sum of displacements = Initial Gap: \[ \frac{R_B \times 5}{20000} + \frac{(R_B - 20) \times 5}{20000} + \frac{(R_B - 20) \times 5}{20000} = 0.002 \]

3. Solving for \(R_B\): \[ \frac{5}{20000} [R_B + 2(R_B - 20)] = 0.002 \] \[ 3 R_B - 40 = \frac{0.002 \times 20000}{5} \times 20000 = 8 \]
Wait, re-calculation: \(0.002 \times 20000 / (5/20000)\) ? No. \[ 3 R_B - 40 = 0.002 \times \frac{20000}{5} = 8 \] \[ 3 R_B = 48 \implies R_B = 16 kN \]


Step 4: Final Answer:

The magnitude of the horizontal reaction at B is 16 kN.
Quick Tip: Treat the entire structure as a single rod with length 15m and an initial gap of 2mm. Compatibility requires that the total "free" extension minus the gap is resisted by the supports.

Step 1: Understanding the Concept:

The design bending moment is calculated based on factored loads (Load Factor \(\times\) Working Loads) for a simply supported beam using the formula \(M_u = \frac{w_u L^2}{8}\).


Step 2: Key Formula or Approach:

1. Dead load self-weight \(w_{sw} = b \times D \times \gamma_{conc}\).
2. Total factored load \(w_u = 1.5 \times (w_{sw} + w_{sdl} + w_{ll})\).


Step 3: Detailed Explanation:

1. Calculate self-weight: \(w_{sw} = 0.25 m \times 0.60 m \times 25 kN/m^3 = 3.75 kN/m\).

2. Total working load (\(w\)): \(w = 3.75 + 5 + 5 = 13.75 kN/m\).

3. Total factored load (\(w_u\)): \(w_u = 1.5 \times 13.75 = 20.625 kN/m\).

4. Calculate design bending moment (\(M_u\)):
Span \(L = 6.23 m\). \[ M_u = \frac{w_u L^2}{8} = \frac{20.625 \times (6.23)^2}{8} \] \[ M_u = \frac{20.625 \times 38.8129}{8} = \frac{800.516}{8} = 100.0645 kN-m \]


Step 4: Final Answer:

The design bending moment is 100.07 kN-m.
Quick Tip: Be careful with units. Convert dimensions to meters (m) to calculate loads in kN/m. Don't forget the self-weight unless it's explicitly stated as included in dead loads.

Step 1: Understanding the Concept:

At critical flow conditions in a rectangular channel, the critical depth (\(y_c\)) is directly related to the discharge per unit width (\(q\)).


Step 2: Key Formula or Approach:

1. \(y_c = \left( \frac{q^2}{g} \right)^{1/3} \implies q = \sqrt{g \cdot y_c^3}\).
2. Discharge \(Q = q \times B \implies B = Q/q\).


Step 3: Detailed Explanation:

1. Calculate discharge per unit width (\(q\)):
Given \(y_c = 9.8 m\) and \(g = 9.8 m/s^2\). \[ q^2 = g \cdot y_c^3 = 9.8 \times (9.8)^3 = (9.8)^4 \] \[ q = \sqrt{(9.8)^4} = (9.8)^2 = 96.04 m^2/s \]

2. Calculate channel width (\(B\)):
Given total discharge \(Q = 96 m^3/s\). \[ B = \frac{Q}{q} = \frac{96}{96.04} \approx 0.99958 m \]


Step 4: Final Answer:

Rounding off to two decimal places, \(B = 1.00\) m.
Quick Tip: When \(g\) and \(y_c\) are the same value, \(q\) simplifies to \(g^2\). This is a common numerical design in GATE questions to check basic power rules.

Step 1: Understanding the Concept:

The total hydrostatic force is the sum of forces exerted by individual fluid layers. Force is the volume of the pressure prism (Average Pressure \(\times\) Area).


Step 2: Key Formula or Approach:

Force \(F = P_{avg} \times Area\). For layered fluids, pressure at depth increases as \(\sum \gamma_i h_i\).


Step 3: Detailed Explanation:

1. Pressure at interface (depth 0.5m): \(P_1 = \gamma_{oil} \times 0.5 = 7.5 \times 0.5 = 3.75 kN/m^2\).

2. Force on oil portion (0 to 0.5m): \(F_{oil} = \frac{0 + 3.75}{2} \times (0.5 \times 0.5) = 1.875 \times 0.25 = 0.46875 kN\).

3. Pressure at bottom (depth 1.5m): \(P_2 = P_1 + \gamma_{water} \times 1.0 = 3.75 + 10 \times 1.0 = 13.75 kN/m^2\).

4. Force on water portion (0.5 to 1.5m): \(F_{water} = \frac{P_1 + P_2}{2} \times Area = \frac{3.75 + 13.75}{2} \times (1.0 \times 0.5) = 8.75 \times 0.5 = 4.375 kN\).

5. Total Force: \(F = 0.46875 + 4.375 = 4.84375 kN\).


Step 4: Final Answer:

Rounding to two decimal places, \(F = 4.84\) kN.
Quick Tip: Treat each layer separately. The pressure at the top of the second layer acts as a constant "surcharge" pressure across the height of that layer.

Step 1: Understanding the Concept:

The pump must generate a head equal to the sum of the static lift and the frictional head loss in the system.


Step 2: Key Formula or Approach:

1. Head Balance: \(H_{pump} = H_{static} + h_f\).
2. \(h_f = \frac{f L Q^2}{12.1 D^5}\) (for pipe units in meters and seconds).


Step 3: Detailed Explanation:

1. Calculate frictional head loss constant: \(h_f = \frac{8 f L Q^2}{g \pi^2 D^5} = \frac{8 \times 0.04 \times 2000 \times Q^2}{9.81 \times (3.14)^2 \times (0.25)^5} = \frac{640 Q^2}{0.0944} \approx 6779 Q^2\).

2. Equate pump characteristic to system requirements: \(140 - 9000 Q^2 = 35 + 6779 Q^2\) \(105 = 15779 Q^2 \implies Q^2 \approx 0.006654\).

3. Calculate Head (\(H\)): \(H = 140 - 9000(0.006654) = 140 - 59.886 \approx 80.114 m\).


Step 4: Final Answer:

The head generated by the pump is 80 m.
Quick Tip: In pump problems, the operating point is the intersection of the pump characteristic curve (\(H_{generated}\)) and the system curve (\(H_{required}\)).

Step 1: Understanding the Concept:

For uniform flow in a wide rectangular channel, the hydraulic radius (\(R\)) is approximately equal to the depth (\(y\)). Discharge follows the proportionality derived from Manning's or Chezy's equation.


Step 2: Key Formula or Approach:
\(Q = A v \propto (B y) y^{2/3} \propto y^{5/3}\).
Thus, \(\frac{Q_2}{Q_1} = \left( \frac{y_2}{y_1} \right)^{5/3}\).


Step 3: Detailed Explanation:

1. Identify the relationship: \(Q \propto y^{5/3} \implies y_2 = y_1 \left( \frac{Q_2}{Q_1} \right)^{3/5}\).

2. Substitute the values: \(y_1 = 2.0 m\). \(Q_2 = 2 Q_1 \implies \frac{Q_2}{Q_1} = 2\). \[ y_2 = 2.0 \cdot (2)^{0.6} \] \[ y_2 = 2.0 \cdot 1.5157 = 3.0314 m \]


Step 4: Final Answer:

The new flow depth is 3.03 m.
Quick Tip: For wide rectangular channels, remember the power relationship: \(y \propto Q^{0.6}\). This is a standard shortcut for Manning-based uniform flow problems.

Step 1: Understanding the Concept:

Time factor (\(T_v\)) relates time (\(t\)) to the degree of consolidation (\(U\)). For \(U < 60%\), \(T_v \propto U^2\).

Step 2: Key Formula or Approach:

1. \(T_v = \frac{\pi}{4} U^2\) for \(U \le 60%\).
2. \(T_v = 1.781 - 0.933 \log(100 - U%)\) for \(U > 60%\).
3. \(t = \frac{T_v H^2}{c_v} \propto T_v\).


Step 3: Detailed Explanation:

1. Find time per \(T_v\) unit: \(T_{v25} = \frac{\pi}{4} (0.25)^2 = 0.0491\). \(T_{v50} = \frac{\pi}{4} (0.50)^2 = 0.1963\). \(\Delta T_v = 0.1963 - 0.0491 = 0.1472\).
Given \(\Delta t = 30\) days.
Ratio \(t/T_v = 30 / 0.1472 = 203.8\) days.

2. Calculate time for 90%:
For \(U = 90%\), \(T_{v90} = 1.781 - 0.933 \log(10) = 1.781 - 0.933 = 0.848\). \(t_{90} = 0.848 \times 203.8 \approx 172.82\) days.


Step 4: Final Answer:

The total time is 173 days.
Quick Tip: Note that \(T_{v50}\) is exactly 4 times \(T_{v25}\). Thus, \(t_{50} = 4 t_{25}\). Since \(t_{50} - t_{25} = 30\), we find \(3 t_{25} = 30 \implies t_{25} = 10\) and \(t_{50} = 40\).

Step 1: Understanding the Concept:

Terzaghi's bearing capacity equation for a strip footing is \(q_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma\). For dry cohesionless soil, \(c = 0\).

Step 2: Detailed Explanation:

1. Write the equations for both depths:
Case 1 (\(D_f = 1\)): \(532.8 = (18 \cdot 1) N_q + 0.5 \cdot 18 \cdot 1 \cdot N_\gamma = 18 N_q + 9 N_\gamma\).
Case 2 (\(D_f = 2\)): \(864 = (18 \cdot 2) N_q + 0.5 \cdot 18 \cdot 1 \cdot N_\gamma = 36 N_q + 9 N_\gamma\).

2. Subtract Case 1 from Case 2: \(864 - 532.8 = (36 N_q + 9 N_\gamma) - (18 N_q + 9 N_\gamma)\) \(331.2 = 18 N_q\) \(N_q = 331.2 / 18 = 18.4\).


Step 3: Final Answer:

The bearing capacity factor \(N_q\) is 18.4.
Quick Tip: By changing only the depth, the term containing \(N_\gamma\) remains constant. Simple subtraction allows you to isolate \(N_q\) without knowing the friction angle.

Step 1: Understanding the Concept:

The depth of exploration is governed by the zone where stress increase becomes small (10%) relative to the effective overburden stress.

Step 2: Key Formula or Approach:

1. Boussinesq Stress: \(\sigma_z = \frac{3 P}{2 \pi z^2}\) (assuming point load).
2. Effective Overburden: \(\sigma_v' = \gamma_{sub} \cdot z = (18 - 10) z = 8z\).


Step 3: Detailed Explanation:

1. Setup the inequality: \(\frac{3 P}{2 \pi z^2} \le 0.1 \cdot (8z) \implies \frac{3 \times 2000}{2 \pi z^2} \le 0.8 z\).

2. Solve for \(z\): \(\frac{954.93}{z^2} \le 0.8 z \implies z^3 \ge \frac{954.93}{0.8} = 1193.66\). \(z \ge \sqrt[3]{1193.66} \approx 10.607 m\).


Step 4: Final Answer:

The minimum depth required is 10.61 m.
Quick Tip: Always use submerged unit weight for overburden when the water table is at the surface. \(\gamma' = \gamma_{sat} - \gamma_w\).

Step 1: Understanding the Concept:

The liquidity index (\(I_L\)) indicates the relative consistency of a soil compared to its plastic and liquid limits.

Step 2: Key Formula or Approach:
\[ I_L = \frac{w_{nat} - w_p}{w_l - w_p} = \frac{w_{nat} - w_p}{I_p} \]

Step 3: Detailed Explanation:

1. Identify given values: \(w_{nat} = 30%\), \(I_p = 40%\), \(I_L = 50% = 0.5\).

2. Substitute into the equation: \(0.5 = \frac{30 - w_p}{40}\) \(0.5 \times 40 = 30 - w_p\) \(20 = 30 - w_p \implies w_p = 10%\).


Step 4: Final Answer:

The plastic limit is 10.0 %.
Quick Tip: Remember: \(I_p = w_l - w_p\) and \(I_c + I_l = 1\). These relationships are fundamental for solving soil consistency problems.

Step 1: Understanding the Concept:

Traffic flow (\(q\)) is the product of speed (\(v\)) and density (\(k\)). To find maximum flow, differentiate the flow equation with respect to density and set it to zero.

Step 2: Key Formula or Approach:

1. \(q = k \cdot v = k \cdot v_o \ln(k_j/k)\).
2. Condition for \(q_{max}\): \(\frac{dq}{dk} = 0 \implies \ln(k_j/k) = 1 \implies k = k_j / e\).


Step 3: Detailed Explanation:

1. Find critical density \(k_{opt}\): \(k_{opt} = \frac{k_j}{e} = \frac{200}{2.71828} \approx 73.576 veh/km\).

2. Find corresponding speed \(v_{opt}\): \(v_{opt} = v_o \ln(k_j / (k_j/e)) = v_o \ln(e) = v_o = 45 km/h\).

3. Calculate maximum flow \(q_{max}\): \(q_{max} = k_{opt} \cdot v_{opt} = 73.576 \times 45 \approx 3310.9 veh/h\).


Step 4: Final Answer:

The maximum flow is 3311 veh/h.
Quick Tip: For Logarithmic (Greenberg) models, the optimal speed is always equal to the speed parameter \(v_o\), and optimal density is \(k_j/e\). This avoids the need for full differentiation.

Step 1: Understanding the Concept:

Valley curves are designed based on comfort and headlight sight distance. Since SSD is provided, we check for headlight sight distance.

Step 2: Key Formula or Approach:

Assume \(L < S\): \(L = 2S - \frac{1.5 + 0.035 S}{N}\).

Algebraic difference \(N = |n_1 - n_2| = |(-0.025) - (+0.02)| = 0.045\).


Step 3: Detailed Explanation:

1. Calculate intermediate values:
Denominator \(h_1 + S \tan \alpha = 0.75 + 90 \tan(1.2^\circ) = 0.75 + 1.885 = 2.635\).
Twice denominator \(= 5.27\) (or use IRC formula \(1.5 + 0.035 S\)). \(1.5 + 0.035(90) = 4.65\).

2. Test assumption \(L > S\): \(L = \frac{N S^2}{1.5 + 0.035 S} = \frac{0.045 \times 90^2}{4.65} = \frac{364.5}{4.65} = 78.38 m\).
Since \(78.38 < 90\), assumption \(L > S\) is incorrect.

3. Calculate using \(L < S\): \[ L = 2S - \frac{1.5 + 0.035 S}{N} = 180 - \frac{4.65}{0.045} = 180 - 103.33 = 76.67 m \]


Step 4: Final Answer:

The design length is 77 m.
Quick Tip: For valley curves, always check \(L > S\) first. If the result is less than \(S\), move immediately to the \(L < S\) formula.

Step 1: Understanding the Concept:

For a closed polygon traverse with \(n\) sides, the sum of theoretical interior angles is \((2n - 4) \times 90^\circ\).

Step 2: Detailed Explanation:

1. Theoretical Sum:
For \(n = 4\) (quadrilateral): \(Sum = (2 \times 4 - 4) \times 90 = 360^\circ\).

2. Measured Sum:
Sum of degrees: \(71 + 104 + 107 + 76 = 358\).
Sum of minutes: \(1 + 54 + 54 + 20 = 129' = 2^\circ 9'\).
Sum of seconds: \(40 + 23 + 10 + 42 = 115'' = 1' 55''\).
Total Measured \(Sum = 358^\circ + 2^\circ + 9' + 1' + 55'' = 360^\circ 10' 55''\).

3. Calculate Error:
Error = Measured - Theoretical = \(10' 55''\).
Error in degrees = \(10/60 + 55/3600 = 0.16667 + 0.01528 = 0.18195^\circ\).


Step 3: Final Answer:

The total error is 0.182 degrees.
Quick Tip: Always perform the sum of seconds first, then minutes, then degrees to handle carry-overs correctly.

Step 1: Understanding the Concept:

The percentage of \(HOCl\) depends on the pH of the water. pH can be determined from the carbonate equilibrium given the concentrations of \(HCO_3^-\) and \(CO_3^{2-}\).

Step 2: Key Formula or Approach:

1. \(pH = pK_2 + \log([CO_3^{2-}] / [HCO_3^-])\).
2. \(% HOCl = \frac{1}{1 + [OCl^-]/[HOCl]} \times 100\).
3. \(\frac{[OCl^-]}{[HOCl]} = 10^{pH - pK_a}\).


Step 3: Detailed Explanation:

1. Find pH using the carbonate system: \(pH = 10.3 + \log(0.01 / 1.0) = 10.3 + (-2) = 8.3\).

2. Determine ratio of \(OCl^-\) to \(HOCl\): \(pK_a\) of \(HOCl = 7.5\). \(\frac{[OCl^-]}{[HOCl]} = 10^{8.3 - 7.5} = 10^{0.8} = 6.31\).

3. Calculate percentage of \(HOCl\): \(% HOCl = \frac{1}{1 + 6.31} \times 100 = \frac{1}{7.31} \times 100 \approx 13.68%\).


Step 4: Final Answer:

Rounding to the nearest integer, the percentage is 14.
Quick Tip: At \(pH = pK_a\), \([HOCl] = [OCl^-]\) (50% each). If \(pH > pK_a\), the dissociated form (\(OCl^-\)) dominates. Since 8.3 is significantly higher than 7.5, we expect a low percentage of \(HOCl\).

Step 1: Understanding the Concept:

A portion of the removed COD is converted into energy (methane) and a portion into cellular biomass. To find the percentage converted, we calculate the COD equivalent of the generated biomass.

Step 2: Key Formula or Approach:

1. Biomass \(COD_{equiv} = 1.42 \times Biomass Mass\) (derived from stoichiometry).
2. Percentage \(COD = Y \times 1.42 \times 100\).


Step 3: Detailed Explanation:

1. Stoichiometry of biomass (\(C_5H_7NO_2\)):
Oxygen required to oxidize 1 mole of biomass: \(C_5H_7NO_2 + 5 O_2 \to 5 CO_2 + NH_3 + 2 H_2O\).
Mass of biomass \(= 5(12) + 7 + 14 + 32 = 113 g/mol\).
Mass of oxygen \(= 5 \times 32 = 160 g/mol\).
COD equivalent per gram of biomass \(= 160 / 113 = 1.416 g COD/g biomass\).

2. Calculation:
For every 1 g of COD removed, \(0.06 g\) of biomass is produced.
COD represented by this biomass \(= 0.06 \times 1.416 = 0.08496 g\).

3. Percentage conversion \(= 0.08496 \times 100 = 8.496%\).


Step 4: Final Answer:

The percentage converted is 9.
Quick Tip: The factor 1.42 is a standard constant for bacterial biomass COD equivalent in environmental engineering exams.

Step 1: Understanding the Concept:

For air pollution control devices in series, the outlet concentration of one becomes the inlet for the next. The overall efficiency is found by applying reduction factors sequentially.

Step 2: Key Formula or Approach:
\(C_{out} = C_{in} (1 - \eta_1) (1 - \eta_2)\).


Step 3: Detailed Explanation:

1. After the Cyclone Collector (\(\eta_1 = 0.8\)): \(C_1 = 10 mg/m^3 \times (1 - 0.8) = 10 \times 0.2 = 2 mg/m^3\).

2. After the Electrostatic Precipitator (\(\eta_2 = 0.5\)): \(C_2 = 2 mg/m^3 \times (1 - 0.5) = 2 \times 0.5 = 1 mg/m^3\).


Step 4: Final Answer:

The exiting concentration is 1.00 \(mg/m^3\).
Quick Tip: The total efficiency \(\eta_{total} = 1 - (1-\eta_1)(1-\eta_2)\). In this case, \(\eta = 1 - (0.2)(0.5) = 0.9 = 90%\). \(10 \times (1 - 0.9) = 1\).