In cement concrete mix design, with the increase in water-cement ratio, which one of the following statements is TRUE?

For the integral $\rm I=\displaystyle\int^1_{-1}\frac{1}{x^2}dx$

which of the following statements is TRUE? 

The following function is defined over the interval [-L, L]:

f(x) = px4 + qx5.

If it is expressed as a Fourier series, 

$\rm f(x)=a_0 +\displaystyle\sum^\infty_{n=1} \left\{a_n \sin\left( \frac{\pi x}{L} \right) +b_n\cos\left( \frac{\pi x}{L} \right) \right\} $,

which options amongst the following are true?

A function f(x), that is smooth and convex-shaped between interval (x_l , s_u) is shown in the figure. This function is observed at odd number of regularly spaced points. If the area under the function is computed numerically, then . 

For the matrix

$[A]= \begin{bmatrix}1&2&3\\\ 3&2&1\\\ 3&1&2 \end{bmatrix} $

which of the following statements is/are TRUE?

The differential equation,

$\rm \frac{du}{dt}+2tu^2=1,$

is solved by employing a backward difference scheme within the finite difference framework. The value of 𝑢 at the (𝑛 − 1) th time-step, for some 𝑛, is 1.75. The corresponding time (t) is 3.14 s. Each time step is 0.01 s long. Then, the value of (un - un -1) _________ is (round off to three decimal places). 

Which one of the following statements is TRUE for Greenhouse Gas (GHG) in the atmosphere?

In the following table, identify the correct set of associations between the entries in Column - 1 and Column-2.

Column-1	Column-2
P: Reverse Osmosis	III: Concentration Polarization
Q: Trickling Filter	I: Ponding
R: Coagulation	IV: Charge Neutralization
S: Adsorption	II: Freundlich Isotherm

The composition and energy content of a representative solid waste sample are given in the table. If the moisture content of the waste is 26%, the energy content of the solid waste on dry-weight basis is ________ MJ/kg (round off to one decimal place).

$$ \begin{array}{|c|c|c|} \hline \text { Component } & \text { Percent by mass } & \text { Energy content as-discarded basis (MJ/kg) } \\ \hline \text { Food waste } & 20 & 4.5 \\ \hline \text { Paper } & 45 & 16.0 \\ \hline \text { Cardboard } & 5 & 14.0 \\ \hline \text { Plastics } & 10 & 32.0 \\ \hline \text { Others } & 20 & 8.0 \\ \hline \end{array} $$

A flocculator tank has a volume of 2800 m3. The temperature of water in the tank is 15°C, and the average velocity gradient maintained in the tank is 100/s. The temperature of water is reduced to 5°C, but all other operating conditions including the power input are maintained as the same. The decrease in the average velocity gradient (in %) due to the reduction in water temperature is ________ (round off to nearest integer).

[Consider dynamic viscosity of water at 15°C and 5°C as 1.139 × 10-3 N - s/m2 and 1.518 × 10-3 N - s/m2 , respectively] 

When a supercritical stream enters a mild-sloped (M) channel section, the type of flow profile would become _________.

G₁ and G₂ are the slopes of the approach and departure grades of a vertical curve, respectively.

Given |G₁| < |G₂| and |G₁| ≠ |G₂| ≠ 0

Statement 1 : +G₁ followed by +G₂ results in a sag vertical curve.

Statement 2 : −G₁ followed by −G₂ results in a sag vertical curve.

Statement 3 : +G₁ followed by −G₂ results in a crest vertical curve.

Which option amongst the following is true?

A very wide rectangular channel carries a discharge (Q) of 70 m³/s per meter width. Its bed slope changes from 0.0001 to 0.0009 at a point P, as shown in the figure (not to scale). The Manning’s roughness coefficient of the channel is 0.01. What water surface profile(s) exist(s) near the point P?

A jet of water having a velocity of 20 m/s strikes a series of plates fixed radially on a wheel revolving in the same direction as the jet at 15 m/s. What is the percentage efficiency of the plates? (round off to one decimal place)

The direct and reversed zenith angles observed by a theodolite are 56° 00' 00" and 303° 00' 00", respectively. What is the vertical collimation correction?

A student is scanning his 10 inch × 10 inch certificate at 600 dots per inch (dpi) to convert it to raster. What is the percentage reduction in number of pixels if the same certificate is scanned at 300 dpi?

Trigonometric levelling was carried out from two stations P and Q to find the reduced level (R. L.) of the top of hillock, as shown in the table. The distance between Stations P and Q is 55 m. Assume Stations P and Q, and the hillock are in the same vertical plane. The R. L. of the top of the hillock (in m) is ____________ (round off to three decimal places).

$$ \begin{array}{|c|c|c|c|} \hline \text { Station } & \begin{array}{c} \text { Vertical } \\ \text { angle of } \\ \text { the top of } \\ \text { hillock } \end{array} & \begin{array}{c} \text { Staff } \\ \text { reading on } \\ \text { benchmark } \end{array} & \begin{array}{c} \text { R.L. of } \\ \text { bench } \\ \text { mark } \end{array} \\ \hline \mathrm{P} & 18^{\circ} 45^{\prime} & 2.340 \mathrm{~m} & \begin{array}{c} 100.000 \\ \mathrm{~m} \end{array} \\ \hline \mathrm{Q} & 12^{\circ} 45^{\prime} & 1.660 \mathrm{~m} & \\ \hline \end{array} $$

The specific gravity of a soil is 2.60. The soil is at 50% degree of saturation with a water content of 15%. The void ratio of the soil is _______.

A group of 9 friction piles are arranged in a square grid maintaining equal spacing in all directions. Each pile is of diameter 300 mm and length 7 m. Assume that the soil is cohesionless with effective friction angle 𝜙′ = 32°. What is the center-to-center spacing of the piles (in m) for the pile group efficiency of 60%?

A possible slope failure is shown in the figure. Three soil samples are taken from different locations (I, II and III) of the potential failure plane. Which is the most appropriate shear strength test for each of the sample to identify the failure mechanism? Identify the correct combination from the following options:

P : Triaxial compression test

Q : Triaxial extension test

R : Direct shear or shear box test

S : Vane shear test

With reference to the compaction test conducted on soils, which of the following is INCORRECT?

Consider that a force P is acting on the surface of a half-space (Boussinesq’s problem). The expression for the vertical stress (σz) at any point (r, z), within the half-space is given as,

$\rm \sigma_z=\frac{3P}{2\pi} \frac{z^3}{_{(r^2+z^2)}\frac{5}{2}}$

where, r is the radial distance, and z is the depth with downward direction taken as positive. At any given r, there is a variation of σ_z along z, and at a specific z, the value of σz will be maximum. What is the locus of the maximum σz ?

A square footing of size 2.5 m × 2.5 m is placed 1.0 m below the ground surface on a cohesionless homogeneous soil stratum. Considering that the groundwater table is located at the base of the footing, the unit weights of soil above and below the groundwater table are 18 kN/m³ and 20 kN/m³, respectively, and the bearing capacity factor N_q is 58, the net ultimate bearing capacity of the soil is estimated as 1706 kPa (unit weight of water = 10 kN/m³).

Earlier, a plate load test was carried out with a circular plate of 30 cm diameter in the same foundation pit during a dry season, when the water table was located beyond the plate influence zone. Using Terzaghi’s bearing capacity formulation, what is the ultimate bearing capacity (in kPa) of the plate?

A soil having the average properties, bulk unit weight = 19 kN/m3 ; angle of internal friction = 25° and cohesion = 15 kPa, is being formed on a rock slope existing at an inclination of 35° with the horizontal. The critical height (in m) of the soil formation up to which it would be stable without any failure is _________ (round off to one decimal place).

[Assume the soil is being formed parallel to the rock bedding plane and there is no ground water effect.]

A smooth vertical retaining wall supporting layered soils is shown in figure. According to Rankine’s earth pressure theory, the lateral active earth pressure acting at the base of the wall is ________ kPa (round off to one decimal place).

The ordinates of a one-hour unit hydrograph for a catchment are given below: 

$$ \begin{array}{|l|l|l|l|l|l|l|l|l|} \hline t(\text { hour }) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline Q\left(\mathrm{~m}^3 / \mathrm{s}\right) & 0 & 9 & 21 & 18 & 12 & 5 & 2 & 0 \\ \hline \end{array} $$

Using the principle of superposition, a D-hour unit hydrograph for the catchment was derived from this one-hour unit hydrograph. The ordinates of the D-hour unit hydrograph were obtained as 3 m3/s at t = 1 hour and 10 m3/s at t = 2 hour. The value of D (in integer) is _____________.

A 12-hour storm occurs over a catchment and results in a direct runoff depth of 100 mm. The time-distribution of the rainfall intensity is shown in the figure (not to scale). The 𝜙-index of the storm is (in mm, rounded off to two decimal places) _____________.

Creep of concrete under compression is defined as the _________. 

A singly reinforced concrete beam of balanced section is made of M20 grade concrete and Fe415 grade steel bars. The magnitudes of the maximum compressive strain in concrete and the tensile strain in the bars at ultimate state under flexure, as per IS 456 ∶ 2000 are ________, respectively. (round off to four decimal places)

Consider a doubly reinforced RCC beam with the option of using either Fe250 plain bars or Fe500 deformed bars in the compression zone. The modulus of elasticity of steel is 2 × 10⁵ N/mm² . As per IS456 : 2000, in which type(s) of the bars, the stress in the compression steel (f_sc) can reach the design strength (0.87 f_y) at the limit state of collapse?

Consider the fillet-welded lap joint shown in the figure (not to scale). The length of the weld shown is the effective length. The welded surfaces meet at right angle. The weld size is 8 mm, and the permissible stress in the weld is 120 MPa. What is the safe load P (in kN, rounded off to one decimal place) that can be transmitted by this welded joint? ___________

A hanger is made of two bars of different sizes. Each bar has a square cross-section. The hanger is loaded by three-point loads in the mid vertical plane as shown in the figure. Ignore the self-weight of the hanger. What is the maximum tensile stress in N/mm² anywhere in the hanger without considering stress concentration effects?

Consider the horizontal axis passing through the centroid of the steel beam cross-section shown in the figure. What is the shape factor (rounded off to one decimal place) for the cross-section? 

Consider the beam shown in the figure (not to scale), on a hinge support at end A and a roller support at end B. The beam has a constant flexural rigidity, and is subjected to the external moments of magnitude M at one-third spans, as shown in the figure. Which of the following statements is/are TRUE?

The infinitesimal element shown in the figure (not to scale) represents the state of stress at a point in a body. What is the magnitude of the maximum principal stress (in N/mm2, in integer) at the point? ________

The cross-section of a girder is shown in the figure (not to scale). The section is symmetric about a vertical axis (Y - Y). The moment of inertia of the section about the horizontal axis (X - X) passing through the centroid is _______ cm4 (round off to nearest integer). 

Consider the following three structures : 

Structure I : Beam with hinge support at A, roller at C, guided roller at E, and internal hinges at B and D

Structure II : Pin-jointed truss, with hinge support at A, and rollers at B and D

Structure III : Pin-jointed truss, with hinge support at A and roller at C

Which of the following statements is/are TRUE?

Consider the pin-jointed truss shown in the figure (not to scale). All members have the same axial rigidity, AE. Members QR, RS, and ST have the same length L. Angles QBT, RCT, SDT are all 90°. Angles BQT, CRT, DST are all 30°. The joint T carries a vertical load P. The vertical deflection of joint T is k$\rm \frac{PL}{AE}$. What is the value of k?

An idealised bridge truss is shown in the figure. The force in Member U2L3 is __________ kN (round off to one decimal place).

A plot of speed-density relationship (linear) of two roads (Road A and Road B) is shown in the figure.

If the capacity of Road A is C_A and the capacity of Road B is C_B, what is $\rm \frac{ C_A}{C_B }$?

Which of the following options match the test reporting conventions with the given material tests in the table?

Test reporting convention	Material test
(P) Reported as ratio	(I) Solubility of bitumen
(Q) Reported as percentage	(II) Softening point of bitumen
(R) Reported in temperature	(III) Los Angeles abrasion test
(S) Reported in length	(IV) Flash point of bitumen
	(V) Ductility of bitumen
	(VI) Specific gravity of bitumen
	(VII) Thin film oven test

“I have not yet decided what I will do this evening; I ______ visit a friend.”

Eject : Insert : : Advance : _______

(By word meaning)

In the given figure, PQRSTV is a regular hexagon with each side of length 5 cm. A circle is drawn with its centre at V such that it passes through P. What is the area (in cm² ) of the shaded region? (The diagram is representative) 

A duck named Donald Duck says “All ducks always lie.”

Based only on the information above, which one of the following statements can be logically inferred with certainty?

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line.

The figure below consists of 20 unit squares arranged as shown. In addition to the given black squares, upto 5 more may be coloured black. Which one among the following options depicts the minimum number of boxes that must be coloured black to achieve two lines of symmetry? (The figure is representative)

Based only on the truth of the statement ‘Some humans are intelligent’, which one of the following options can be logically inferred with certainty?

Which one of the options can be inferred about the mean, median, and mode for the given probability distribution (i.e. probability mass function), 𝑃(𝑥), of a variable x?

The James Webb telescope, recently launched in space, is giving humankind unprecedented access to the depths of time by imaging very old stars formed almost 13 billion years ago. Astrophysicists and cosmologists believe that this odyssey in space may even shed light on the existence of dark matter. Dark matter is supposed to interact only via the gravitational interaction and not through the electromagnetic-, the weak- or the strong-interaction. This may justify the epithet “dark” in dark matter.

Based on the above paragraph, which one of the following statements is FALSE?

Let a = 30! , b = 50! , and c = 100! . Consider the following numbers:

log_a c, log_c a, log_b a, log_a b

Which one of the following inequalities is CORRECT?

A square of side length 4 cm is given. The boundary of the shaded region is defined by one semi-circle on the top and two circular arcs at the bottom, each of radius 2 cm, as shown.

The area of the shaded region is __________ cm².