M20 concrete as per IS 456: 2000 refers to concrete with a design mix having _______.

Let 𝜙 be a scalar field, and 𝒖 be a vector field. Which of the following identities is true for div(𝜙𝒖)?

Which of the following probability distribution functions (PDFs) has the mean greater than the median?

A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000. Out of the total vehicles, 30% are without pollution clearance certificate. Further, even- and odd-numbered vehicles are operated on even- and odd-numbered dates, respectively.

If 100 vehicles are chosen at random on an even-numbered date, the number of vehicles expected without pollution clearance certificate is ________.

For the matrix

$\rm [A]=\begin{bmatrix}1&-1&0\\\ -1&2&-1\\\ 0&-1&1\end{bmatrix}$

which of the following statements is/are TRUE?

The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100°C and the other three sides are kept at a temperature of 0°C. Ignoring the effect of discontinuities in the boundary conditions at the corners, the steady-state temperature at the center of the plate is obtained as T₀°C. Due to symmetry, the steady-state temperature at the center will be same (T₀°C), when any one side of the square is kept at a temperature of 100°C and the remaining three sides are kept at a temperature of 0°C. Using the principle of superposition, the value of T₀ is _________ (rounded off to two decimal places).

The solution of the differential equation

$\rm \frac{d^3y}{dx^3}-5.5\frac{d^2y}{dx^2}+9.5\frac{dy}{dx}-5y=0$

is expressed as 𝑦 = 𝐶₁𝑒^2.5𝑥 + 𝐶₂𝑒^𝛼𝑥 + 𝐶₃𝑒^𝛽𝑥 , where 𝐶₁, 𝐶₂, 𝐶₃, 𝛼, and 𝛽 are constants, with α and β being distinct and not equal to 2.5. Which of the following options is correct for the values of 𝛼 and 𝛽?

Two vectors [2 1 0 3]^𝑇 and [1 0 1 2]^𝑇 belong to the null space of a 4 × 4 matrix of rank 2. Which one of the following vectors also belongs to the null space?

Cholesky decomposition is carried out on the following square matrix [𝐴]. 

$\rm [A]=\begin{bmatrix}8&-5\\\ -5&a_{22}\end{bmatrix}$

Let 𝑙_ij and 𝑎ij be the (i, j)^th elements of matrices [𝐿] and [𝐴], respectively. If the element 𝑙₂₂ of the decomposed lower triangular matrix [𝐿] is 1.968, what is the value (rounded off to the nearest integer) of the element 𝑎₂₂?

In the context of Municipal Solid Waste Management, ‘Haul’ in ‘Hauled Container System operated in conventional mode’ includes the _________.

Which of the following is/are NOT active disinfectant(s) in water treatment?

For the elevation and temperature data given in the table, the existing lapse rate in the environment is _____ ℃/100 m (round off to two decimal places).

$$ \begin{array}{|c|c|} \hline \text { Elevation from ground level (m) } & \text { Temperature }\left({ }^{\circ} \mathbf{C}\right) \\ \hline 5 & 14.2 \\ \hline 325 & 16.9 \\ \hline \end{array} $$

Match the following air pollutants with the most appropriate adverse health effects :

Air pollutant	Health effect to human and/or test animal
(P) Aromatic hydrocarbons	(IV) Carcinogenic effect
(Q) Carbon monoxide	(I) Reduce the capability of the blood to carry oxygen
(R) Sulfur oxides	(II) Bronchitis and pulmonary emphysema
(S) Ozone	(III) Damage of chromosomes

In the context of water and wastewater treatments, the correct statements are:

Which of the following statements is/are TRUE for the aerobic composting of sewage sludge?

The theoretical aerobic oxidation of biomass (C₅H₇O₂N) is given below:

C₅H₇O₂N + 5O₂ → 5CO₂ + NH₃ + 2H₂O

The biochemical oxidation of biomass is assumed as a first-order reaction with a rate constant of 0.23/d at 20ºC (logarithm to base e). Neglecting the second-stage oxygen demand from its biochemical oxidation, the ratio of BOD₅ at 20ºC to total organic carbon (TOC) of biomass is _______ (round off to two decimal places).

[Consider the atomic weights of C, H, O and N as 12 g/mol, 1 g/mol, 16 g/mol and 14 g/mol, respectively

A system of seven river segments is shown in the schematic diagram. The R_i’s, Q_i’s, and C_i’s (i = 1 to 7) are the river segments, their corresponding flow rates, and concentrations of a conservative pollutant, respectively. Assume complete mixing at the intersections, no additional water loss or gain in the system, and steady state condition. Given: Q₁ = 5 m³/s ; Q₂ = 15 m³/s ; Q₄ = 3 m³/s ; Q₆ = 8 m³/s ; C₁ = 8 kg/m³; C₂ = 12 kg/m³; C₆ = 10 kg/m³. What is the steady state concentration (in kg/m³, rounded off to two decimal place) of the pollutant in the river segment 7 ? ______

The pressure in a pipe at X is to be measured by an open manometer as shown in figure. Fluid A is oil with a specific gravity of 0.8 and Fluid B is mercury with a specific gravity of 13.6. The absolute pressure at X is ________ kN/m2 (round off to one decimal place).

[Assume density of water as 1000 kg/m3 and acceleration due to gravity as 9.81 m/s2 and atmospheric pressure as 101.3 kN/m2]

A compound symmetrical open channel section as shown in the figure has a maximum of _______ critical depth(s).

B_m – Bottom width of main channel

B_f – Bottom width of flood channel

y_m – Depth of main channel

y – Total depth of the channel

n_m – Manning’s roughness of the main channel

n_f – Manning’s roughness of the flood channel

The critical flow condition in a channel is given by _______.

[Note: 𝛼 – kinetic energy correction factor; 𝑄 – discharge; A_c – cross-sectional area of flow at critical flow condition; T_c – top width of flow at critical flow condition; 𝑔 – acceleration due to gravity]

Which of the following statements is/are TRUE?

The cross-section of a small river is sub-divided into seven segments of width 1.5 m each. The average depth, and velocity at different depths were measured during a field campaign at the middle of each segment width. The discharge computed by the velocity area method for the given data is _____ m³/s (round off to one decimal place).

Segment	Average depth (D) (m)	Velocity (m/s) at 0.2D	Velocity (m/s) at 0.6D	Velocity (m/s) at 0.8D
1	0.40	--	0.40	--
2	0.70	0.76	--	0.70
3	1.20	1.19	--	1.13
4	1.40	1.25	--	1.10
5	1.10	1.13	--	1.09
6	0.80	0.69	--	0.65
7	0.45	--	0.42	--

The magnetic bearing of the sun for a location at noon is 183˚ 30ˊ. If the sun is exactly on the geographic meridian at noon, the magnetic declination of the location is ________.

If the size of the ground area is 6 km × 3 km and the corresponding photo size in the aerial photograph is 30 cm × 15 cm, then the scale of the photograph is 1 : ________ (in integer).

A delivery agent is at a location R. To deliver the order, she is instructed to travel to location P along straight-line paths of RC, CA, AB and BP of 5 km each. The direction of each path is given in the table below as whole circle bearings. Assume that the latitude (L) and departure (D) of R is (0, 0) km. What is the latitude and departure of P (in km, rounded off to one decimal place)? 

$$ \begin{array}{|c|c|c|c|c|} \hline \text { Paths } & \text { RC } & \text { CA } & \text { AB } & \text { BP } \\ \hline \begin{array}{c} \text { Directions } \\ \text { (in degrees) } \end{array} & 120 & 0 & 90 & 240 \\ \hline \end{array} $$

A standard penetration test (SPT) was carried out at a location by using a manually operated hammer dropping system with 50% efficiency. The recorded SPT value at a particular depth is 28. If an automatic hammer dropping system with 70% efficiency is used at the same location, the recorded SPT value will be _________.

A vertical sheet pile wall is installed in an anisotropic soil having coefficient of horizontal permeability, 𝑘_𝐻 and coefficient of vertical permeability, 𝑘_𝑉. In order to draw the flow net for the isotropic condition, the embedment depth of the wall should be scaled by a factor of ______, without changing the horizontal scale.

The reason(s) of the nonuniform elastic settlement profile below a flexible footing, resting on a cohesionless soil while subjected to uniform loading, is/are: 

An unconfined compression strength test was conducted on a cohesive soil. The test specimen failed at an axial stress of 76 kPa. The undrained cohesion (in kPa, in integer) of the soil is _____

In the given figure, Point O indicates the stress point of a soil element at initial non-hydrostatic stress condition. For the stress path (OP), which of the following loading conditions is correct?

The figure shows a vertical retaining wall with backfill consisting of cohesive-frictional soil and a failure plane developed due to passive earth pressure. The forces acting on the failure wedge are: P as the reaction force between the wall and the soil, R as the reaction force on the failure plane, C as the cohesive force along the failure plane and W as the weight of the failure wedge. Assuming that there is no adhesion between the wall and the wedge, identify the most appropriate force polygon for the wedge.

A square footing is to be designed to carry a column load of 500 kN which is resting on a soil stratum having the following average properties: bulk unit weight = 19 kN/m³; angle of internal friction = 0° and cohesion = 25 kPa. Considering the depth of the footing as 1 m and adopting Meyerhof’s bearing capacity theory with a factor of safety of 3, the width of the footing (in m) is _________ (round off to one decimal place)

[Assume the applicable shape and depth factor values as unity; ground water level at greater depth.]

A circular pile of diameter 0.6 m and length 8 m was constructed in a cohesive soil stratum having the following properties: bulk unit weight = 19 kN/m³; angle of internal friction = 0° and cohesion = 25 kPa.

The allowable load the pile can carry with a factor of safety of 3 is __________ kN (round off to one decimal place).

[Adopt: Adhesion factor, α = 1.0 and Bearing capacity factor, N_c = 9.0]

For the flow setup shown in the figure (not to scale), the hydraulic conductivities of the two soil samples, Soil 1 and Soil 2, are 10 mm/s and 1 mm/s, respectively. Assume the unit weight of water as 10 kN/m³ and ignore the velocity head. At steady state, what is the total head (in m, rounded off to two decimal places) at any point located at the junction of the two samples? _____

A consolidated drained (CD) triaxial test was carried out on a sand sample with the known effective shear strength parameters, 𝑐′ = 0 and 𝜙′ = 30°. In the test, prior to the failure, when the sample was undergoing axial compression under constant cell pressure, the drainage valve was accidentally closed. At the failure, 360 kPa deviatoric stress was recorded along with 70 kPa pore water pressure. If the test is repeated without such error, and no back pressure is applied in either of the tests, what is the deviatoric stress (in kPa, in integer) at the failure? ___________

Which one of the following options provides the correct match of the terms listed in Column-1 and Column-2?

Column-1	Column-2
P: Horton equation	IV: Infiltration
Q: Muskingum method	V: Channel routing
R: Penman method	III: Evapotranspiration
	I: Precipitation
	II: Flood frequency

A catchment may be idealized as a circle of radius 30 km. There are five rain gauges, one at the center of the catchment and four on the boundary (equi-spaced), as shown in the figure (not to scale).

The annual rainfall recorded at these gauges in a particular year are given below.

$$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Gauge } & \mathrm{G}_1 & \mathrm{G}_2 & \mathrm{G}_3 & \mathrm{G}_4 & \mathrm{G}_5 \\ \hline \text { Rainfall }(\mathrm{mm}) & 910 & 930 & 925 & 895 & 905 \\ \hline \end{array} $$

Using the Thiessen polygon method, what is the average rainfall (in mm, rounded off to two decimal places) over the catchment in that year? ________

Identify the cross-drainage work in the figure.

With regard to the shear design of RCC beams, which of the following statements is/are TRUE?

Consider the singly reinforced section of a cantilever concrete beam under bending, as shown in the figure (M25 grade concrete, Fe415 grade steel). The stress block parameters for the section at ultimate limit state, as per IS 456: 2000 notations, are given. The ultimate moment of resistance for the section by the Limit State Method is ________ kN.m (round off to one decimal place).

[Note: Here, As is the total area of tension steel bars, b is the width of the section, d is the effective depth of the bars, f_ck is the characteristic compressive cube strength of concrete, f_y is the yield stress of steel, and x_u is the depth of neutral axis.]

Two plates are connected by fillet welds of size 10 mm and subjected to tension, as shown in the figure. The thickness of each plate is 12 mm. The yield stress and the ultimate stress of steel under tension are 250 MPa and 410 MPa, respectively. The welding is done in the workshop (partial safety factor, 𝛾_𝑚𝑤 = 1.25). As per the Limit State Method of IS 800: 2007, what is the minimum length (in mm, rounded off to the nearest higher multiple of 5 mm) required of each weld to transmit a factored force P equal to 275 kN? 

A circular solid shaft of span L = 5 m is fixed at one end and free at the other end. A torque T = 100 kN.m is applied at the free end. The shear modulus and polar moment of inertia of the section are denoted as G and J, respectively. The torsional rigidity GJ is 50,000 kN.m² /rad. The following are reported for this shaft:

Statement i) The rotation at the free end is 0.01 rad

Statement ii) The torsional strain energy is 1.0 kN.m

With reference to the above statements, which of the following is true?

When a simply-supported elastic beam of span L and flexural rigidity EI (E is the modulus of elasticity and I is the moment of inertia of the section) is loaded with a uniformly distributed load w per unit length, the deflection at the mid-span is

$\rm \Delta_0=\frac{5}{384}\frac{wL^4}{El}$

If the load on one half of the span is now removed, the mid-span deflection _______.

In the frame shown in the figure (not to scale), all four members (AB, BC, CD, and AD) have the same length and same constant flexural rigidity. All the joints A, B, C, and D are rigid joints. The midpoints of AB, BC, CD, and AD, are denoted by E, F, G, and H, respectively. The frame is in unstable equilibrium under the shown forces of magnitude 𝑃 acting at E and G. Which of the following statements is/are TRUE?

In a two-dimensional stress analysis, the state of stress at a point is shown in the figure. The values of length of PQ, QR, and RP are 4, 3, and 5 units, respectively. The principal stresses are ________. (round off to one decimal place)

A 5 cm long metal rod AB was initially at a uniform temperature of T0°C. Thereafter, temperature at both the ends are maintained at 0°C. Neglecting the heat transfer from the lateral surface of the rod, the heat transfer in the rod is governed by the one-dimensional diffusion equation $\rm\frac{\partial T}{\partial t}=D\frac{\partial^2T}{\partial x^2}$, where D is the thermal diffusivity of the metal, given as 1.0 cm2/s.

The temperature distribution in the rod is obtained as

$\rm T(x,t)=\Sigma_{n=1,3,5...}^{\infty}C_n\sin\frac{n\pi x}{5}e^{-\beta n^2t}$,

where x is in cm measured from A to B with 𝑥 = 0 at A, t is in s, 𝐶𝑛 are constants in °C, T is in °C, and β is in s−1 .

The value of β (in 𝑠−1 , rounded off to three decimal places) is ________.

A beam is subjected to a system of coplanar forces as shown in the figure. The magnitude of vertical reaction at Support P is ______ N (round off to one decimal place). 

A 2D thin plate with modulus of elasticity, 𝐸 = 1.0 N/m², and Poisson’s ratio, 𝜇 = 0.5, is in plane stress condition. The displacement field in the plate is given by 𝑢 = 𝐶𝑥²𝑦 and 𝑣 = 0, where 𝑢 and 𝑣 are displacements (in m) along the 𝑋 and 𝑌 directions, respectively, and 𝐶 is a constant (in m⁻²). The distances x and y along X and Y, respectively, are in m. The stress in the 𝑋 direction is 𝜎_𝑋𝑋 = 40𝑥𝑦 N/m², and the shear stress is 𝜏_𝑋𝑌 = 𝛼𝑥² N/m². What is the value of 𝛼 (in N/m⁴, in integer)? ________

Muller-Breslau principle is used in analysis of structures for ________.

For the frame shown in the figure (not to scale), all members (AB, BC, CD, GB, and CH) have the same length, 𝐿 and flexural rigidity, 𝐸𝐼. The joints at B and C are rigid joints, and the supports A and D are fixed supports. Beams GB and CH carry uniformly distributed loads of 𝑤 per unit length. The magnitude of the moment reaction at A is 𝑤𝐿2⁄𝑘. What is the value of 𝑘 (in integer)? ____________

An idealised frame supports a load as shown in the figure. The horizontal component of the force transferred from the horizontal member PQ to the vertical member RS at P is _______ N (round off to one decimal place).

Which of the following is equal to the stopping sight distance?

As per the Indian Roads Congress guidelines (IRC 86: 2018), extra widening depends on which of the following parameters? 

The figure presents the time-space diagram for when the traffic on a highway is suddenly stopped for a certain time and then released. Which of the following statements are true?

Consider the Marshall method of mix design for bituminous mix. With the increase in bitumen content, which of the following statements is/are TRUE? 

The line ran _______ the page, right through the centre, and divided the page into two.

Kind : ________ :: Often : Seldom

(By word meaning)

In how many ways can cells in a 3 × 3 grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown. 

There are 4 red, 5 green, and 6 blue balls inside a box. If 𝑁 number of balls are picked simultaneously, what is the smallest value of 𝑁 that guarantees there will be at least two balls of the same colour?

One cannot see the colour of the balls until they are picked.

Consider a circle with its centre at the origin (O), as shown. Two operations are allowed on the circle.

Operation 1: Scale independently along the x and y axes.

Operation 2: Rotation in any direction about the origin.

Which figure among the options can be achieved through a combination of these two operations on the given circle? 

Elvesland is a country that has peculiar beliefs and practices. They express almost all their emotions by gifting flowers. For instance, if anyone gifts a white flower to someone, then it is always taken to be a declaration of one’s love for that person. In a similar manner, the gifting of a yellow flower to someone often means that one is angry with that person.

Based only on the information provided above, which one of the following sets of statement(s) can be logically inferred with certainty?

(i) In Elvesland, one always declares one’s love by gifting a white flower.

(ii) In Elvesland, all emotions are declared by gifting flowers.

(iii) In Elvesland, sometimes one expresses one’s anger by gifting a flower that is not yellow.

(iv) In Elvesland, sometimes one expresses one’s love by gifting a white flower.

Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?

Based only on the following passage, which one of the options can be inferred with certainty?

When the congregation sang together, Apenyo would also join, though her little screams were not quite audible because of the group singing. But whenever there was a special number, trouble would begin; Apenyo would try singing along, much to the embarrassment of her mother. After two or three such mortifying Sunday evenings, the mother stopped going to church altogether until Apenyo became older and learnt to behave.

At home too, Apenyo never kept quiet; she hummed or made up silly songs to sing by herself, which annoyed her mother at times but most often made her become pensive. She was by now convinced that her daughter had inherited her love of singing from her father who had died unexpectedly away from home.

If x satisfies the equation $\rm 4^{8^x}=256$, then x is equal to ______.

Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds v_p, v_Q, and V_R, respectively.

Which one of the following options is CORRECT?