In the following partial differential equation, θ is a function of t and z, and D and K are functions of θ

$D\left(\theta \right)\frac{{\delta }^2\theta }{\delta z^2}+\frac{\delta K\left(\theta \right)}{\delta z}-\frac{\delta \theta }{\delta t}=0$

Consider the planar truss shown in the figure (not drawn to the scale)

Neglecting self-weight of the members, the number of zero-force members in the truss under the action of the load P is

A reinforcing steel bar, partially embedded in concrete, is subjected to a tensile force P. The figure that appropriately represents the distribution of the magnitude of bond stress (represented as hatched region). Along embedded length of the bar, is

In a two-dimensional stress analysis, the state of stress at a point P is

$\left[\sigma \right]=\left[\begin{array}{cc}{\sigma }_{xx}& {\tau }_{xy}\\ {\tau }_{xy}& {\sigma }_{yy}\end{array}\right]$

Uniform flow with velocity u makes an angle θ with the y-axis, as shown in the figure

The velocity potential (φ), is

The data for an agricultural field for a specific month are given below;

Pan Evaporation = 100 mm

Effective Rainfall = 20 mm (after deducting losses due to runoff and deep percolation)

Crop Coefficient = 0.4

Irrigation Efficiency = 0.5

During the chlorination process, aqueous chlorine reacts rapidly with water to form Cl^-, HOCl, and H⁺ as shown below

CI₂ (aq) + H₂O ⇋ HOCI + CI^- + H⁺

A planar elastic structure is subjected to uniformly distributed load, as shown in the figure (not drawn to the scale)

Neglecting self-weight the maximum bending moment generated in in the structure (in kN.m round off to the nearest integer), is

A river has a flow of 1000 million litres per day (MLD), BOD₅ of 5 mg/litre and Dissolved Oxygen (DO) level of 8 mg/litre before receiving the wastewater discharge at a location. For the existing environmental conditions, the saturation DO level is 10 mg/litre in the river. Wastewater discharge of 100 MLD with the BOD₅ of 200 mg/litre and DO level of 2 mg/litre fall at that location. Assuming complete mixing of wastewater and river water, the immediate DO deficit (in me/litre, round off to two decimal places), is

Distributed load(s) of 50 kN/m may occupy any position(s) (either continuously or in patches) on the girder PQRST as shown in the figure (not drawn to the scale)

A rigid weightless platform PQRS shown in the figure (not drawn to the scale) can slide freely in the vertical direction. The platform is held in position by the weightless member OJ and four weightless, frictionless rollers. Points O and J are pin connections. A block of 90 kN rests on the platform as shown in the figure.

The magnitude of horizontal component of reaction (in kN) at pin O, is f

A cantilever beam PQ of uniform flexural rigidity (EI) is subjected to a concentrated moment M at R as shown in the figure.

The deflection at the free end Q is

The relationship between traffic flow rate (q) and density (D) is shown in the fugure

The shock wave condition is depicted by

The soil profile at a site up to a depth of 10 m is show in the figure (not drawn to the scale). The soil is preloaded with a uniform surcharge (q) of 70 kN/m² at the ground level. The water table is at a depth of 3 m below ground level. The soil unit weight of the respective layers is shown in the figure. Consider unit weight of water as 9.81 kN/m³ and assume that the surcharge (q) is applied instantaneously

The total stress paths corresponding to different loading conditions, for a soil specimen under the isotropically consolidated stress state (O), are shown below :

A water supply scheme transports 10 MLD (Million Litres per Day) water through a 450 mm diameter pipeline for a distance of 2.5 km. A chlorine dose of 3.50 mg/litre is applied at the starting point of the pipeline to attain a certain level of disinfection at the downstream end. It is decided to increase the flow rate from 10 MLD to 13 MLD in the pipeline. Assume exponent for concentration, n = 0.86. With this increased flow, to attain the same level of disinfection, the chlorine dose (in mg/litre) to be applied at the starting point should be

An open traverse PQRST is surveyed using theodolite and the consecutive coordinates obtained are given in the table.

If the independent coordinates (Nothing, Easting) of station P are (400 m, 200m), the independent coordinates (in m) of station T, are

If C represents a line segment between (0,0,0) and (1, 1, 1) in the Cartesian coordinate system, the value (expressed as an integer) of the line integral

${\int }_X^{}\left[\left(y=z\right)dx+\left(x+z\right)dy+\left(x+y\right)dz\right]$ 

Consider the system of equations $\left[\begin{array}{ccc}1& 3& 2\\ 2& 2& -3\\ 4& 4& -6\\ 2& 5& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ 2\\ 1\end{array}\right]$ The value of x₃ (round off to the nearest integer), is ______.

A rigid, uniform, weightless, horizontal bar is connected to three vertical members P. Q and R as shown in the figure (not drawn to the scale). All three members have identical axial stiffness of 10 kN/mm. The lower ends of bars P and R rest on a rigid horizontal surface. When NO load is applied. a gap of 2 mm exists between the lower end of the bar Q and the rigid horizontal surface. When a vertical load W is placed on the horizontal bar in the downward direction, the bar still remains horizontal and gets displaced by 5 mm in the vertically downward direction.

The magnitude of load W (in kN, round off to the nearest integer), is 

The flange and web plates of the doubly symmetric built-up section are connected by continuous 10 mm thick fillet welds as shown in the figure (not drawn to the scale). The moment of inertia of the section about its principal axis X—-X is 7.73 x 10⁶ mm⁴. The permissible shear stress in the fillet welds is 100 N/mm². The design shear strength of the section is governed by the capacity of the fillet welds.

The maximum shear force (in kN, round of to one decimal place) that can be carried by the section, is

The singly reinforced concrete beam section shown in the figure (not drawn to the scale) is made of M25 grade concrete and Fe500 grade reinforcing steel. The total cross-sectional area of the tension steel is 942 mm²

As per Limit State Design of IS 456:2000, the design moment capacity (in kKN.m, round off to two decimal places) of the beam section, is ,

Traffic volume count has been collected on a 2-lane road section which needs upgradation due to severe traffic flow condition. Maximum service flow rate per lane is observed as 1280 veh/h at level of service "C”. The Peak Hour Factor i

s reported as 0.78125. Historical traffic volume count provides Annual Average Daily Traffic as 12270 veh/day. Directional split of the traffic flow is observed to be 60:40. Assuming that traffic stream consists of ‘All Cars’ and all drivers are

 ‘Regular Commuters’, the number of extra lane(s) (round off to the next higher integer) to be provided, is

Water flows in the upward direction in a tank through 2.5 m thick sand layer as shown in the figure. The void ratio and specific gravity of sand are 0.58 and 2.7, respectively. The sand is fully saturated. Unit weight of water is 10 kN/m³

The effective stress (in kPa, round off to two decimal places) at point A, located1 m above the base of tank, is

A 10 m thick clay layer is resting over a 3 m thick sand layer and is submerged.

A fill of 2 m thick sand with unit weight of 20 kN/m³ is placed above the clay layer to accelerate the rate of consolidation of the clay layer. Coefficient of consolidation of clay is 9 × 10^-2 m²/year and coefficient of volume compressibility of clay is 2.2 × 10^-4 m² /KN. Assume Taylor’s relation between time factor and average degree of consolidation.

The settlement (in mm. round off to two decimal places) of the clay layer, 10 years after the construction of the fill, is

Three reservoirs P, Q, and R are interconnected by pipes as shown in the figure (not drawn to the scale). Piezometric head at the junction S of the pipes is 100 m. Assume acceleration due to gravity as 9.81 m/s² and density of water as 1000 kg/m³. The length of the pipe from junction S to the inlet of reservoir R is 180 m.

Considering head loss only due to friction (with friction factor of 0.03 for all the pipes), the height of water level in the lowermost reservoir R (in m, round off to one decimal place) with respect to the datum, is

The lengths and bearings of a traverse PQRS are:

The length of line segment SP (in m, round off to two decimal places), is