The activities of a project are given in the following table along with their durations and dependency.

The total float of the activity E (in day) is _________. (in integer)

Match all the possible combinations between Column X (Cement compounds) and Column Y (Cement properties) :

Which one of the following combinations is correct?

Let y be a non-zero vector of size 2022 $$\times$$ 1. Which of the following statements is/are TRUE?

A pair of six-faced dice is rolled twice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is _________ (round off to three decimal places).

Consider the polynomial f(x) = x³ $$-$$ 6x² + 11x $$-$$ 6 on the domain S, given by 1 $$\le$$ x $$\le$$ 3. The first and second derivatives are f'(x) and f''(x).

Consider the following statements :

I. The given polynomial is zero at the boundary points x = 1 and x = 3.

II. There exists one local maxima of f(x) within the domain S.

III. The second derivative f''(x) > 0 throughout the domains S.

IV. There exists one local minima f(x) within the domain S.

The components of pure shear strain in a sheared are given in the matrix form:

$$\varepsilon = \left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right]$$

Here, Trace ($$\varepsilon $$) = 0. Given, P = Trace ($$\varepsilon$$⁸) and Q = Trace ($$\varepsilon $$¹¹).

The numerical value of (P + Q) is ___________. (in integer)

The function f(x, y) satisfies the Laplace equation

$$\Delta$$²f(x, y) = 0

on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0. The value of this function on the circular boundary of this domain is equal to 3. The numerical value of f/(0, 0) is :

P and Q are two square matrices of the same order. Which of the following statements is/are correct?

$$\int {\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 3} - {{{x^4}} \over 4} + ....} \right)dx} $$ is equal to :

An undamped spring-mass system with mass m and spring stiffness k is shown in the figure. The natural frequency and natural period of this system are $$\omega$$ rad/s and T s, respectively. If the stiffness of the spring is doubled and the mass is halved, then the natural frequency and the natural period of the modified system, respectively, are

A uniform rod KJ of weight w shown in the figure rests against a frictionless vertical wall at the point K and a rough horizontal surface at point J. It is given that w = 10 kN, a = 4 m and b = 3 m.

The minimum coefficient of static friction that is required at the point J to hold the rod in equilibrium is __________. (round off to three decimal places)

A 100 mg of HNO₃ (strong acid) is added to water, bringing the final volume to 1.0 liter. Consider the atomic weights of H, N, and O, as 1 g/mol. 14 g mol. and 16 g/mol, respectively. The final pH of this water is (Ignore the dissociation of water.)

In a city, the chemical formula of biodegradable fraction of municipal solid waste (MSW) is C₁₀₀H₂₅₀O₈₀N. The waste has to be treated by forced-aeration composting process for which air requirement has to be estimated.

Assume oxygen in air (by weight) = 23% and density of air = 1.3 kg/m³. Atomic mass : C = 12, H = 1, O = 16, N = 14.

C and H are oxidized completely whereas N is converted only into NH₃ during oxidation.

For oxidative degradation of 1 tonne of the waste, the required theoretical volume of air (in m³/tonne) will be (round off the nearest integer)

A sewage treatment plant receives sewage at a flow rate of 5000 m³/day. The total suspended solids (TSS) concentration in the sewage at the inlet of primary clarifier is 200 mg/L. After the primary treatment, the TSS concentration in sewage is reduced by 60%. The sludge from the primary clarifier contains 2% solids concentration. Subsequently, the sludge is subjected to gravity thickening process to achieve a solids concentration of 6%. Assume that the density of sludge, before and after thickening, is 1000 kg/m³. The daily volume of the thickened sludge (in m³/day) will be _________. (round off to the nearest integer)

Match the following attributes of a city with the appropriate scale of measurements.

Which one of the following combinations is correct?

To design an optimum municipal solid waste collection route, which of the following is/are NOT desired :

A sample of air analyzed at 25$$^\circ$$C and 1 atm pressure is reported to contain 0.04 ppm of SO₂. Atomic mass of S = 32. O = 16. The equivalent SO₂ concentration (in $$\mu$$g/m²) will be __________. (round off to the nearest integer)

A process equipment emits, 5 kg/h of volatile organic compounds (VOCs). If a hood placed over the process equipment captures 95% of the VOCs, then the fugitive emission in kg/h is

In a solid waste handling facility, the moisture contents (MC) of food waste, paper waste, and glass waste were found to be MCf, MCp, and MCg, respectively. Similarly, the energy contents (EC) of plastic waste, food waste, and glass waste were found to be ECpp, ECf, and ECg. respectively. Which of the following statements is/are correct?

At a municipal waste handling facility, 30 metric ton mixture of food waste, yard waste, and paper waste was available. The moisture content of this mixture was found to be 10%. The ideal moisture content for composting this mixture is 50%. The amount of water to be added to this mixture to bring its moisture content to the ideal condition is _________ metric ton. (in integer)

The concentration s(x, t) of pollutants, in a one-dimensional reservoir at position x and time t satisfies the diffusion equation

$${{\partial s(x,t)} \over {\partial t}} = D{{{\partial ^2}s(x,t)} \over {\partial {x^2}}}$$

on the domain 0 $$\le$$ x $$\le$$ L, where D is the diffusion coefficient of the pollutants. The initial condition s(x, 0) is defined by the step-function shown in the figure.

The boundary conditions of the problem are given by $${{\partial s(x,t)} \over {\partial x}}$$ = 0 at the boundary points x = 0 and x = L at all times. Consider D = 0.1 m²/s, s₀ = 5 $$\mu$$mol/m and L = 10 m. The steady-state concentration $$\overline s \left( {{L \over 2}} \right) = s\left( {{L \over 2},\infty } \right)$$ at the center x = $${{L \over 2}}$$ of the reservoir (in $$\mu$$mol/m) is __________. (in integer)

Two discrete spherical particles (P and Q) of equal mass density are independently released in water. Particle P and particle Q have diameters of 0.5 mm and 1.0 mm, respectively. Assume Stoke's law is valid.

The drag force on particle Q will be ___________ times the drag force on particle P. (round off to the nearest integer)

A hydraulic jump takes place in a 6 in wide rectangular channel at a point where the upstream depth is 0.5 m (just before the jump). If the discharge in the channel is 30 m³/s and the energy loss in the jump is 1.6 m, then the Froude number computed at the end of the jump is _________. (round off to two decimal places)

(Consider the acceleration due to gravity as 10 m/s².)

Water is flowing in a horizontal, frictionless, rectangular channel. A smooth hump is built on the channel floor at a section and its height is gradually increased to reach choked condition in the channel. The depth of water at this section is y₂ and that at its upstream section is y₁. The correct statements for the chocked and un-chocked conditions in the channel is/are

The dimension of dynamic viscosity is :

If the magnetic bearing of the Sun at a place at noon is S 2$$^\circ$$E. The magnetic declination (in degrees) at that place is

The error in measuring the radius of a 5 cm circular rod was 0.2%. If the cross-sectional area of the rod was calculated using this measurement, then the resulting absolute percentage error in the computed area is __________. (round off to two decimal places)

Match the following in Column X with Column Y :

Which one of the following combinations is correct?

A saturated compressible clay layer of thickness h is sandwiched between two sand layers, as shown in the figure. Initially, the total vertical stress and pore water pressure at point P, which is located at the mid-depth of the clay layer were 150 kPa and 25 kPa, respectively. Construction of a building caused an additional total vertical stress of 100 kPa at P. When the vertical effective stress at P is 175 kPa. The percentage of consolidation in the clay layr at P is _________. (in integer)

A concentrically loaded isolated square footing of size 2 m $$\times$$ 2 m carries a concentrated vertical load of 1000 kN. Considering Boussinesq's theory of stress distribution, the maximum depth (in m) of the pressure bulb corresponding to 10% of the vertical load intensity will be _________. (round off to two decimal all places)

A soil sample is underlying a water column of height h₁, as shown in the figure. The vertical effective stresses at point A, B and C are $$\sigma {'_A}$$, $$\sigma {'_B}$$ and $$\sigma {'_C}$$ respectively. Let $$\gamma$$_sat and $$\gamma$$' be the saturated and submerged unit weights of the soil sample, respectively and $$\gamma$$_w be the unit weight of water. Which one of the following expressions correctly represents the sum ($$\sigma {'_A}$$ + $$\sigma {'_B}$$ + $$\sigma {'_C}$$) ?

Read the following statements :

(P) While designing a shallow footing in sandy soil, monsoon season is considered for critical design in terms of bearing capacity.

(Q) For slope stability of an earthen dam, sudden drawdown is never a critical condition.

(R) In a sandy sea beach, quicksand condition can arise only if the critical hydraulic gradient exceeds the existing hydraulic gradient.

(S) The active earth thrust on a rigid retaining wall supporting homogenous cohesionless backfill will reduce with the lowering of water table in the backfill.

Which one of the following combinations is correct?

The inside diameter of a sampler tube is 50 mm. The inside diameter of the cutting edge is kept such that the Inside Clearance Ratio (ICR) is 1.0% to minimize the friction on the sample as the sampler tube enters into the soil. The inside diameter (in mm) of the cutting edge is _________. (round off to two decimal places)

A group of total 16 piles are arranged in a square grid format. The center-to-center spacing (s) between adjacent piles is 3 m. The diameter (d) and length of embedment of each pile are 1 m and 20 m, respectively. The design capacity of each pile is 1000 kN in the vertical downward direction. The pile group efficiency $${\eta _g}$$ is given by

$${\eta _g} = 1 - {\theta \over {90}}\left[ {{{(n - 1)m + (m - 1)n} \over {mn}}} \right]$$

where m and n are number of rows and columns in the plan grid of pile arrangement and $$\theta = {\tan ^{ - 1}}\left( {{d \over s}} \right)$$.

The design value of the pile group capacity (in kN) in the vertical downward direction is _________. (round off to the nearest integer)

In a triaxial unconsolidated undrained (UU) test on a saturated clay sample, the cell pressure was 100 kPa. If the deviatoric stress at failure was 150 kPa, then the undrained shear strength of the soil is _________ kPa. (in integer)

Match Column X with Column Y :

Which one of the following combinations is correct?

A pump with an efficiency of 80% is used to draw groundwater from a well for irrigating a flat field of area 108 hectares. The base period and delta for paddy crop on this field are 120 days and 144 cm, respectively. Water application efficiency in the field is 80%. The lowest level of water in the well is 10 m below the ground. The minimum required horse power (h.p.) of the pump is _________. (round off to two decimal places)

(Consider 1 h.p. = 746 W; unit weight of water = 9810 N/m³

In a certain month, the reference crop evapotranspiration at a location is 6 mm day. If the crop coefficient and soil coefficient are 1.2 and 0.8, respectively, the actual evapotranspiration in mm/day is

Read the following statements relating to flexure of reinforced concrete beams :

I. In over-reinforced sections, the failure strain in concrete reaches earlier than the yield strain in steel.

II. In under-reinforced sections, steel reaches yielding at a load lower than the load at which the concrete reaches failure strata.

III. Over-reinforced beams are recommended in practice as compared to the under-reinforced beams.

IV. In balanced sections, the concrete reaches failure strain earlier than the yield strain in tensile steel.

Each of the above statements is either True or False.

Which one of the following combinations is correct?

A post-tensioned concrete member of span 15 m and cross-section of 450 mm $$\times$$ 450 mm is prestressed with three steel tendons, each of cross-sectional area 200 mm². The tendons are tensioned one after another to a stress of 1500 MPa. All the tendons are straight and located at 125 mm from the bottom of the member. Assume the prestress to be the same in all tendons and the modular ratio to be 6. The average loss of prestress due to elastic deformation of concrete, considering all three tendons is

Consider two linearly elastic rods HI and IJ. Each of length b, as shown in the figure. The rods are co-linear, and confined between two fixed supports at if and J. Both the rods are initially stress free. The coefficient of linear thermal expansion is a for both the rods. The temperature of the rod IJ is raised by $$\Delta$$T whereas the temperature of rod HI remains unchanged. An external horizontal force P is now applied at node I. It is given that a = 10^$$-$$6 $$^\circ$$C^$$-$$1, $$\Delta$$T = 50$$^\circ$$C, b = 2m, AE = 10⁶N. The axial rigidities of the rods HI and U are 2 AE and AE, respectively.

To make the axial force in rod HI equal to zero, the value of the external force P (in N) is _________. (rounded off to the nearest integer).

Stresses acting on an infinitesimal soil element are shown in the figure (with $$\sigma$$_z > $$\sigma$$_x). The major and minor principal stresses are $$\sigma$$₁ and $$\sigma$$₃, respectively. Considering the compressive stresses as positive, which one of the following expressions correctly represents the angle between the major principal stress plane and the horizontal plane?

For a linear elastic and isotropic material, the correct relationship among Young's modulus of elasticity (E), Poisson's ratio (v), and shear modulus (G) is

The linearly elastic planar structure shown in the figure is acted upon by two vertical concentrated forces. The horizontal beams UV and WX are connected with the help of the vertical linear spring with spring constant k = 20 kN/m. The fixed supports are provided at U and X. It is given that flexural rigidity EI = 10⁵ kN-m², P = 100 kN, and a = 5 m. Force Q is applied at the center of beam WX such that the force in the spring VW becomes zero.

The magnitude of force Q (in kN) is __________. (round off to the nearest integer)

For the square steel beam cross-section shown in the figure, the shape factor about Z - Z axis is S and the plastic moment capacity is M_p. Consider yield stress f_y = 250 MPa and a = 100 mm.

The values of S and M_p (rounded off to one decimal place) are

Which of the following statements is/are correct?

Consider a beam PQ fixed at P, hinged at Q, and subjected to a load F as shown in figure (not drawn to scale). The static and kinematic degrees of indeterminacy, respectively are

A single-lane highway has a traffic density of 40 vehicles km. If the time-mean speed and space-mean speed are 40 kmph and 30 kmph, respectively, the average headway (in seconds) between the vehicles is

The lane configuration with lane volumes in vehicles per hour of a four-arm signalized intersection is shown in the figure. There are only two phases: the first phase is for the East-West and the West-East through movements, and the second phase is for the North-South and South-North through movements. There are no turning movements. Assume that the saturation flow is 1800 vehicles per hour per lane for each lane and the total lost time for the first and the second phases together is 9 seconds.

The optimum cycle length (in second), as per the Webster's method, is __________. (round off to the nearest integer).

Assuming that traffic on a highway obeys the Greenshields model, the speed of a shockwave between two traffic streams (P) and (Q) as shown in the schematic is __________ kmph. (in integer)

It is given that an aggregate mix has 260 grams of coarse aggregates and 240 grams of fine aggregates. The specific gravities of the coarse and fine aggregates are 2.6 and 2.4 respectively. The bulk specific gravity of the mix is 2.3. The percentage air voids in the mix is _________. (round off to the nearest integer).

For a traffic stream, V is the space mean speed, k is the density, q is the flow. V_f is the free flow speed, and k_f is the jam density. Assume that the speed decreases linearly with density.

Which of the following relations is/are correct?

The base length of the runway at the mean sea level (MSL) is 1500 m. If the runway is located at an altitude of 300 m above the MSL, the actual length (in m) of the runway to be provided is _________. (round off to the nearest integer)

A parabolic vertical crest curve connects two road segments with grades +1.0% and $$-$$2.0%. If a 200 m stopping sight distance is needed for a driver at a height of 1.2 m to avoid an obstacle of height 0.15 m, then the minimum curve length should be _________ m. (round off to the nearest integer)

The movie was funny and I ____________.

For the picture shown above, which one of the following is the correct picture representing reflection with respect to the mirror shown as the dotted line?

An ant walks in a straight line on a plane leaving behind a trace of its movement. The initial position of the ant is at point P facing East.

The ant first turns 72$$^\circ$$ anticlockwise at P, and then does the following two steps in sequence exactly FIVE times before halting.

1. moves forward for 10 cm.

2. turns 144$$^\circ$$ clockwise

The pattern made by the trade left behind by the ant is

In a partnership business the monthly investment by three friends for the first six months is in the ratio 3 : 4 : 5. After six months, they had to increase their monthly investments by 10%, 15% and 20%, respectively, of then initial monthly investment. The new investment ratio was kept constant for the next six months.

What is the ratio of their shares in the total profit (m the same order) at the end of the year such that the share is proportional to their individual total investment over the year?

A survey of 450 students about their subjects of interest resulted in the following outcome.

$$\bullet$$ 150 students are interested in Mathematics.

$$\bullet$$ 200 students are interested in Physics.

$$\bullet$$ 175 students are interested in Chemistry.

$$\bullet$$ 50 students are interested in Mathematics and Physics.

$$\bullet$$ 60 students are interested in Physics and Chemistry.

$$\bullet$$ 40 students are interested in Mathematics and Chemistry.

$$\bullet$$ 30 students are interested in Mathematics, Physics and Chemistry.

$$\bullet$$ Remaining students are interested in Humanities.

Based on the above information, the number of students interested in Humanities is

$$x:y:z = {1 \over 2}:{1 \over 3}:{1 \over 4}$$

What is the value of $${{x + z - y} \over y}$$ = ?

Both the numerator and the denominator of $${3 \over 4}$$ are increased by a positive $${15 \over 17}$$ integer, x, and those of _____ are decreased by the same integer. This operation results in the same value for both the fractions. What is the value of x?

Consider the following equations of straight lines :

Line L1 : 2x $$-$$ 3y = 5

Line L2 : 3x + 2y = 8

Line L3 : 4x $$-$$ 6y = 5

Line L4 : 6x $$-$$ 9y = 6

Which one of the following is the correct statement?

In the last few years, several new shopping malls were opened in the city. The total number of visitors in the malls is impressive. However, the total revenue generated through sales in the shops in these malls is generally low.

Which one of the following is the CORRECT logical inference based on the information in the above passage?

Given below are two statements and four conclusions drawn based on the statements.

Statement 1 : Some soaps are clean.

Statement 2 : All clean objects are wet.

Conclusion I : Some clean objects are soaps.

Conclusion II : No clean object is a soap.

Conclusion III : Some wet objects are soaps.

Conclusion IV : All wet objects are soaps.

Which one of the following options can be logically inferred?