Potential function $$\phi $$ is given as $$\phi \, = \,{x^2}\, - \,{y^2}$$. What will be the stream function $$\psi $$ with the condition $$\psi \, = \,0$$ at x = 0, y = 0?

Given that one root of the equation $$\,{x^3} - 10{x^2} + 31x - 30 = 0\,\,$$ is $$5$$ then other roots are

The following equation needs to be numerically solved using the Newton $$-$$ Raphson method $${x^3} + 4x - 9 = 0.\,\,$$ The iterative equation for this purpose is ($$k$$ indicates the iteration level)

A body originally at $${60^ \circ }$$ cools down to $$40$$ in $$15$$ minutes when kept in air at a temperature of $${25^ \circ }$$c. What will be the temperature of the body at the and of $$30$$ minutes?

The inverse of $$2 \times 2$$ matrix $$\left[ {\matrix{ 1 & 2 \cr 5 & 7 \cr } } \right]$$ is

The minimum and maximum eigen values of matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr } } \right]$$ are $$-2$$ and $$6$$ respectively. What is the other eigen value?

For what values of $$\alpha $$ and $$\beta $$ the following simultaneous equations have an infinite number of solutions $$$x+y+z=5,$$$ $$$x+3y+3z=9,$$$ $$$x + 2y + \alpha z = \beta $$$

The velocity vector is given as $${\mkern 1mu} \vec V = 5xy\widehat i + 2{y^2}\widehat j + 3y{z^2}\widehat k.{\mkern 1mu} {\mkern 1mu} $$ The divergence of this velocity vector at $$(1,1,1)$$ is

If the standard deviation of the speed of vehicles in a highway is $$8.8$$ kmph and the mean speed of the vehicles is $$33$$ kmph, the coefficient of variation in speed is

The solution for the differential equation $$\,{{d\,y} \over {d\,x}} = {x^2}\,y$$ with the condition that $$y=1$$ at $$x=0$$ is

The degree of the differential equation $$\,{{{d^2}x} \over {d{t^2}}} + 2{x^3} = 0\,\,$$ is

The presence of hardness in excess of permissible limit causes

The alkalinity and the hardness of a water sample are 250 mg/L and 350 mg/L as $$CaCo_3$$, respectively. The water has

The plan of a map was photo copied to a reduced size such that a line originally 100mm, measures 90mm. The original scale of the plan was 1:1000. The revised scale is

The plan of a survey plotted to a scale of 10m to 1cm is reduced in such a way that a line originally 10cm long now measures 9cm. the area of the reduced plan is measured as 81cm2. the actual (m2) of the survey is

The magnetic bearing of a line AB is S 45° E and the declination is 5° west. The true bearing of the line AB is

The water content of a saturated soil and the specific gravity of soil solids were found to be 30% and 2.70, respectively. Assuming the unit weight of water to be 10 $$kN /m^3$$, the saturated unit weight ($$kN /m^3$$), and the void ratio of the soil are

The consumptive use of water for a crop during a particular stage of growth is 2.0mm / day. The maximum depth of available water is the root zone is 60mm. Irrigation is required when the amount of available water is 50% of the maximum available water in the root zone. Frequency of irrigation should be

The culturable command area for a distributary channel is 20,000 hectares. Wheat grown in the entire area and the intensity of irrigation is 50%. The kor period for wheat is 30 days and the kor water depth is 120mm. The outlet discharge for the distributary should be

A singly reinforced rectangular concrete beam has a width of $$150$$ $$mm$$ and an effective depth of $$330$$ $$mm.$$ The characteristic compressive strength of concrete is $$20$$ $$MPa$$ and the characteristic tensile strength of steel is $$\,\,\,\,\,\,\,415\,\,MPa.$$

Adopt the stress block for concrete as given in $$IS$$ $$456$$-$$2000$$ and take limiting value of depth of neutral axis as $$0.48$$ times the effective depth of the beam.

The limiting value of the moment of resistance of the beam in $$kN$$-$$m$$ is

A singly reinforced rectangular concrete beam has a width of $$150$$ $$mm$$ and an effective depth of $$330$$ $$mm.$$ The characteristic compressive strength of concrete is $$20$$ $$MPa$$ and the characteristic tensile strength of steel is $$\,\,\,\,\,\,\,415\,\,MPa.$$

Adopt the stress block for concrete as given in $$IS$$ $$456$$-$$2000$$ and take limiting value of depth of neutral axis as $$0.48$$ times the effective depth of the beam.

The limiting area of tension steel in $$m{m^2}$$ is

A concrete beam of rectangular cross-section of size $$120$$ $$mm$$ (width) and $$200$$ $$mm$$ (depth) is prestressed by a straight tendon to an effective force of $$150$$ $$kN$$ at an eccentricity of $$20$$ $$mm$$ (below the centroidal axis in the depth direction). The stresses at the top and bottom fibres of the section are

Consider the following statements:

1. The compressive strength of concrete decreases with increase in water-cement ratio of the concrete mix.

2. Water is added to the concrete mix for hydration of cement and workability.

3. Creep and shrinkage of concrete are independent of the water-cement ratio in the concrete mix.

The true statements are

Consider the following statements:

I. Modulus of elasticity concrete increases with increase in compressive strength of concrete.

II. Brittleness of concrete increases with decrease in compressive strength of concrete.

III. Shear strength of concrete increases with increase in compressive strength of concrete.

The TRUE statements are

A bracket connection is made with four bolts of $$10$$ $$mm$$ diameter and supports a load of $$10$$ $$kN$$ a an eccentricity of $$100$$ $$mm.$$ The maximum force to be resisted by any bolt will be

A steel flat of rectangular section of size $$70 \times 6\,\,mm$$ is connected to a gusset plate by three bolts each having a shear capacity of $$15$$ $$kN$$ in holes having diameter $$11.5$$ $$mm.$$ If the allowable tensile stress in the flat is $$150$$ $$MPa,$$ the maximum tension that can be applied to the flat is

An axially loaded bar is subjected to a normal stress of $$173$$ $$MPa.$$ The maximum shear stress in the bar is

The shear stress at the neutral axis in a beam of triangular section with a base of $$40$$ $$mm$$ and height $$20$$ $$mm,$$ subjected to a shear force of $$3$$ $$kN$$ is

The maximum and minimum shear stresses in a hollow circular shaft of outer diameter $$20$$ $$mm$$ and thickness $$2$$ $$mm,$$ subjected to a torque of $$92.7$$ $$N.m$$ will be

A two span continuous beam having equal spans each of length $$L$$ is subjected to a uniformly distributed load $$\omega $$ per unit length. The beam has constant flexural rigidity.

The bending moment at the middle support is

A two span continuous beam having equal spans each of length $$L$$ is subjected to a uniformly distributed load $$\omega $$ per unit length. The beam has constant flexural rigidity.

The reaction at the middle support is

A steel column, pinned at both ends, has a buckling load of $$200$$ $$kN.$$ If the column is restrained against lateral movement at its mid-height, its buckling load will be

A metal bar of length 100mm is inserted between two rigid supports and its temperature is increased by $$10^o$$C. If the coefficient of thermal expansion is 12 x$$10^{-6}$$ per $$^o$$C and the Young’s modulus is 2 x $$10^{5}$$ MPa, the stress in the bar is

For an isotropic material, the relationship between the young’s modulus (E), shear modulus (G) and Poisson’s ratio (μ) is given by

The right triangular truss is made of members having equal cross sectional area of $$1550$$ $$m{m^2}$$ and Young's Modulus of $$2 \times {10^5}\,\,MPa.$$

The horizontal deflection of the joint $$Q$$ is

The influence line diagram $$(ILD)$$ shown is for the member

A two span continuous beam having equal spans each of length $$L$$ is subjected to a uniformly distributed load $$w$$ per unit length. The beam has constant flexural rigidity.

The reaction at the middle support is

A two span continuous beam having equal spans each of length $$L$$ is subjected to a uniformly distributed load $$w$$ per unit length. The beam has constant flexural rigidity.

The bending moment at the middle support is

The plastic collapse load $${{W_P}}$$ for the propped cantilever supporting two point loads as shown in the figure in terms of plastic moment capacity, $${{M_P}}$$ is given by

The stiffness coefficient $${k_{ij}}$$ indicates

While designing a hill road with a ruling gradient of 6%, if a sharp horizontal curve of 50m radius is encountered, the compensated gradient at the curve as per the Indian Roads Congress specifications should be