There is no value of $$x$$ that can simultaneously satisfy both the given equations. Therefore, find the 'least squares error' solution to the two equations, i.e., find the value of $$x$$ that minimizes the sum of squares of the errors in the two equations
$$2x=3$$
$$4x=1$$

Find the value of $$\lambda $$ such that the function $$f(x)$$ is a valid probability density function ________.

The solution $$\int\limits_0^{\pi /4} {{{\cos }^4}3\theta {{\sin }^3}\,6\theta d\theta \,\,} $$ is :

What is the minimum number of multiplications involved in computing the matrix product $$PQR?$$ Matrix $$P$$ has $$4$$ rows and $$2$$ columns, matrix $$Q$$ has $$2$$ rows and $$4$$ columns and matrix $$R$$ has $$4$$ rows and $$1$$ column ______.

In its natural condition, a soil sample has a mass of 1.980 kg and a volume of 0.001 m³. After being completely dried in an oven, the mass of the sample is 1.800 kg. Specific gravity G is 2.7. Unit weight of water is 10 kN/m³. The degree of saturation of soil is

An isohyet is a line joining points of

A 1-hour rainfall of 10 cm has a return period of 50 years. The probability that 1 hour of rainfall 10 cm or more will occur in each of two successive years is

The transplantation of rice requires 10 days and total depth of water required during transplantation is 48 cm. During transplantation, there is an effective rainfall (useful for irrigation) of 8 cm. The duty of irrigation water (in hectare/ cumec) is:

A rectangular concrete beam $$250$$ $$mm$$ wide and $$600$$ $$mm$$ deep is pre-stressed by means of $$16$$ high tensile wires, each of $$7$$ $$mm$$ diameter, located at $$200$$ $$mm$$ from the bottom face of the beam at a given section. If the effective pre-stress in the wires is $$700$$ $$MPa,$$ what is the maximum sagging bending moment (in $$kN$$-$$m$$) (correct to $$1$$-decimal place) due to live load that this section of the beam can with stand without causing tensile stress at the bottom face of the beam? Neglect the effect of dead load of beam.

The creep strains are

As per $$IS$$ $$456:2000$$ for $$M20$$ grade concrete and plain barsin tension the design bond stress $${\tau _{bd}} = 1.2\,\,MPa.$$ Further, $$IS$$ $$456:2000$$ permits this design bond stress value to be increased by $$60\% $$ for $$HYSD$$ bars. The stress in the $$HYSD$$ reinforcing steel bars in tension $${\sigma _s} = 360\,MPa.$$ Find the required development length, $${L_d},$$ for $$HYSD$$ bars in terms of the bar diameter, $$\phi $$ ______________.

Maximum possible value of compaction factor for fresh (green) concrete is

A symmetric $${\rm I}$$-section (with width of each flange $$=50$$ $$mm$$, thickness of each flange $$=10$$ $$mm,$$ depth of web $$= 100$$ $$mm,$$ and thickness of web $$=10$$ $$mm$$ ݉݉) of steel is subjected to a shear force of $$100$$ $$kN.$$ Find the magnitude of the shear stress (in ܰ$$N/m{m^2}$$) in the web at its junction with the top flange. ___________

Two steel columns $$P$$ (Length $$L$$ and yield strength $${f_y} = 250\,\,MPa$$) and $$Q$$ (length $$2L$$ and yield strength $${f_y} = 500\,\,MPa$$) have the same cross-sections and end-conditions. The ratio of buckling load of column $$P$$ to that of column $$Q$$ is:

Two steel columns $$P$$ (Length $$L$$ and yield strength $${f_y} = 250\,\,MPa)$$ and $$Q$$ (length $$2L$$ and yield strength $${f_y} = 500\,\,MPa$$) have the same cross-sections and end-conditions. The ratio of buckling load of column $$P$$ to that of column $$Q$$ is

The ‘'plane section remains plane’' assumption in bending theory implies:

A symmetric $${\rm I}$$-section (with width of each flange $$=50mm$$, thickness of each flange $$=10mm,$$ depth of web $$=100mm,$$ and thickness of web $$=10mm$$) of steel is subjected to a shear force of $$100$$ $$kN.$$ Find the magnitude of the shear stress (in $$N/m{m^2}$$) in the web at its junction with top flange __________.

A uniform beam weighing $$1800$$ $$N$$ is supported at $$E$$ and $$F$$ by cable $$ABCD$$. Determine the tension (in $$N$$) in segment $$AB$$ of this cable (correct to $$1$$-decimal place). Assume the cables $$ABCD$$, $$BE$$ and $$CF$$ to be weightless ________

The state of $$2D$$-stress at a point is given by the following matrix of stresses: $$$\left[ {\matrix{ {{\sigma _{xx}}} & {{\sigma _{xy}}} \cr {{\sigma _{xy}}} & {{\sigma _{yy}}} \cr } } \right] = \left[ {\matrix{ {100} & {30} \cr {30} & {20} \cr } } \right]MPa$$$

What is the magnitude of maximum shear stress in $$MPa$$ ?

The pin-jointed $$2$$-$$D$$ truss is loaded with a horizontal force of $$15$$ ݇ܰ at joint S and another $$15$$ ݇ܰ vertical force at joint $$U,$$ as shown.Find the force in member $$RS$$ (in $$kN$$ ݇ܰ) and report your answer taking tension as positive and compression as negative. _____________

A propped cantilever made of a prismatic steel beam is subjected to a concentrated load $$P$$ at mid span as shown.

If load $$P=80$$ $$kN,$$ find the reaction ܴ$$R$$ (in $$kN$$) (correct to $$1$$-decimal place)using elastic analysis ___________

A propped cantilever made of a prismatic steel beam is subjected to a concentrated load $$P$$ at mid span as shown.

If the magnitude of loadܲ is increased till collapse and the plastic moment carrying capacity of steel beam section is $$90$$ $$kN$$-$$m$$, ݇ܰ݉determine reaction ܴ$$R$$(in ݇ܰ$$kN$$)(correct to $$1$$-decimal place) using plastic analysis. ____________

As per $$IS$$ $$800:2007$$ the cross-section in which the extreme fiber can reach the yield stress, but cannot develop the plastic moment of resistance due to failure by local buckling is classified as

All members in the rigid-jointed frame shown are prismatic and have the same flexural stiffness $$EI$$. Find the magnitude of the bending moment at $$Q$$ (in ݇ܰ݉$$kN$$-$$m$$) due to the given loading _______________

A uniform beam ($$EI=$$ constant) $$PQ$$ in the form of a quarter-circle of radius $$R$$ is fixed at end $$P$$ and free at the end $$Q,$$ where a load $$W$$ is applied as shown. The vertical downward displacement, $${\delta _q},$$ at the loaded point $$Q$$ is given by : $${\delta _q} = \beta \left( {{{W{R^3}} \over {EI}}} \right).$$

Find the value of $$\beta $$(correct to $$4-$$decimal places). _____________________

Beam $$PQRS$$ has internal hinges in span $$PQ$$ and $$RS$$ as shown. The beam may be subjected to a moving distributed vertical load of maximum intensity $$4$$ $$kN/m$$ of any length anywhere on the beam. The maximum absolute value of the shear force (in $$kN$$) that can occur due to this loading just to the right of support $$Q$$ shall be: