Solve $${{{d^4}y} \over {d{x^4}}} - y = 15\,\cos \,\,2x$$

$${\left( {s + 1} \right)^{ - 2}}$$ is laplace transform of

The Laplace Transform of a unit step function $${u_a}\left( t \right),$$ defined as
$$\matrix{ {{u_a}\left( t \right) = 0} & {for\,\,\,t < a\,} \cr { = 1} & {for\,\,\,t > a,} \cr } $$ is

The continuous function $$f(x, y)$$ is said to have saddle point at $$(a, b)$$ if

The Taylor's series expansion of sin $$x$$ is ______.

A discontinuous real function can be expressed as

Obtain the eigen values and eigen vectors of $$A = \left[ {\matrix{ 8 & -4 \cr 2 & { 2 } \cr } } \right].$$

The real symmetric matrix $$C$$ corresponding to the quadratic form $$Q = 4{x_1}{x_2} - 5{x_2}{x_2}$$ is

If $$A$$ is a real square matrix then $$A{A^T}$$ is

In matrix algebra $$AS=AT$$ ($$A,S,T,$$ are matrices of appropriate order) implies $$S=T$$ only if

The microbial quality of treated piped water supplies is monitored by

Water distribution systems are sized to meet the

The return period for the annual maximum flood of a given magnitude is 8 years. The probability that this flood magnitude will be executed once during the next 5 years is

The cross-section of a pretensioned pre-stressed concrete beam is shown in Figure. The reinforcement is placed concentrically. If the stress in steel at transfer is $$1000$$ $$MPa,$$ compute the stress in steel immediately after transfer. The modular ratio is $$6.$$

An isolated $$T$$ beam is used as a walkway. The beam is simply supported with an effective span of $$6$$ $$m.$$ The effective width of flange, for the cross-section shown in figure, is

The cross-section of a simply supported plate girder is shown in figure. The loading on the girder is symmetrical. The bearing stiffeners at supports are the sole means of providing restraint against torsion. Design the bearing stiffeners at supports, with minimum moment of inertia about the center line of web plate only as the sole design criterion. The flat sections available are: $$250 \times 25,\,\,250 \times 32,\,\,\,200 \times 28$$ and $$200 \times 32\,\,mm.$$ Draw a sketch

A cantilever beam is shown in the figure. The moment to be applied at free end for zero vertical deflection at the point is

The stiffness matrix of a beam element is given as $${{2EI} \over L}\left[ {\matrix{ 2 & { + 1} \cr { + 1} & 2 \cr } } \right].$$ Then the flexibility matrix is

The steel portal frame shown in figure is subjected to an imposed service load of $$15$$ $$kN$$. Compute the required plastic moment capacity of the members. All the members are of the same cross-section. Draw the collapse load.

The plastic modulus of a section is $$4.8 \times {10^{ - 4}}\,\,{m^3}.$$ The shape factor is $$1.2$$. The plastic moment capacity of the section is $$120$$ $$kN$$-$$m.$$ The yield stress of the material is

The strain energy stored in member $$'AB'$$ of the pin jointed truss shown aside when $$'E'$$ and $$'A'$$ are same for all members is

A three hinged arch shown in Figure is quarter of a circle. If the vertical and horizontal components of reaction at $$A$$ are equal, the value of $$\theta $$ is

For a highway with a design speed of 100kmph, the safe OSD is assume acceleration as 0.53 $$m/sec^2$$. the road is two lane with two way traffic. User other specifications as per IRC