The inverse Laplace transform of $$1/\left( {{s^2} + 2s} \right)$$ is

The number of boundary conditions required to solve the differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} = 0\,\,$$ is

The solution for the following differential equation with boundary conditions $$y(0)=2$$ and $$\,\,{y^1}\left( 1 \right) = - 3$$ is where $${{{d^2}y} \over {d{x^2}}} = 3x - 2$$

Limit of the following series as $$x$$ approaches $${\pi \over 2}$$ is
$$f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + - - - - - $$

The product $$\left[ P \right]\,\,{\left[ Q \right]^T}$$ of the following two matrices $$\left[ P \right]\,$$ and $$\left[ Q \right]\,$$
where $$\left[ P \right]\,\, = \left[ {\matrix{ 2 & 3 \cr 4 & 5 \cr } } \right],\,\,\left[ Q \right] = \left[ {\matrix{ 4 & 8 \cr 9 & 2 \cr } } \right]$$ is

The eigen values of the matrix $$\left[ {\matrix{ 5 & 3 \cr 2 & 9 \cr } } \right]$$ are

The determinant of the following matrix $$\left[ {\matrix{ 5 & 3 & 2 \cr 1 & 2 & 6 \cr 3 & 5 & {10} \cr } } \right]$$

A 15 cm length of steel rod with relative density of 7.4 is submerged in a two layer fluid. The bottom layer is mercury and the top layer is water. The height of top surface of the rod above the liquid interface in 'cm' is

The effective spans for a simple one-way slab system, with an overhang are indicated in the figure below. The specified ultimate design loads on the slab are $$6.0\,kN/{m^2}$$ and $$4.5\,kN/{m^2}$$ for dead loads and live loads respectively. Considering the possibility of live loads not occurring simultaneously on both spans, determine the maximum spacing (in $$mm$$ units) of $$8$$ $$mm$$ diameter bars required as bottom reinforcement in the span $$AB,$$ assuming an effective depth of $$125$$ $$mm.$$ Assume $$M20$$ concrete and $$Fe$$ $$415$$ steel.

Consider the following two statements related to reinforced concrete design, and identify whether they are TRUE/FALSE:

$${\rm I}.$$ Curtailment of bars in the flexural tension zone in beams reduces the shear strength at the cut-off locations
$${\rm II}.$$ When a rectangular column section is subject to biaxially eccentric compression, the neutral axis will be parallel to the resultant axis of bending.

Identify the FALSE statement from the following, pertaining to the design of concrete structures.

The relevant cross-sectional details of a compound beam comprising a symmetric $${\rm I}$$-section and a channel section (with welded connections), proposed for a steel gantry girder, are given below (all dimensions are in $$mm$$)

$$(a)$$ Determine the depth of the centroidal axis $$\overline y $$ and the second moment of area, $${{\rm I}_{xx}}$$ and $${{\rm I}_{yy\,\,eff}}$$ of the compound section. For computing $${{\rm I}_{yy\,\,eff}}$$ include the full contribution of the channel section, but only the top flange of the $${\rm I}$$-section.

$$(b)$$ Determine the maximum compressive stress that develops at a top corner location on account of a vertical bending moment of $$550.0$$ $$kN$$-$$m,$$ combined with a horizontal bending moment of $$15.0$$ $$kN$$-$$m.$$

Consider the following two statements related to structural steel design, and identify whether they are True or FALSE.

$${\rm I}.\,\,\,\,\,\,$$ The Euler buckling load of a slender steel column depends on the yield strength of steel.
$${\rm II}.\,\,\,\,\,\,$$ In the design of laced column, the maximum spacing of the lacing does not depend on the slenderness of column as a whole.

Identify the most efficient butt joint (with double cover plates) for a plate in tension from the patterns (plan views) shown below, each comprising $$6$$ identical bolts with the same pitch and gauge.

The bending moment (in $$kN$$-$$m$$ units) at the mid-span location $$X$$ in the beam with overhangs shown below is equal

The degree of static indeterminacy, $${N_s}$$ and the degree of kinematic indeterminacy, $${N_k}$$ for the plane frame shown below, assuming axial deformations to be negligible, are given by:

The two-span continuous beam shown below is subject to a clockwise rotational slip $${\theta _A} = 0.004$$ radian at the fixed end $$A.$$ Applying the slope-deflection method of analysis, determine the slope $${\theta _B}$$ at $$B.$$ Given that the flexural rigidity $$EI = 25000\,kN$$ - $${m^2}$$ and span $$L=5$$ $$m,$$ determine the end moments (in $$kN$$-$$m$$ units ) in the two spans, and draw the bending moment diagram.

The frame below shows three beam elements $$OA, OB$$ and $$OC$$, with identical length $$L$$ and flexural rigidity $$EI$$, subject to an external moment $$M$$ applied at the rigid joint $$O.$$ The correct set of bending moments $$\left\{ {{M_{OA}},\,{M_{O{\bf{B}}}},{M_{OC}}} \right\}$$

Identify the FALSE statement from the following, pertaining to the effects due to a temperature rise $$\Delta T$$ of the bar $$BD$$ alone in the plane truss shown below :

Identify, from the following, the correct value of the bending moment $${M_A}\,\,$$ (in $$kNm$$ units) at the fixed end $$A$$ in the statically determinate beam shown below (with internal hinges at $$B$$ and $$D),$$ when a uniformly distributed load of $$10$$ $$kN/m$$ is placed on the spans. (Hint: Sketching the influence line for $${M_A}\,\,$$ or applying the Principle of Virtual Displacements makes the solution easy).

The design value of lateral friction coefficient on highway is