If $$F\left( s \right) = L\left\{ {f\left( t \right)} \right\} = {{2\left( {s + 1} \right)} \over {{s^2} + 4s + 7}}$$ then the initial and final values of $$f(t)$$ are respectively

For an analytic function $$f\left( {x + i\,y} \right) = u\left( {x,y} \right) + i\,v\left( {x,y} \right),\,u$$ is given by $$u = 3{x^2} - 3{y^2}.$$ The expression for $$v,$$ considering $$k$$ is to be constant is

Given two continuous time signals $$x\left( t \right) = {e^{ - t}}$$ and $$y\left( t \right) = {e^{ - 2t}}$$ which exists for $$t>0$$ then the convolution $$z\left( t \right) = x\left( t \right) * y\left( t \right)$$ is ____________.

The square root of a number $$N$$ is to be obtained by applying the Newton $$-$$ Raphson iteration to the equation $$\,{x^2} - N = 0.\,\,$$ If $$i$$ denotes the iteration index, the correct iterative scheme will be

The solution of the differential equation $${{dy} \over {dx}} + {y \over x} = x$$ with the condition that $$y=1$$ at $$x=1$$ is

There are two containers with one containing $$4$$ red and $$3$$ green balls and the other containing $$3$$ blue balls and $$4$$ green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be ___________.

If $$\overrightarrow a $$ and $$\overrightarrow b $$ are two arbitrary vectors with magnitudes $$a$$ and $$b$$ respectively, $${\left| {\overrightarrow a \times \overrightarrow b } \right|^2}$$ will be equal to

What should be the value of $$\lambda $$ such that the function defined below is continuous at $$x = {\pi \over 2}$$?
$$f\left( x \right) = \left\{ {\matrix{ {{{\lambda \,\cos x} \over {{\pi \over 2} - x}},} & {if\,\,x \ne {\pi \over 2}} \cr {1\,\,\,\,\,\,\,\,\,\,,} & {if\,\,x = {\pi \over 2}} \cr } } \right.$$

What is the value of the definite integral? $$\,\,\int\limits_0^a {{{\sqrt x } \over {\sqrt x + \sqrt {a - x} }}dx\,\,} $$?

For a body completely submerged in a fluid, the centre of gravity (G) and centre of Buoyancy (O) are known. The body is considered to be in stable equilibrium is

Consider a bar of diameter $$'D'$$ embedded in a large concrete block as shown in the adjoining figure, with a pull out force $$P$$ being applied. Let $${\sigma _b}$$ and $${\sigma _{st}},$$ be the bond strength (between the bar and concrete) and the tensile strength of the bar, respectively. If the block is held in position and it is assumed that the material of the block does not fail, which of the following options represents the maximum value of $$P?$$

Consider two $$RCC$$ beams, $$P$$ and $$Q$$, each having the section $$400$$ $$mm$$ $$ \times \,\,750\,\,mm$$ (effective depth, $$d=750$$ $$mm$$) made with concrete having a $${\tau _{c\max }} = 2.1\,\,N/m{m^2}.$$ For the reinforcement provided and the grade of concrete used, it may be assumed that the $$\,{\tau _c} = 0.75\,\,N/m{m^2}.$$ The design shear in beam $$P$$ is $$400$$ $$kN$$ and in beam $$Q$$ is $$750$$ $$kN.$$ Considering the provisions of $$IS$$ $$456$$-$$2000.$$

Which of the following statements is TRUE?

The cross-section of a thermo-mechanically treated (TMT) reinforcing bar has

Consider a reinforcing bar embedded in concrete. In a marine environment this bar undergoes uniform corrosion, which leads to deposition of corrosion products on its surface and an increase in apparent volume of the bar. This subjects the surrounding concrete to expansive pressure. As a result, corrosion cracks appear at surface of concrete. Which of the following statement is true?

The adjoining figure shows a schematic representation of a steel plate girder to be used as a simply supported beam with a concentrated load. For stiffeners, $$PQ$$ (running along the beam axis) and $$RS$$ (running between the top and bottom flanges) which of the following pairs of statements will be TRUE?

For the fillet weld of size $$'S'$$ shown in the adjoining figure the effective throat thickness is

A disc of radius $$'r'$$ has a hole of radius $$'r/2'$$ cut-out as shown. The centroid of the remaining disc (shaded portion) at a radial distance from the center $$''O''$$ is

Consider a simply supported beam with a uniformly distributed load having a neutral axis $$(NA)$$ as shown. For points $$P$$ (on the neutral axis) and $$Q$$ (at the bottom of the beam) the state of stress is best represented by which of the following pairs? (elements $$P$$ & $$Q$$ are taken slightly right of mid span).

For the cantilever bracket, $$PQRS,$$ loaded as shown in the adjoining figure $$(PQ=RS=L,$$ and $$QR=2L),$$ which of the following statements is FALSE ?

The value of $$W$$ that results in the collapses of the beam shown in the adjoining figure and having a plastic moment capacity of $${M_P}$$ is