If $$\left\{ x \right\}$$ is a continuous, real valued random variable defined over the interval $$\left( { - \infty ,\,\, \pm \infty } \right)$$ and its occurrence is defined by the density function given as: $$f\left( x \right) = {1 \over {\sqrt {2\pi * b} }}{e^{ - {1 \over 2}{{\left( {{{x - a} \over b}} \right)}^2}}}$$ where $$'a'$$ and $$'b'$$ are the statistical attributes of the random variable $$\left\{ x \right\}$$. The value of the integral $$\int\limits_{ - \infty }^a {{1 \over {\sqrt {2\pi * b} }}{e^{ - {1 \over 2}{{\left( {{{x - a} \over b}} \right)}^2}}}} dx\,\,\,$$ is

$$z = {{2 - 3i} \over { - 5 + i}}$$ can be expressed as

Water is following at a steady rate through a homogeneous and saturated horizontal soil strip of $$10$$m length. The strip is being subjected to a constant water head $$(H)$$ of $$5$$m at the beginning and $$1$$m at the end. If the governing equation of flow in the soil strip is $$\,\,{{{d^2}H} \over {d{x^2}}} = 0\,\,$$ (where $$x$$ is the distance along the soil strip), the value of $$H$$ (in m) at the middle of the strip is _______.

The integrating factor for the differential equation $${{dP} \over {dt}} + {k_2}\,P = {k_1}{L_0}{e^{ - {k_1}t}}\,\,$$ is

An observer counts $$240$$veh/h at a specific highway location. Assume that the vehicle arrival at the location is Poisson distributed, the probability of having one vehicle arriving over a $$30$$-second time interval is _______.

A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes; (i) Head, (ii) Head, (III) Head, (iv) Head. The probability of obtaining a ''Tail'' when the coin is tossed again is

The expression $$\mathop {Lim}\limits_{a \to 0} \,{{{x^a} - 1} \over a}\,\,$$ is equal to

The rank of the matrix $$\left[ {\matrix{ 6 & 0 & 4 & 4 \cr { - 2} & {14} & 8 & {18} \cr {14} & { - 14} & 0 & { - 10} \cr } } \right]$$ is

The determinant of matrix $$\left[ {\matrix{ 0 & 1 & 2 & 3 \cr 1 & 0 & 3 & 0 \cr 2 & 3 & 0 & 1 \cr 3 & 0 & 1 & 2 \cr } } \right]$$ is

The survey carried out to delineate natural features, such as hills, rivers, forests and manmade features, such as towns, villages, buildings, roads, transmission lines and canals is classified as

A certain soil has the following properties: G_s = 2.71, n = 40% and w = 20%. The degree of saturation of the soil (rounded off to the nearest percent) is _________

The modulus of elasticity, E = 5000$$\sqrt{{\mathrm f}_\mathrm{ck}}$$ where f_ck is the characteristic compressive strength of concrete, specified in IS:456-2000 is based on

The target mean strength f_cm for concrete mix design obtained from the characteristic strength (f_ck) and standard deviation $$\sigma$$, as defined in IS:456-2000, is

The flexural tensile strength of M25 grade of concrete, in N/mm², as per IS:456-2000 is __________.

The axial load (in $$kN$$) in the member $$PQ$$ for the arrangement/assembly shown in the figure given below is ___________

Polar moment of inertia $$\left( {{{\rm I}_p}} \right),$$ in $$c{m^4},$$ of a rectangular section having width, $$b=2$$ $$cm$$ and depth $$d=6$$ $$cm$$ is __________

The beam of an overall depth $$250$$ $$mm$$ (shown below) is used in a building subjected to two different thermal environments. The temperatures at the top and bottom surfaces of the beam are $${36^ \circ }C$$ and $${72^ \circ }C$$ respectively. Considering coefficient of thermal expansion $$\left( \alpha \right)$$ as $$1.50 \times {10^{ - 5}}\,\,$$ per$${}^ \circ C,$$ the vertical deflection of the beam (in $$mm$$) at its mid-span due to temperature gradient is ______________

The first moment of area about the axis of bending for a beam cross-section is

The values of axial stress $$\left( \sigma \right)$$ in $$kN/{m^2},$$ bending moment $$(M)$$ in $$kNm,$$ and shear force $$(V)$$ in $$kN$$ acting at point $$P$$ for the arrangement shown in the figure are respectively.

For the state of plane stress (in $$MPa$$) shown in the figure below, the maximum shear stress (in $$MPa$$) is ___________

A prismatic beam (as shown below) has plastic moment capacity of $${{M_P}}$$, then the collapse load $$P$$ of the beam is

Considering the symmetry of a rigid frame as shown below, the magnitude of the bending moment (in $$kN$$-$$m$$) at $$P$$ (preferably using the moment distribution method) is

The tension (in $$kN$$) in a $$10$$ $$m$$ long cable, shown in the figure, neglecting its self-weight is

The static indeterminacy of the two-span continuous beam with an internal hinge, shown below, is ______________.

A road is being designed for a speed of 110 km/hr on a horizontal curve with a super elevation of 8%. If the coefficient of side friction is 0.10, the minimum radius of the curve (in m) required for safe vehicular movement is