Consider the following second order linear differential equation $${{{d^2}y} \over {d{x^2}}} = - 12{x^2} + 24x - 20$$
The boundary conditions are: at $$x=0, y=5$$ and at $$x=2, y=21$$
The value of $$y$$ at $$x=1$$ is

In Newton-Raphson iterative method, the initial guess value $$\left( {{x_{ini}}} \right)$$ is considered as zero while finding the roots of the equation: $$\,f\left( x \right) = - 2 + 6x - 4{x^2} + 0.5{x^3}.\,\,\,$$ The correction, $$\Delta x,$$ to be added to $${{x_{ini}}}$$ in the first iteration is __________.

For step-size, $$\Delta x = 0.4,$$ the value of following integral using Simpson's $$1/3$$ rule is ______

Four cards are randomly selected from a pack of $$52$$ cards. If the first two cards are kings, what is the probability that the third card is a king?

The probability density function of a random variable, $$x$$ is $$$\matrix{ {f\left( x \right) = {x \over 4}\left( {4 - {x^2}} \right)} & {for\,\,0 \le x \le 2 = 0} \cr { = 0} & {otherwise} \cr } $$$
The mean, $${\mu _x}$$ of the random variable is __________.

Given $$i = \sqrt { - 1} ,$$ the value of the definite integral, $$\,{\rm I} = \int\limits_0^{\pi /2} {{{\cos x + \sin x} \over {\cos x - i\,\sin x}}dx\,\,} $$ is :

$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^{2x}}\,\,$$ is equal to

The two Eigen Values of the matrix $$\left[ {\matrix{ 2 & 1 \cr 1 & p \cr } } \right]$$ have a ratio of $$3:1$$ for $$p=2.$$ What is another value of $$'p'$$ for which the Eigen values have the same ratio of $$3:1$$?

Let $$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$ with $$n \ge 3$$ and $${{a_{ij}} = i.j.}$$
The rank of $$A$$ is :

A groundwater sample was found to contain 500 mg/L total dissolved solids (TDS). TDS (in %) present in the sample is ________.

A triangular gate with a base width of 2 m and a height of 1.5 m lies in a vertical plane. The top vertex of the gate is 1.5 m below the surface of a tank which contains oil of specific gravity 0.8. Considering the density of water and acceleration due to gravity to be 1000 $$kg/m^3$$ and 9.81 $$m/s^2$$, respectively, the hydrostatic force (in kN) exerted by the oil on the gate is __________.

A 588 $$cm^3$$ volume of moist sand weighs 1010 gm. Its dry weight is 918 gm and specific gravity of solids, G is 2.67. Assuming density of water as 1 gm/$$cm^3$$, the void ratio is __________.

If the water content of a fully saturated soil mass is 100%, the void ratio of the sample is

The average surface area of a reservoir in the month of June is 20 km². In the same month, the average rate of inflow is 10 m³/s, outflow rate is 15 m³/s, monthly rainfall is 10 cm, monthly seepage loss is 1.8 cm and the storage change is 16 million m³. The evaporation (in cm) in that month is

In a pre-stressed concrete beam section shown in the figure, the net loss is $$10\% $$ and the final prestressing force applied at $$X$$ is $$750$$ $$kN.$$ The initial fiber stresses (in $$N/m{m^2}$$ ) at the top and bottom of the beam were:

A column of size $$450\,mm \times 600\,mm$$ has unsupported length of $$3.0$$ $$m$$ and is braced against side sway in both directions. According to $$IS$$ $$456: 2000,$$ the minimum eccentricities (in $$mm$$) with respect to major and minor principal axes are:

According to the concept of Limit State Design as per IS 456: 2000, the probability of failure of a structure is __________.

A steel member ‘M’ has reversal of stress due to live loads, whereas another member ‘N’ has reversal of stress due to wind load. As per $$IS$$ $$800: 2007,$$ the maximum slenderness ratio permitted is:
$$\left( 1 \right)\,\,\,\,\,$$ Lesser for member $$‘M’$$ than that of member $$‘N’$$
$$\left( 2 \right)\,\,\,\,\,$$ More for member $$‘M’$$ than for member $$‘N’$$
$$\left( 3 \right)\,\,\,\,\,$$ Same for both the members
$$\left( 4 \right)\,\,\,\,\,$$ Not specified in the Code

Prying forces are

A steel strip of length, $$L = 200$$ $$mm$$ is fixed at end $$A$$ and rests at $$B$$ on a vertical spring of stiffness, $$k = 2$$ $$N/mm.$$ The steel strip is $$5$$ $$mm$$ wide and $$10$$ $$mm$$ thick. $$A$$ vertical load, $$P = 50$$ $$N$$ is applied at $$B,$$ as shown in the figure. Considering $$E = 200$$ $$GPa,$$ the force (in $$N$$) developed in the spring is _______________

In a system, two connected rigid bars $$AC$$ and $$BC$$ are of identical length $$L$$ with pin supports at $$A$$ and $$B.$$ The bars are interconnected at $$C$$ by a frictionless hinge. The rotation of the hinge is restrained by a rotational spring of stiffness, $$k.$$ The system initially assumes a straight line configuration, $$ACB.$$ Assuming both the bars as weightless, the rotation at supports, $$A$$ and $$B,$$ due to a transverse load, $$P$$ applied at $$C$$ is:

A simply supported reinforced concrete beam of length $$10$$ $$m$$ sags while undergoing shrinkage. Assuming a uniform curvature of $$0.004\,\,{m^{ - 1}}$$ along the span, the maximum deflection (in $$m$$) of the beam at mid-span is ________

For the plane stress situation shown in the figure, the maximum shear stress and the plane on which it acts are:

A guided support as shown in the figure below is represented by three springs (horizontal, vertical and rotational) with stiffness $${k_x},\,\,{k_y}$$ and $${k_\theta }$$ respectively. The limiting values of $${k_x},\,\,{k_y}$$ and $${k_\theta }$$ are :

A fixed end beam is subjected to a load, W at $$1/3rd$$ span from the left support as shown in the figure. The collapse load of the beam is

A horizontal beam $$ABC$$ is loaded as shown in the figure below. The distance of the point of contraflexure from end $$A$$ (in $$m$$) is

A simply supported beam $$AB$$ of span, $$L = 24$$ $$m$$ is subjected to two wheel loads acting at a distance, $$d = 5$$ $$m$$ apart as shown in the figure below. Each wheel transmits a load, $$P = 3$$ $$kN$$ and may occupy any position along the beam. If the beam is an $$I$$-section having section modulus, $$S = 16.2\,c{m^3},$$ the maximum bending stress (in $$GPa$$) due to the wheel loads is __________________

A superspeedway in New Delhi has among the highest super-elevation rates of any track on the Indian Grand Prix circuit. The track requires drivers to negotiate turns with a radius of 335 m and 33° banking. Given this information, the coefficient of side friction required in order to allow a vehicle to travel at 320 km/h along the curve is