The table below gives values of a function $$f(x)$$ obtained for values of $$x$$ at intervals of $$0.25$$

The value of the integral of the function between the limits $$0$$ to $$1,$$ using Simpson's rule is

The modulus of the complex number $${{3 + 4\,i} \over {1 - 2\,i}}$$ is

The partial differential equation that can be formed from $$z=ax+by+ab$$ has the form $$\,\,\left( {p = {{\partial z} \over {\partial x}},q = {{\partial z} \over {\partial y}}} \right)\,\,$$

The solution to the ordinary differential equation $${{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} - 6y = 0\,\,\,$$ is

The order and degree of a differential equation $${{{d^3}y} \over {d{x^3}}} + 4\sqrt {{{\left( {{{dy} \over {dx}}} \right)}^3} + {y^2}} = 0$$ are respectively

Two coins are simultaneously tossed. The probability of two heads simultaneously appearing

Given a function $$f\left( {x,y} \right) = 4{x^2} + 6{y^2} - 8x - 4y + 8,$$ the optimal values of $$f(x,y)$$ is

The $$\mathop {Lim}\limits_{x \to 0} {{\sin \left( {{2 \over 3}x} \right)} \over x}\,\,\,$$ is

A parabolic cable is held between two supports at the same level. The horizontal span between the supports is $$L.$$ The sag at the mid-span is $$h.$$ The equation of the parabola is $$y = 4h{{{x^2}} \over {{L^2}}},\,\,$$ where $$x$$ is the horizontal coordinate and $$y$$ is the vertical coordinate with the origin at the centre of the cable. The expanssion for the total length of the cable is

The inverse of the matrix $$\left[ {\matrix{ {3 + 2i} & i \cr { - i} & {3 - 2i} \cr } } \right]$$ is

Ion concentrations obtained for a groundwater sample (having pH=8.1) are given below.

Ion	$$Ca^{2+}$$	$$Mg^{2+}$$	$$Na^+$$	$$HCO_3^-$$	$$SO_4^{2-}$$	$$Cl^{-}$$
Ionconcentration(mg/L)	100	6	15	250	45	39
Atomic Weight	Ca=40	Mg=24	Na=23	H=1,C=12,O=16	S=32,O=16	Cl=35.5

Carbonate hardness (mg/L as $$CaCO_3$$) present in the above water sample is

Ion concentrations obtained for a groundwater sample (having pH=8.1) are given below.

Ion	$$Ca^{2+}$$	$$Mg^{2+}$$	$$Na^+$$	$$HCO_3^-$$	$$SO_4^{2-}$$	$$Cl^{-}$$
Ionconcentration(mg/L)	100	6	15	250	45	39
Atomic Weight	Ca=40	Mg=24	Na=23	H=1,C=12,O=16	S=32,O=16	Cl=35.5

Total hardness (mg/L as $$CaCO_3$$) present in the above water sample is

The moisture holding capacity of the soil in a 100 hectare farm is 18 cm/m. the field is to be irrigated when 50 percent of the available moisture in the root zone is depleted. The irrigation water is to be supplied by a pump working for 10hours a day, and water application efficiency is 75%. Details of crops planned for cultivation are as follows:

Crop	Root zone depth (m)	Peak rate of moisture use(mm/day)
X	1.0	5.0
Y	0.8	4.0

The area of crop Y that can be irrigated when the available capacity of irrigation system is 40 liters/sec is

The moisture holding capacity of the soil in a 100 hectare farm is 18cm/m. the field is to be irrigated when 50 percent of the available moisture in the root zone is depleted. The irrigation water is to be supplied by a pump working for 10hours a day, and water application efficiency is 75%. Details of crops planned for cultivation are as follows:

Crop	Root zone depth (m)	Peak rate of moisture use(mm/day)
X	1.0	5.0
Y	0.8	4.0

The capacity of irrigation system required to irrigate crop X in 36 hectares is

A doubly reinforced rectangular concrete beam has a width of $$300$$ $$mm$$ and an effective depth of $$500$$ $$mm.$$ the beam is reinforced with $$2200\,m{m^2}$$ of steel in tension and $$628\,m{m^2}$$ of steel in compression. The effective cover for compression steel is $$50$$ $$mm$$. Assume that both tension and compression steel yield. The grades of concrete and steel used are $$M20$$ and $$Fe250$$ respectively. The stress lock parameters (rounded off to first two decimal places) for concrete shall be as per IS $$456:200.$$

The moment of resistance of the section is

A doubly reinforced rectangular concrete beam has a width of $$300$$ $$mm$$ and an effective depth of $$500$$ $$mm.$$ the beam is reinforced with $$2200\,m{m^2}$$ of steel in tension and $$628\,m{m^2}$$ of steel in compression. The effective cover for compression steel is $$50$$ $$mm$$. Assume that both tension and compression steel yield. The grades of concrete and steel used are $$M20$$ and $$Fe250$$ respectively. The stress lock parameters (rounded off to first two decimal places) for concrete shall be as per IS $$456:200.$$

The depth of neutral axis is

Two plates, subjected to direct tension, each of $$10$$ $$mm$$ thickness and having widths of $$100$$ $$mm$$ and $$175$$ $$mm,$$ respectively are to be fillet welded with an overlap of $$200$$ $$mm.$$ Given that the permissible weld stress is $$110$$ $$MPa$$ and the permissible stress in steel is $$150$$ $$MPa,$$ then length of the weld required using the maximum permissible weld size as per $$IS:$$ $$800$$ -$$1984$$ is

A double cover butt riveted joint is used to connect two flat plates of 200 mm width and 14 mm thickness as show in the figure. There are twelve power driven rivets of 20 mm diameter at a pitch of 50 mm in both directions on either side of the plate. Two cover plates of 10 mm thickness are used. The capacity of the joint in tension considering bearing and shear ONLY, with permissible bearing and shear stresses as 300MPa respectively is

The effective length of a column of length $$L$$ fixed against rotation and translation at one end is

In the cantilever beam $$PQR$$ shown in figure below, the segment $$PQ$$ has flexural $$EI$$ and the segment $$QR$$ has infinite flexural rigidity

The deflection of the beam at $$' R '$$ is

In the cantilever beam $$PQR$$ shown in figure below, the segment $$PQ$$ has flexural $$EI$$ and the segment $$QR$$ has infinite flexural rigidity

The deflection and slope of the beam at $$'Q'$$ are respectively

A solid circular shaft of diameter d and length $$L$$ is fixed at one end and free at the other end. A torque t is applied at the free end. The shear modulus of the material is $$G.$$ The angle of twist at three free ends is

For the simply supported beam of length $$L,$$ subjected to a uniformly distributed moment $$(M)$$ $$kN$$-$$m$$ per unit length as shown in the figure, the bending moment (in $$kN$$-$$m$$) at the mid-span of the beam is

Two people weighing W each are sitting on a plank of length $$L$$ floating on water at $$L/4$$ from either end. Neglecting the weight of the plank, the bending moment at the center of the plank is

The major and minor principal stresses at a point are $$3MPa$$ and $$-3MPa$$ respectively. The maximum shear stress at the point is

The number of independent elastic constants for a linear elastic isotropic and homogeneous material is

For the truss shown in the figure, the force in the member QR is

A three hinged parabolic arch having a span of $$20$$ $$m$$ and a rise of $$5$$ $$m$$ carries a point load of $$10$$ $$kN$$ at quarter span from the left end as shown in the figure. The resultant reaction at the left support and its inclination with the horizontal are respectively