A hydraulic structure has four gates which operate independently. The probability of failure of each gate is $$0.2.$$ Given that gate $$1$$ has failed, the probability that both gates $$2$$ and $$3$$ will fail is

Real matrices $$\,\,{\left[ A \right]_{3x1,}}$$ $$\,\,{\left[ B \right]_{3x3,}}$$ $$\,\,{\left[ C \right]_{3x5,}}$$ $$\,\,{\left[ D \right]_{5x3,}}$$ $$\,\,{\left[ E \right]_{5x5,}}$$ $$\,\,{\left[ F \right]_{5x1,}}$$ are given. Matrices $$\left[ B \right]$$ and $$\left[ E \right]$$ are symmetric. Following statements are made with respect to their matrices.
$$(I)$$ Matrix product $$\,\,{\left[ F \right]^T}\,\,$$ $$\,\,{\left[ C \right]^T}\,\,$$ $$\,\,\left[ B \right]\,\,$$ $$\,\,\left[ C \right]\,\,$$ $$\,\,\left[ F \right]\,\,$$ is a scalar.
$$(II)$$ Matrix product $$\,\,{\left[ D \right]^T}\,\,$$ $$\,\left[ F \right]\,\,$$ $$\,\left[ D \right]\,\,$$ is always symmetric.
With reference to above statements which of the following applies?

The eigen values of the matrix $$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$ are

The function $$f\left( x \right) = 2{x^3} - 3{x^2} - 36x + 2\,\,\,$$ has its maxima at

The value of the function, $$f\left( x \right) = \mathop {Lim}\limits_{x \to 0} {{{x^3} + {x^2}} \over {2{x^3} - 7{x^2}}}\,\,\,$$ is

Biotransformation of an organic compound having concentration $$(x)$$ can be modeled using an ordinary differential equation $$\,{{d\,x} \over {dt}} + k\,{x^2} = 0,$$ where $$k$$ is the reaction rate constant. If $$x=a$$ at $$t=0$$ then solution of the equation is

A delayed unit step function is defined as $$$u\left( {t - a} \right) = \left\{ {\matrix{ {0,} & {t < a} \cr {1,} & {t \ge a} \cr } } \right.$$$

Its Laplace transform is ____________.

The present population of a community is 28000 with an average water consumption of 4200 $$m^3/d$$. The existing water treatment plant has a design capacity of 6000 $$m^3/d$$. It is expected that the population will increase to 44000 during the next 20 years. The number of years from now when the plant will reach it's design capacity, assuming an arithmetic rate of population growth, will be

Most of the turbidity meters work on the scattering principle. The turbidity value obtained is expressed in

The ratio of saturated unit weight to dry unit weight of a soil is 1.25. If the specific gravity of solids (G_s) is 2.65, the void ratio of the soil is

A sprinkler irrigation system is suitable when

In the limit state design method of concrete structures, the recommended partial material safety factor $$\left( {{\gamma _m}} \right)$$ for steel according to $$IS:456$$-$$2000$$ is

At the limit state of collapse, an $$RC$$ beam is subjected to flexural moment $$200$$ $$kN$$-$$m$$, shear force $$20$$ $$kN$$ and torque $$9$$ $$kN$$-$$m$$. The beam is $$300$$ $$mm$$ wide and has a gross depth of $$425$$ $$mm,$$ with an effective cover of $$25$$ $$mm$$. The equivalent nominal shear stress $$\left( {{\tau _{ve}}} \right)$$ as calculated by using the design code turns out to the lesser than the design shear strength $$\left( {{\tau _c}} \right)$$ of the concrete.

The equivalent shear force $$\left( {{V_e}} \right)$$ is

At the limit state of collapse, an $$RC$$ beam is subjected to flexural moment $$200$$ $$kN$$-$$m$$, shear force $$20$$ $$kN$$ and torque $$9$$ $$kN$$-$$m$$. The beam is $$300$$ $$mm$$ wide and has a gross depth of $$425$$ $$mm,$$ with an effective cover of $$25$$ $$mm$$. The equivalent nominal shear stress $$\left( {{\tau _{ve}}} \right)$$ as calculated by using the design code turns out to the lesser than the design shear strength $$\left( {{\tau _c}} \right)$$ of the concrete.

The equivalent flexural moment $${M_{{e_1}}}$$ for designing the longitudinal tension steel is

An $$RC$$ short column with $$300\,\,mm \times 300\,mm$$ square cross-section is made of $$M20$$ grade concrete and has $$4$$ numbers, $$20$$ $$mm$$ diameter longitudinal bars of $$Fe$$-$$415$$ steel. It is under the action of a concentric axial compressive load. Ignoring the reduction in the area of concrete due to steel bars, the ultimate axial load carrying capacity of the column is

An $$RC$$ square footing of side length $$2$$ $$m$$ and uniform effective depth $$200$$ $$mm$$ is provided for a $$300$$ $$mm \times 300\,mm$$ column. The line of action of the vertical compressive load passes through the centroid of the footing as well as of the column. If the magnitude of the load is $$320$$ $$kN,$$ the nominal transverse (one way) shear stress in the footing is

A simply supported prestressed concrete beam is $$6$$ $$m$$ long and $$300$$ $$mm$$ wide. Its gross depth is $$600$$ $$mm.$$ It is prestressed by horizontal cable tendons at a uniform eccentricity of $$100$$ $$mm.$$ The prestressing tensile force in the cable tendons is $$1000$$ $$kN$$. Neglect the self weight of beam. The maximum normal compressive stress in the beam at transfer is

Two equal angles $$ISA$$ $$100\,\,mm \times 100\,\,mm$$ of thickness $$10$$ $$mm$$ are placed back-to-back and connected to the either side of a gusset plate through a single row of $$16$$ $$mm$$ diameter rivets in double shear. The effective areas of the connected and unconnected legs of each of these angles are $$775$$ $$m{m^2}$$ and $$950$$ $$m{m^2},$$ respectively. If these angles are NOT Tack riveted, the net effective area of this pair of angle is

A moment $$M$$ of magnitude $$50$$ $$kN$$-$$m$$ is transmitted to a column flange through a bracket by using four $$20$$ $$mm$$ diameter rivets as shown in the figure. The shear force induced in the rivet $$A$$ is

A strut in a steel truss is composed of two equal angles $$ISA$$ $$150mm \times 150mm$$ of thickness $$100$$ $$mm$$ connected back-to back to the same side of a gusset plate. The cross sectional area of each angle is $$2921$$ $$m{m^2}$$ and moment of inertia $$\left( {{{\rm I}_{XX}} = {{\rm I}_{YY}}} \right)$$ is $$6335000$$ $$m{m^4}.$$ The distance of the centroid of the angle from its surface $$\left( {{C_x} = {C_y}} \right)$$ is $$40.8$$ $$mm.$$ The minimum radius of gyration of the strut is

In a plate girder, the web plate is connected to the flange plates by fillet welding. The size of the fillet welds is designed to safely resist

Rivet value is defined as

For the linear elastic beam shown in the figure, the flexural rigidity, $$EI$$ is $$781250\,\,kN$$ - $${m^2}.$$ When $$w=10$$ $$kN/m,$$ the vertical reaction $${R_A}$$ at $$A$$ is $$50$$ $$kN.$$ The value of $${R_A}$$ for $$w = 100\,kN/m$$ is

$${U_1}$$ and $${U_2}$$ are the strain energies stored in a prismatic bar due to axial tensile forces $${P_1}$$ and $${P_2},$$ respectively. The strain energy $$U$$ stored in the same bar due to combined action of $${P_1}$$ and $${P_2}$$ will be

If principle stresses in two-dimensional case are $$\left( - \right)\,\,10\,MPa$$ and $$20$$ $$MPa$$ respectively, then maximum shear stress at the point is

In a two dimensional analysis, the state of stress at a point is shown below.

If $$\sigma = 120\,\,MPa$$ and $${\tau _{xy}} = 70\,\,MPa,\,\,{\sigma _x}$$ and $${\sigma _y},$$ are respectively

A three-span continuous beam has an internal hinge at $$B.$$ Section $$B$$ is at the mid-span of $$A.C.$$ Section $$E$$ is at the mid-span of $$CG$$. The $$20$$ $$kN$$ load is applied at section $$B$$ whereas $$10$$ $$kN$$ loads are applied at sections $$D$$ and $$F$$ as shown in the figure. Span $$GH$$ is subjected to uniformly distributed load of magnitude $$5$$ $$kN/m$$. For the loading shown, shear force immediate to the right of section $$E$$ is $$9.84$$ $$kN$$ upwards and the sagging moment at section $$E$$ is $$10.31$$ $$kN$$-$$m.$$

The magnitude of the shear force immediate to the left and immediate to the right of section $$B$$ are, respectively

A three-span continuous beam has an internal hinge at $$B.$$ Section $$B$$ is at the mid-span of $$A.C.$$ Section $$E$$ is at the mid-span of $$CG$$. The $$20$$ $$kN$$ load is applied at section $$B$$ whereas $$10$$ $$kN$$ loads are applied at sections $$D$$ and $$F$$ as shown in the figure. Span $$GH$$ is subjected to uniformly distributed load of magnitude $$5$$ $$kN/m$$. For the loading shown, shear force immediate to the right of section $$E$$ is $$9.84$$ $$kN$$ upwards and the sagging moment at section $$E$$ is $$10.31$$ $$kN$$-$$m.$$

The vertical reaction at support $$H$$ is

A homogeneous, simply supported prismatic beam of width $$B,$$ depth $$D$$ and span $$L$$ is subjected to a concentrated load of magnitude $$P.$$ The load can be placed anywhere along the span of the beams. The maximum flexural stress developed in beam is

A circular solid shaft of span $$L=5m$$ is fixed at one end and free at the other end. A twisting moment $$T=100$$ $$kN$$-$$m$$ is applied at the free end. The torsional rigidity $$GJ$$ per unit angular twist is $$50000$$ $$kN$$-$${m^2}/rad.$$ Following statements are made for this shaft:

$$1.$$ The maximum rotation is $$0.01$$ radius
$$2.$$ The torsional strain energy is $$1$$ $$kN$$-$$m.$$

With reference to the above statements, which of the following applies?

For the plane frame with an overhang as shown below, assuming negligible axial deformation, the degree of static indeterminacy, $$d,$$ and the degree of kinematic indeterminacy, $$k,$$ are

A homogeneous, simply supported prismatic beam of width $$B,$$ depth $$D$$ and span $$L$$ is subjected to a concentrated load of magnitude $$P.$$ The load can be placed anywhere along the span of the beam. The maximum flexural stress developed in beam is

A three hinged parabolic arch $$ABC$$ has a span of $$20$$ $$m$$ and a central rise of $$4$$ $$m.$$ The arch has hinges at the ends and at the center. A train of two point loads of $$20$$ $$kN$$ and $$10$$ $$kN$$, $$5$$ $$m$$ apart, crosses this arch from left to right, with $$20$$ $$kN$$ load leading. The maximum thrust induced at the supports is

The unit load method used in Structural analysis is

For a linear elastic structural system, minimization of potential energy yields

The plane frame below is analyzed by neglecting axial deformations. Following statements are made with respect to the analysis :

$$1.$$$$\,\,\,\,\,\,\,$$ Column $$AB$$ carries axial force only.
$$2.$$$$\,\,\,\,\,\,\,$$ Vertical deflection at the center of beam $$BC$$ is $$1mm.$$

Which of the following applies?

For the plane truss shown in the figure, the number of zero force members for the given loading is

For linear elastic frame, if stiffness matrix is doubled with respect to the existing stiffness matrix, the deflection of the resulting frame will be

The star and grid pattern of road networks are adopted in