The Newton-Raphson iteration $${x_{n + 1}} = {1 \over 2}\left( {{x_n} + {R \over {{x_n}}}} \right)$$ can be used to compute

If the interval of integration is divided into two equal intervals of width $$1.0,$$ the value of the definite integral $$\,\,\int\limits_1^3 {\log _e^x\,\,dx\,\,\,\,} $$ using simpson's one $$-$$ third rule will be

Three values of $$x$$ and $$y$$ are to be fitted in a straight line in the form $$y=a+bx$$ by the method of least squares. Given $$\,\,\,\sum x = 6,\,\,\sum y = 21,\,\,\sum {{x^2} = 14,\,\,\sum {xy} = 46,\,\,\,\,} $$ the values of $$a$$ and $$b$$ are respectively

The following system of equations $$$x+y+z=3,$$$ $$$x+2y+3z=4,$$$ $$$x+4y+kz=6$$$
will not have a unique solution for $$k$$ equal to

The eigenvalues of the matrix $$\left[ P \right] = \left[ {\matrix{ 4 & 5 \cr 2 & { - 5} \cr } } \right]$$ are

The product of matrices $${\left( {PQ} \right)^{ - 1}}P$$ is

A wastewater sample contains $$10^{-5.6}$$ mol/l of OH^- ions at $$25^o$$ C. The pH of this sample is

The base width of an elementary profile of gravity dam of height H is b. The specific gravity of the material of the dam is G and uplift pressure coefficient is K. the correct relationship for no tension at the heel is given by

An outlet irrigates an area of 20 ha. The discharge (lit/sec) required at this outlet to meet the evapotranspiration requirement of 20mm occurring uniformly in 20 days neglecting other field losses is

A pre-tensioned concrete member of section $$200\,\,mm \times 250\,\,mm$$ contains tendons of area $$500\,\,m{m^2}$$ at the centre of gravity of the section. The prestress in tendons is $$1000\,\,N/m{m^2}.$$. Assuming modular ratio as $$10,$$ the stress $$\left( {N/m{m^2}} \right)$$ in concrete is

In the design of a reinforced concrete bean the requirement for bond is not getting satisfied. The economical option to satisfy the requirement for bond is by

A reinforced concrete column contains longitudinal steel equal to $$1$$ percent of net cross-sectional area of the column. Assume modular ration as $$10.$$ the loads carried (using the elastic theory) by the longitudinal steel and the net area of concrete, are $${P_s}$$ and $${P_c}$$ respectively. The ration $${P_s}$$/$${P_c}$$ expressed as percent is

A reinforced concrete beam of rectangular cross section of breadth $$230$$ $$mm$$ and effective depth $$400$$ $$mm$$ is subjected to maximum factored shear force of $$120$$ $$kN.$$ The grades of concrete, main steel and stirrup steel are $$M20,$$ $$Fe415$$ and $$Fe250$$ respectively. For the area of main steel provided, the design shear strength τc is per $$IS: 456$$-$$2000$$ is $$0.48$$ $$N/m{m^2}.$$ The beam is designed for collapse limit state.

In addition, the beam is subjected to a torque whose factored value is $$10.90$$ $$kN$$-$$m.$$ The stirrups have to be provided to carry a shear $$(kN)$$ equal to

A reinforced concrete beam of rectangular cross section of breadth $$230$$ $$mm$$ and effective depth $$400$$ $$mm$$ is subjected to maximum factored shear force of $$120$$ $$kN.$$ The grades of concrete, main steel and stirrup steel are $$M20,$$ $$Fe415$$ and $$Fe250$$ respectively. For the area of main steel provided, the design shear strength τc is per $$IS: 456$$-$$2000$$ is $$0.48$$ $$N/m{m^2}.$$ The beam is designed for collapse limit state.

The spacing $$(mm)$$ of $$2$$-legged $$8$$ $$mm$$ stirrups to be provided is

Un-factored maximum bending moments at a section of a reinforced concrete beam resulting from a frame analysis are 50, 80, 120 and 180 kN-m under dead, live, wind and earthquake loads respectively. The design moment (kNm) as per IS: 456- 2000 for the limit state of collapse (flexure) is

A reinforced concrete structure has to be constructed along a sea coast. The minimum grade of concrete to be used as per IS: 456-2000 is

Rivets and bolts subjected to both shear stress $$\left( {{\tau _{vf}},\,cal} \right)$$ and axial tensile stress $$\left( {{\sigma _{tf,cal}}} \right)$$ shall be so proportioned that the stresses do not exceed the respective allowable stresses $${\tau _{vf,}}$$ and $${{\sigma _{tf,}}}$$ and the value of $$\left( {{{{\tau _{vf.cal}}} \over {{\tau _{vf}}}} + {{{\sigma _{tf,cal}}} \over {{\sigma _{tf}}}}} \right)$$ does not exceed

Beam $$GHI$$ is supported by these pontoons as shown in the figure below. The horizontal cross sectional area of each pontoon is $$8\,\,{m^2},$$ the flexural rigidity of the beam is $$10000$$ $$kN$$-$${m^2}$$ and the unit weight of water is $$10$$ $$kN$$/$${m^3}$$.

When the middle pontoon is removed, the deflection at $$H$$ will be

A rigid bar $$GH$$ of length $$L$$ is supported by a hinge and a spring of stiffness $$K$$ as shown in the figure below. The buckling load, $${P_{cr}}'$$ for the bar will be

Cross-section of a column consisting of two steel strips, each of thickness $$t$$ and width $$b$$ is shown in the figure below. The critical loads of the column with perfect bond and without bond between the strips are $$P$$ and $${P_0}$$ respectively. The ratio $$P/{P_0}$$ is

A mild steel specimen is under uniaxial tensile stress. Young's modulus and yield stress for mild steel are $$2 \times {10^5}\,\,MPa$$ and $$250$$ $$MPa$$ respectively. The maximum amount of strain energy per unit volume that can be stored in this specimen without permanent set is

A thin walled cylindrical pressure vessel having a radius of $$0.5$$ $$m$$ and wall thickness of $$25$$ $$mm$$ is subjected to an internal pressure of $$700$$ $$kPa$$. The hoop stress developed is

Beam $$GHI$$ is supported by these pontoons as shown in the figure below. The horizontal cross sectional area of each pontoon is $$8\,\,{m^2},$$ the flexural rigidity of the beam is $$10000$$ $$kN$$-$${m^2}$$ and the unit weight of water is $$10$$ $$kN$$/$${m^3}$$.

When the middle pontoon is brought back to its position as shown in the figure above, the reaction at $$H$$ will be

The stepped cantilever is subjected to moments, $$M$$ as shown in the figure below. The vertical deflection at the free end (neglecting the self weight) is

The maximum shear stress in a solid shaft of circular cross-section having diameter subjected to a torque $$T$$ is $$\tau $$ . If the torque is increased by four times and the diameter of the shaft is increased by two times, the maximum shear stress in the shaft will be

A continuous beam is loaded as shown in the figure below. Assuming a plastic moment capacity equal to $${{M_P}}$$, the minimum load at which the beam would collapse is

The shape of the cross-section, which has the largest shape factor, is

The members $$EJ$$ and $$IJ$$ of a steel truss shown in the figure below are subjected to a temperature rise of $${30^ \circ }C.$$ The coefficient of thermal expansion of steel is $$0.000012$$ per $${^ \circ }C.$$ per unit length. The displacement ($$mm$$) of joint $$E$$ relative to joint $$H$$ along the direction $$HE$$ of truss, is

The span$$(s)$$ to be loaded uniformly for maximum positive (upward) reaction at support $$P,$$ as shown in the figure below, is $$(are)$$

The degree of static indeterminacy of the rigid frame having two internal hinges as shown in the figure below, is