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 type of surveying in which the curvature of the earth is taken into account is called

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

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 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

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