Consider the following two statements.

$$(I)$$ The maximum number of linearly independent column vectors of a matrix $$A$$ is called the rank of $$A.$$

$$(II)$$ If $$A$$ is $$nxn$$ square matrix then it will be non-singular if rank of $$A=n$$

If $$A,B,C$$ are square matrices of the same order then $${\left( {ABC} \right)^{ - 1}}$$ is equal be

The Taylor series expansion of sin $$x$$ about $$x = {\pi \over 6}$$ is given by

Limit of the function
$$f\left( x \right) = {{1 - {a^4}} \over {{x^4}}}\,\,as\,\,x \to \infty $$ is given by

If $$f\left( {x,y,z} \right) = $$
$${\left( {{x^2} + {y^2} + {z^2}} \right)^{{\raise0.5ex\hbox{$\scriptstyle { - 1}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}},$$ $${{{\partial ^2}f} \over {\partial {x^2}}} + {{{\partial ^2}f} \over {\partial {y^2}}} + {{{\partial ^2}f} \over {\partial {z^2}}}$$ is equal to _______.

Consider the following integral $$\mathop {Lim}\limits_{x \to 0} \int\limits_1^a {{x^{ - 4}}} dx$$ ________.

Let $$F\left( s \right) = L\left[ {f\left( t \right)} \right]$$ denote the Laplace transform of the function $$f(t)$$. Which of the following statements is correct?

Two Samples of Water A and B have pH values of a 4.4 and 6.4 respectively. How many times more acidic sample A is than sample B

Cavitation is caused by

A soil sample has a void ratio of 0.5 and its porosity will be closed to

A borrow pit soil has a dry density of 17 kN/m³. How many cubic meters of this soils will be required to construct an embankment of 100 m³ volume with a dry density of 16 kN/m³.

If duty (D) is 1428 hectares/cumec and base period (B) is 120 days for an irrigated crop, then delta ($$\Delta$$) in meters is given by

A tube well having a capacity of 4 $$m^3 /hour$$ operates for 20 hours each day during the irrigation season. How much area can be commanded if the irrigation interval is 20 days and depth of irrigation is 7 cm?

The working stress method of design specifies the value of modular ratio, $$m = 280/\left( {3{\sigma _{cbc}}} \right),$$ where $${{\sigma _{cbc}}}$$ is the allowable stress in bending compression in concrete. To what extent does the above value of $$'m'$$ make any allowance for the creep of concrete?

The following two statements are made with reference to a simply supported under - reinforced $$RCC$$ beam:

$${\rm I}.\,\,\,\,\,\,\,\,\,\,\,\,$$ Failure takes place by crushing of concrete before the steel has yielded.
$${\rm II}.\,\,\,\,\,\,\,\,\,\,$$ The neutral axis moves up as the load is increased.

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

Assume straight line instead of parabola for stress-strain curve of concrete as given below and partial factor of safety as $$1.0.$$

A rectangular under-reinforced concrete section of $$300$$ $$mm$$ width and $$500$$ $$mm$$ effective depth is reinforced with $$3$$ bars of grade $$Fe$$-$$415,$$ each of $$16$$ $$mm$$ diameter. Concrete mix is $$M20.$$

The depth of the neutral axis from the compression fibre is

Assume straight line instead of parabola for stress-strain curve of concrete as given below and partial factor of safety as $$1.0.$$

A rectangular under-reinforced concrete section of $$300$$ $$mm$$ width and $$500$$ $$mm$$ effective depth is reinforced with $$3$$ bars of grade $$Fe$$-$$415,$$ each of $$16$$ $$mm$$ diameter. Concrete mix is $$M20.$$

The depth of the neutral axis obtained as per $$IS:$$ $$456$$-$$2000$$ differs from the depth of neutral axis obtained in $$Q.08$$ by

A continuous beam $$250\,mm \times 450\,mm$$ carries $$6$$ numbers of $$12$$ $$mm$$ diameter longitudinal bars as shown. The factored shear force at the point of inflection is $$200$$ $$kN.$$ Check if the beam is safe in bond. Assume $$M15$$ mix with $${\sigma _{ck}} = 15\,\,N/m{m^2}$$ and mild steel with $${\sigma _y} = 250\,N/m{m^2}.$$ A clear cover of $$25$$ $$mm$$ can be assumed. The design bond stress for mild steel bars in $$M15$$ concrete is specified to be $$1.0\,N/m{m^2}.$$

A simply supported rectangular prestressed concrete beam $$200$$ $$mm$$ wide and $$400$$ $$mm$$ deep has an effective span of $$12$$ $$m.$$ The prestressing cable has a triangular profile with zero eccentricity at ends and $$70$$ $$mm$$ at the midspan as shown in the figure below. The effective prestress is $$800$$ $$kN$$ after all losses. Determine the value of point load, the beam can support at the midspan if the pressure line passes through the upper kern of the section. The weight density of the material of the beam can be taken to be $$25$$ $$kN/{m^3}$$

A $$12$$ $$mm$$ bracket plate is connected to a column flange as shown in the figure below. The bracket transmits a load of $$W = 200\,\,kN$$ to the column flange. $$A$$ $$10$$ $$mm$$ fillet weld is provided along $$AB,$$ $$BC$$ and $$CD.$$ If $$e=350$$ $$mm,$$ $$b=200$$ $$mm$$ and $$d=600$$ $$mm,$$ Verify if the size of the weld provided is adequate. Allowable shearing stress in the fillet weld can be taken to be $$108$$ $$MPa.$$

A two span beam with an internal hinge is shown below.

Conjugate beam corresponding to this beam is

For the structure shown below, the vertical deflection at point $$A$$ is given by

Four column of the same material having identical geometric properties are supported in different ways as shown below

It is required to order these four beams in the increasing order of their respective first buckling loads. The correct order is given by

A frame $$ABCD$$ is supported by a roller at $$A$$ and is on a hinge at $$C$$ as shown below:

The reaction at the roller end $$A$$ is given by

Pick the incorrect statement from the following four statements:

The stress-strain diagram for two materials A and B is shown below: The following statements are made based on this diagram:
I. Material A is more brittle than material B.
II. The ultimate strength of material B is more than that of A.
With reference to the above statements, which of the following applies?

The four cross sections shown below are required to be ordered in the increasing order of their respective shape factors.

The following two statements are made with reference to the planar truss shown below :

$${\rm I}.$$ The truss is statically determinate
$${\rm II}.$$ The truss is kinematically determinate

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

Identify the FALSE statement from the following, pertaining to the methods of structural analysis.

A simply supported beam with an overhang is traversed by a unit concentrated moment from left to the right as shown below:

The influence line for reaction at $$B$$ is given by

For the structure shown below, the vertical deflection at point $$A$$ is given by

A frame $$ABCD$$ is supported by a roller at $$A$$ and is on a hinge at $$C$$ as shown below. The reaction at the roller end $$A$$ is given by

Identify the correct deflection diagram corresponding to the loading in the plane frame shown below :

A cantilever beam of length $$L$$ and a cross section with shape factor $$'f'$$ supports a concentrated load $$P$$ as shown below:

The length $${L_P}$$ of the plastic zone, when the maximum bending moment, equals the plastic moment $${M_P}$$, given by

The ruling minimum radius of horizontal curve of a national highway in plain terrain for a ruling design speed of 100 km/hour with e = 0.07 and f = 0.15 is close to

Width of carriage way for a single lane is recommended to be

Design rate of super elevation for horizontal highway curve of radius 450 m for a mixed traffic condition, having a speed of 125 km/hour is