The line integral $$\int {\,\,V.dr\,\,} $$ of the vector function $$V\left( r \right) = 2xyz\widehat i + {x^2}z\widehat j + {x^2}y\widehat k\,\,$$ from the origin to the point $$P(1,1,1)$$

Consider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system will be

Consider the following system of equations in three real variable $${x_1},$$ $${x_2}$$ and $${x_3}:$$ $$$2{x_1} - {x_2} + 3{x_3} = 1$$$ $$$3{x_1} + 2{x_2} + 5{x_3} = 2$$$ $$$ - {x_1} + 4{x_2} + {x_3} = 3$$$

This system of equations has

Consider the matrices $$\,{X_{4x3,}}\,\,{Y_{4x3}}$$ $$\,\,{P_{2x3}}.$$ The order of $$\,{\left[ {P{{\left( {{X^T}Y} \right)}^{ - 1}}{P^T}} \right]^T}$$ will be

Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda $$ is a scalar. Let $$\left( {{\lambda _i},\,\,{X_i}} \right)$$ be an eigen value and its corresponding eigen vector for real matrix $$A$$. Let $${{\rm I}_{nxn}}$$ be unit matrix. Which one of the following statement is not correct?

Value of the integral $$\,\,\oint {xydy - {y^2}dx,\,\,} $$ where, $$c$$ is the square cut from the first quadrant by the line $$x=1$$ and $$y=1$$ will be (Use Green's theorem to change the line integral into double integral)

Transformation to linear form by substituting $$v = {y^{1 - n}}$$ of the equation $${{dy} \over {dt}} + p\left( t \right)y = q\left( t \right){y^n},\,\,n > 0$$ will be

The solution $${{{d^2}y} \over {d{x^2}}} + 2{{dy} \over {dx}} + 17y = 0;$$ $$y\left( 0 \right) = 1,{\left( {{{d\,y} \over {d\,x}}} \right)_{x = {\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}}} = 0\,\,$$ in the range $$0 < x < {\pi \over 4}$$ is given by

Which one of the following is not true for the complex number z₁ and z₂ ?

Consider likely applicability of Cauchy's Integral theorem to evaluate the following integral counterclockwise around the unit circle C.

$$I\, = \,\oint\limits_C {\sec z\,dz} $$, z being a complex variable. The value of I will be

Given $$a>0,$$ we wish to calculate it reciprocal value $${1 \over a}$$ by using Newton - Raphson method for $$f(x)=0.$$ The Newton - Raphson algorithm for the function will be

Given $$a>0,$$ we wish to calculate its reciprocal value $${1 \over a}$$ by using Newton - Raphson method for $$f(x)=0.$$ For $$a=7$$ and starting with $${x_0} = 0.2\,\,$$ the first two iterations will be

The Laplace transform of a function $$f(t)$$ is $$$F\left( s \right) = {{5{s^2} + 23s + 6} \over {s\left( {{s^2} + 2s + 2} \right)}}$$$
As $$t \to \propto ,\,\,f\left( t \right)$$ approaches

Laplace transform of $$f\left( t \right) = \cos \left( {pt + q} \right)$$ is

1 TCU is equivalent to color produced by

If tomato juice is having a pH of 4.1, the hydrogen ion concentration will be

A rectangular column section of $$250\,\,mm \times 400\,mm$$ is reinforced with five steel bars of grade $$Fe$$ $$500,$$ each of $$20$$ $$mm$$ diameters. Concrete mix is $$M30.$$ Axial load on the column section with minimum eccentricity as per $$IS: 456$$ - $$2000$$ using limit state method can be applied upto

$$IS:$$ $$1343 – 1980$$ limits the minimum characteristic strength of pre-stressed concrete for post tensioned works and pretension work as

As per Indian standard code of practice for prestressed concrete $$(IS:1343-1980)$$ the minimum grades of concrete to be used for post-tensioned and pre-tensioned structural elements are respectively

A concrete beam of rectangular cross section of $$200\,\,mm \times 400\,\,mm$$ is pre-stressed with a force $$400$$ $$kN$$ at eccentricity $$100$$ $$mm.$$ the maximum compressive stress in the concrete is

The partial factor of safety for concrete as per IS: 456-2000 is

The flexural strength of M30 concrete as per IS: 456-2000 is

A fillet-welded joint of $$6$$ $$mm$$ size is shown in the figure. The welded surfaces meet at $$60$$ - $$90$$ degree and permissible stress in the fillet weld is $$108$$ $$MPa.$$ The safe load that can be transmitted by the joint is

The permissible stress in axial tension in steel member on the net effective area of the section shall not exceed the following value ($${{f_y}}$$ is the yield stress)

Which of the following is NOT correct for steel sections as per $$IS:$$ $$800-1984$$?

An unstiffened web $${\rm I}$$ section is fabricated from a $$10$$ $$mm$$ thick plate by fillet welding as shown in the figure. If yield stress of steel is $$250$$ $$MPa,$$ the maximum shear load that section can take is

The components of strain tensor at a point in the plane strain case can be obtained by measuring longitudinal strain in following directions.

The bending Moment diagram for a beam is given below:

The shear force at sections $$aa'$$ and $$bb'$$ respectively are of the magnitude

If a beam of rectangular cross-section is subjected to a vertical shear force $$V,$$ the shear force carried by the upper one-third of the cross-section is

A circular shaft shown in the figure is subject to torsion $$T$$ at two points $$A$$ and $$B.$$ The torsional rigidity of portions $$CA$$ and $$BD$$ is $$G{J_1}$$ and that portion $$AB$$ is $$G{J_2}$$. The rotations of shaft at points $$A$$ and $$B$$ are $${\theta _1}$$ and $${\theta _2}$$. The rotation $${\theta _1}$$ is

The maximum tensile stress at the section $$X$$ - $$X$$ shown in the figure below is

The symmetry of stress tensor at a point in the body under equilibrium is obtained from

All members of the frame shown below have the same flexural rigidity $$EI$$ and length $$L.$$ If a moment $$M$$ is applied at joint $$B,$$ the rotation of the joint is

Considering beam as axially rigid, the degree of freedom of a plane frame shown below is

Match List - $${\rm I}$$ with List - $${\rm II}$$ and select the correct answer using the codes given below the lists :

A truss is shown in the figure. Members are to equal cross section $$A$$ and same modulus of elasticity $$E.$$ $$A$$ vertical force $$P$$ is applied at point $$C.$$

Deflection of the point $$C$$ is

A truss is shown in the figure. Members are to equal cross section $$A$$ and same modulus of elasticity $$E.$$ $$A$$ vertical force $$P$$ is applied at point $$C.$$

Force in the member $$AB$$ of the truss is

A cantilever beam of length $$l,$$ width $$b$$ and depth $$d$$ is loaded with a concentrated vertical load at the tip. If yielding starts at a load $$P,$$ the collapse load shall be

Pradhan Mantri Gram Sadak Yojana (PMGSY), launched in the year 2000 aims to provide rural connectivity with all weather roads. It is proposed to connect the habitations and plain areas of populations more than 500 persons by the year