The activity details of a project are given below: The estimated minimum time (in days) for the completion of the project will be________

Consider the following second $$-$$order differential equation : $$\,y''\,\, - 4y' + 3y = 2t - 3{t^2}\,\,\,$$
The particular solution of the differential equation is

The solution of the equation $$\,{{dQ} \over {dt}} + Q = 1$$ with $$Q=0$$ at $$t=0$$ is

Consider the equation $${{du} \over {dt}} = 3{t^2} + 1$$ with $$u=0$$ at $$t=0.$$ This is numerically solved by using the forward Euler method with a step size. $$\,\Delta t = 2.$$ The absolute error in the solution at the end of the first time step is __________

Consider the following partial differential equation: $$\,\,3{{{\partial ^2}\phi } \over {\partial {x^2}}} + B{{{\partial ^2}\phi } \over {\partial x\partial y}} + 3{{{\partial ^2}\phi } \over {\partial {y^2}}} + 4\phi = 0\,\,$$ For this equation to be classified as parabolic, the value of $${B^2}$$ must be ____________.

For the function $$\,f\left( x \right) = a + bx,0 \le x \le 1,\,\,$$ to be a valid probability density function, which one of the following statements is correct?

The number of parameters in the univariate exponential and Gaussian distributions, respectively, are

$$\mathop {Lim}\limits_{x \to 0} \left( {{{\tan x} \over {{x^2} - x}}} \right)$$ is equal to _________.

Let $$x$$ be a continuous variable defined over the interval $$\left( { - \infty ,\infty } \right)$$, and $$f\left( x \right) = {e^{ - x - {e^{ - x}}}}.$$
The integral $$g\left( x \right) = \int {f\left( x \right)dx\,\,} $$ is equal to

The matrix $$P$$ is the inverse of a matrix $$Q.$$ If $${\rm I}$$ denotes the identity matrix, which one of the following options is correct?

Consider the matrix $$\left[ {\matrix{ 5 & { - 1} \cr 4 & 1 \cr } } \right].$$ Which one of the following statements is TRUE for the eigenvalues and eigenvectors of this matrix?

A particle of mass 2 kg is traveling at a velocity of 1.5 m/s. A force f(t)=3t² (in N) is applied to it in the direction of motion for a duration of 2 seconds, where t denotes time in seconds. The velocity (in m/s, up to one decimal place) of the particle immediately after the removal of the force is________

A pre-tensioned rectangular concrete beam $$150$$ $$mm$$ wide and $$300$$ $$mm$$ depth is prestressed with three straight tendones, each having a cross-sectional area of $$50$$ $$m{m^2},$$ to an initial stress of $$1200$$ $$N/m{m^2}.$$ The tendons are located at $$100$$ $$mm$$ from the soffit of the beam. If the modular ratio is $$6,$$ the loss of prestressing force (in $$kN,$$ up to one decimal place) due to the elastic deformation of concrete only is ________

According to $$IS$$ $$456$$- $$2000,$$ which one of the following statements about the depth of natural axis $${x_u},\,\,bal$$ for a balanced reinforced concrete section is correct?

A column is subjected to a load through a bracket as shown in figure

The resultant force (in $$kN,$$ up to one decimal place) in the bolt $$1$$ is _______________

Consider the stepped bar made with a linear elastic material and subjected to an axial load of $$1$$ $$kN$$, as shown in the figure

Segment $$1$$ and $$2$$ have cross-sectional area of $$100\,\,m{m^2}$$ and $$60\,\,m{m^2}$$, Young's modulus of $$2 \times {10^5}\,\,MPa$$ and $$3 \times {10^5}\,\,MPa,$$ and length of $$400$$ $$mm$$ and $$900$$ $$mm,$$ respectively. The strain energy (in $$N$$-$$mm,$$ up to one decimal place) in the bar due to the axial load is _________

A simply supported beam is subjected to a uniformly distributed load. Which one of the following statements is true?

Consider two axially loaded columns, namely. $$1$$ and $$2,$$ made of a linear elastic material with Young's modulus $$2 \times {10^5}\,\,MPa,$$ square cross-section with side $$10$$ $$mm$$, and length $$1$$ $$m.$$ For Column $$1,$$ one end is fixed and the other end is free. For column $$2,$$ one end is fixed and the other end is pinned. Based on the Euler's theory, the ratio (up to one decimal place) of the buckling load of Column $$2$$ to the buckling load of column $$1$$ is ___________

An elastic bar of length L, uniform cross sectional area A, coefficient of thermal expansion a, and Young’s modulus E is fixed at the two ends. The temperature of the bar is increased by T, resulting in an axial stress $$\sigma$$. Keeping all other parameters unchanged, if the length of the bar is doubled, the axial stress would be

The figure shows a two $$-$$hinged parabolic arch of span $$L$$ subjected to a uniformly distributed load of intensity $$q$$ per unit length.

The maximum bending moment in the arch is equal to

The value of $$M$$ in the beam $$ABC$$ shown in the figure is such that the joint $$B$$ does not rotate.

The value of support reaction (in $$kN$$) at $$B$$ should be equal to _____________

Consider the beam $$ABCD$$ shown in figure

For a moving concentrated load of $$50$$ $$kN$$ on the beam, the magnitude of the maximum bending moment (in $$kN$$-$$m$$) obtained at the support $$C$$ will be equal to _________

A planar truss tower structure is shown in the figure:
Consider the following statements about the external and internal determinacies of the truss.
(P) Externally Determinate
(Q) External Static Indeterminacy = 1
(R) External Static Indeterminacy = 2
(S) Internally Determinate
(T) Internal Static Indeterminacy = 1
(U) Internal Static Indeterminacy = 2
Which one of the following options is correct?

A super-elevation e is provided on a circular horizontal curve such that a vehicle can be stopped on the curve without sliding. Assuming a design speed v and maximum coefficient of side friction f_max, which one of the following criteria should be satisfied?