Question 1 |

Consider the following expression:

z=\sin(y+it)+\cos(y-it)

where z, y, and t are variables, and i=\sqrt{-1} is a complex number. The partial differential equation derived from the above expression is

z=\sin(y+it)+\cos(y-it)

where z, y, and t are variables, and i=\sqrt{-1} is a complex number. The partial differential equation derived from the above expression is

\frac{\partial^2 z}{\partial t^2}+\frac{\partial^2 z}{\partial y^2}=0 | |

\frac{\partial^2 z}{\partial t^2}-\frac{\partial^2 z}{\partial y^2}=0 | |

\frac{\partial z}{\partial t}-i\frac{\partial z}{\partial y}=0 | |

\frac{\partial z}{\partial t}+i\frac{\partial z}{\partial y}=0 |

Question 1 Explanation:

\begin{aligned}
z&=\sin(y+it)+ \cos (y-it)\\
\frac{\partial z}{\partial y}&=\cos (y+it)-\sin (y-it)\\
\frac{\partial ^2 z}{\partial ^2 y^2}&=-\sin(y+it)- \cos (y-it)\\
\frac{\partial ^2 z}{\partial ^2 y^2}&=-z \;\;...(i)\\
\frac{\partial z}{\partial t}&=i \cos (y+it)+i\sin (y-it)\\
\frac{\partial ^2 z}{\partial ^2 t^2}&=+\sin(y+it)+ \cos (y-it)\\
\frac{\partial ^2 z}{\partial ^2 t^2}&=z\;\;...(ii)\\
&\text{Adding (i) and (ii)}\\
&\frac{\partial ^2 z}{\partial ^2 y^2}+\frac{\partial ^2 z}{\partial ^2 t^2}=0
\end{aligned}

Question 2 |

For the equation

\frac{d^3y}{dx^3}+x\left ( \frac{dy}{dx} \right )^{\frac{3}{2}}+x^2y=0

the correct description is

\frac{d^3y}{dx^3}+x\left ( \frac{dy}{dx} \right )^{\frac{3}{2}}+x^2y=0

the correct description is

an ordinary differential equation of order 3 and degree 2. | |

an ordinary differential equation of order 3 and degree 3. | |

an ordinary differential equation of order 2 and degree 3. | |

an ordinary differential equation of order 3 and degree 3/2. |

Question 2 Explanation:

\frac{d^3y}{dx^3}+x\left ( \frac{dy}{dx} \right )^{3/2}+x^2y=0

Power of \left ( \frac{dy}{dx} \right ) is fractional, make it integer.

\frac{d^3y}{dx^3}+x^2y=-x\left ( \frac{dy}{dx} \right )^{3/2}

\left (\frac{d^3y}{dx^3}+x^2y \right )^2=x^2\left ( \frac{dy}{dx} \right )^{3}

Now order = 3 and degree = 2

Power of \left ( \frac{dy}{dx} \right ) is fractional, make it integer.

\frac{d^3y}{dx^3}+x^2y=-x\left ( \frac{dy}{dx} \right )^{3/2}

\left (\frac{d^3y}{dx^3}+x^2y \right )^2=x^2\left ( \frac{dy}{dx} \right )^{3}

Now order = 3 and degree = 2

Question 3 |

The hoop stress at a point on the surface of a thin cylindrical pressure vessel is
computed to be 30.0 MPa. The value of maximum shear stress at this point is

7.5 MPa | |

15.0 MPa | |

30.0 MPa | |

22.5 MPa |

Question 3 Explanation:

Given,

Hoop stress (\sigma _h)=\frac{pd}{2t}=30MPa

Maximum shear stress in plane (\tau _{max})_{\text{in plane}}=\frac{\frac{pd}{2t}-\frac{pd}{4t}}{2}=7.5MPa

Hoop stress (\sigma _h)=\frac{pd}{2t}=30MPa

Maximum shear stress in plane (\tau _{max})_{\text{in plane}}=\frac{\frac{pd}{2t}-\frac{pd}{4t}}{2}=7.5MPa

Question 4 |

In the context of elastic theory of reinforced concrete, the modular ratio is
defined as the ratio of

Young's modulus of elasticity of reinforcement material to Young?s modulus of elasticity of concrete. | |

Youngs modulus of elasticity of concrete to Young?s modulus of elasticity of reinforcement material. | |

shear modulus of reinforcement material to the shear modulus of concrete. | |

Young's modulus of elasticity of reinforcement material to the shear modulus of concrete. |

Question 4 Explanation:

This is a question of working stress method i.e. elastic theory.

Modular ratio

=\frac{E_s}{E_c}=\frac{\text{Young's modulus of steel}}{\text{Young's modulus of concrete}}

Modular ratio

=\frac{E_s}{E_c}=\frac{\text{Young's modulus of steel}}{\text{Young's modulus of concrete}}

Question 5 |

Which of the following equations is correct for the Pozzolanic reaction?

Ca(OH)_2 + Reactive Superplasticiser + H_2O \rightarrow C-S-H | |

Ca(OH)_2 + Reactive Silicon dioxide + H_2O \rightarrow C-S-H | |

Ca(OH)_2 + Reactive Sulphates + H_2O \rightarrow C-S-H | |

Ca(OH)_2 + Reactive Sulphur + H_2O \rightarrow C-S-H |

Question 5 Explanation:

Pozzolanic materials have no cementing properties itself but have the property of combining with lime to produce stable compound.

Pozzolana is considered as siliceous and aluminous materials and when added in cement it have SiO_2 and Al_2O_3 form.

So, pozzolanic reaction :

H_2O + Reactive slilica-di-oxide + H_2O \rightarrow C-S-H gel or tobermonite gel

Pozzolana is considered as siliceous and aluminous materials and when added in cement it have SiO_2 and Al_2O_3 form.

So, pozzolanic reaction :

H_2O + Reactive slilica-di-oxide + H_2O \rightarrow C-S-H gel or tobermonite gel

Question 6 |

Consider the cross-section of a beam made up of thin uniform elements having
thickness t(t \lt \lt a) shown in the figure. The (x,y) coordinates of the points along
the center-line of the cross-section are given in the figure.

The coordinates of the shear center of this cross-section are:

The coordinates of the shear center of this cross-section are:

x = 0, y = 3a | |

x = 2a, y = 2a | |

x = -a, y = 2a | |

x = -2a, y = a |

Question 6 Explanation:

Shear centre of section consisting of two intersecting narrow rectangles always lies at the intersection of centrelines of two rectangles.

Coordinate of shear centre (0, 3a).

Coordinate of shear centre (0, 3a).

Question 7 |

Four different soils are classified as CH, ML, SP, and SW, as per the Unified
Soil Classification System. Which one of the following options correctly
represents their arrangement in the decreasing order of hydraulic conductivity?

SW, SP, ML, CH | |

SW, SP, ML, CH | |

SP, SW, CH, ML | |

ML, SP, CH, SW |

Question 7 Explanation:

Hydraulic conductivity Order.

Gravel \gt Sand \gt silt \gt lay

Gravel \gt Sand \gt silt \gt lay

Question 8 |

Let \sigma _v' and \sigma _h' denote the effective vertical stress and effective horizontal stress, respectively. Which one of the following conditions must be satisfied for a soil
element to reach the failure state under Rankine?s passive earth pressure
condition?

\sigma ' _v \lt\sigma ' _h | |

\sigma ' _v \gt\sigma ' _h | |

\sigma ' _v = \sigma ' _h | |

\sigma ' _v + \sigma ' _h =0 |

Question 8 Explanation:

We know, \sigma _h'=K\sigma _v'

For passive earth pressure,

\begin{aligned} k&=K_P \gt 1\\ \Rightarrow \frac{\sigma _h'}{\sigma _v'}&=K_P \gt 1\\ \Rightarrow \sigma _h' \gt \sigma _v' \end{aligned}

For passive earth pressure,

\begin{aligned} k&=K_P \gt 1\\ \Rightarrow \frac{\sigma _h'}{\sigma _v'}&=K_P \gt 1\\ \Rightarrow \sigma _h' \gt \sigma _v' \end{aligned}

Question 9 |

With respect to fluid flow, match the following in Column X with Column Y:

\begin{array}{|l|l|}\hline \text{Column X}& \text{Column Y}\\ \hline \text{(P) Viscosity} & \text{(I) Mach number}\\ \hline \text{(Q) Gravity}&\text{(II) Reynolds number}\\ \hline \text{(R) Compressibility}&\text{(III) Euler number}\\ \hline \text{(S) Pressure} &\text{(IV) Froude number}\\ \hline \end{array}

Which one of the following combinations is correct?

\begin{array}{|l|l|}\hline \text{Column X}& \text{Column Y}\\ \hline \text{(P) Viscosity} & \text{(I) Mach number}\\ \hline \text{(Q) Gravity}&\text{(II) Reynolds number}\\ \hline \text{(R) Compressibility}&\text{(III) Euler number}\\ \hline \text{(S) Pressure} &\text{(IV) Froude number}\\ \hline \end{array}

Which one of the following combinations is correct?

(P) - (II), (Q) - (IV), (R) - (I), (S) - (III) | |

(P) - (III), (Q) - (IV), (R) - (I), (S) - (II) | |

(P) - (IV), (Q) - (II), (R) - (I), (S) - (III) | |

(P) - (II), (Q) - (IV), (R) - (III), (S) - (I) |

Question 9 Explanation:

Reynold's number (R_e) is defined when apart from inertial force, viscous forces are dominant.

R_e=\frac{\text{Inertial force}}{\text{Viscous force}}

Froude?s number (F_e): It is used when in addition to inertial force, gravity forces are important.

F_e=\frac{\text{Inertial force}}{\text{Gravity force}}

Euler number (E_u): It is used when apart from inertial force, only pressure forces are dominant.

E_u=\frac{\text{Inertial force}}{\text{Pressure force}}

Mach number (M): It is used when in addition to inertial force, compressibility forces are dominant

M=\frac{\text{Inertial force}}{\text{Elastic force}}

R_e=\frac{\text{Inertial force}}{\text{Viscous force}}

Froude?s number (F_e): It is used when in addition to inertial force, gravity forces are important.

F_e=\frac{\text{Inertial force}}{\text{Gravity force}}

Euler number (E_u): It is used when apart from inertial force, only pressure forces are dominant.

E_u=\frac{\text{Inertial force}}{\text{Pressure force}}

Mach number (M): It is used when in addition to inertial force, compressibility forces are dominant

M=\frac{\text{Inertial force}}{\text{Elastic force}}

Question 10 |

Let \psi represent soil suction head and K represent hydraulic conductivity of the
soil. If the soil moisture content \theta increases, which one of the following
statements is TRUE?

\psi decreases and K increases. | |

\psi increases and K decreases. | |

Both \psi and K decrease. | |

Both \psi and K increase. |

Question 10 Explanation:

h_c\propto \frac{1}{R}

K\propto S

Water content \uparrow \rightarrow R\uparrow \rightarrow h_c \downarrow \rightarrow \psi \downarrow

Water content \uparrow \rightarrow S\uparrow \rightarrow K \uparrow

K\propto S

Water content \uparrow \rightarrow R\uparrow \rightarrow h_c \downarrow \rightarrow \psi \downarrow

Water content \uparrow \rightarrow S\uparrow \rightarrow K \uparrow

There are 10 questions to complete.