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

There are 5 questions to complete.