# GATE Civil Engineering 2022 SET-1

 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
 A $\frac{\partial^2 z}{\partial t^2}+\frac{\partial^2 z}{\partial y^2}=0$ B $\frac{\partial^2 z}{\partial t^2}-\frac{\partial^2 z}{\partial y^2}=0$ C $\frac{\partial z}{\partial t}-i\frac{\partial z}{\partial y}=0$ D $\frac{\partial z}{\partial t}+i\frac{\partial z}{\partial y}=0$
Engineering Mathematics   Partial Differential Equation
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
 A an ordinary differential equation of order 3 and degree 2. B an ordinary differential equation of order 3 and degree 3. C an ordinary differential equation of order 2 and degree 3. D an ordinary differential equation of order 3 and degree 3/2.
Engineering Mathematics   Ordinary Differential Equation
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

 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
 A 7.5 MPa B 15.0 MPa C 30.0 MPa D 22.5 MPa
Solid Mechanics   Bending and Shear Stresses
Question 3 Explanation:
Given,
Hoop stress $(\sigma _h)=\frac{pd}{2t}=30MPa$
Maximum shear stress in plane $(\tau _{max})_{\text{in plane}}=\frac{\frac{pd}{2t}-\frac{pd}{4t}}{2}=7.5MPa$
 Question 4
In the context of elastic theory of reinforced concrete, the modular ratio is defined as the ratio of
 A Young's modulus of elasticity of reinforcement material to Young?s modulus of elasticity of concrete. B Youngs modulus of elasticity of concrete to Young?s modulus of elasticity of reinforcement material. C shear modulus of reinforcement material to the shear modulus of concrete. D Young's modulus of elasticity of reinforcement material to the shear modulus of concrete.
RCC Structures   Working Stress and Limit State Method
Question 4 Explanation:
This is a question of working stress method i.e. elastic theory.
Modular ratio
$=\frac{E_s}{E_c}=\frac{\text{Young's modulus of steel}}{\text{Young's modulus of concrete}}$
 Question 5
Which of the following equations is correct for the Pozzolanic reaction?
 A $Ca(OH)_2$ + Reactive Superplasticiser + $H_2O \rightarrow C-S-H$ B $Ca(OH)_2$ + Reactive Silicon dioxide + $H_2O \rightarrow C-S-H$ C $Ca(OH)_2$ + Reactive Sulphates + $H_2O \rightarrow C-S-H$ D $Ca(OH)_2$ + Reactive Sulphur + $H_2O \rightarrow C-S-H$
RCC Structures   Concrete Technology
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

There are 5 questions to complete.