# 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
 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:
 A x = 0, y = 3a B x = 2a, y = 2a C x = -a, y = 2a D x = -2a, y = a
Solid Mechanics   Theory of Columns and Shear Centre
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).
 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?
 A SW, SP, ML, CH B SW, SP, ML, CH C SP, SW, CH, ML D ML, SP, CH, SW
Geotechnical Engineering   Classification of Soils and Clay Minerals
Question 7 Explanation:
Hydraulic conductivity Order.
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?
 A $\sigma ' _v \lt\sigma ' _h$ B $\sigma ' _v \gt\sigma ' _h$ C $\sigma ' _v = \sigma ' _h$ D $\sigma ' _v + \sigma ' _h =0$
Geotechnical Engineering   Retaining Wall-Earth Pressure Theories
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}
 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?
 A (P) - (II), (Q) - (IV), (R) - (I), (S) - (III) B (P) - (III), (Q) - (IV), (R) - (I), (S) - (II) C (P) - (IV), (Q) - (II), (R) - (I), (S) - (III) D (P) - (II), (Q) - (IV), (R) - (III), (S) - (I)
Fluid Mechanics and Hydraulics   Flow Through Pipes
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
$M=\frac{\text{Inertial force}}{\text{Elastic force}}$
 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?
 A $\psi$ decreases and K increases. B $\psi$ increases and K decreases. C Both $\psi$ and K decrease. D Both $\psi$ and K increase.
Geotechnical Engineering   Properties of Soils
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 $\uparrow \rightarrow S\uparrow \rightarrow K \uparrow$
There are 10 questions to complete.