# GATE Civil Engineering 2021 SET-2

 Question 1
The value of $\lim _{x \rightarrow \infty} \frac{x \ln (x)}{1+x^{2}}$ is
 A 0 B 1 C 0.5 D $\infty$
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} &\lim _{x \rightarrow \infty}\left(\frac{x \ln x}{x^{2}+1}\right) \qquad \qquad \qquad \qquad \qquad \left(\frac{\infty}{\infty} \text { form }\right)\\ &=\lim _{x \rightarrow \infty}\left(\frac{x\left(\frac{1}{x}\right)+\ln x}{2 x}\right) \qquad \qquad \qquad \left(\frac{\infty}{\infty} \text { form }\right)\\ \lim _{x \rightarrow \infty}\left(\frac{0+\frac{1}{x}}{2}\right)&=\lim _{x \rightarrow \infty}\left(\frac{1}{2 x}\right)=\frac{1}{2 \times \infty}=0 \end{aligned}
 Question 2
The rank of the matrix $\left[\begin{array}{cccc} 5 & 0 & -5 & 0 \\ 0 & 2 & 0 & 1 \\ -5 & 0 & 5 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right]$ is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} \left[\begin{array}{cccc} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ -5 & 0 & -1 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right] & \stackrel{R_{1} \longleftrightarrow R_{1}+R_{3}}{\longrightarrow}\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right] \\ & \stackrel{R_{4} \longleftrightarrow R_{4}-\frac{1}{2} R_{2}}{\longrightarrow}\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2} \end{array}\right]\\ &R_{3} \longleftrightarrow R_{4}\left[\begin{array}{llll}5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2} \\ 0 & 0 & 0 & 0\end{array}\right] \end{aligned}
Rank(A) = 3
 Question 3
The unit normal vector to the surface $X^{2}+Y^{2}+Z^{2}-48=0$ at the point (4,4,4) is
 A $\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ B $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$ C $\frac{2}{\sqrt{2}}, \frac{2}{\sqrt{2}}, \frac{2}{\sqrt{2}}$ D $\frac{1}{\sqrt{5}}, \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{5}}$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \phi &=x^{2}+y^{2}+z^{2}-48, P(4,4,4) \\ \operatorname{grad} \phi &=\vec{\nabla} \phi=\hat{i} \frac{\partial \phi}{\partial x}+\hat{j} \frac{\partial \phi}{\partial y}+\hat{k} \frac{\partial \phi}{\partial z} \\ &=(2 x) \hat{i}+(2 y) \hat{j}+(2 z) \hat{k} \\ \vec{n} &=(\operatorname{grad} \phi)_{P}=8 \hat{i}+8 \hat{j}+8 \hat{k} \\ \hat{n} &=\frac{\vec{n}}{|\vec{n}|}=\frac{8 \hat{i}+8 \hat{j}+8 \hat{k}}{\sqrt{64+64+64}}=\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}} \\ & \simeq\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}},\right) \end{aligned}
 Question 4
If A is a square matrix then orthogonality property mandates
 A $A A^{T}=I$ B $A A^{T}=0$ C $A A^{T}=A^{-1}$ D $A A^{T}=A^{2}$
Engineering Mathematics   Linear Algebra
Question 4 Explanation:
$\text { If, } \qquad \qquad A A^{\top}=I \quad \text { or } A^{-1}=A^{T}$
The matrix is orthogonal.
 Question 5
In general, the CORRECT sequence of surveying operations is
 A Field observations$\rightarrow$ Reconnaissance$\rightarrow$ Data analysis$\rightarrow$ Map making B Data analysis$\rightarrow$ Reconnaissance$\rightarrow$ Field observations $\rightarrow$ Map making C Reconnaissance$\rightarrow$ Field observations $\rightarrow$ Data analysis $\rightarrow$ Map making D Reconnaissance$\rightarrow$ Data analysis $\rightarrow$ Field observations $\rightarrow$ Map making
Geometics Engineering   Fundamental Concepts of Surveying
Question 5 Explanation:
Reconnaissance$\rightarrow$Field observations$\rightarrow$Data analysis$\rightarrow$Map making
 Question 6
Strain hardening of structural steel means
 A experiencing higher stress than yield stress with increased deformation B strengthening steel member externally for reducing strain experienced C strain occurring before plastic flow of steel material D decrease in the stress experienced with increasing strain
Solid Mechanics   Properties of Metals, Stress and Strain
Question 6 Explanation:
Strain hardening is experiencing higher stress than yield stress with increased deformation
In the figure AB = Strain hardening zone
OA = Linear elastic zone
Stress corresponding to point 'A' is yield stress. Question 7
A single story building model is shown in the figure. The rigid bar of mass 'm' is supported by three massless elastic columns whose ends are fixed against rotation. For each of the columns, the applied lateral force (P) and corresponding moment (M) are also shown in the figure. The lateral deflection $(\delta)$ of the bar is given by $\delta=\frac{P L^{3}}{12 E I}$, where L is the effective length of the column, E is the Young's modulus of elasticity and I is the area moment of inertia of the column cross-section with respect to its neutral axis. For the lateral deflection profile of the columns as shown in the figure, the natural frequency of the system for horizontal oscillation is
 A $6 \sqrt{\frac{E I}{m L^{3}}} \mathrm{rad} / \mathrm{s}$ B $\frac{1}{L} \sqrt{\frac{2 E I}{m}} \mathrm{rad} / \mathrm{s}$ C $6 \sqrt{\frac{6 E I}{m L^{3}}} \mathrm{rad} / \mathrm{s}$ D $\frac{2}{L} \sqrt{\frac{E I}{m}} \mathrm{rad} / \mathrm{s}$
Solid Mechanics   Deflection of Beams
Question 7 Explanation: As the deflection will be same in all the 3 columns, so it represents a parallel connection. \begin{aligned} k_{e q} &=3 k=\frac{36 E I}{L^{3}} \\ \text { Natural frequency }(\omega) &=\sqrt{\frac{k}{m}} \\ &=\sqrt{\frac{36 E I}{m L^{3}}}=6 \sqrt{\frac{E I}{m L^{3}}} \mathrm{rad} / \mathrm{s} \end{aligned}
 Question 8
Seasoning of timber for use in construction is done essentially to
 A increase strength and durability B smoothen timber surfaces C remove knots from timber logs D cut timber in right season and geometry
Construction Materials and Management
Question 8 Explanation:
Option 1 Increase strength and durability.
The process of drying of timber is known as seasoning.
Natural tree has more the 50% weight of water of its dry weight.
If we directly use this timber the because of irregular drying internal stresses will develop between fibres of timber and it will develop lots of defects (warps, shakes etc).
 Question 9
In case of bids in Two-Envelop System, the correct option is
 A Technical bid is opened first B Financial bid is opened first C Both (Technical and Financial) bids are opened simultaneously D Either of the two (Technical and Financial) bids can be opened first
Construction Materials and Management
Question 9 Explanation:
Option 1 technical bid is opened first

Opening of Tender
First technical bid is opened and after ensuring that all the technical aspects of a contractor are in order than only financial bid is opened
1. Envelope 1 ( Technical bid )

1. Cover letter
2. Registration Details
3. Pre-qualification documents
4. Earnest money deposit
5. Assumptions & Deviations in making of tender
6. Drawings

2. Envelope 2 (Financial Bid)

1. Forms of tender
 Question 10
The most appropriate triaxial test to assess the long-term stability of an excavated clay slope is
 A consolidated drained test B unconsolidated undrained test C consolidated undrained test D unconfined compression test
Geotechnical Engineering   Shear Strength of Soil
Question 10 Explanation:
To assess the long term stability of clayey soil, the results of consolidated drained (CD) test are used.
There are 10 questions to complete. 