# 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

There are 5 questions to complete.