# GATE Mechanical Engineering 2022 SET-1

 Question 1
The limit
$p=\lim_{x \to \pi}\left ( \frac{x^2+\alpha x+2\pi ^2}{x-\pi+2 \sin x } \right )$
has a finite value for a real $\alpha$. The value of $\alpha$ and the corresponding limit $p$ are
 A $\alpha =-3\pi, \text{ and }p= \pi$ B $\alpha =-2\pi, \text{ and }p= 2\pi$ C $\alpha =\pi, \text{ and }p= \pi$ D $\alpha =2\pi, \text{ and }p= 3\pi$
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} p&=\lim_{x \to \pi}\left ( \frac{x^2+\alpha x+2 \pi ^2}{x-\pi+2 \sin x} \right ) \\ p&=\left ( \frac{\pi ^2+\alpha \pi +2 \pi ^2}{\pi-\pi+2 \sin \pi } \right ) \\ &= \frac{2 \pi ^2+\alpha \pi}{0}\\ \therefore \;\; \alpha &= -3 \pi\\ p&=\lim_{x \to \pi}\left ( \frac{x^2- 3 \pi x+2 \pi ^2}{x-\pi+2 \sin x} \right ) \\ &=\lim_{x \to \pi}\left ( \frac{2x- 3 \pi }{1+2 \cos \pi} \right ) \\ &= \frac{2 \pi-3 \pi}{1-2}=\frac{-\pi}{-1}=\pi\\ \therefore \; \alpha &=-3 \pi \text{ and }p= \pi \end{aligned}
 Question 2
Solution of $\triangledown^2T=0$ in a square domain ($0 \lt x \lt 1$ and $0 \lt y \lt 1$) with boundary conditions:
$T(x, 0) = x; T(0, y) = y; T(x, 1) = 1 + x; T(1, y) = 1 + y$ is
 A $T(x,y)=x-xy+y$ B $T(x,y)=x+y$ C $T(x,y)=-x+y$ D $T(x,y)=x+xy+y$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
T(x, 0) = x $\Rightarrow$ option (c) is not correct.
T(0, y) = y $\Rightarrow$ all options satisfied.
T(x, 1) = 1 + x; $\Rightarrow$ only option (b) is satisfied.
T(1, y) = 1 + y is $\Rightarrow$ only option (b) is satisfied.

 Question 3
Given a function $\varphi =\frac{1}{2}(x^2+y^2+z^2)$ in threedimensional Cartesian space, the value of the surface integral
$\oiint_{S}{\hat{n}.\triangledown \varphi dS}$
where S is the surface of a sphere of unit radius and $\hat{n}$ is the outward unit normal vector on S, is
 A $4 \pi$ B $3 \pi$ C $4 \pi/3$ D $0$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \varphi &=\frac{1}{2}(x^2+y^2+z^2)\\ \triangledown \varphi &=(x\hat{i}+y\hat{j}+z\hat{k})=\bar{F}\\ \oiint_{S}(\triangledown \varphi\cdot \bar{n})dS&=\int \int _v\int Div\; \bar{F} dv\\ &=\int \int \int 3dv\\ &=3v\\ &=3\left ( \frac{4}{3} \pi \right )=4\pi \end{aligned}
 Question 4
The Fourier series expansion of $x^3$ in the interval $-1\leq x\leq 1$ with periodic continuation has
 A only sine terms B only cosine terms C both sine and cosine terms D only sine terms and a non-zero constant
Engineering Mathematics   Calculus
Question 4 Explanation:
$f(x)=x^3, \;\; -1 \leq x \leq 1$
It is an odd function
Fourier series contains only sine terms.
 Question 5
If $A=\begin{bmatrix} 10 &2k+5 \\ 3k-3 & k+5 \end{bmatrix}$ is a symmetric matrix, the value of $k$ is ___________.
 A 8 B 5 C -0.4 D $\frac{1+\sqrt{1561}}{12}$
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
$A=\begin{bmatrix} 10 & 2k+5\\ 3k-3 & k+5 \end{bmatrix}$
$(2k + 5) = (3k - 3)$
$k=8$

There are 5 questions to complete.