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.

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