Engineering Mathematics

Question 1
Given z=x+iy,i=\sqrt{-1} is a circle of radius 2 with the centre at the origin. If the contour C is traversed anticlockwise, then the value of the integral \frac{1}{2}\int_{c}^{}\frac{1}{(z-i)(z+4i)}dz is _______(round off to one decimal place).
A
0.2
B
0.4
C
0.6
D
0.8
GATE ME 2022 SET-2      Complex Variables
Question 1 Explanation: 
Given contour is a circle at centre (0,0) and radius 2 given function is \frac{1}{(z-i)(z+4i)} here z = i is a singular point lies now by caucus integral formula

\begin{aligned} &\int \frac{1}{(z-i)(z+4i)}dz=\int \frac{\left ( \frac{1}{z+4} \right )}{z-i}dz=2 \pi i \times f(i)\\ &f(z)=\frac{1}{z+4i}\\ &f(i)=\frac{1}{5i}\\ &2 \pi i \times f(i)=2 \pi i \times \frac{1}{5i}=\frac{2\pi}{5}\\ &\text{Now }\frac{1}{2 \pi}\int_{c}\frac{1}{(z-1)(z+4i)}dz=\frac{1}{2 \pi}+\frac{2 \pi}{5}=\frac{1}{5} \end{aligned}
Question 2
If the sum and product of eigenvalues of a 2 \times 2 real matrix \begin{bmatrix} 3 & p\\ q & p \end{bmatrix} are 4 and -1 respectively, then |p| is ______ (in integer).
A
4
B
2
C
6
D
8
GATE ME 2022 SET-2      Linear Algebra
Question 2 Explanation: 
From the property of eigen values,
Sum of eigen values = Trace of matrix
4=3+q
q=1

Product of eigen values = Determinant
\begin{aligned} -1&=\begin{vmatrix} 3 &p \\ p& q \end{vmatrix}\\ 3q-p^2&=-1\\ 3-p^2&=-1\\ p^2&=4\\ p&=\pm 2\\ \Rightarrow |p|&=2 \end{aligned}
Question 3
A is a 3\times 5 real matrix of rank 2. For the set of homogeneous equations Ax = 0, where 0 is a zero vector and x is a vector of unknown variables, which of the following is/are true?
A
The given set of equations will have a unique solution.
B
The given set of equations will be satisfied by a zero vector of appropriate size.
C
The given set of equations will have infinitely many solutions.
D
The given set of equations will have many but a finite number of solutions.
GATE ME 2022 SET-2      Linear Algebra
Question 3 Explanation: 
Zero solution is always a solution of Ax = 0.
Option (b) is correct.
Given A is 3x5 real matrix and r(A) = 2 and Ax = 0 is a system of homogeneous linear equations since r(A) \lt number of unknown the system has infinite solution option (c) is also correct.
Question 4
For the exact differential equation, \frac{du}{dx}=\frac{-xu^2}{2+x^2u} , which one of the following is the solution ?
A
u^2+2x^2=\text{Constant}
B
xu^2+u=\text{Constant}
C
\frac{1}{2}x^2 u^2+2u=\text{Constant}
D
\frac{1}{2} u x^2+2x=\text{Constant}
GATE ME 2022 SET-2      Differential Equations
Question 4 Explanation: 
Given D.E. is \frac{du}{dx}=\frac{-xu^2}{2+x^2u}
\Rightarrow (2+x^2u)du+xu^2dx=0
Here M=xu^2 \;\;\;\;N=2+x^2u
\frac{\partial N}{\partial x}=2xu \;\;\;\;\;\frac{\partial M}{\partial x}=2xu
This is exact D.E
General solution is
\int_{\text{u constant}}Mdx+\int_{\text{term without 'x'}}Ndx=c
\Rightarrow \int (xu^2)dx+\int 2du=c
\therefore \; \frac{x^2u^2}{2}+2u=constant
Question 5
A polynomial \phi (s)=a_{n}s^{n}+a_{n-1}s^{n-1}+...+a_{1}s+a_0 of degree n \gt 3 with constant real coefficients a_n, a_{n-1},...a_0 has triple roots at s=-\sigma . Which one of the following conditions must be satisfied?
A
\phi (s)=0 at all the three values of s satisfying s^3+\sigma ^3=0
B
\phi (s)=0, \frac{d \phi (s)}{ds}=0, \text{ and }\frac{d^2 \phi (s)}{ds^2}=0 \text{ at }s=-\sigma
C
\phi (s)=0, \frac{d \phi (s)}{ds}=0, \text{ and }\frac{d^4 \phi (s)}{ds^4}=0 \text{ at }s=-\sigma
D
\phi (s)=0, \text{ and }\frac{d^3 \phi (s)}{ds^3}=0 \text{ at }s=-\sigma
GATE ME 2022 SET-2      Differential Equations
Question 5 Explanation: 
Since \varphi (s) has a triple roots at s=-\sigma
Therefore, \varphi (s)=(s+\sigma )^3\psi (s)
It satisfies all the conditions in option (B) is correct.
Question 6
Given \int_{-\infty }^{\infty }e^{-x^2}dx=\sqrt{\pi}
If a and b are positive integers, the value of \int_{-\infty }^{\infty }e^{-a(x+b)^2}dx is ___.
A
\sqrt{\pi a}
B
\sqrt{\frac{\pi}{a}}
C
b \sqrt{\pi a}
D
b \sqrt{\frac{\pi}{a}}
GATE ME 2022 SET-2      Calculus
Question 6 Explanation: 
\begin{aligned} &\text{ Let }(x+b)=t\\ &\Rightarrow \; dx=dt\\ &\text{When ,} x=-\infty ;t=-\infty \\ &\int_{-\infty }^{-\infty }e^{-n(x+b)^2}dx=\int_{-\infty }^{-\infty }e^{-at^2}dt\\ &\text{Let, }at^2=y^2\Rightarrow t=\frac{y}{\sqrt{a}}\\ &2at\;dt=3y\;dy\\ &dt=\frac{ydy}{at}=\frac{ydy}{a\frac{y}{\sqrt{a}}}=\frac{y}{\sqrt{a}}\\ &\int_{-\infty }^{-\infty }e^{-at^2}dt=\int_{-\infty }^{-\infty }e^{-y^2}\cdot \frac{dy}{\sqrt{a}}=\sqrt{\frac{\pi}{a}} \end{aligned}
Question 7
Consider the definite integral
\int_{1}^{2}(4x^2+2x+6)dx
Let I_e be the exact value of the integral. If the same integral is estimated using Simpson's rule with 10 equal subintervals, the value is I_s . The percentage error is defined as e=100 \times (I_e-I_s)/I_e . The value of e is
A
2.5
B
3.5
C
1.2
D
0
GATE ME 2022 SET-2      Numerical Methods
Question 7 Explanation: 
Exact value
\begin{aligned} &=\int_{1}^{2} (4x^2+2x+6)dx\\ &=\frac{4x^3}{3}+\frac{2x^2}{2}+6x\\ &=\frac{4}{3}(\pi) \times (3)+6\\ &=\frac{28}{3}+9=\frac{55}{3} \end{aligned}
Approximate value
Here f(x) is a polynomial of degree 2, so Simpsons rule gives exact value with zero error
\begin{aligned} \therefore \;\; \text{Approx value}&=\frac{55}{3}\\ \frac{I_e-I_s}{I_e}&=0\\ \therefore \;\;e=\left (\frac{I_e-I_s}{I_e} \right )\times 100&=0 \end{aligned}
Question 8
Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral \int_{A}^{}\vec{F}.d\vec{A} of a vector field \vec{F}=3x\hat{i}+5y\hat{j}+6z\hat{k} over the entire surface A of the cube is ______.
A
14
B
27
C
28
D
31
GATE ME 2022 SET-2      Calculus
Question 8 Explanation: 
Given,
\begin{aligned} \bar{F} &=3x\bar{i}+5y\bar{j}+6z\bar{k} \\ \triangledown \cdot \bar{F}&= \frac{\partial }{\partial x}(3x)+\frac{\partial }{\partial x}(5y)+\frac{\partial }{\partial x}(6z)\\ &= 3+5+6=14 \end{aligned}
By gauss divers once Theorem
\begin{aligned} \int_{A}^{}\bar{F}\cdot dA &=\int \int \int (\triangledown \cdot F)dV =\int \int \int 14\; dv\\ &=14 \times \text{volume of a cube of side 1 unit } \\ &=14 \times (1)^3=14\end{aligned}
Question 9
F(t) is a periodic square wave function as shown. It takes only two values, 4 and 0, and stays at each of these values for 1 second before changing. What is the constant term in the Fourier series expansion of F(t)?

A
1
B
2
C
3
D
4
GATE ME 2022 SET-2      Calculus
Question 9 Explanation: 
The constant term in the Fourier series expansion of F(t) is the average value of F(t) in one fundamental period i.e.,
\frac{\int_{0}^{1}4dt+\int_{1}^{2}0dt}{2}=\frac{4}{2}=2
Question 10
Consider two vectors:
\vec{a}=5i+7j+2k
\vec{b}=3i-j+6k
Magnitude of the component of \vec{a} orthogonal to \vec{b} in the plane containing the vectors \vec{a} and \vec{b} is __________ (round off to 2 decimal places).
A
2.95
B
8.32
C
12.65
D
5.23
GATE ME 2022 SET-1      Calculus
Question 10 Explanation: 


Component of \vec{a} parallel to \vec{b}=\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}=\frac{20}{\sqrt{46}}=2.95
Now,
\begin{aligned} |\vec{a}|^2&=2.95^2+(a\perp )^2\\ 78&=2.95^2+(a\perp )^2\\ a\perp&=8.32 \end{aligned}


There are 10 questions to complete.

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