Engineering Mathematics

Question 1
For the solid S shown below, the value of \int \int_S \int x dxdydz (rounded off to two decimal places) is ______.
A
3.52
B
1.25
C
2.25
D
4.25
GATE EC 2020      Calculus
Question 1 Explanation: 
x : 0 to 3
y : 0 to 1
z : 0 to 1-y
=\int_{y=0}^{1}\int_{z=0}^{1-y}\int_{x=0}^{3}x\, dx\, dy\, dz=\int_{y=0}^{1}\int_{0}^{1-y}\left ( \frac{x^{2}}{2} \right )^{3}_{0}dz\, dy
=\int_{0}^{1}\frac{9}{2}(z)_{0}^{1-y}dy=\frac{9}{2}\int_{0}^{1}(1-y)dy=\frac{9}{2}\left ( y-\frac{y^{2}}{2} \right )_{0}^{1}
=\frac{9}{2}\left ( 1-\frac{1}{2} \right )=\frac{9}{4}= 2.25
Question 2
Which one of the following options contains two solutions of the differential equation \frac{dy}{dx}=(y-1)x?
A
ln |y-1|=0.5x^2 +C and y=1
B
ln |y-1|=2x^2 +C and y=1
C
ln |y-1|=0.5x^2 +C and y=-1
D
ln |y-1|=2x^2 +C and y=-1
GATE EC 2020      Differential Equations
Question 2 Explanation: 
\begin{aligned} \frac{dy}{dx}&=(y-1)x\\ \frac{dy}{y-1}&=x dx\\ \text{Integrating}&\text{ both side, we get}\\ \int \frac{1}{y-1}dy&=\int xdx\\ \ln |y-1|&=\frac{x^2}{2}+C\; \; \;...(i)\\ \text{Now, }&\\ \frac{dy}{dx}&=0 \text{ when y=1}\\ \text{It means }y&= \text{constant C}\\ \therefore \; y&=1\;\;\;...(ii) \end{aligned}
So, equation (i) and (ii) are two different solution of given differential equation.
Question 3
Consider the following system of linear equation.

x_1+2x_2=b_1;
2x_1+4x_2=b_2;
3x_1+7x_2=b_3;
3x_1+9x_2=b_4;

Which one of the following conditions ensures that a solution exists for the above system?
A
b_2=2b_1 and 6b_1-3b_3+b_4=0
B
b_3=2b_1 and 6b_1-3b_3+b_4=0
C
b_2=2b_1 and 3b_1-6b_3+b_4=0
D
b_3=2b_1 and 3b_1-6b_3+b_4=0
GATE EC 2020      Linear Algebra
Question 3 Explanation: 
Given:
x_{1}+2x_{2}=b_{1} \; \; ...(i)
2x_{1}+4X_{2}=b_{2}\; \; ...(ii)
3x_{1}+7x_{2}=b_{3}\; \; ...(iii)
3x_{1}+9x_{2}=b_{4}\; \; ...(iv)
From equations (ii) and (i)
We can write,
b_{2}=2[x_{1}+2x_{2}]=2b_{1}
From option (C):
3b_{1}-6b_{3}+b_{4}=3[x_{1}+2x_{2}]-6[3x_{1}+7x_{2}]+3x_{1}+9x_{2}\neq 0
From option (A):
b_{2}=2b_{1}
and b_{1}-3b_{3}+b_{4}=6[x_{1}+2x_{2}]-3[3x_{1}+7x_{2}]+[3x_{1}+9x_{2}]=0
6b_{1}-3b_{3}+b_{4}=0
Hence, answer is option (A).
Question 4
The two sides of a fair coin are labelled as 0 and 1. The coin is tossed two times independently. Let M and N denote the labels corresponding to the outcomes of those tosses. For a random variable X, defined as X = min(M, N), the expected value E(X) (rounded off to two decimal places) is _________.
A
0.15
B
0.75
C
0.55
D
0.25
GATE EC 2020      Probability and Statistics
Question 4 Explanation: 
s={(H,H),(H,T),(T,H),(T,T)}
M=\begin{bmatrix} 1 & 1 & 0 & 0 \\ H & H & T & T \end{bmatrix} of first toss


N=\begin{bmatrix} H & T & H& T \\ 1 & 0 & 1 & 0 \end{bmatrix} of second toss
Now,X=Min{M,N}
\therefore
X=Min{H,H}=Min{1,1}=1
X=Min{H,T}=Min{1,0}=0
X=Min{T,H}=Min{0,1}=0
X=Min(T,T)=Min(0,0)=0
\thereforeP(X=1)=\frac{1}{4},P(x=0)=\frac{3}{4}
We Know that, E(X)=\sum_{i}X_{i}P(x_{i})=1\times \frac{1}{4}+0\times \frac{3}{4}=\frac{1}{4}=0.25
Question 5
The general solution of \frac{d^2y}{dx^2}-6\frac{dy}{dx}+9y=0 is
A
y=C_1e^{3x}+C_2e^{-3x}
B
y=(C_1+C_2x)e^{-3x}
C
y=(C_1+C_2x)e^{3x}
D
y=C_1e^{3x}
GATE EC 2020      Differential Equations
Question 5 Explanation: 
Taking \frac{\mathrm{d} }{\mathrm{d} x}=D
Given, D^{2}-6D+9=0
(D-3)^2=0
D=3,3
So, Solution of the given Differential equation
y=(C_{1}+C_{2}x)e^{3x}
Question 6
The partial derivative of the function

f(x,y,z)=e^{1-x \cos y}+xze^{-1/(1+y^2)}

with respect to x at the point (1,0,e) is
A
-1
B
0
C
1
D
\frac{1}{e}
GATE EC 2020      Calculus
Question 6 Explanation: 
\begin{aligned} \text{Given, } f(x,y,z)&=e^{1-x\cos y}+xze^{-1/(1+y^{2})} \\ \frac{\partial f}{\partial x}&=e^{1-x\cos y}(0-\cos y)+ze^{-1/1+y^{2}} \\ \left ( \frac{\partial f }{\partial x} \right )_{(1,0,e)}&=e^{0}(0-1)+e\cdot e^{-1/(1+0)} \\ &=-1+1=0 \end{aligned}
Question 7
For a vector field \vec{A}, which one of the following is False?
A
\vec{A} is solenoidal if \bigtriangledown \cdot \vec{A}=0
B
\bigtriangledown \times \vec{A} is another vector field.
C
\vec{A} is irrotational if \bigtriangledown ^2 \vec{A}=0.
D
\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}
GATE EC 2020      Calculus
Question 7 Explanation: 
Divergence and curl operator is performed on a vector field \vec{A}
Curl operation provides a vector orthogonal to the given vector field \vec{A}
\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}
If a vector field is irrortational then \bigtriangledown \times \vec{A}=0
If a vector field is solenoidal then \bigtriangledown \cdot \vec{A}=0
If a field is scalar A, then \bigtriangledown ^2 \vec{A}=0, is a laplacian equation.
Hence option (C) is incorrect
Question 8
If v_1, v_2,..., v_6 are six vectors in \mathbb{R}^4, which one of the following statements is False?
A
It is not necessary that these vectors span \mathbb{R}^4.
B
These vectors are not linearly independent.
C
Any four of these vectors form a basis for \mathbb{R}^4.
D
If {v_1, v_3,v_5, v_6} spans \mathbb{R}^4, then it forms a basis for \mathbb{R}^4.
GATE EC 2020      Calculus
Question 8 Explanation: 
v_1, v_2,..., v_6 are six vectors in \mathbb{R}^4.
For a 4-dimensional vector space,
(i) any four linearly independent vectors form a basis (or)
(ii) Any set of four vectors in \mathbb{R}^4 spans \mathbb{R}^4, then it forms a basis.
Therefore, clearly options (A), (B), (D) are true.
Option (C) is FALSE
Question 9
Consider the homogeneous ordinary differential equation

x^2\frac{d^2y}{dx^2}-3x\frac{dy}{dx}+3y=0, \; x \gt 0

with y(x) as a generalsolution. Given that
y(1)=1 and y(2)=14
the value of y(1.5), rounded off to two decimal places, is ______.
A
1.25
B
4.25
C
3.75
D
5.25
GATE EC 2019      Differential Equations
Question 9 Explanation: 
\begin{aligned} \left(x^{2} D^{2}-3 x D+3\right)&=0 \\ (0(0-1)-30+3) y&=0 \\ \left(0^{2}-40+3\right) y&=0 \\ \text { AE is }\quad m^{2}-4 m+3&=0 \\ m&=1.3 \\ CF&=C_{1}x+C_{2}x^{3} \\ \text { Solution is } \quad y&=C_{1}x+C_{2}x^{3} \\ y(1)=1\quad 1&=C_{1}+C_{2} \\ y(2)=14\quad 14&=2 C_{1}+8 C_{2} \\ \text{From (il) and (ii). }C_{1}&=-1, \quad C_{2}=2 \\ \therefore\quad y&=-x+2 x^{3} \\ y(1.5)&=-1.5+2(1.5)^{3}=5.25 \end{aligned}
Question 10
Consider the line integral

\int _c (xdy-ydx)

the integral being taken in a counterclockwise direction over the closed curve C that forms the boundary of the region R shown in the figure below. The region R is the area enclosed by the union of a 2x3 rectangle and a semi-circle of radius 1. The line integral evaluates to
A
6+\pi/2
B
8+\pi
C
12+\pi
D
16+2 \pi
GATE EC 2019      Calculus
Question 10 Explanation: 
\begin{array}{l} \text { Given, } \int-y d x+x d y \\ \qquad \begin{aligned} \text { here, } \quad F_{1}&=-y \text { and } \frac{\partial F_{1}}{\partial y}=-1 \\ F_{2} &=x \text { and } \frac{\partial F_{2}}{\partial x}=1 \\ \therefore \int F_{1} d x+F_{2} d y &=\iint\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\right) d x d y \\ \int-y d x+x d y &=\iint 1-(-1) d x d y \\ &=2( \text { Area of region R}) \\ &=2\left(6+\frac{\pi}{2}\right)=12+\pi \end{aligned} \end{array}


There are 10 questions to complete.
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