# 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 $b_{1}-3b_{3}+b_{4}=6[x_{1}+2x_{2}]-3[3x_{1}+7x_{2}]+[3x_{1}+9x_{2}]=0$
$6b_{1}-3b_{3}+b_{4}=0$
 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
$\therefore$P(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.