Engineering Mathematics

 Question 1
A quadratic function of two variables is given as

$f\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{2}^{2}+3 x_{1}+3 x_{2}+x_{1} x_{2}+1$

The magnitude of the maximum rate of change of the function at the point $(1,1)$ is ____(Round off to the nearest integer).
 A 10 B 12 C 8 D 16
GATE EE 2023      Differential Equations
Question 1 Explanation:
Given :
$\mathrm{F}\left(\mathrm{x}, \mathrm{x}_{2}\right)=\mathrm{x}_{1}^{2}+2 \mathrm{x}_{2}^{2}+\mathrm{x}_{1}+\mathrm{x}_{1} \mathrm{x}_{2}+1$
$\nabla \cdot F=\left(2 x_{1}+3+x_{2}\right) \hat{a}_{x_{1}}+\left(4 x_{2}+3+x_{1}\right) \hat{a}_{x_{z}}$

At $(1,1)$
\begin{aligned} \nabla \cdot F & =6 \hat{a}_{x_{1}}+7 \hat{a}_{x_{2}} \\ |\nabla \cdot F| & =\sqrt{6^{2}+8^{2}}=10 \end{aligned}
 Question 2
The closed curve shown in the figure is described by

$r=1+\cos \theta$, where $r=\sqrt{x^{2}+y^{2}}$ $x=r \cos \theta, y=r \sin \theta$

The magnitude of the line integral of the vector field $F=-y \hat{i}+x \hat{j}$ around the closed curve is ___(Round off to 2 decimal places).

 A 9.42 B 6.36 C 2.45 D 7.54
GATE EE 2023      Calculus
Question 2 Explanation:
\begin{aligned} I & =\int_{0}^{2 \pi} \vec{F} \cdot \overrightarrow{d l} \\ & =\int_{0}^{2 \pi}(-y \hat{i}+x)(d x \hat{i}+d y) \\ & =\int_{0}^{2 \pi}(-y d x+x d y) \end{aligned}
Given : $\quad x=r \cos \theta$ and $y=r \sin \theta$
\begin{aligned} \therefore I&=\int_{0}^{2 \pi}[(-r \sin \theta)(-r \sin \theta) d \theta+(r \cos \theta)(r \cos \theta) d \theta] \\ &=\int_{0}^{2 \pi} r^{2} d \theta \\ &=\int_{0}^{2 \pi}(1+\cos \theta)^{2} d \theta \\ &=\int_{0}^{2 \pi}\left(1+\cos ^{2} \theta+2 \cos \theta\right) d \theta \\ &=\int_{0}^{2 \pi}\left(1+\frac{1+\cos 2 \theta}{2}+2 \cos \theta\right) \mathrm{d} \theta \\ &=3 \pi=9.425 \end{aligned}

 Question 3
The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is ___ (Round off to 3 decimal places).
 A 0.125 B 0.188 C 0.254 D 0.564
GATE EE 2023      Probability and Statistics
Question 3 Explanation:
Let probability of head $=\mathrm{P}$
Let probability of Tail $=q=P-1$
$\begin{array}{|c|c|c|c|c|} \hline Trial & 1 & 2 & 3 & ... \\ \hline Probability & [\mathrm{P} & \mathrm{qP} & [latex]\mathrm{q}^{2} \mathrm{P} & .... \\ \hline \end{array}$
$\therefore$ Expected No. of trail
\begin{aligned} & =\sum_{n=1}^{\infty} p+2 P q+3 q^{2} P+\ldots \\ & =P\left(1+2 q+3 q^{2}+\ldots\right) \\ & =P(1-q)^{-2} \\ & =\frac{P}{P^{2}}=\frac{1}{P} \end{aligned}

Given : Trial $=4$
$\frac{1}{P}=4 \Rightarrow P=\frac{1}{4}$
$\therefore \quad \mathrm{q}=\frac{3}{4}$

Now, probability of head for second trail. \begin{aligned} & =\mathrm{qP} \\ & =\frac{3}{4} \times \frac{1}{4}=\frac{3}{16}=0.1875 \end{aligned}
 Question 4
Consider the following equation in a 2-D realspace.

$\left|x_{1}\right|^{p}+\left|x_{2}\right|^{p}=1$ for $p \gt 0$

Which of the following statement(s) is/are true.
 A When $\mathrm{p}=2$, the area enclosed by the curve is $\pi$. B When $p$ tends to $\infty$, the area enclosed by the curve tends to 4. C When $p$ tends to 0 , the area enclosed by the curve is 1. D When $p=1$, the area enclosed by the curve is 2.
GATE EE 2023      Calculus
Question 4 Explanation:
Check option (A),
put $P=2$
$x_{1}^{2}+x_{2}^{2}=1$

Which is equation of circle whose radius is 1 .
$\therefore \quad$ Area $=\pi r^{2}=\pi$

Check option (B),
If $x_{2} \lt 1$ Then, $P \rightarrow \infty,\left|x_{2}\right|^{P} \rightarrow 0$
$\therefore$ For equation $\left|x_{1}\right|^{\mathrm{P}}+\left|\mathrm{x}_{2}\right|^{\mathrm{P}}=1$ satisfaction,
$\mathrm{x}_{1} \rightarrow 1 \text { or } \dot{Y}$

Curve :

$\therefore$ Area tends to 4 .

Check option (D),
Put $P=1$,
$\left|x_{1}\right|+\left|\mathrm{x}_{2}\right|=1$

Curve:

$\therefore$ Area of curve (square) $=(\sqrt{2})^{2}=2$
 Question 5
Three points in the $x-y$ plane are $(-1,0.8)$ $(0,2.2)$ and $(1,2.8)$. The value of the slope of the best fit straight line in the least square sense is ___ (Round off to 2 decimal places).
 A 0.25 B 0.5 C 0.75 D 1
GATE EE 2023      Differential Equations
Question 5 Explanation:
Straight line equation, $y=a x+b$ [Let]
where, $a=$ slope
By lest approximation,
$\sum \mathrm{y}_{i}=a \sum \mathrm{x}_{\mathrm{i}}+\mathrm{bn}$
and $\quad \sum x_{i} y_{i}=a \sum x_{1}^{2}+b \sum x_{i}$
$\begin{array}{|c|c|c|c|} \hline x & y & x^{2} & xy \\ \hline -1 & 0.8 & 1 & -0.8 \\ 0 & 2.2 & 0 & 0 \\ 1 & 2.8 & 1 & 2.8 \\ \hline \sum x=0 & \sum y=5.8 &\sum x^{2}=2 & \sum x y=2 \\ \hline \end{array}$
From eqn. (1), we get
\begin{aligned} 5.8 & =a(0)+3 b \Rightarrow 5.8=3 b \\ 2 & =a(2)+b(0) \\ \Rightarrow \quad a & =1 \end{aligned}

There are 5 questions to complete.