GATE Electronics and Communication 2023

 Question 1
Let $v_{1}=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right]$ and $v_{2}=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]$ be two vectors. The value of the coefficient $\alpha$ in the expression $v_{1}=\alpha v_{2}+e$, which minimizes the length of the error vector $e$, is
 A $\frac{7}{2}$ B $-\frac{2}{7}$ C $\frac{2}{7}$ D $-\frac{7}{2}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} e & =V_{1}-\alpha V_{2} \\ e & =(i+2 k+0 k)-\alpha(2 i+j+3 k) \\ \hat{e} & =(1-2 \alpha) \hat{i}+(2-\alpha) \hat{j}+(0-3 \alpha) \hat{k} \\ |\hat{e}| & =\sqrt{(1-2 \alpha)^{2}+(2-\alpha)^{2}+(-3 \alpha)^{2}} \\ |\hat{e}|^{2} & =5+14 \alpha^{2}-8 \alpha \text { to be minimum at } \frac{\partial e^{2}}{\partial \alpha}=28 \alpha-8=0 \\ \alpha & =\frac{2}{7} \text { stationary point } \end{aligned}
 Question 2
The rate of increase, of a scalar field $f(x, y, z)=x y z$ in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is
 A $\frac{2}{3}$ B $\frac{4}{3}$ C $2$ D $4$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} f(x, y, z) & =x y z \\ \overline{\nabla f} & =\hat{i} f_{x}+\hat{j} f_{y}+\hat{k} f_{z} \\ & =\hat{i}(y z)+\hat{j}(x z)+\hat{k}(x y) \\ \overline{\nabla f}_{(0,2,1)} & =\hat{i}(2)+0 \hat{j}+0 \hat{k} \end{aligned}

Directional derivative,
\begin{aligned} D \cdot D & =\overline{\nabla f} \cdot \frac{\bar{a}}{|\bar{a}|} \\ & =(2 \hat{i}+0 \hat{j}+0 \hat{k}) \cdot \frac{(2 \hat{i}+\hat{j}+2 \hat{k})}{\sqrt{2^{2}+1^{2}+2^{2}}}=\frac{4}{\sqrt{9}}=\frac{4}{3} \end{aligned}

 Question 3
Let $w^{4}=16 j$. Which of the following cannot be a value of $w$ ?
 A $2 e^{\frac{j 2 \pi}{8}}$ B $2 e^{\frac{j \pi}{8}}$ C $2 e^{\frac{j 5 \pi}{8}}$ D $2 e^{\frac{j 9 \pi}{8}}$
Engineering Mathematics   Complex Variables
Question 3 Explanation:
$w=(2) j^{1 / 4}$
$w=2(0+j)^{1 / 4}$
$w=2\left[e^{j(2 n+1) \pi / 2}\right]^{1 / 4}$ $=2\left[e^{j(2 n+1) \pi / 8}\right]$

For $n=0$, $w=e^{j \pi / 8}$
For $n=2$, $w=2 e^{5 \pi j / 8}$
For $n=4$, $w=2 e^{9 \pi j / 8}$
 Question 4
The value of the contour integral, $\oint_{c}\left(\frac{z+2}{z^{2}+2 z+2}\right) d z$, where the contour $C$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is
 A $-\pi(1+j)$ B $\pi(1+j)$ C $\pi(1-j)$ D $-\pi(1-j)$
Engineering Mathematics   Calculus
Question 4 Explanation:
$I=\oint_{c} \frac{z+2}{z^{2}+2 z+2} d z ; \quad c=\left|z+1-\frac{3}{2} i\right|=1$

Poles are given $(z+1)^{2}+1=0$
\begin{aligned} & z+1= \pm \sqrt{-1} \\ & z=-1+j,-1-j \end{aligned}
where $-1-i$ lies outside ' $c$ '
$z=(-1,1) \text { lies inside } 'c'$.

by $\mathrm{CRT}$
\begin{aligned} \oint_{c} f(z) d z & =2 \pi i \operatorname{Res}(f(z), z=-1+j) \\ & =2 \pi i\left(\frac{z+2}{2(z+1)}\right)_{z=-1+i}\\ & =2 \pi i\left(\frac{-1+j+2}{2(-1+j+1)}\right) \\ & =\pi(1+j) \end{aligned}
 Question 5
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1} A B$, respectively, are
 A $\left\{\lambda_{k} \operatorname{det}\mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ B $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$ C $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ D $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P^{-1} v_{k} \mid 1 \leq k \leq n\right\}$
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
\begin{aligned} & B & =P^{-1} A P \\ A & =P B P^{-1}\end{aligned}

$\Rightarrow A, B$ are called matrices similar.
$\Rightarrow$ Both $A, B$ have same set 7 eigen values
But eigen vectors of $A, B$ are different.

Let $B X=\lambda X$
$\Rightarrow \quad\left(P^{-1} A P\right) X=\lambda X$
$\Rightarrow \quad A(P X)=\lambda(P X)$

$\therefore$ Eigen vectors of $A$ are $P X$.

There are 5 questions to complete.