# GATE Electrical Engineering 2021

 Question 1
Let p and q be real numbers such that $p^{2}+q^{2}=1$. The eigenvalues of the matrix $\begin{bmatrix} p & q\\ q& -p \end{bmatrix}$ are
 A 1 and 1 B 1 and -1 C j and -j D pq and -pq
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Characteristic equation of A
\begin{aligned} \left|A_{2 \times 2}-\lambda I\right|&=(-1)^{2} \lambda^{2}+(-1)^{1} \text{Tr}(A) \lambda+|A|=0 \\ \lambda^{2}-(p-p) \lambda+\left(-p^{2}-q^{2}\right) &=0 \\ \Rightarrow \qquad \qquad\lambda^{2}-1 &=0 \\ \Rightarrow \qquad\qquad \lambda &=\pm 1 \end{aligned}
 Question 2
Let $p\left ( z\right )=z^{3}+\left ( 1+j \right )z^{2}+\left ( 2+j \right )z+3$, where z is a complex number.
Which one of the following is true?
 A $\text{conjugate}\:\left \{ p\left ( z \right ) \right \}=p\left ( \text{conjugate} \left \{ z \right \} \right )$ for all z B The sum of the roots of $p\left ( z \right )=0$ is a real number C The complex roots of the equation $p\left ( z \right )=0$ come in conjugate pairs D All the roots cannot be real
Engineering Mathematics   Complex Variables
Question 2 Explanation:
Since sum of the roots is a complex number
$\Rightarrow$ absent one root is complex
So all the roots cannot be real.

 Question 3
Let $f\left ( x \right )$ be a real-valued function such that ${f}'\left ( x_{0} \right )=0$ for some $x _{0} \in\left ( 0,1 \right )$, and ${f}''\left ( x \right )> 0$ for all $x \in \left ( 0,1 \right )$. Then $f\left ( x \right )$ has
 A no local minimum in (0,1) B one local maximum in (0,1) C exactly one local minimum in (0,1) D two distinct local minima in (0,1)
Engineering Mathematics   Calculus
Question 3 Explanation:
$x_{0} \in(0,1)$, where $f(x)=0$ is stationary point
and $f^{\prime \prime}(x)>0 \qquad \qquad \forall x \in(0,1)$
So $\qquad \qquad f^{\prime}\left(x_{0}\right)=0$
and $\qquad \qquad f^{\prime}(0)>0, \text { where } x_{0} \in(0,1)$
Hence, f(x) has exactly one local minima in $(0,1)$
 Question 4
For the network shown, the equivalent Thevenin voltage and Thevenin impedance as seen across terminals 'ab' is

 A $\text{10 V}$ in series with $12\:\Omega$ B $\text{65 V}$ in series with $15\:\Omega$ C $\text{50 V}$ in series with $2\:\Omega$ D $\text{35 V}$ in series with $2\:\Omega$
Electric Circuits   Network Theorems
Question 4 Explanation:
Given circuit can be resolved as shown below,

$V_{T H}=15+50=65 \mathrm{~V}$

\begin{aligned} V_{x} &=2+3+10=15 \mathrm{~V} \\ R_{\mathrm{TH}} &=\frac{V_{x}}{1}=15 \Omega \end{aligned}
 Question 5
Which one of the following vector functions represents a magnetic field $\overrightarrow{B}$?
($\hat{X}$, $\hat{Y}$ and $\hat{Z}$ are unit vectors along x-axis, y-axis, and z-axis, respectively)
 A $10x\hat{X}+20y\hat{Y}-30z\hat{Z}$ B $10y\hat{X}+20x\hat{Y}-10z\hat{Z}$ C $10z\hat{X}+20y\hat{Y}-30x\hat{Z}$ D $10x\hat{X}-30z\hat{Y}+20y\hat{Z}$
Electromagnetic Theory   Magnetostatic Fields
Question 5 Explanation:
If $\vec{B}$ is magnetic flux density then $\vec{\nabla} \cdot \vec{B}=0$
\begin{aligned} &\vec{\nabla} \cdot \vec{B}=\frac{\partial B x}{\partial x}+\frac{\partial B y}{\partial y}+\frac{\partial B z}{\partial z}\\ &\frac{\partial}{\partial x}(10 x)+\frac{\partial}{\partial y}(20 y)+\frac{\partial}{\partial z}(-30 z)=\vec{\nabla} \cdot \vec{B}\\ &\qquad \qquad \vec{\nabla} \cdot \vec{B}=10+20-30=0 \end{aligned}

There are 5 questions to complete.