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

\frac{7}{2} | |

-\frac{2}{7} | |

\frac{2}{7} | |

-\frac{7}{2} |

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

\frac{2}{3} | |

\frac{4}{3} | |

2 | |

4 |

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}

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 ?

2 e^{\frac{j 2 \pi}{8}} | |

2 e^{\frac{j \pi}{8}} | |

2 e^{\frac{j 5 \pi}{8}} | |

2 e^{\frac{j 9 \pi}{8}} |

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}

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

-\pi(1+j) | |

\pi(1+j) | |

\pi(1-j) | |

-\pi(1-j) |

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}

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

\left\{\lambda_{k} \operatorname{det}\mid 1 \leq k \leq n\right\} and \left\{P v_{k} \mid 1 \leq k \leq n\right\} | |

\left\{\lambda_{k} \mid 1 \leq k \leq n\right\} and \left\{v_{k} \mid 1 \leq k \leq n\right\} | |

\left\{\lambda_{k} \mid 1 \leq k \leq n\right\} and \left\{P v_{k} \mid 1 \leq k \leq n\right\} | |

\left\{\lambda_{k} \mid 1 \leq k \leq n\right\} and \left\{P^{-1} v_{k} \mid 1 \leq k \leq n\right\} |

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.

\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.