1.

If (i) A=\(\begin{bmatrix} \cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha \end{bmatrix}\),then verify that A'A=I
(ii) A= \(\begin{bmatrix} \sin\alpha & \cos\alpha\\ -\cos \alpha & \sin\alpha \end{bmatrix}\),then verify that A'A=I

2.

Find the vector and the cartesian equations of the lines that pass through the origin and(5,-2,3).

3.

A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is

• $0.1\,kgm^2s^{-1}$
• $0.04\,kgm^2s^{-1}$
• $0.025\,kgm^2s^{-1}$
• $0.01\,kgm^2s^{-1}$

4.

A series LCR circuit with R = 20 W, L = 1.5 H and C = 35 μF is connected to a variable-frequency 200 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?

5.

If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix}  -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that 
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

6.

Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 &  3\\ 5 & 7 \end{bmatrix}\)