Basics of Frequency Response

Q. In the system shown in figure, the input $x\left(t\right)=sint$. In the steady-state, the response $y\left(t\right)$ will be

1. $\frac{1}{\sqrt{2}}sin(t-{45}^{\circ })$
2. $\frac{1}{\sqrt{2}}sin(t+{45}^{\circ })$
3. $sin(t-{45}^{\circ })$
4. $sin(t+{45}^{\circ })$

Q. A system with transfer function:
$G\left(s\right)=\frac{(s^2+9)(s+2)}{(9s+1)(s+3)(s+4)}$
is excited $sin\left(\omega t\right)$. The steady-state output of the system is zero at

1. $\omega =1rad/s$
2. $\omega =2rad/s$
3. $\omega =3rad/s$
4. $\omega =4rad/s$

Q. The closed loop transfer function of a control system is given by
$\frac{C\left(s\right)}{R\left(s\right)}=\frac{1}{(1+s)}$
For the input $r\left(t\right)=sint,$ the steady state value of $c\left(t\right)$ is equal to

1. $\frac{1}{\sqrt{2}}cost$
2. 1
3. $\frac{1}{\sqrt{2}}sint$
4. $\frac{1}{\sqrt{2}}sin(t-\frac{\pi }{4})$