Frequency Response of First Order System

Q. A system with the transfer function $\frac{Y\left(s\right)}{X\left(s\right)}=\frac{s}{s+p}$ has an output $y\left(t\right)=\cos (2t-\frac{\pi }{3})$ for the input signal $x\left(t\right)=p\cos (2t-\frac{\pi }{2}).$ Then, the system parameter ${}^{\prime }{p}^{\prime }$ is

1. $\sqrt{3}$
2. $\frac{2}{\sqrt{3}}$
3. $1$
4. $\frac{\sqrt{3}}{2}$

Q. In the system shown below, $x\left(t\right)=\left(\sin t\right)u\left(t\right)$. In steady-state, the response $y\left(t\right)$ will be

1. $\frac{1}{\sqrt{2}}\sin (t-\frac{\pi }{4})$
2. $\frac{1}{\sqrt{2}}\sin (t+\frac{\pi }{4})$
3. $\frac{1}{\sqrt{2}}{e}^{-t}\sin t$
4. $\sin t-\cos t$

Q. A system with zero initial conditions has the closed loop transfer function
$T\left(s\right)=\frac{s^2+4}{(s+1)(s+4)}.$
The system output zero at the frequency.

1. 0.5 rad/sec
2. 1 rad/sec
3. 2 rad/sec
4. 4 rad/sec

Q. The transfer function of a system is given by, $\frac{V_0\left(s\right)}{V_i\left(s\right)}=\frac{1-s}{1+s}.$ Let the output of the system be $V_0\left(t\right)=V_{m}sin(\omega t+j)$ for the input, $V_i\left(t\right)=V_{m}sin\left(\omega t\right)$ then the minimum and maximum values of $j$(in radians) are respectively

1. $\frac{-\pi }{2}and\frac{\pi }{2}$
2. $\frac{-\pi }{2}and0$
3. $0and\frac{\pi }{2}$
4. $-\pi and0$

Q. A stable real linear time-invariant system with single pole at p, has a transfer function $H\left(s\right)=\frac{s^2+100}{s-p}$
with a dc gain of 5. The smallest positive frequency, in rad/s at unity gain is closed to:

1. 11.08
2. 78.13
3. 8.84
4. 122.87

Q. The Closed loop transfer function of a control system is given by $\frac{C\left(s\right)}{R\left(s\right)}=\frac{1}{s+1}$. For the input r(t) = sin t, the steady response c(t)______.

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