Roots on Imaginary Axis

Q. A system with the open loop transfer function:

$G\left(s\right)=\frac{K}{s(s+2)(s^2+2s+2)}$ is connected in a negative feedback configuration with a feedback gain of unity. For the closed-loop system to be marginally stable, the value of k is

1. 5

Q.

The loop transfer function of a negative feedback system is
$G\left(s\right)H\left(s\right)=\frac{K(s+11)}{s(s+2)(s+8)}$
The value of $K$ , for which the system is marginally stable is

1. 160

Q. The first two rows of Routh's tabulation of a third order equation are as follows:

S${}^3$ 2 2
S${}^2$ 4 4

This means there are :

1. two roots at s = $\pm$ j and one root in right half s-plane
2. two roots at s = $\pm$ j2 and one root in left half s-plane
3. two roots at s = $\pm$ j2 and one root in right half s-plane
4. two roots at s = $\pm$ j and one root in left half s-plane

Q. Which one of the following options correctly describes the locations of the roots of the equation $s^4+s^2+1=0$ on the complex plane?

1. Four left half plane (LHP) roots
2. One right half plane (RHP) root , one LHP root and two roots on the imaginary axis
3. Two RHP roots and two LHP roots
4. All four roots are on the imaginary axis

Q.

Consider a standard negative feedback configuration with
$G\left(s\right)=\frac{1}{(s+1)(s+2)}andH\left(s\right)=\frac{s+\alpha }{s}.$ For the closed loop system to have poles on the imaginary axis, the value of $\alpha$ should be equal to (up to one decimal place)

1. 9