A stable real linear time-invariant system with single pole at p, has a transfer function H(s)=s2+100/s−p with a dc gain of 5. The smallest positive frequency, in rad/s at unity gain is closest to:

The correct option is 8.84.

To find the smallest positive frequency at unity gain, we follow these steps:

Step 1: Determine the pole $p$ using the DC gain
The transfer function of the system is given by:
$$H\left(s\right)=\frac{s^2+100}{s-p}$$ The DC gain of a system is the value of the transfer function at $s=0$:
$$H\left(0\right)=\frac{0^2+100}{0-p}=-\frac{100}{p}$$ Given that the DC gain is 5:
$$-\frac{100}{p}=5\Rightarrow p=-20$$ Since the system is stable, the pole $p=-20$ lies in the left half of the s-plane, which confirms stability. The transfer function is:
$$H\left(s\right)=\frac{s^2+100}{s+20}$$

Step 2: Find the magnitude at frequency $\omega$
Substitute $s=j\omega$ into the transfer function:
$$H\left(j\omega \right)=\frac{{\left(j\omega \right)}^2+100}{j\omega +20}=\frac{100-{\omega }^2}{20+j\omega }$$ The magnitude of the transfer function is:
$$|H(j\omega )|=\frac{|100-{\omega }^2|}{\sqrt{{20}^2+{\omega }^2}}$$

Step 3: Set magnitude to unity gain
We set $|H(j\omega )|=1$:
$$\frac{|100-{\omega }^2|}{\sqrt{400+{\omega }^2}}=1$$ Squaring both sides:
$${(100-{\omega }^2)}^2=400+{\omega }^2$$ Let $x={\omega }^2$ (where $x>0$):
$${(100-x)}^2=400+x$$ $$10000-200x+x^2=400+x$$ $$x^2-201x+9600=0$$

Step 4: Solve the quadratic equation
Using the quadratic formula:
$$x=\frac{201\pm \sqrt{{201}^2-4\left(1\right)\left(9600\right)}}{2}$$ $$x=\frac{201\pm \sqrt{40401-38400}}{2}$$ $$x=\frac{201\pm \sqrt{2001}}{2}$$ Since $\sqrt{2001}\approx 44.73$:
$$x_1=\frac{201-44.73}{2}\approx 78.13$$ $$x_2=\frac{201+44.73}{2}\approx 122.87$$

Step 5: Solve for the smallest positive frequency $\omega$
Since $x={\omega }^2$, the positive frequencies are:
$${\omega }_1=\sqrt{78.13}\approx 8.84\text{ rad/s}$$ $${\omega }_2=\sqrt{122.87}\approx 11.08\text{ rad/s}$$ The smallest positive frequency is ${\omega }_1\approx 8.84\text{ rad/s}$.