The Routh Array is as below :

\(\begin{array}{lllll} \mathrm{S}^{6} & 1 & 8 & 20 & 16 \\ \mathrm{S}^{5} & 2 & 12 & 16 & \\ \mathrm{S}^{4} & 2 & 12 & 16 & \\ \mathrm{S}^{3} & 0 & 0 & & \end{array}\)

The row of zero of this array will be replaced by coefficients of 

Let's analyze the given Routh array and determine how to replace the row of zeros.

Understanding the Routh Array and Row of Zeros:

• The Routh array is used to determine the stability of a linear time-invariant system.
• A row of zeros in the Routh array indicates the presence of roots on the imaginary axis (jω-axis) or roots that are symmetrical about the origin.
• To continue the Routh array and determine stability, we need to form an auxiliary polynomial from the row above the row of zeros.

Forming the Auxiliary Polynomial:

• The row above the row of zeros is s⁴.
• The coefficients in this row are used to form the auxiliary polynomial.
• The auxiliary polynomial is formed using only even powers of 's'.

In this case, the row s⁴ has coefficients 2, 12, and 16.

Therefore, the auxiliary polynomial is:

A(s) = 2s⁴ + 12s² + 16

Finding the Derivative of the Auxiliary Polynomial:

To replace the row of zeros, we need to find the derivative of the auxiliary polynomial with respect to 's'.

dA(s)/ds = 8s³ + 24s

Replacing the Row of Zeros:

The coefficients of the derivative will replace the row of zeros in the Routh array.

The coefficients of the derivative are 8 and 24.

We can simplify these coefficients by dividing by 8:

8/8 = 1 24/8 = 3

So, the simplified coefficients are 1 and 3.

The row of zeros will be replaced by the coefficients of s³ + 3s.

Therefore, the correct answer is option 2.