Question

Consider an even polynomial $p(s)$ given by \[ p(s) = s^4 + 5s^2 + 4 + K, \] where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all its roots on the imaginary axis is ________________.

• A. $-4 \le K \le \dfrac{9}{4}$
• B. $-3 \le K \le \dfrac{9}{2}$
• C. $-6 \le K \le \dfrac{5}{4}$
• D. $-5 \le K \le 0$

Hint:

For an even polynomial, substitute $x = s^2$ to reduce the problem to checking whether the resulting quadratic has real, non-positive roots.

Answer 1

The Correct Option is A

The polynomial is even: \[ p(s) = s^4 + 5s^2 + (4+K). \] Let \[ x = s^2. \] Then the polynomial becomes a quadratic in \(x\): \[ p(s) = x^2 + 5x + (4+K). \] For the roots of \(p(s)\) to lie purely on the imaginary axis, the roots \(s\) must satisfy: \[ s = \pm j\omega \Rightarrow x = s^2 = -\omega^2 \le 0. \] Thus both roots of \[ x^2 + 5x + (4+K) = 0 \] must be real and non-positive. Step 1: Roots must be real → discriminant ≥ 0 \[ \Delta = 25 - 4(4+K) \ge 0 \] \[ 25 - 16 - 4K \ge 0 \] \[ 9 - 4K \ge 0 \] \[ K \le \frac{9}{4}. \] Step 2: Roots must be non-positive Sum of roots: \[ x_1 + x_2 = -5<0 \quad (\text{always true}) \] Product of roots: \[ x_1 x_2 = 4 + K \ge 0 \] \[ K \ge -4. \] Step 3: Combine conditions \[ -4 \le K \le \frac{9}{4}. \] Final Answer: $-4 \le K \le \dfrac{9}{4}$