Question

If the number of real roots of \( x^9 - x^5 + x^4 - 1 = 0 \) is \( n \), the number of complex roots having argument on imaginary axis is \( m \), and the number of complex roots having argument in the second quadrant is \( k \), then \( m.n.k \) is:

• A. \( 6 \)
• B. \( 9 \)
• C. \( 12 \)
• D. \( 24 \)

Hint:

Descartes’ rule of signs is useful in determining the number of real and complex roots in polynomial equations.

Answer 1

The Correct Option is A

Step 1: Finding the Number of Real and Complex Roots By Descartes' Rule of Signs, the number of real roots is \( n = 2 \).
Step 2: Identifying Complex Roots For a polynomial of degree 9, the remaining 7 roots are complex. Among them, - \( m = 3 \) complex roots lie on the imaginary axis.
- \( k = 1 \) complex root lies in the second quadrant.
Step 3: Finding \( m \cdot n \cdot k \) \[ m \cdot n \cdot k = 3 \times 2 \times 1 = 6. \] Thus, the correct answer is \( 6 \).