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 \( 6 \).

Answer 2

Step 1: Analyze the Nature of the Roots
According to Descartes' Rule of Signs or given information, the polynomial has:
- \( n = 2 \) real roots.
Since the polynomial is of degree 9, the total number of roots (real + complex) must be 9.
Therefore, the number of complex roots = \( 9 - 2 = 7 \).

Step 2: Breakdown of Complex Roots
Among these 7 complex roots:
- \( m = 3 \) complex roots lie on the imaginary axis.
- \( k = 1 \) complex root lies in the second quadrant.
(Note: Complex roots not on the real axis appear in conjugate pairs. So, if there's one root in the second quadrant, its conjugate lies in the third quadrant.)

Step 3: Compute the Required Expression
Multiply the values to get the desired result:
\[ m \cdot n \cdot k = 3 \times 2 \times 1 = 6. \]
Final Answer:
Hence, the required value is:
\(\boxed{6}\).