Question

If the roots of x⁵ – 40x⁴ + Px³ + Qx² + Rx + S = 0 are in G.P. and sum of their reciprocals is 10, then |S| is –

• A. 4
• B. 6
• C. 8
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Roots of the equation

x⁵ – 40x⁴ + Px³ + Qx² + Rx + S = 0

are in G.P. Let root be a, ar, ar², ar³, ar⁴.

\ a + ar + ar² + ar³ + ar⁴ = 40

and $\frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^{2}} + \frac{1}{ar^{3}} + \frac{1}{ar^{4}}$

= $\frac{1 + r + r^{2} + r^{3} + r^{4}}{ar^{4}}$ = 10

Ž a²r⁴ = 4 or ar² = ±2.

Now –S = product of roots a⁵r¹⁰ = (ar²)⁵ = ± 32.

\ |S| = 32.