Question

The greatest value of the term independent of x, as a varies over R, in the expansion of $\left( x\cos\alpha + \frac{\sin\alpha}{x} \right)^{20}$ is –

• A. ²⁰C₁₀
• B. ²⁰C₁₅
• C. ²⁰C₁₉
• D. None of these

Answer 1

Answer: (A)

²⁰C₁₀

Answer 2

The general term in the expansion of

$\left( x\cos\alpha + \frac{\sin\alpha}{x} \right)^{20}$is

Ž ²⁰C_r (x cos a)^{20 –r} $\left( \frac{\sin\alpha}{x} \right)^{r}$

= ²⁰C_r x^{20 – 2r} (cos a)^{20 – r} (sin a)^r

For this term to be independent of x, we get

20 – 2r = 0

Ž r = 10.

Let b = Term independent of x

= ²⁰C₁₀ (cos a)¹⁰ (sin a)¹⁰

= ²⁰C₁₀ (cos a sin a)¹⁰

Thus the greatest possible value is ²⁰C₁₀ $\left( \frac{1}{2} \right)^{10}$.