Find the greatest value of the term independent of in the expansion of (x sin a + cos a/ x )^10 where a ∈ R

Question

Find the greatest value of the term independent of in the expansion of \((x sina+\frac{cosa}{x})^{10}\) , where a ∈ R

Answer 1

Let ( r + 1)^th term be independent of x 

We have, 

T_{r + 1} = ¹⁰C_r (x sin a)^{10 – r} \((\frac{cosa}{x})\) ^r

= ¹⁰C_r x^{10 – 2r} (sin a )^{10 – r} (cos a ) ^r

 If it is independent of x , then r = 5

∴ term independent of 

x = T⁶ = ¹⁰C₅ (sin a cos a)⁵ 

= ¹⁰C₅ × 2 ^{– 5} (sin 2a) ⁵

Clearly, It is greatest when 2 = \(\frac{π}{2}\)and its greatest

Value is ¹⁰C₅ x 2^-5 = \(\frac{101}{2^5(5)^2}\)