First slide

Binomial theorem for positive integral Index

Question

The greatest value of the term independent of X, as $α$ varies over R, in the expansion of ${(xcos α + \frac{\sin α}{x})}^{20} i s$

Moderate

A

$^{20} C_{10}$
B

$^{20} C_{15}$
C

$^{20} C_{19}$
D

None of these

Solution

The general term in the expansion of ${(xcos α + \frac{\sin α}{x})}^{20}$
$\begin{array}{l} {is}^{20} C_{r} (xcos α)^{20 - r} {(\frac{\sin α}{x})}^{r} \\ =^{20} C_{r} x^{20 - 2 r} (\cos α)^{20 - r} (\sin α)^{r} \end{array}$
For this term to be independent of X ,we get
20-2r=0 + r=10
Let $β$ = Term independent of x
=²⁰C₁₀(cos $α$ )¹⁰ (sin $α$ )¹⁰
=²⁰C₁₀(cos $α$ sin $α$ )¹⁰=²⁰C₁₀ ${(\frac{\sin 2 α}{2})}^{10}$
Thus, the greatest possible value of $β$ is ²⁰C₁₀ ${(\frac{1}{2})}^{10}$ .

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