Question

The greatest value of the term independent of x in the expansion of (x sin p + x^–1 cos p)¹⁰, p ∈ R is ______.

Options

• 2⁵

• `(10!)/(2^5 (5!)^2)`

• `(10!)/(5!)^2`

• None of these

Answer 1

The greatest value of the term independent of x in the expansion of (x sin p + x^–1 cos p)¹⁰, p ∈ R is `underlinebb((10!)/(2^5 (5!)^2))`.

Explanation:

(x sin p + x^–1 cos p)¹⁰, general term is

T_r+1 = ¹⁰C_r(x sin p)^10–r(x^–1 cos p)^r.

For the term independent of x we have 10 – 2r = 0

or r = 5

Hence, independent term is

¹⁰C₅ sin⁵P cos⁵P = `""^10C_5 (sin^5 2p)/32`

which is greatest when sin 2p = 1.