Find the greatest value of the term independent of `x` in the expansion of `(xsinalpha+(cosalpha)/x)^(10)` , where `alpha in R`.

Options

A. `2^(5)`
B. `(10!)/((5!)^(2))`
C. `(10!)/(2^(5) xx(5 !))^(2)`
D. none of these

Answer

c

Explanation

Let `(r + 1)^(th)` term be indepandent of x.
We have,
`T_(r + 1) = ""^(10)C_(r) (x sin alpha)^(10 - r) ((cos alpha)/(x))^(r)`
`rArr T_(r +1) = ""^(10)C_(r) x ^(10 - 2r) (sin alpha)^(10 - r) (cos alpha )^(r)`
If it is independent of x, then r = 5
`therefore ` Term independent of x,
`= T_(6)= ""^(10)C_(5)(sin alpha cos alpha)^(5) = ""^(10)C_(5) xx2 ^(-5) (sin 2 alpha)^(5)`
Clearly , it is greatest when 2 `alpha = pi 2` and its greatest value is
`""^(10)C_(5) xx2^(-5) = (10!)/(2^(5) (5!)^(2))`