The positive value of `lambda` for which the coefficient of `x^(2)` in the expression `x^(2)(sqrt(x)+(lambda)/(x^(2)))^(10)` is 720 is

Options

A. 3
B. `sqrt(5)`
C. `2 sqrt(2)`
D. 4

Explanation

The general term in the expansion of binomial expression `(a + b)^(n)` is `T_(r + 1) = .^(n)C_(r ) a^(n - r) b^(r )`, so the general term in the expansio of binomial expression
`x^(2) (sqrt(x) + (lambda)/(x^(2)))^(10)` is
`T_(r + 1) = x^(2) (.^(10)C_(r ) (sqrt(x))^(10-r) ((lambda)/(x^(2)))^(r )) = .^(10)C_(r ) x^(2). x^((10 - r)/(2) lambda_(r ) x^(-2r)`
`= .^(10)C_(r ) lambda^(r ) x^(2 + (10 - 2)/(2) - 2r)`
No, for the coefficeint of `x^(2)`, put `2 + (10 - r)/(2) - 2r = 2`
`implies (10 - r)/(2) - 2r = 0`
`implies 10 - r = 4r implies r = 2`
So, the coefficient of `x^(2)` is `.^(10)C_(2) lambda^(2) = 720` [given]
`implies (10 !)/(2!8) lambda_(2) = 720 implies (10.9.8)/(2.8) lambda^(2) = 720`
`implies 45 lambda^(2) = 720`
`implies lambda^(2) = 16 implies lambda = +- 4`
`:. lambda = 4" "[lambda gt 0]`