Question

If the sum of the coefficients in the expansion of
`(b + c)^(20) {1 +(a -2) x}^(20)` is equal to square of the sum of the
coefficients in the expansion of `[2 bcx - (b + c)y]^(10)`, where a, b, c are
positive constants, then
A. ` ge sqrt((a c)`
B. `(b +c)/(2) ge a`
C. c, a and b are in G. P
D. `(1)/(c),(1)/(a),(1)/(b)` are in H.P

Answer 1

Correct Answer - b
The sum of the coefficients in the expansion of
`(b + c )^(20){1 + (a - 2)x}^(20)` is given by
`S_(1) = ( b + c)^(20) (a - 1)^(20)` [ Putting x = 1]
The sum of the coefficients in the expansion of
`[2 bcx - (b + c)y]^(10)` is given by
`S_(2) = [ 2 bc - (b + c)]^(10)` [ Putting x = y = 1]`
If is given that
`S_(1) = S_(2)""^(2)`
`rArr (b + c)^(20) (a -1)^(20) = (2bc - b - c )^(20)`
`rArr (b + c ) (a -1) - =2bc - b - c`
`rArr a - 1 = (2bc)/(b +c) - 1`
`rArr a= (2 bc)/(b+c)`
`rArr a ` is the H.M of of b and c.
But, `(b +c)/(2)` is the A. M. of b and c. Therefore,
A. M. `ge` H.M.`rArr (b+c)/(2) ge a` .