Coefficient of `x^(2)`in the expansion of `(x^(3) + 2x^(2) + x + 4)^(15)` is

Options

A. Prime
B. Composite
C. 0
D. Perfect square

Answer

D

Explanation

Let `(x^(3)+2x^(2)+x+4)^(15) = a_(0)+a_(1)x+a_(2)x^(2)+"….."`
Differentiating w.r.t we get
`15(x^(3)+2x^(2)+x+4)^(14)(3x^(2)+4x+1)`
`= a_(1) + 2a_(2)x+3a_(3)x^(2) + "…."`
Again , differentiating w.r.tx, we get
`15{(x^(3)+2x^(2)+x+4)^(14)(6x+4)+14(x^(3)+2x^(2)+x+4)^(13)(3x^(2)+4x+1)^(2)}`
`=2a_(2) + 6a_(3)x+"...."`
Now, putting `x = 0`, we get
`15(4xx4^(14)+14xx4^(13)) = 2a_(2)`
`rArr 2a_(2) = 15 xx 4^(13) xx (16+14)`
`rArr a_(2) = (15)^(2) xx 2^(26)`
which is perfect squar.