Statement 1 If one root of `Ax^(3)+Bx^(2)+Cx+D=0 A!=0`, is the arithmetic mean of the other two roots, then the relation `2B^(3)+k_(1)ABC+k_(2)A^(2)D=0` holds good and then `(k_(2)-k_(1))` is a perfect square.
Statement -2 If a,b,c are in AP then `b` is the arithmetic mean of a and c.

Options

A. Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement-1
B. Statement -1 is true, Statement -2 is true, Statement -2 is not a correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false, Statement -2 is true

Answer

A

Explanation

Let roots of `Ax^(3)+Bx^(2)+Cx+D=0`……….i
are `alpha-beta, alpha, alpha+beta` (in AP)
Then `(alpha-beta)+alpha+(alpha+beta)=-B/A`
`impliesalpha=-B/(3A)`, which is a root of Eq. (i)
Then `A alpha^(3)+B alpha^(2)+C alpha +D=0`
`impliesA(-B/(3A))^(3)+B(-B/(3A))^(2)+C(-B/(3A))+D=0`
`implies-(B^(3))/(27A^(2))+(B^(3))/(9A^(2))-(BC)/(3A)+D=0`
`implies2B^(3)-9ABC+27A^(2)D=0`
Now comparing with `2B^(3)+k_(1)ABC+k_(2)A^(2)D=0` we get
`k_(1)=-9,k_(2)=27`
`:.k_(2)-k_(1)=27-(-9)=36=6^(2)`
Hence both statement are true and Statement 2 is a correct explanation of Statement -1.