Question

Q. Let p and q real number such that `p!= 0`,`p^2!=q` and `p^2!=-q`. if `alpha` and `beta` are non-zero complex number satisfying `alpha+beta=-p` and `alpha^3+beta^3=q`, then a quadratic equation having `alpha/beta` and `beta/alpha` as its roots is
A. `(p^(3) + q )x^(2) - (p^(3) + 2p ) x + (p^(3) + q ) = 0 `
B. `(p^(3) + q )x^(2) - (p^(3) - 2p ) x + (p^(3) + q ) = 0 `
C. `(p^(3) - q )x^(2) - (5p^(3) - 2p ) x + (p^(3) - q ) = 0 `
D. `(p^(3) - q )x^(2) - (5p^(3) + 2p ) x + (p^(3) - q ) = 0 `

Answer 1

Correct Answer - 2
` alpha ^(3) + beta ^(3) = q `
` rArr (alpha + beta)^(3) - 3 alpha beta (alpha + beta) = q `
`rArr - P^(3) + 3 p alpha beta = q rArr alpha beta = (q + p^(3))/(3p)`
Required equation is
`x^(2) - ((alpha )/(beta)+(beta)/(alpha )) x + (alpha )/(beta).(beta)/(alpha ) = 0 `
`x^(2) - ((alpha ^(2) + beta^(2)))/(alpha beta) x + 1 = 0 `
`rArr x^(2) -( ((alpha + beta)^(2) - 2 alpha beta)/(alpha beta)) x + 1 = 0 `
`rArr x^(2) - (P^(2) ((p^(3) + q)/(3p)))/((p^(3) + q)/(3p))x + 1= 0 `
`rArr (p^(3) + q )x^(2) - (3p^(3) - 2p^(3) - 2p^(3) - 2q)x + (p^(3) + q ) = 0 `
`rArr (p^(3) + q) x^(2) - (p^(3) - 2p ) x + (p^(3) + =q ) = 0` .