If two roots of the equation `x^3 - px^2 + qx - r = 0` are equal in magnitude but opposite in sign, then:

Options

A. r = pq
B. `r = 2p^(3) + pq`
C. `r = p^(2)q`
D. none of these

Answer

A

Explanation

Let `alpha, beta, gamma` be the roots of the given equation such that `alpha = - beta`. Then, `alpha + beta + gamma = p rArr gamma = p`.
Since `gamma` is a root of the given equation, so it satisfies the equation i.e. `gamma^(3) - p gamma^(2) + q gamma - r = 0 rArr p^(3) - p^(3) + pq - r = 0 rArr r = pq`