If `p, q, r` each are positive rational number such tlaht `p gt q gt r` and the quadratic equation `(p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0` has a root in `(-1 , 0)` then which of the following statement hold good? (A) `(r + p)/(q) lt 2` (B) Both roots of given quadratic are rational (C) The equation `px^(2) + 2qx + r = 0` has real and distinct roots (D) The equation `px^(2) + 2qx + r = 0` has no real roots

Options

A. `(r + p)/(q) lt 2`
B. Both roots of given quadratic are rational
C. The equation `px^(2) + 2qx + r = 0` has real and distinct roots
D. The equation `px^(2) + 2qx + r = 0` has no real roots

Answer

A, B, C

Explanation

`:' f(1) = 0`
`f(x) = (p + q -2r)x^(2) (q + r - 2p)x + (r + p - 2q)`
`:.` one root is `n = 1`
`:. f(0) lt 0 rArr r + p - 2q lt 0 rArr (r + p)/(q) lt 2` (A)
Also product of roots `= (r + p - 2q)/(p + q - 2p) = alpha, 1`
(`:' p, q, r` are rational) `:. alpha` is rational (B)
Also `(r + p)lt2q rArr (r + p)^(2)lt4q^(2)`
`rArr ubrace(4q^(2) - 4pr)_("Discriminant")gtubrace((p - r)^(2))_("positive")`
`rArr` root of `px^(2) + 2qx + tau = 0`are real and distinct (C)