Question

Let m, n be the roots of the equation x² + qx + r = 0 and let s, t be the roots of the equation x² + bx + c = 0.

1.  If  m/n = s/t, then

(A)  r²c = qb²

(B)  r²b = qc² 

(C)  rb² = cq² 

(D)  rc² = bq²

2.  If mn = st, then q² − b² is equal to

(A)  [(m + t) + (n + s)][(m + s) − (n + t)]

(B)  [(m + t) + (n + s)][(m + s) + (n + t)]

(C)  [(m + t) − (n + s)][(m + s) + (n + t)]

(D)  [(m + t) − (n + s)][(m + s) − (n + t)]

3. If m = s and rq = bc, then n and t are the roots of the equation 

(A)  x² − (b + q)x + bq = 0

(B)  x² − (b + r)x + rb = 0 

(C)  x² − (c + q)x + cq = 0

(D)  x² − (c + r)x + rc = 0

Answer 1

Correct option  1. (C), 2.  (D)  3.  (A)

1.  We have

2.  For mn = st, 

q² − b² = (m + n)² − (s + t) = (m − n) ² − (s − t)²

= [(m + t) − (n + s)][(m + s) − (n + t)]

3.   With m = s, rq = bc, s + n = − q, sn = r and s + t = − b, st = c, we have 

 

n − t = b − q and n/t = r/c

⇒ rt/c - t = b - q

⇒ t(r − c) = cb - cq - q(r - c)

⇒ t = q ⇒ n = b

Hence, n and t are the roots of the equation x² − (b + q)x + bq = 0.