Question

If 0 < a < b < c < d, then the quadratic equation ax² + {1 – a(b + c)} x + abc – d = 0 (1)has –

• A. Real and distinct roots out of which one lines between c and d
• B. Real and distinct roots out of which one lines between a and b
• C. Real and distinct roots out of which one lines between b and c
• D. Non-real roots

Answer 1

Answer: (A)

Real and distinct roots out of which one lines between c and d

Answer 2

We can rewrite (1) as

ax² – a(b + c)x + abc + x – d = 0

or a(x – b) (x – c) + x – d = 0

Let ƒ(x) = a(x – b) (x – c) + x – d

As a > 0, y = ƒ(x) represents a parabola which open upwards. See Fig.

Also ƒ(2) = b – d < 0

ƒ(3) = c – d < 0, and ƒ(4) = a(d – b) (d – c) > 0

Thus, ƒ(x) = 0 has root between – and b and between c and d.