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Question: The coefficients of a quadratic equation ax<sup>2</sup> + bx + c = 0 (a ≠ b ≠ c) are chosen from fir...

The coefficients of a quadratic equation ax2 + bx + c = 0 (a ≠ b ≠ c) are chosen from first three prime numbers, the probability that roots of the equations are real is

A

2/3

B

¼

C

1/3

D

φ

Answer

1/3

Explanation

Solution

First three prime numbers are 2, 3 and 5.

For roots to be real, D ≥ 0

⇒ b2 – 4ac ≥ 0,

Clearly if b = 2 or 3,

B2 – 4ac < 0,

Thus real roots are obtained by

b = 5, a = 2, c= 3

b = 5, a = 3, c = 2 i.e., in two ways.

Total ways of choosing a, b, c = 3 x 2 x 2 = 6

∴ Required probability = 26=13\frac { 2 } { 6 } = \frac { 1 } { 3 }