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

# The coefficients of a quadratic equation ${\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0\left(\mathrm{a}\ne \mathrm{b}\ne \mathrm{c}\right)$ are chosen from first three prime numbers, the probability that roots ofthe equation are real is

1. A

2. B

3. C

¼

4. D

¾

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### Solution:

First three prime numbers are 2, 3 and 5
Total ways of choosing a, b, c = 3 x 2 x 1= 6   $\left[\because \mathrm{a}\ne \mathrm{b}\ne \mathrm{c}\right]$
For roots to be real ,${\mathrm{b}}^{2}-4\mathrm{ac}\ge 0$

Value of b              Possible values of a and c

2                             $\begin{array}{l}4-4\mathrm{ac}\ge 0⇒\mathrm{ac}\le 1\\ \mathrm{a}=\mathrm{c}=1\end{array}$

(No values of a and c)

3                             $\begin{array}{l}9-4\mathrm{ac}\ge 0⇒\mathrm{ac}\le \frac{9}{4}\\ \mathrm{ac}=1,\mathrm{ac}=2\\ \mathrm{a}=\mathrm{c}=1,\mathrm{a}=2,\mathrm{c}=1\\ \mathrm{a}=1,\mathrm{c}=2\end{array}$

(No values of a and c)

5

Favorable ways = 2

Required probability

$=\frac{2}{6}=\frac{1}{3}$

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