If m is a natural such that $m \leq 5,$ then the probability that the quadratic equation $x^{2} + m x + \frac{1}{2} + \frac{m}{2} = 0$ has real roots is

1/5

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2/3

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3/5

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Solution

The correct option is C
3/5

Discriminant D of the quadratic equation
$x^{2} + m x + \frac{1}{2} + \frac{m}{2} = 0$
is given by
$D = m^{2} - 4 (\frac{1}{2} + \frac{m}{2}) = m^{2} - 2 m - 2 = (m - 1)^{2} - 3 N o w, D \geq 0 \Leftrightarrow (m - 1)^{2} \geq 3$
This is possible for m=3, 4 and 5. Also, the total number of ways of choosing m is 5.
$∴$ Probability of the required event= 3/5.

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If m is a natural such that m≤5, then the probability that the quadratic equation x2+mx+12+m2=0 has real roots is

The correct option is C 3/5 Discriminant D of the quadratic equation x2+mx+12+m2=0 is given by D=m2−4(12+m2)=m2−2m−2=(m−1)2−3Now, D≥0⇔(m−1)2≥3 This is possible for m=3, 4 and 5. Also, the total number of ways of choosing m is 5. ∴ Probability of the required event= 3/5.

If m is a natural such that $m \leq 5,$ then the probability that the quadratic equation $x^{2} + m x + \frac{1}{2} + \frac{m}{2} = 0$ has real roots is