If m is a natural number such that m ≤5 , then the probability that the quadratic equationx2+mx+12+m2=0 has real roots is

S=1,2,3,4,5 ⇒n(S)=5, and x2+mx+12+m2=0 has real roots Δ≥0 ⇒m2-411+m2≥0 ⇒m2-2m-2≥0 ⇒m={3,4,5},n(E)=3 ⇒PE=35

If m is a natural number such that m ≤5 , then the probability that the quadratic equationx2+mx+12+m2=0 has real roots is

S=1,2,3,4,5 ⇒n(S)=5, and x2+mx+12+m2=0 has real roots Δ≥0 ⇒m2-411+m2≥0 ⇒m2-2m-2≥0 ⇒m={3,4,5},n(E)=3 ⇒PE=35

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

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$\frac{3}{5}$

$\frac{1}{5}$

$\frac{2}{5}$

answer is C.

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Detailed Solution

$S = \{1, 2, 3, 4, 5\} \Rightarrow n (S) = 5, and x^{2} + mx + \frac{1}{2} + \frac{m}{2} = 0 has real roots Δ \geq 0 \Rightarrow m^{2} - 4 (1) (\frac{1 + m}{2}) \geq 0 \Rightarrow m^{2} - 2 m - 2 \geq 0 \Rightarrow m = {3, 4, 5}, n (E) = 3 \Rightarrow P (E) = \frac{3}{5}$