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

m=1, 2, 3, 4, 5n(S)=5E=eq x2+mx+1+m2=0 has real roots∆≥0 ⇒ m2-41+m2≥0 ⇒ m2-2m-1≥0 ⇒ m=3, 4, 5n(E)=3 ⇒ P(E)=35

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

m=1, 2, 3, 4, 5n(S)=5E=eq x2+mx+1+m2=0 has real roots∆≥0 ⇒ m2-41+m2≥0 ⇒ m2-2m-1≥0 ⇒ m=3, 4, 5n(E)=3 ⇒ P(E)=35

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

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$\frac{1}{5}$

$\frac{2}{5}$

$\frac{3}{5}$

answer is C.

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Detailed Solution

$\begin{array}{rcl} m & = & \{1, 2, 3, 4, 5\} \\ n (S) & = & 5 \end{array}$

$E =$ eq $x^{2} + mx + \frac{1 + m}{2} = 0$ has real roots

$\begin{array}{rcl} ∆ & \geq & 0 \Rightarrow m^{2} - 4 (\frac{1 + m}{2}) \geq 0 \\ \Rightarrow & m^{2} - 2 m - 1 \geq 0 \Rightarrow m = \{3, 4, 5\} \\ n (E) & = & 3 \Rightarrow P (E) = \frac{3}{5} \end{array}$

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

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If m is a natural number 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