Question

If $m$ is selected at random from set $\{1,2,\dots 10\}$ and the probability that the quadratic equation $2x^2+2mx+m+1=0$ has real root is $k$ then value of $5k$ is ?

Answer 1

$2x^2+2mx+m+1=0$ has real roots

$\Rightarrow 4m^2\ge \mathrm{4.2.}(m+1)$

$\Rightarrow 4m^2-8m-8\ge 0$

$\Rightarrow m^2-2m-2\ge 0$

$[m-(1-\sqrt{m})][m-(1+\sqrt{3})]\ge 0$

$\Rightarrow m\le (1-\sqrt{3})$ Or $m\ge (1-\sqrt{3})$

$m\le -0.73$ Or $m\ge 2.7$

$\therefore m=3,4,5,6,7,8,9,10$

$\therefore$Probability $k=\frac{{}^8{C}_1}{{}^{10}{C}_1}=\frac{8}{10}=\frac{4}{5}$

$\therefore 5k=4$