If a ∈ [-20, 0], the probability that the equation $16x^2 + 8 (a + 5) x-7a =0 $ has imaginary roots, is

$\frac{13}{20}$

If the equation $16x^2 + 8 (a + 5) x -7a -5 =0$ has imaginary roots, then

Discriminant < 0

$⇒ 64 (a+5)^2 + 64 (71+5) < 0$

$⇒ a^2 + 17a + 30 < 0 ⇒ (a + 15) (a + 2) < 0 ⇒-15 <a< -2 $

Hence, required probability $=\frac{\int\limits^{-2}_{-15}dx}{\int\limits^{0}_{-20}dx}=\frac{13}{20}$