Question

If $a\in [-20,0]$, then find the probability that the graph of the function $16x^2+8(a+5)x-7a-5$ is strictly above the x-axis.

• A. Required probability $=\frac{13}{20}$
• B. Required probability $=\frac{12}{20}$
• C. Required probability $=\frac{7}{20}$
• D. Required probability $=\frac{8}{20}$

Answer 1

The correct option is A Required probability $=\frac{13}{20}$

For the function to be always positive,the roots should be imaginery $\Rightarrow$ discriminant is negative $\Rightarrow$ $D={\left[8(a+5)\right]}^2-\left(4\right)\left(16\right)(-7a-5)=64(a^2+25+10a)+64(7a+5)=64(a^2+30+17a)<0\Rightarrow a^2+30+17a<0\Rightarrow a\in (-15,-2)$
Hence,probability $=\frac{-2-(-15)}{0-(-20)}=\frac{13}{20}$