• # question_answer If $a\in \left[ -\text{ }20,0 \right]$, then probability that the graph of the function $y=16{{x}^{2}}+8\left( a+5 \right)\text{ }x-7a-5$ is strictly above the $x-$axis is A) $\frac{7}{20}$ B) $\frac{13}{20}$ C) $\frac{17}{20}$ D) $\frac{3}{20}$

 the graph of $y=16{{x}^{2}}+8(a+5)x-7a-5$is strictly above the x-axis $\Rightarrow$$y>0\,\forall \,x\in \mathbf{R}$$\Rightarrow$$16{{x}^{2}}+8(a+5)x-7a-5>0\,\forall \,x\in R$ The above inequality holds if discriminant <0 [$\because$coefficient ${{x}^{2}}>0$] $\Rightarrow$$64{{\left( a+5 \right)}^{2}}-4.16\left( -7a-50 \right)<0$$\Rightarrow$${{a}^{2}}+17a+30<0$$\Rightarrow$$\left( a+2 \right)\left( a+15 \right)<0$$\Rightarrow$$-15  Given \[-20\le a\le 0$ and favourable cases \[-15