JEE Main 2021 (Online) 20th July Morning ShiftQuestion 1
The probability of selecting integers $a\in[-5,30]$ such that $x^{2}+2(a+4)x-5a+64>0$, for all $x\in R$, is :
Quadratic Equation · Sign, Range and Inequalities of Quadratic Expressions
Solution
Concept. An upward quadratic is positive for all $x$ exactly when $D<0$. Formulas used. (1) $x^2+2Bx+C>0\ \forall x\iff B^2-C<0$Step 1 - Condition. By (1), $$(a+4)^2-(64-5a)=a^2+13a-48<0.$$ The roots are $\frac{-13\pm19}{2}=3,\ -16$, so $-16<a<3$. Step 2 - Count. Integers in $[-5,30]$: $36$. Favourable: $a=-5,\dots,2$, i.e. $8$. Step 3 - Probability. $\frac{8}{36}=\frac29$. Answer: (B)