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  4. If the numbers appeared on the two throws of a fair six faced die are α and β, then the probability that x2+α x+β>0, for all x ∈ R, is :

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Q. If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x^2+\alpha x+\beta>0$, for all $x \in R$, is :

JEE MainJEE Main 2022Probability - Part 2

Solution:

$ x ^2+\alpha x +\beta>0, \forall x \in R$
$ D =\alpha^2-4 \beta<0 $
$ \alpha^2<4 \beta$
Total cases$ = 6 \times 6=36 $
Fav. cases $=\beta=1, \alpha=1 $
$ \beta=2, \alpha=1,2 $
$ \beta=3, \alpha=1,2,3 $
$ \beta=4, \alpha=1,2,3$
$ \beta=5, \alpha=1,2,3,4 $
$ \beta=6, \alpha=1,2,3,4$
Total favourable cases $=17 $
$ P ( x )=\frac{17}{36}$