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(A) For the quadratic expression $x^2 + bx + c > 0$ to hold for all $x \in R$, the discriminant $D$ must be less than $0$.$D = b^2 - 4c < 0 \Rightarro

Vedclass · Accepted Answer

$\frac{29}{72}$

Vedclass · Answer

(A) For the quadratic expression $x^2 + bx + c > 0$ to hold for all $x \in R$, the discriminant $D$ must be less than $0$.$D = b^2 - 4c Since $b$ and $c$ are chosen from $\{1, 2, \ldots, 9\}$ without replacement $(b 
eq c)$, we count the pairs $(b, c)$ satisfying $b^2 $b$Possible $c$ values $(c 
eq b)$Count$1$$2, 3, 4, 5, 6, 7, 8, 9$$8$$2$$3, 4, 5, 6, 7, 8, 9$$7$$3$$4, 5, 6, 7, 8, 9$$6$$4$$5, 6, 7, 8, 9$$5$$5$$7, 8, 9$$3$$6$None $(6^2=36, 4c \leq 32)$$0$Total favorable outcomes $= 8 + 7 + 6 + 5 + 3 = 29$.Total possible outcomes $= 9 	imes 8 = 72$.Probability $= \frac{29}{72}$.

Two numbers $b$ and $c$ are chosen at random in succession without replacement from the set $\{1, 2, 3, \ldots, 9\}$. Then the probability that $x^2 + bx + c > 0, \forall x \in R$ is

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