Q55 / 160 · TS EAMCET 2023 (12 May Shift 1) · Mathematics

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 $\mathrm{x}^2+\mathrm{bx}+\mathrm{c}>0, \forall \mathrm{x} \in \mathbb{R}$ is

  • Correct answer
    A
    $\frac{29}{72}$
  • B
    $\frac{32}{81}$
  • C
    $\frac{45}{143}$
  • D
    $\frac{82}{125}$

Solution

$x^2+b x+c>0, \forall x \in R$ $\begin{aligned} \Rightarrow & D < 0 \\ & b^2-4 c < 0 \\ & b^2 < 4 c \end{aligned}$ Possible outcomes : \begin{array}{|c|c|c|} \hline \boldsymbol{b} & \boldsymbol{c} & Number of pairs \\ \hline 1 & 2,3,4,5,6,7,8,9 & 8 \\ \hline 2 & 3,4,5,6,7,8,9 & 7 \\ \hline 3 & 4,5,6,7,8,9 & 6 \\ \hline 4 & 5,6,7,8,9 & 5 \\ \hline 5 & 7,8,9 & 3 \\ \hline 6 & Not possible & \\ \hline \end{array} $\begin{array}{ll}\therefore \quad & \text { Total no. of possible pairs }=29 \\ & \text { Maximum number of pairs }=9 \times 8=72 \\ \therefore \quad & \text { Probability }=\frac{29}{72} .\end{array}$

Hence, the correct answer is option: 'A'.

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