The coefficients $$a, b, c$$ in the quadratic equation $$a x^2+b x+c=0$$ are chosen from the set $$\{1,2,3,4,5,6,7,8\}$$. The probability of this equation having repeated roots is :
JEE · Math · previous-year question
- A.$$\frac{1}{128}$$
- B.$$\frac{1}{64}$$correct
- C.$$\frac{3}{256}$$
- D.$$\frac{3}{128}$$
Answer
B. $$\frac{1}{64}$$
Explanation
Given quadratic equation is $$a x^2+b x+c=0 \text { where } a, b, c \in\{1,2,3, \ldots, 8\}$$ For repeated roots, $$\begin{aligned} & b^2-4 a c=0 \\ & \Rightarrow b^2=4 a c \end{aligned}$$ $$\Rightarrow a c$$ must be a perfect square $$(a, c) \in\{(1,1),(1,4),(2,2),(2,8),(3,3),(4,1),(4,4),(5,5),(6,6),(7,7),(8,2),(8,8)\}$$ Corresponding $$b$$ must lie in set $$\{1,2,3, \ldots 8\}$$ $$\begin{aligned} & (a, b, c) \in\{(1,2,1),(1,2,4),(2,4,2),(2,8,8) \\ & (3,6,3),(4,4,1),(4,8,4),(8,8,2)\} \\ & \therefore \text { probability }=\frac{8}{8^3} \\ & =\frac{1}{64} \\ & \end{aligned}$$
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