JEE Mains · Maths · STD 12 - 13. probability
Let \(a, b\) and \(c\) denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked \(1,2,3,4\). If the probability that \(a x^2+b x+c=0\) has all real roots is \(\frac{m}{n}\), \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(\mathrm{m}+\mathrm{n}\) is equal to ..........
- A \(19\)
- B \(20\)
- C \(6\)
- D \(71\)
Answer & Solution
Correct Answer
(A) \(19\)
Step-by-step Solution
Detailed explanation
\(a, b, c \in\{1,2,3,4\}\) Tetrahedral dice \(a x^2+b x+c=0\) has all real roots \( \Rightarrow \mathrm{D} \geq 0 \) \( \Rightarrow \mathrm{b}^2-4 \mathrm{ac} \geq 0\) Let \(b=1 \Rightarrow 1-4 a c \geq 0\) (Not feasible) \( \mathrm{b}=2 \Rightarrow 4-4 \mathrm{ac} \geq 0 \)…
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