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JEE Main 20249 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
9 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Probability

JEE Main 2024 · 9 April · Morning Shift · Mathematics · Probability

Let a, b and c denote the outcome of three independent rolls of a...

Let a,b\mathrm{a}, \mathrm{b} and c\mathrm{c} denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1,2,3,41,2,3,4. If the probability that ax2+bx+c=0a x^2+b x+c=0 has all real roots is mn\frac{m}{n}, gcd⁡(m,n)=1\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1, then m+n\mathrm{m}+\mathrm{n} is equal to ________

Correct answer

0

Step-by-step solution

\mathrm{a}, \mathrm{b}, \mathrm{c} \in\{1,2,3,4\}TetrahedraldiceTetrahedral dicea x^2+b x+c=0hasallrealrootshas all real roots⇒D≥0⇒b2−4ac≥0\begin{aligned} & \Rightarrow \mathrm{D} \geq 0 \\ & \Rightarrow \mathrm{b}^2-4 \mathrm{ac} \geq 0 \end{aligned}LetLetb=1 \Rightarrow 1-4 a c \geq 0(Notfeasible)(Not feasible)b=2⇒4−4ac≥01≥ac⇒a=1,c=1,b=3⇒9−4ac≥094≥ac⇒a=1,c=1⇒a=1,c=2⇒a=2,c=1b=4⇒16−4ac≥04≥ac\begin{aligned} & b=2 \Rightarrow 4-4 a c \geq 0 \\ & 1 \geq a c \Rightarrow a=1, c=1, \\ & b=3 \Rightarrow 9-4 a c \geq 0 \\ & \frac{9}{4} \geq a c \\ & \Rightarrow a=1, c=1 \\ & \Rightarrow a=1, c=2 \\ & \Rightarrow a=2, c=1 \\ & b=4 \Rightarrow 16-4 a c \geq 0 \\ & 4 \geq a c\end{aligned}$\begin{aligned} & \Rightarrow a=1, c=1 \\ & \Rightarrow a=1, c=2 \quad \Rightarrow a=2, c=1 \\ & \Rightarrow a=1, c=3 \quad \Rightarrow a=3, c=1 \\ & \Rightarrow a=1, c=4 \quad \Rightarrow a=4, c=1 \\ & \Rightarrow a=2, c=2 \\ & \text{ Probability }=\frac{12}{(4)(4)(4)}=\frac{3}{16}=\frac{m}{m} \\ & m+n=19\end{aligned}$
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