Probability
Geometric progression condition with dice sum
nta_pyq_2023_jan
Grade 12
Question:
Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N-2$, $\sqrt{3N}$, $N+2$ are in geometric progression be $\frac{k}{48}$. Then the value of k is
Step-by-Step Solution
Key Concept: GP condition: $3N = (N-2)(N+2) = N^2-4$; solve for N and count favorable outcomes
$3N = (N-2)(N+2) = N^2-4 \Rightarrow N^2-3N-4=0 \Rightarrow (N-4)(N+1)=0 \Rightarrow N=4$ (N=-1 rejected). Sum = 4 can occur as: $(1,3),(3,1),(2,2)$: 3 ways. $P = 3/36 = 1/12 = 4/48 \Rightarrow k=4$. Answer: (2)
Correct Answer: 4