Complex Numbers
Probability & Complex
MMTS_Full_Test_14
Grade 12
Question:
Let $\alpha_1,\alpha_2,\ldots,\alpha_8$ be roots of $1+z+z^2+\cdots+z^8=0$. Three dice thrown; sum is $n$. Probability that $\sum_{i=1}^8(\alpha_i)^n=8$ is $a/b$ (coprime). Then $|7a-b|$
Step-by-Step Solution
Key Concept: $\sum\alpha_i^n=8$ when $n$ is multiple of 9; find probability with 3 dice
$\sum\alpha_i^n=8$ when $9|n$. With 3 dice (sum 3-18): $9|n\Rightarrow n=9$ or $n=18$. $P(n=9)=25/216$, $P(n=18)=1/216$. $a/b=26/216=13/108$. $|7\cdot13-108|=|91-108|=17$.
Correct Answer: 4