Binomial Theorem
Binomial Sum with Complex Numbers
nta_pyq_2025_apr
Grade 11
Question:
If $\alpha = 1 + \displaystyle\sum_{r=1}^{6}(-3)^{r-1} \cdot {}^{12}C_{2r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x - \sqrt{3}y + 1 = 0$ is ___
Step-by-Step Solution
Key Concept: Relate the sum to the imaginary part of the binomial expansion $(1 + \sqrt{3}i)^{12}$, where $i = \sqrt{-1}$, to evaluate $\alpha$.
Using $(1+\sqrt{3}i)^{12}$: $\alpha = 1 + \frac{1}{\sqrt{3}i}\cdot\frac{(1+\sqrt{3}i)^{12}-(1-\sqrt{3}i)^{12}}{2} = 1$. Distance of $(12,\sqrt{3})$ from line $x - \sqrt{3}y + 1 = 0$ is $\frac{|12 - 3 + 1|}{\sqrt{1+3}} = \frac{10}{2} = 5$.
Correct Answer: 5