Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
Let $a = e^{2\pi i/1}, \lambda = a^k, \mu = a^{j}, \beta = a^2$. Then:
$\text{Re}\left(\lambda + \lambda^2 + \lambda^3 + \lambda^4 + \lambda^5\right) = -\frac{1}{2}$
$(\mu - \beta)(\mu - \beta^2)(\mu - \beta^3)\ldots(\mu - \beta^{10}) = 0$
$(i - \beta)(i - \beta^2)(i - \beta^3)\ldots(i - \beta^{10}) = i$
None of these
Step-by-Step Solution
Key Concept: Use multinomial coefficients to count arrangements when some items are identical.
The number of ways to arrange four items $C_1, C_2, C_3, C_4$ with values 2, 2, 2, 3 respectively is $\frac{9}{(2!)^3 \times 3} = \frac{9}{(2!)^3}$. This accounts for identical items using the multinomial coefficient.
Correct Answer: 1,2,3