Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
If $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ are complex numbers such that $|z_1|=1, |z_2|=2$ and $\text{Re}(z_1\overline{z_2})=0$, then the pair of complex numbers $\omega_1 = a_1 - \frac{ia_2}{2}$ and $\omega_2 = 2b_1 + ib_2$ satisfy:
$|\omega_1| = 1$
$|\omega_2| = 2$
$\text{Re}(\omega_1\omega_2) = 0$
$\text{Im}(\omega_1\omega_2) = 0$
Step-by-Step Solution
Key Concept: Expressing permutation formulas in terms of products of odd numbers by factoring out powers of 2 from the numerator.
The expression $^{2n}p_n$ represents permutations of $2n$ items taken $n$ at a time. This equals $\frac{(2n)!}{n!} = \frac{2^n \cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdots 2n}{[n]} = 2^n \cdot \frac{(2n)!}{n!} = 2^n \cdot [1 \cdot 3 \cdot 5 \cdots]$, which simplifies to $2^n \cdot 1 \cdot 3 \cdot 5 \cdots$ representing the product of all odd numbers up to $2n-1$.
Correct Answer: 1,2,3,4