Complex Numbers
Complex Numbers
nta_abhyas_2025
Grade 11
Question:
If $\alpha + 1 + i = a$, $\alpha(1 + i) = a$, $\alpha + \alpha i = \alpha + 1 + i$, $\alpha = \frac{1+i}{i} = 1 - i$, $\alpha = 1 - i + 1 + i = 2$, and $\alpha^4 = (\alpha^2)^2 = (1 + (-1))^2 = 0^2 = (2i)^2 = -4$, then find the value of $\alpha^4$.
Step-by-Step Solution
Key Concept: To find powers of complex numbers, first compute the square, then use it to find higher powers.
Working through the given conditions: $\alpha + 1 + i = a$ and $\alpha(1 + i) = a$ imply $\alpha + 1 + i = \alpha(1 + i)$, which gives $\alpha + 1 + i = \alpha + \alpha i$, so $1 + i = \alpha i$. Therefore $\alpha = \frac{1+i}{i} = \frac{(1+i)(-i)}{i(-i)} = \frac{-i + 1}{1} = 1 - i$. Now $\alpha^2 = (1-i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i$. Thus $\alpha^4 = (-2i)^2 = 4i^2 = -4$.
Correct Answer: -4