Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
Let $f(z)$ be a polynomial function of a complex number $z$. On division by $z-i, z+i$ and $z^2+1$ we obtain remainder as $\alpha, \beta$ and $g(z)$, ($\alpha, \beta \in \mathbb{C}$). Then:
$\alpha = i$ and $\beta = 1-i \Rightarrow g(z) = i\left(\frac{z}{2}+1\right) + \frac{1}{2}$
$\alpha = i$ and $\beta = 1-i \Rightarrow g(z) = \left(z-\frac{1}{2}\right)\frac{iz}{2}$
$\alpha = i$ and $\beta = 1-i \Rightarrow g(z) = i\left(z+\frac{1}{2}\right) + \frac{1}{2}$
$\alpha = i$ and $\beta = 1-i \Rightarrow g(z) = \left(z+\frac{1}{2}\right)\frac{iz}{2}$
Step-by-Step Solution
Key Concept: Calculating powers of 2 and their differences to find the precise numerical answer.
The expression $2^{10} - 2$ equals 1024 - 2 = 1022. Option (A) gives $2^{10} - 2 = 1022$. Option (B) gives $2^{10} - 1 = 1023$. Option (C) also gives $2^{10} - 1 = 1023$. Option (D) gives 10. The correct answer requires evaluating which expression matches the given problem context.
Correct Answer: 1,4