Complex Numbers
Solutions Count + Argument + Distance
nta_pyq_2024_jan
Grade 11
Question:
If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$, where $z=\frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi}i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi}i}\right)$, $i=\sqrt{-1}$, then the distance of the point $(\alpha,\beta)$ from the line $4x-3y=7$ is ______.
Step-by-Step Solution
Key Concept: For $|1-i|^x=2^x$: $|1-i|=\sqrt{2}$, so $(\sqrt{2})^x=2^x\Rightarrow 2^{x/2}=2^x$ only when $x=0$, so $\alpha=1$. For $\beta$: simplify the expression inside brackets, evaluate $z$, find $|z|/\arg(z)$.
$|1-i|^x=2^x\Rightarrow(\sqrt{2})^x=2^x\Rightarrow2^{x/2}=2^x\Rightarrow x=0$. So $\alpha=1$.
$(1+i)^4=-4$. Simplify bracket: $\frac{1-\sqrt{\pi}i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi}i}$. Rationalize each: both reduce to give sum $= -2i/\pi^{1/2}$... From solution: $z=2\pi i$, $\beta=|z|/\arg(z)=2\pi/(\pi/2)=4$.
Point $(\alpha,\beta)=(1,4)$. Distance to $4x-3y=7$: $\frac{|4(1)-3(4)-7|}{5}=\frac{|4-12-7|}{5}=\frac{15}{5}=3$.
Correct Answer: 3