1993 Paper 2 Q9

Year: 1993
Paper: 2
Question Number: 9

Course: LFM Stats And Pure
Section: Complex Numbers (L8th)

Difficulty: 1600.0 Banger: 1500.0

complex numbers argument inequality geometric proof Argand diagram proof by contradiction AM-GM inequality

Problem

\textit{In this question, the argument of a complex number is chosen to satisfy \(0\leqslant\arg z<2\pi.\)} Let \(z\) be a complex number whose imaginary part is positive. What can you say about \(\arg z\)? The complex numbers \(z_{1},z_{2}\) and \(z_{3}\) all have positive imaginary part and \(\arg z_{1}<\arg z_{2}<\arg z_{3}.\) Draw a diagram that shows why \[ \arg z_{1}<\arg(z_{1}+z_{2}+z_{3})<\arg z_{3}. \] Prove that \(\arg(z_{1}z_{2}z_{3})\) is never equal to \(\arg(z_{1}+z_{2}+z_{3}).\)

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1500.0

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Problem source

\textit{In this question, the argument of a complex number is chosen
to satisfy $0\leqslant\arg z<2\pi.$}

Let $z$ be a complex number whose imaginary part is positive. What
can you say about $\arg z$?

The complex numbers $z_{1},z_{2}$ and $z_{3}$ all have positive
imaginary part and $\arg z_{1}<\arg z_{2}<\arg z_{3}.$ Draw a diagram
that shows why 
\[
\arg z_{1}<\arg(z_{1}+z_{2}+z_{3})<\arg z_{3}.
\]
Prove that $\arg(z_{1}z_{2}z_{3})$ is never equal to $\arg(z_{1}+z_{2}+z_{3}).$

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