JEE Complex Numbers - Advanced Conceptual Understanding
This page will teach you how to master JEE Complex Numbers up to JEE Advanced level. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.
JEE Conceptual Theory
To have mastery over Complex Numbers for JEE Advanced, the main concepts you should be confident of are square root of a complex number, logarithm of a complex number, modular inequalities, De Moivre's theorem and rotation of a complex number.
Square root of a complex number
- \(\sqrt{a+ib}=x+iy \Rightarrow x+iy=\pm \left[ \left( \frac{\sqrt{a^2+b^2}+a}{2} \right)^\frac{1}{2}+i \left( \frac{\sqrt{a^2+b^2}-a}{2} \right)^\frac{1}{2} \right] \)
Logarithm of a complex number
- \(\log_ez=\log_e|z|+i(2k\pi+\theta)\) where \(\theta=arg(z)\), \(k \in \mathbb Z\)
Modular inequalities
Minimum value of \(|z-z_1|+|z-z_2|\) is \(|z_1-z_2|\)
\(|z_1-z_2| \geq \left| |z_1|-|z_2| \right| \)
\( |z_1+z_2|=|z_1-z_2| \Leftrightarrow arg(z_1)-arg(z_2)=\pm \frac{\pi}{2}\)
\(|z_1+z_2|^2=|z_1|^2+|z_2|^2 \Leftrightarrow \frac{z_1}{z_2}\) is purely imaginary number
De Moivre's theorem
\((\cos \theta +i \sin \theta)^n=\cos n\theta+i \sin n\theta\), when \(n\) is an integer
\(\cos n\theta+i \sin n\theta\) is one of the values of \((\cos \theta+i\sin \theta)^n\), when \(n\) is a fraction
Rotation of a complex number
When we rotate \(z_1\) through an angle \(\theta\) in anticlockwise direction about origin \((0,0)\), say we get \(z_2\), then \(\frac{z_2-0}{z_1-0}=\frac{|z_2-0|}{|z_1-0|} e^{i \theta}\)
When we rotate \(z_1\) through an angle \(\theta\) in anticlockwise direction about another complex number \(z_0\), say we get \(z_2\), then \(\frac{z_2-z_0}{z_1-z_0}=\frac{|z_2-z_0|}{|z_1-z_0|} e^{i \theta}\)
JEE Mains Problems
\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]
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\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]
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JEE Advanced Problems
\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]
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\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]
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Once you are confident of Complex Numbers, move on to JEE Binomial Theorem.