If $\left|z^2-1\right|={\left|z\right|}^2+1,$ then $z$ lies on

Important Questions on Complex Numbers

EASY

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 39

If $z=x+iy \left(x, y\in R, x\ne -\frac{1}{2}\right)$, the number of values of $z$ satisfying ${\left|z\right|}^n=z^2 {\left|z\right|}^{n-2}+z{\left|z\right|}^{n-2}+1. \left(n\in N, n>1\right)$ is

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 40

The number of solutions of the equation $z^3+\frac{3{\left(\overline{z}\right)}^2}{\left|z\right|}=0$ (where, $z$ is a complex number) are

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 41

Number of ordered pairs(s) $\left(\mathrm{a},\mathrm{b}\right)$ of real numbers such that ${\left(\mathrm{a}+\mathrm{ib}\right)}^{2008} = \mathrm{a}-\mathrm{ib}$ holds good is

EASY

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 42

The equation $a{z}^3+b{z}^2+\overline{b}z+\overline{a}=0$ has a root $\alpha ;$ where $a,b,z$ and $\alpha$ belong to the set of complex numbers. The value of $\left|\alpha \right|$

EASY

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 43

If $k>0, \left|z\right|=\left|w\right|=k$ and $\alpha =\frac{z-\overline{w}}{k^2+z\overline{w}},$ then $\mathrm{Re}\left(\alpha \right)$ equals

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 44

${\mathrm{z}}_1$ and ${\mathrm{z}}_2$ are two distinct points in an argand plane. If $a\left|z_1\right|=b\left|z_2\right|,$ where $a,b\in R,$ then the point $\frac{a{z}_1}{b{z}_2}+\frac{b{z}_2}{a{z}_1}$ is a point on the

EASY

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 45

If $z$ is a complex number such that $-\frac{\pi }{2}\le \arg z\le \frac{\pi }{2},$ then which of the following inequalities is true?

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 46

If $\cos \alpha +2\cos \beta +3\cos \gamma =\sin \alpha +2\sin \beta +3\sin \gamma =0,$ then the value of $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma$ is