If $\alpha$ is the fifth root of unity, then

Important Questions on Complex Numbers

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 35

If $1,{\alpha }_1,{\alpha }_2,{\alpha }_3,\dots \dots ,{\alpha }_{n-1}$ be the $n^{\mathrm{th}}$ roots of unity, then which of the following are true

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 36

Let $\mathrm{z},\omega \mathrm{z}$ and $\mathrm{z}+\omega \mathrm{z}$ represent three vertices of $∆ABC$, where $\omega$ is cube root unity, then

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 37

Centre of square $ABCD$ is at $z=0$. If affix of vertex $A$ is $z_1$, centroid of triangle $ABC$ can be:

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 38

The possible values of the real parameter $m$ for which $\left|z-\left(m^2-7m+11+i\right)\right|=1$ and $\arg \left(z\right)\ge \frac{\pi }{2}$ is satisfied by at least one $z$ is/are

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 39

Let $z_1, z_2$ and $z_3$ are the vertices of $△ABC$, respectively, such that $\frac{z_3-z_2}{z_4-z_2}$ is purely an imaginary number. A square on the side $AC$ is drawn outwardly and $P\left(z_4\right)$ is the center of the square, then

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 40

If $\left|az+b\overline{z}+c\right|=\mathit{\alpha }\left|\mathit{a}\right|\left|z-z_1\right|$, $a\ne 0, a=\overline{b}, c\in R$, where $z_1$ is a given non-zero complex number and $\alpha \in R^{+},$ then the locus of $z$ maybe

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 41

If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$ and $z_1, z_2, z_3$ are represented by the vertices of an equilateral triangle then

HARD

Mathematics

IMPORTANT

Mathematics Crash Course JEE Advanced>Chapter 12 - Complex Numbers>Exercise 1>Q 42

Let $s, t, r$ be non-zero complex numbers and$L$ be the set of solutions $z=x+iy$  $(x,y\in R,i=\sqrt{-1})$of the equation $sz+t\overline{z}+r=0$, where $\overline{z}=x-iy$. Then, which of the following statement(s) is (are) TRUE ?