What is the real part of $\left(1+i\right){}^{50}?$

Important Questions on Complex Numbers

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 3

The complex number $z$ satisfies $z+\left|z\right|=2+8i$, then the value of $\left|z\right|$ is

EASY

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 4

If $z+z^3=0,$ then which of the following must be true on the complex plane?

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 5

The sequences $S=i+2i^2+3i^3+\dots$ upto $100$ terms simplifies to, where $i=\sqrt{-1}$

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 6

Given $i=\sqrt{-1},$ then the value of the sum $\frac{1}{1+i}+\frac{1}{1-i}+\frac{1}{-1+i}+\frac{1}{-1-i}+\frac{2}{1+i}+\frac{{\displaystyle 2}}{{\displaystyle 1-i}}+\frac{{\displaystyle 2}}{{\displaystyle -1+i}}+\frac{{\displaystyle 2}}{{\displaystyle -1-i}}+\frac{{\displaystyle 3}}{{\displaystyle 1+i}}+\frac{{\displaystyle 3}}{{\displaystyle 1-i}}+\frac{{\displaystyle 3}}{{\displaystyle -1+i}}+\frac{{\displaystyle 3}}{{\displaystyle -1-i}}+\dots +\frac{{\displaystyle n}}{{\displaystyle 1+i}}+\frac{{\displaystyle n}}{{\displaystyle 1-i}}+\frac{{\displaystyle n}}{{\displaystyle -1+i}}+\frac{{\displaystyle n}}{{\displaystyle -1-i}}$ is 

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 7

Let $i=\sqrt{-1}$. The product of the real part of the roots of $z^2-z=5-5i$ is

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 8

Number of complex numbers $z$ satisfying $z^3=\overline{z}$ is

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 9

Number of real solution of the equation, $z^3+iz-1=0$ is

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 1>Chapter 4 - Complex Numbers>Work Book Exercise 4.1>Q 10

The diagram shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of $F$, which is