If z 1 = 2 - i and z 2 = - 2 + i , find R e 1 z 1 z ¯ 1 .

Given, z 1 = 2 - i ⇒ z ¯ 1 = 2 + i and z 2 = - 2 + i Now, 1 z 1 z ¯ 1 = 1 2 - i 2 + i ⇒ 1 z 1 z ¯ 1 = 1 4 - i 2 ⇒ 1 z 1 z ¯ 1 = 1 4 + 1 ∵ i 2 = - 1 ⇒ 1 z 1 z ¯ 1 = 1 5 + i . 0 Therefore, this is of the form x + i y , x , y ∈ R and x and y are real and imaginary parts of a complex number. ∴ R e 1 z 1 z ¯ 1 = 1 5 .

Find the real value of θ such that 1 + i cos θ 1 - 2 i cos θ is a real.

Let, z = 1 + i cos θ 1 - 2 i cos θ ⇒ z = 1 + i cos θ 1 - 2 i cos θ × 1 + 2 i cos θ 1 + 2 i cos θ (Multiply and divide by conjugate of a denominator) ⇒ z = 1 + 2 i cos θ + i cos θ - 2 i 2 cos 2 θ 1 - 4 i 2 cos 2 θ ⇒ z = 1 + 3 i cos θ + 2 cos 2 θ 1 + 4 cos 2 θ ∵ i 2 = - 1 ⇒ z = 1 + 2 cos 2 θ 1 + 4 cos 2 θ + 3 i cos θ 1 + 4 cos 2 θ Since z real, the imaginary part is zero. ∴ 3 cos θ 1 + 4 cos 2 θ = 0 ⇒ 3 cos θ = 0 ⇒ cos θ = 0 ⇒ cos θ = cos ( 2 n + 1 ) π 2 ⇒ θ = ( 2 n + 1 ) π 2 , n ∈ I .

E M B I B E

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1

EXERCISE 5.1

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for EXERCISE 5.1

Simple step-by-step solutions to EXERCISE 5.1 questions of Complex Numbers from Understanding ISC Mathematics Class 11 Volume 1. Also get 3D topic explainers, cheat sheets, and unlimited doubts solving on EMBIBE.

Questions from EXERCISE 5.1 with Hints & Solutions

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 30

Express the complex number $i^{55} + i^{60} + i^{65} + i^{70}$ , in standard form $a + i b$ .

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 35

Find the modulus of $\frac{(1 + 3 i) (2 - 5 i)}{(2 - \sqrt{6} i) (- 3 + 2 \sqrt{5} i)}$ .

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 36

If $z_{1} = 2 - i and z_{2} = - 2 + i$ , find $I m (\frac{z_{1} z_{2}}{{\bar{z}}_{1}})$ .

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 36

If $z_{1} = 2 - i and z_{2} = - 2 + i$ , find $R e (\frac{1}{z_{1} {\bar{z}}_{1}})$ .

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 36

If $z_{1} = 2 - i and z_{2} = - 2 + i$ , find $I m (\frac{1}{z_{1} \bar{z_{1}}})$ .

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 43

Find the least positive integral value of $n$ for which ${(\frac{1 + i}{1 - i})}^{n}$ is real number.

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 44

Find the real value of $θ$ such that $\frac{1 + i \cos θ}{1 - 2 i \cos θ}$ is a real.

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.1>Q 45

If $z_{1}, z_{2} and z_{3}$ are complex numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1$ , find the value of $|z_{1} + z_{2} + z_{3}|$ .

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