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

M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 1: EXERCISE 5.1

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 1: EXERCISE 5.1

Attempt the practice questions on Chapter 5: Complex Numbers, Exercise 1: EXERCISE 5.1 with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 1 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 1: 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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