If A , B , and C are angles of a triangle such that x = cis A , y = cis B , z = cis C , then find the value of x y z

Given that, A , B , C are angles of a triangle and x = cis A , y = cis B , z = cis C So, from angle sum property, we have A + B + C = 180 0 = π → 1 Now, x y z = cis A · cis B · cis C ⇒ x y z = cis A + B + C [ ∵ cis x · cis y · cis z = cis x + y + z ] ⇒ x y z = cis π {Using 1 } ⇒ x y z = cosπ + i sinπ = - 1 + 0 = - 1 Hence, the value of x y z = - 1 .

If x = cis θ , then find the value of x 6 + 1 x 6 .

Given that, x = cis θ = cos θ + i sin θ = e i θ Now consider, x 6 + 1 x 6 = e 6 i θ + 1 e 6 i θ = e 6 i θ + e - 6 i θ ∵ 1 x = x - 1 = cos 6 θ + i sin 6 θ + cos - 6 θ + i sin - 6 θ = cos 6 θ + i sin 6 θ + cos 6 θ - i sin 6 θ ∵ cos - θ = cos θ a n d sin - θ = - sin θ = 2 cos 6 θ Therefore, x 6 + 1 x 6 = 2 cos 6 θ .

If the real part of z + 1 z + i is 1 , then find the locus of z where z = x + i y .

Given, z + 1 z + i Let, z = x + i y ⇒ z + 1 z + i = ( x + 1 ) + i y ( x ) + ( y + 1 ) i = [ ( x + 1 ) + i y ] [ x - ( y + 1 ) i ] [ x + ( y + 1 ) i ] [ x - ( y + 1 ) i ] = x ( x + 1 ) + y ( y + 1 ) + i ( x y - ( x + 1 ) ( y + 1 ) ) x 2 + ( y + 1 ) 2 = x ( x + 1 ) + y ( y + 1 ) x 2 + ( y + 1 ) 2 + i ( x y - ( x + 1 ) ( y + 1 ) ) x 2 + ( y + 1 ) 2 Given, real part = 1 ⇒ x ( x + 1 ) + y ( y + 1 ) x 2 + ( y + 1 ) 2 = 1 ⇒ x 2 + y 2 + x + y = x 2 + ( y + 1 ) 2 ⇒ x 2 + y 2 + x + y = x 2 + y 2 + 2 y + 1 ⇒ x - y = 1 ∴ Locus of z is x - y = 1 .

Write z = - 3 + i in modulus-amplitude form.

Given that z = - 3 + i We know that the modulus-amplitude form of z = a + i b is z = r cos θ + i sin θ , where r = a 2 + b 2 and θ is the principal argument of complex number. Modulus of the given complex number is, r = - 3 2 + 1 2 = 3 + 1 = 2 The complex number z = - 3 + i is in the second quadrant. When a complex number z = x + i y is belongs to the second quadrant, the principal argument is θ = π - tan - 1 y x . For given complex number z = - 3 + i , θ = π - tan - 1 1 - 3 = π - tan - 1 1 3 ⇒ θ = π - π 6 = 5 π 6 Hence, the required modulus-amplitude form is z = 2 cos 5 π 6 + i sin 5 π 6 .

Show that the points in the Argand plane represented by the complex numbers - 2 + 7 i . - 3 2 + 1 2 i , 4 - 3 i , 7 2 1 + i are the vertices of a rhombus.

Given, - 2 + 7 i , - 3 2 + 1 2 i , 4 - 3 i , 7 2 1 + i Let the four points represented by the complex numbers be A - 2 , 7 , B - 3 2 , 1 2 , C 4 , - 3 , D 7 2 , 7 2 The distance between the given points are, A B = - 2 + 3 2 2 + 7 - 1 2 2 = 1 4 + 169 4 = 170 2 units B C = 4 + 3 2 2 + - 3 - 1 2 2 = 121 4 + 49 4 = 170 2 units C D = 4 - 7 2 2 + - 3 - 7 2 2 = 1 4 + 169 4 = 170 2 units A D = - 2 - 7 2 2 + 7 - 7 2 2 = 121 4 + 49 4 = 170 2 units So, all sides are equal, A B = B C = C D = A D . Length of diagonal A C = - 2 - 4 2 + 7 + 3 2 = 36 + 100 = 136 units Length of diagonal B D = - 3 2 - 7 2 2 + 1 2 - 7 2 2 = 100 4 + 36 4 = 136 4 units We know that the diagonals of a rhombus are not equal. ∴ A B C D is a rhombus.

E M B I B E

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Algebra>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2020

Embibe Experts Solutions for Exercise 1: Andhra Pradesh Board-2018

Author:Embibe Experts

Embibe Experts Mathematics Solutions for Exercise - Embibe Experts Solutions for Exercise 1: Andhra Pradesh Board-2018

Attempt the free practice questions from Exercise 1: Andhra Pradesh Board-2018 with hints and solutions to strengthen your understanding. EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS solutions are prepared by Experienced Embibe Experts.

Questions from Embibe Experts Solutions for Exercise 1: Andhra Pradesh Board-2018 with Hints & Solutions

EASY

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2018>Q 3

If $A, B,$ and $C$ are angles of a triangle such that $x y z = - 1$ , $z = cis C,$ then find the value of $x y z$

EASY

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2019>Q 5

Find the multiplicative inverse of : $7 + 24 i$

EASY

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2019>Q 6

Simplify $i^{2} + i^{4} + i^{6} + . . . . . . . . . . + (2 n + 1)$ terms.

EASY

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2019>Q 7

If $x = cis θ$ , then find the value of $x^{6} + \frac{1}{x^{6}}$ .

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2019>Q 8

If the real part of $(\frac{z + 1}{z + i})$ is $1$ , then find the locus of $z$ where $z = x + i y$ .

EASY

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2020>Q 9

Find the complex conjugate of $(3 + 4 i) (2 - 3 i)$ .

EASY

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2020>Q 10

Write $z = - \sqrt{3} + i$ in modulus-amplitude form.

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 1 - Complex Numbers>Andhra Pradesh Board-2020>Q 11

Show that the points in the Argand plane represented by the complex numbers $- 2 + 7 i . \frac{- 3}{2} + \frac{1}{2} i, 4 - 3 i, \frac{7}{2} (1 + i)$ are the vertices of a rhombus.

Learn and Prepare for any exam you want !

|

|

|

|

|

|

|

|