complex_prob.pdf

7/29/2019 complex_prob.pdf

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Q. 1 Find the least positive value ofn, if1

11

n

i

i

+ =

Q. 2 Let zbe an arbitrary complex number. If 1 izwz i=

(a)1 1 2z i= + (b) 2 2 5z i=

(c) 3 4 1z i= + (d) 41

1

iz

i

=

+

1 2

( ) ( )1 21 2 1 21 2

2z z

z z z zz z

+ + +

1 2 n

1 2 ...... 1,nz z z= = = = prove that

1 21 2

1 1 1...... .....

n

n

z z zz z z

+ + + = + + +

2z iz=

3 2 0,iz z z i+ + = show that | | 1z =

and | w | = 1, show that zis purely real.

Q. 3 Find the modulus and arguments of the following:

Q. 4 For any two non-zero complex numbers z , z , show that

Q. 5 Ifz z, ......z are complex numbers such that

Q. 6 Find all the non-zero complex numbers satisfying

Q. 7 If

MARKSMAN COACHING CIRCLE, KASHIPUR

COMPLEX NUMBER

ASSIGNMENT


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(a) | 1 | 2 and 2z i z i < + < (b) ( )| | 3, arg6

z z

< .

To solve this question, it is advisable to use a geometrical approach.

Q. 4 For any two non-zero complex numbers z1

and z2, prove that if

2 2 2 11 2 1 2

2

, zz z z zz

+ = + will be purely

Q. 1 Plot the regions represented by the following.

be given by andz be given by Find max

Q. 3 Ifzand w be two complex numbers such that then prove that

imaginary.


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21is .

2z iz z+

Q. 2 If the complex numbers z1, z2 and the origin form an equilateral triangle, show that

2 2

1 2 1 2 0z z z z+ =

Q. 3 If the complex numbersz1,z

2and the origin form an isosceles triangle with vertical angle

2,

3

show that

2 2

1 2 1 2 0z z z z+ + =

Q. 4 If2

4,z =1 3

.2 2

w i= +

verticesz1is known.

Q. 6 Letz1andz

2be roots of the equation 2 0z pz q+ + = where the coefficientsp andq may be complex numbers.

LetA andB representz1andz

2in the complex plane, If 0AOB = and ,OA OB= where O is the origin,

prove that

2 24 cos / 2p q =

Q. 7 If the vertices of a square are 1 2 3 4

( ) ( )3 1 2 4 1 21 and 1 .z iz i z z i z iz= + + = + .

Q. 1 Show that the area of the triangle formed by the complex numbersz, izand

find the area of the triangle formed byz, wzandz+ wzas its sides, where

Q. 5 Find the vertices of a regular polygon ofn sides if its centre is located atz= 0 and one of its

z z, ,z andz taken in the anticlockwise order, prove that


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1 2 1n

thn

1 2 1(1 )(1 )...(1 )n + + +

1 1 1

1 2 2 1 +

+ + +

ifis the complex cube root of unity.

1 2 3

1 2 3z z z A+ + =

2

1 2 3z z z B + + =

21 2 3z z z C + + =

whereA,B, Care constants, express 1 2 3, andz z z independently in terms ofA,B and C.

Q. 4 If 1 2 1nthn

1 2 1( )( )...( )n .

Q. 5 Let a complex number , 1

1 0,p q p qz z z+ + =

wherep and q are distinct primes. Show that either 2 11 ... 0p + + + + = or 2 11 ... 0q + + + + = ,

but not both together.

5 2z = .

Q. 1 If1, , ,..., are the roots of unity, then find the value of

Q. 2 Find the value of

Q. 3 Ifz z, ,z are three complex numbers which satisfy

1, , ..., are the roots of unity, find the value of

, be a root of the equation

Q. 6 Evaluate the fifth roots of 2, i.e., solve the equation


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Q. 1 Let1( )A z and 2( )B z be arbitrary points in the complex plane. Find the equation of the circle having AB as a

Q. 2 Show that the triangles whose vertices are 1 2 3 1 2 3

1 1

2 2

3 3

1

1 0

1

z Z

z Z

z Z

=

00az az b+ + = ( )b !

0 0

2

az az b

a

+ +

12 5

8 3

z

z i

=

and

41

8

z

z

=

Q. 5 Assume that ( 1, 2... )i

A i n= are the vertices of a regular polygon inscribed in a circle of radius unity. Find

the value of2 2 2

1 2 1 3 1... nA A A A A A+ + +

diameter.

z z, ,

zand

Z,

Z,

Zare directly similar if

Q. 3 Show that the perpendicular distance of a pointz from the line is

Q .4 Find the complex numbers which simultaneously satisfy


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Q. 1 Among the complex numberszsatisfying 25 15z i , find the one with the minimum argument and the one

Q. 2 Prove that if ,p ! the sum of thepth powers of the nth roots of unity is 0 unlessp is a multiple ofn. What is

the sum in that case? You can use the following fact:

2 1 11 ....1

nn xx x x

x

+ + + + =

Q. 3 Ifsin 2sin 3sin 0 + + = and cos 2cos 3cos 0, + + = simplify the expression

cos3 8cos3 27cos3 . + +You can use the following fact:

If 3 3 30, then 3a b c a b c abc+ + = + + =

Q. 4 If 1 1,z =

( )2

tan argz

i zz

=

Q. 5 For all complex numbers1 2,z z satisfying 1 212 and 3 4 5,z z i= = find the minimum value of 1 2 .z z

Q. 6 Evaluate ( )32 10

1 1

2 23 2 sin cos

11 11

p

p p

q qp i

= =

+

Hint:First rewrite2 2

sin cos11 11

q qi

in a simpler form:

2 2 2

sin cos cos sin11 11 11 11

q q q qi i i

= +

2 /11i qie = qi=

where 2 /11ie = is the first non-real 11th root of unity. Now evaluate

( )10 10

1 1

q q

q q

i i = =

=

with the maximum argument.

prove that

The rest is straight forward.

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complex_prob.pdf