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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
7/29/2019 complex_prob.pdf
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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.