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Propof
HISTORY
The solution in radicals without trigonometric functions of a general cubic
equation contains the square roots of negative numbers when all three roots
are real numbers a situation that cannot be rectified by factoring aided by
the rational root test if the cubic is irreducible the socalled casus
irreducibilis. This conundrum led Italian mathematician “Gerolamo
Cardano”to conceive of complex numbers in around 1545.
Work on the problem of general polynomials ultimately led to the fundamental
theorem of algebra which shows that with complex numbers a solution
exists to every polynomial equation of degree one or higher. Complex numbers
thus form an algebraically closed field where any polynomial equation has
a root.
Many mathematicians contributed to the full development of complex
numbers. The rules for addition subtraction multiplication and division of
complex numbers were developed by the Italian mathematician “Rafael
Bombelli”.
Complex numbers have practical applications in many fields including
Physics Chemistry Electrical Engineering and Statistics.
In the 18th century complex numbers gained wider use as it was noticed that
formal manipulation of complex expressions could be used to simplify
calculations involving trigonometric functions. For instance in 1730 Abraham
de Moivre noted that the complicated identities relating trigonometric
functions of an integer multiple of an angle to powers of trigonometric
functions of that angle could be simply reexpressed by the following well
known formula which bears his name “DeMoivres formula”
Definition
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A number in the form where x and y are real numbers and i is the
imaginary unit defined as is called Complex Number and it is denoted
by
Where x is called real part y is called imaginary part of complex number z
The complex number is purely real if imaginary part is zero and purely
imaginary if real part zero.
i.e. If y0 then is purely real.
If x0 then is purely imaginary.
ALGEBRA OF COMPLEX NUMBER
A Equality of Complex Number
Two Complex numbers and are said to be equal if their real and
imaginary parts are respectively equal
E.g. For and If and
i.e. Or have no meaning.
For and
B Addition and Subtraction
To add or subtract two complex numbers we add or subtract their real parts
separately and imaginary part separately.
C Multiplication
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D Division
As such division by an imaginary quantity has no meaning. Therefore to
make a meaningful quantity we multiply numerator and denominator by
conjugate of denominator.
i.e.
which is a complex number.
MODULUS OR MAGNITUDE OF COMPLEX NUMBER
Modulus of z is
We write
AMPLITUDE OR ARGUMENT OF COMPLEX NUMBER
Amplitude or Argument of z is denoted by amp z or arg z .
To find argument we have the following four cases depending upon the
position of a point corresponding to given complex number in a particular
quadrant.
1. Given: if
If corresponding points x y lie in first quadrant
Let ‘ ’ be the angle
For first quadrant argument of
2. Given: if
If the corresponding points x y lie in second quadrant
Let ‘ ’ be the angle
For second quadrant argument of
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3. Given: if
If the corresponding points x y lie in third quadrant
Let ‘ ’ be the angle
For third quadrant argument of
4. Given: if
If the corresponding points x y lie in fourth quadrant
Let ‘ ’ be the angle
For fourth quadrant argument of
POLAR FORM OF COMPLEX NUMBER
Polar form of complex number is
EXPONENTIAL FORM OF COMPLEX NUMBER
DEMOIVRE’S THEOREM
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Circular Functions of a Complex Number
Hyperbolic functions
1.
2.
3.
4.
5.
6.
7.
8.
9.
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10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
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27.
28.
29.
Relationship between Hyperbolic Circular Functions
1.
2.
3.
4.
5.
6.
X 0
0
1
0 1 1
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Formulae
1. If then
2. If then
and
3. Expansion of in powers of :
4. Expansion of in Terms of sines or cosines of Multiples of :
5. Roots of a Complex Number:
6.
7.
8.
9.
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10.
11.
12.
13.
14.
15. Inverse Hyperbolic Functions:
a
b
c
● Proof
a Let
This is a quadratic in
Conventionally we take positive sign.
b We leave this as an exercise.
c Let
By componendo and dividendo
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16.
17.
18. The general value of is denoted with capital by and
is given by
i.e.
19.
20.
21.
22.
23.
24.
25.
26.
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CLASSWORK PROBLEMS
Part I : Basics of Complex Number
1. If prove that
i
ii .
2. Find the complex conjugate of .
3. Find .
4. Find the modulus and argument of . Modulus of z 1
Argument of
5. If show that .
6. If and are any two complex numbers prove that
.
7. If and are two complex numbers such that prove the
difference of their amplitudes is .or prove that arg .
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Part II : DeMoivre’s Theorem
1. If n is a positive integer prove that
2. Prove that is equal to 1 if and 2 if
where is an integer.
3. Show that 4nth power of is equal to Where n is a positive
integer.
4. If prove that
i
ii
iii
iv
5. If are the roots of the equation prove that
6. If are the roots of find the equation whose roots are
7. If are the roots of the equation prove that
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8. If then show that
the general value of is
9. If then show that and .
10. By using De Moivre’s Theorem show that
.
11. Evaluate
12. If n is a positive integer and prove that
i
ii
iii
13. Use De Moivre’s Theorem to show that
Hence deduce that .
14. Show that .
15. Using De Moivre’s Theorem prove that where
16. Expand in a series of cosines of multiples of .
17. If prove that
.
18. Using De Moivre’s Theorem prove that
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19. Find the cube roots of unity. If is a complex cube root of unity prove
that
20. If is a complex fourth root of unity prove that .
21. Prove that the n nth roots of unity are in geometric progression.
22. Show that the sum of the n nth roots of unity is zero.
23. Prove that the product of the n nth roots of unity is .
24. Solve
25. Solve completely the equation .
26. If is a root of the equation find all the other
roots of the equation.
27. Find all the values of and show that their continued product is
28. Show that the roots of are given by
29. Solve the equation and show that the real part of all roots is 
1/2.
30. If are the roots of find them and show that
31. Separate into real and imaginary parts .
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32. If prove that .
Part III : Exponential form of Complex Number
1. Prove that .
2. If find tan hx.
3. Express in terms of hyperbolic sines of multiples of x.
4. If show that
i
ii
5. Prove that .
6. If prove that i ii
iii iv
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7. If prove that and .
8. If prove that
i
ii
iii .
9. If prove that
.
10. If prove that .
11. If prove that
12. If prove that .
13. Separate into real and imaginary parts .
14. If show that .
15. If or if express x and y in terms of
and . Hence show that are the roots of the equation
16. If prove that . Further if
and n is an integer prove that .
17. Prove that .
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18. Prove that .
19. If where a b are real prove that .
20. If prove that
.
21. Show that .
22. If show that
i r 1 ii iii
Part IV : Logarithmic Form Of Complex Number
1. Show that .
2. Prove that
3. If prove that .
4. Find the principal value of and show that its real part is
.
5. If find .
6. If prove that where .
7. Prove that where .
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8. If prove that when
.
9. Considering only the principal value if is real prove that its
value is .
10. If prove that the general value of x is given by
where
and .
11. If prove that where n is any positive integer.
12. Prove that .
13. Prove that .
HOMEWORK PROBLEMS
Part I : Basics of Complex Number
1. Express the following in the form x + iy
i ii
2. Find the modulus and the principal value of the argument of
i ii
3. Find the square root of
i ii
4. If prove that .
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5. If and find z. Ans : z 2
6. Prove that .
7. If find
i ii iii
iv v vi
8. If z x + iy prove that .
9. If prove that .
Part II : DeMoivre’s Theorem
1.
Show that i
ii
iii
2.
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3.
Hint:
4. Prove that
5. If Prove that
6. If then show that
Hint:
Also
7. If and in the Argand’s diagram if
where then prove that
Hint:
8. If are three complex numbers with modulus each and
.
Prove that i ii
9. If
Prove that i
ii
10. Using De Moivre’s Theorem prove the following.
i
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ii
iii
iv
11. If
Find the values of
12. Prove that
13. Prove that . Hence deduce that
.
14. Prove that .
Hint: Now
Now Find
Similarly Find
15. i Prove that
ii Expand as a series of cosines of multiples of .
iii Expand as a series of sines of multiples of .
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16. i Express in terms of .
ii Show that
17. Show that
18. If Prove that .
19. If . Show that
20. If x+1/x 2 cos α y+1/y 2 cos β z+1/z 2 cos
Show that xyz + √ +
1
√
2 cos
21. If x 1/x 2i sin y 1/y 2i sin show that
√ √
+
√
√ 2cos
−
22. If show that
23. If then by using De Moivre’s theorem simplify
24. If n is the + ve integer show that
25. If α β are the roots of quadratic equation x2 2x+ 4 0 then
i Prove that α
n
+ β
n
2
n+1
cos
3
ii Find the value of α
15
+ β
15
Ans : 2
16
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26. Find all the values of
where k 0123
27. Solve :
i
ii
28. i x
7
+ x
4
+ x
3
+ 1 0 Ans: 1 1/2 ±
1√ 3
2
1
√2
±
1
√2
−1
√2
±
1
√2
ii x
10
+ 11x
5
+ 10 0 Ans: 10
1/5
1 cos
5
±i sin
5
cos
3 5
±i sin
3 5
iii x
9
 x
5
+ x
4
 1 0. Ans:± 1 ±i cos
5
±i sin
5
cos
3 5
±i sin
3 5
iv x
14
+ 127x
7
 128 0 Ans: 2 cos 2k+ 1
7
+ i sin 2k+ 1
7
k o to 6
v x
7
+ x
4
+ i x
3
+1 0
Ans: 1 1/2 ± i
√3
2
± cos
8
 i sin
8
± cos
3 8
+ i sin
3 8
29. Solve
i x
4
 x
2
+ 1 0 M96 Ans:±
√ 3
2
±
2
.
ii x
4
 x
3
+ x
2
 x+ 1 0. Ans: cos
5
±i sin
5
cos
3 5
±i sin
3 5
30. Find the continued product of all the values of
i 1+ i
2/3
Ans: 2i
ii 1+ i
1/5
Ans: 1+i
iii 1+ i√3
1/4
Ans:  1+ i√3
31. Show that the n
th
roots of unity are given by where
cos 2 / + i sin 2 / . Show that continued products of the all these n
th
roots is 1
n+1
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32. Prove that n
th
roots of unity are in geometric progression. Also find sum of
n
th
root of unity.
33. Find the roots of and show that the real part of all the roots is 
1/2
34. Solve
Hint :
Ans : where k 0 1 2
35. Obtain the solution of the equation
Hint: Ans:
where
Solve where k 01 2 3 4.
36. If then .State true of false. Ans: True
37. If arg z+ 1 and arg z 1 find z.
38. Find z if amp z+ 2i amp z 2i Ans: z 2+ i0
39. If represents a point on the line 3x+ y 0 in Argand’s diagram
find a. Ans: a 1 or ¾
40. Find two complex numbers whose sum is 4 and product is 8.
Ans : z1 2+ 2i z2 2 2i
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41. If where . Find polar form of
.
Hint: Divide N D by z1 where
42. a Express in the form . Find value of a b in terms of
x and y.
b If prove that
43. If Prove that
44. Prove that
Hint: Let where and
45. If prove that
46. If prove that
47. Prove that
48. If . Prove that z lies on imaginary axis where z is a complex
number.
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Part III : Exponential form of Complex Number
1. If z x+ iy and 2
a+ ib. Find the a and b.
Hint : a + ib 2
+
2
2
− 2
+12
Ans: a 2
− 2
cos 2xy b 2
− 2
sin 2xy
2. If find R and .
Ans:
3. If p a + ib q a  ib where a and b are real then prove that pe
p
+ qe
q
is
real.
4. Prove that 1 
1/2
+ 1 
1/2
1 + cosec /2
1/2
.
5. Prove that 1  sec /2
1/2
1 +
1/2
 1 +
1/2
6. Show that
2
+
2 2
2
+
3 2
3
+ …………..
2 5−4
7. Solve the equation 7 cosh x + 8 sinh x 1 for real values of x. Ans: log 3
8. If tanh x 1/2 find sinh 2x and cosh 2x Ans: 4/3 5/3
9. If x tanh
1
0.5. show that sinh 2x 4/3 Hint: sinh 2x 2 tanh x/ 1
tanh
2
x
10. Prove that tanh log√3 1/2. Hint: use definition of tanhx.
11. Prove that 16 sinh
5
x sinh 5 x – 5 sinh 3x + 10 sinh x.
12. Prove that 32 cosh
6
x 10 cosh 6x+ 6 cosh 4x+ 15 cosh 3x.
13. If cosh
6
x a cosh 6x + b cosh 4x + c cosh 2x + d prove that
5a+ 5b+ 3c 4d 0
14. Prove that
15. Prove that i
1+ ℎ
1− ℎ ℎ
n
cosh2nx + sinh2nx
ii cos hx – sin hx
n
cosh nx – sinh nx
16. Prove that
ℎ ℎ + ℎ ℎ
ℎ ℎ − ℎ ℎ
n
cosh 2nx + sinh 2nx
17. If log tan x y prove that
i sinh ny 1/2 tan
n
x – cot
n
x
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ii 2 cosh ny cosec 2x cosh n+ 1 y + cosh n 1 y
18. If sin + i
prove that sin ± cos
2
±sinh
2
19. If cosh + i
prove that sin
2
sin
4
sinh
4
20. If prove that
21. If cos x+iy e
iπ/6
Prove that
i 3sin
2
xcos
2
x 4sin
2
x.cos
2
x
ii 3sinh
2
y + cosh
2
y 4sinh
2
y.cosh
2
y
22. If log cosxiy α + iβ prove that α log and find β.
23. If sin
1
α+iβ λ + iμ. Prove that sin
2
λ and cosh
2
μ are the roots of the
equations x
2
– 1+ α
2
+ β
2
x + α
2
0
24. Let Pz where z sinα+iβ. If α is variable show that the locus of the Pz is
an ellipse . Also show that x
2
cosec
2
α – y
2
sec
2
α 1 if β is
variable.
25. If sinh x+ iy e
iπ/3
prove that
i 3cos
2
y – sin
2
y 4sin
2
y cos
2
y
ii 3sinh
2
x + cosh2 x 4sinh
2
x.cosh
2
x
26. If u+ i v cosh + i /4 .Find the value of u
2
– v
2
Ans : ½
27. If x+ iy 2 cosh + i /3 prove that 3x
2
 y
2
3
28. If x 2 sin cosh β y 2 cos sinh β Show that
i cosec − β + cosec + β
4 2
+ 2
ii cosec − β  cosec + β
4
2
+ 2
29. If tan
6
+ x+ iy prove that x
2
+ y
2
+ 2x/√3 1.
30. If cot
6
+ x+ iy prove that x
2
+ y
2
 2x/√3 1
31. Show that tan
32. If tan h +i β x+ iy prove that x
2
+ y
2
 2x cot 2 1 x
2
+ y
2
+ 2y coth
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2 β + 1 0.
33. If cot +i β i. Prove that β
4
0
34. If +i β tan h x + i
4
prove that 2
+ 2
1
35. If tan h a+ ib x+ iy Prove that x
2
+ y
2
 2x coth 2 + 1 0x
2
+ y
2
+ 2y coth
2  1 0
36. If Show that
37. If . Prove that
38. Separate into real and imaginary parts i sec x+ iy ii tanh x+ iy
39. Show that
i sinh
1
x cosh
1
√1 + 2
ii tanh
1
sinh
1
√1− 2
iii Prove that tanh
1
sin cosh
1
sec .
40. Show that sech
1
sin log cot /2
41. Show that sinh
1
tan x log tan
4
+
2
42. Prove that cosech
1
z log
1+√1+ 2
.Is defined for all values of z
43. Show that cos
1
z  i log z±√ 2
− 1
44. If cosh
1
a + cosh
1
b cosh
1
x then prove that
a √ 2
− 1 + b √ 2
− 1 √ 2
− 1.
45. If cosh
1
x+ iy + cosh
1
x iy cosh
1
a prove that 2a 1 x
2
+ 2a+ 1
y
2
a
2
 1.
46. If A+ iB C tan x+ iy prove that tan 2x
2
2
− 2
− 2
47. Separate tan
1
cos + i sin into real and imaginary parts
48. If tan + i cos + i sin show that
2
+
4
¼ log
1+ 1−
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49. If tan + i
show that n+ 1/2 /2 and 1/2 log tan
/4 + /2
50. Separate into real and imaginary parts : tan
1
a+ iyor Prove that tan
1
a+ iy 1/2 tan
1
2a/1 a
2
 y
2
+ i/4 log 
1+
2
+ 2
1−
2
+ 2

51. Prove that one value of tan
1
x+ iy/x iy is /4 + /2 log x+ y/x y where
x y 0.
52. If tan x+ iy i x y ∈R. Show that x is indeterminate and y is infinite.
Hint: tanx iy – I then tan 2xtan x+ iy+x iy tan 2iy tan
x+ iyxiy
53. If tan u+ iv x+ iy then prove that curves u constant and v constant
are families of circles.
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Part IV : Logarithmic Form of Complex Number
1. Show that
2. Solve for z if
3.
4. Prove that
5. Prove that .
6. Show that
7. Prove that
8. Show that
9. Show that
10. If prove that i
ii
11. Separate into real and imaginary parts :
i
ii
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iii
iv
12. Separate into real and imaginary parts consider principal
values only
13. Prove that the real value of principal of is
14. Prove that the general values of is
Hence find the principal value.
15. If show that
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EXTRA SOLVED PROBLEMS
Q.1. If Z1 Z2 are nonzero complex numbers of equal modulus and Z1 ≠ Z2
then prove that is purely imaginary.
Solution:
Since Z1 and Z2 are two complex numbers with equal modulus say r
Let
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Also
Dividing i by ii we get
which is purely imaginary.
Q.2. If prove that
i ii
Solution:
Now
i Then
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For G.P. Sum
ii Also
Q.3. If prove that and
Solution:
i L.H.S.
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ii L.H.S.
Multiplying Numerator Denominator by i
Q.4. If
Prove that i ii
Solution:
Now
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Comparing both sides we get
Squaring and adding i ii we get
Dividing ii by i we get
Q.5. If and prove that
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Solution:
Now
Similarly
As above
Subtracting ii from i we get
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Q.6. Prove that
Solution:
Let
Then
Multiplying N D by i
From i ii we get
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Q.7. If find the value of a b c.
Hence show that
Solution:
Now
Comparing imaginary part on both sides we get
Comparing above equation with the given equation we get
Deduction:
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Q.8. If prove that
Solution:
Let
Also
Then
Hence
Comparing this with the given equation we get
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Q.9. Show that the power of is where n is a positive integer.
Solution:
Now
Hence
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Q.10. Find the roots common to and .
Solution:
We have
General polar form
Putting k012345 we get the roots as
Also
General polar form
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Putting k0123 we get the roots as
From I and II we get the common roots as
Q.11. If are the roots of find their values and show
that
Solution: Now
Multiplying both sides by x1
where k01234.
When k0 Root
When k1 Root say
When k2 Root say
When k3 Root say
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When k4 Root say
Since are the roots of we have
Putting x1 we get
Note:
Hence
Q.12. Prove that
Solution: Consider
When k 0
k 1
k2
k3
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k 4
Also
FromI and II we have
Q.13. If then prove that i ii
Solution:
i Now
Then
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ii Now
Comparing both sides we get
Then
Also
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From I and II we have
Q.14. If then show that
Solution:
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Now
Adding two equations we get
……………………i
Similarly subtracting we get
as above
……………………ii
From i ii we get
Alternately
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Comparing both sides we get
……………………i
And
……………………ii
From i and ii we get
Q.15. If log tan x y then prove that
i
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ii
Solution:
Now log
i sinh ny
ii
Alternately
But
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Subs .in I we get
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Q.16. If Prove that
Solution:
Let
Now
Then
Also
i tanh
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Subs. in I we get z
Q.17. Find the sum of the series
Solution:
Let
………
.....
………
for a Geom. Series
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Equating the imaginary parts we get
Equating the real parts we get
Q.18. If u + iv prove that and
Solution:
Now
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………i
But
Subs. from i
Comparing both sides we get
Q.19. Find the value of log sinx+ iy
Solution:
sin x+ iy sin x.cos iy + cos x. sin iy
sin x. cosh y + i cos x. sinh y
sinx+ iy log sin x. cosh y + i cos x. sinh y
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Q.20. If find and
Solution:
Now
…… I
But
Subs in I we get
say
Hence where
Q.21. Prove that
Solution:
Let a – b x a + b y
Then
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Q.22. Show that if has real values then one of them is
Solution:
For the given expression to be real we must have
………i
Then value of expression
Subs. fromi
Q.23. If
Then show that the general value of θ is
Solution:
Now
……By comparing imaginary parts
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Q.24. If Z1 Z2 and Z3 Z4 are two pairs of conjugate complex numbers
Then show that i ii
Solution:
Let and
and
Then
……i
Also
………ii
Hence from i ii
1
Q.25. If Show that
Solution:
Let
But …..given
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∴
Q.26. If
Show that i
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Putting n p – q
ii
Solution:
Let
Where
p123…..n
Let
Where
Now given
i Comparing amplitude we get
i.e.
ii Comparing modulii we get
∴ Squaring both thet sides
∴
Q.27. Prove that
Solution:
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Let
Then
Hence
Q.28. If then prove that
Solution:
Let
Comparing both the sides
Then
Q.29. If and then prove that
Solution: Now
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Then
But
By equal ratio theorem
From i ii we get
Q.30. Find two complex numbers such that their difference is 10i and their
product is 29
Solution:
Since that difference between two complex numbers is imaginary and
their product is real
The two numbers must be conjugates
Let the numbers be and
Now
Also
Hence the two numbers are Or
Q.31. If and find z
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Solution:
Let
Now
Also
From i ii we get
Then
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Q.32. If then Show that
i
ii
Solution:
Now
Putting we get
i Comparing real parts we get
ii comparing imaginary part we get
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Q.33. If then prove that
Solution:
Now
Hence
Then
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Q.34. If and show that
Solution: Now
Let
But
………i and
………ii
Then
using i and ii
Q.35. If
Show that
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Solution:
Let
Comparing both that sides we get
Q.36. If
Prove that
Solution:
Now
Hence
Then i + ii gives
Q.37. If then show that
Solution:
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Q.38. Prove that
Solution:
L.H.S.
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Q.39. If prove that
Solution:
Now
Using componendo  dividendo
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Dividing N D by
Q.40. Using De Moivre’s Theorem show that where
Solution:
Now
Expanding R.H.S. by Binomial Theorem and Comparing real parts we
get
Squaring both the sides
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Q.41. Show that
Solution:
Now Multiplying N D by i
Where
. Hence
Q.42. If are the root of the equation prove that
Hence deduce that
Solution:
Now
are its roots we have
Let
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Similarly
Hence
Putting We get
Q.43. If are the roots of . Prove that
Solution:
Now
Adding we get
Q.44. Find the continued product of
Solution:
Now
When
When
When
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When
Continued product od all the values
Now 1+3+5+……. is an A.P. with
Its
Hence Required Product of Values
Q.45. Find the cube root of
Solution:
Let
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When
When
When
Q.46. Solve
Solution:
Now Multiply by x+1 on both the side
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When k0 Root
When k1 Root
When k2 Root
When k3 Root
When k4 Root
Discarded as we have taken it in the equation
Also
Hence Required Roots are
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Q.47. Given that is one root of the equation . Find
the other roots.
Solution:
Since is one root of the equation.
is the other root.
The equation with this root is
For finding the other factor we have to divide
Then
Hence the required roots are
.
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Q.48. Show that all the roots of are given by
Solution:
By ComponendoDividendo
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Multiplying N D by  1
When not defined Hence discard
Hence the solution are given by where
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Q.49. Show that the points representing the roots of the equation
on Argand’s diagram are collinear.
Solution:
Now
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Multiplying N D byi
For k 0 1 2 we get three values of z.
All these values have the same real parts i.e.
Hence the points represented by the 3 numbers are collinear.
Q.50. If and n is an integer prove that is not a
multiple of 3.
Solution:
Now
Similarly
Hence
If n is not a multiple then
Let where k is an integer
Where
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Value of expression
When
Values of expression
Subs in i we get
Q.51. Show that
Solution:
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Q.52. Show that
Solution: L.H.S.
Q.53. If Prove that
Solution:
Now
i Now
But
eq. of ellipse is is constant
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ii Also
But
eq. of hyperbola is is constant
Q.54. Show that
Solution:
Let
Comparing both sides we get
From i
Also
Hence
Q.55. Prove that
Solution:
iLet
From i ii we have
ii Now
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From i ii we have
Q.56. Prove that
Solution: Let
From i ii we have
Q.57. Prove that hence deduce that
Solution:
Let
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By Componendo  Dividendo
From i ii we get
Putting and resp. in i and then adding we get
Q.58. If then prove that
Solution:
Now
Let
Then R.H.S.
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Q.59. If a prove that
Solution:
Let
Adding we get
Subtracting we get
Also given
T.P.T.
i.e. Dividing by
Q.60. If Prove that
i
ii
iii
Solution:
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i Now
ii
iii Now
Q.61. If Prove that i
ii iii
Solution:
i Now
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ii from i
Where
Hence
iii Now
By componendoDividendo
Hence
Q.62. Find the sum of the series
……… to n terms
Solution:
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Let n terms
terms
By Binomial Expression of
By De Moivre’s Theorem
Equating imaginary parts we get
Q.63. Prove that
Solution: Let
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Comparing real parts we get
Q.64. If
Solution:
Now
Comparing both the sides
Dividing ii by i
Q.65. Prove that
Solution:
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Now
Then
Q.66. Prove that
Solution:
Now
Hence
Q.67. If prove that
Solution: Now
Taking log general of both sides we get
Hence
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Q.68. If prove that
Solution:
Now
But
Q.69. If prove that
Solution:
Now
Comparing imaginary parts of both the side we get
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Q.70. If prove that
Solution:
Now
i Comparing imaginary parts we get
ii Comparing imaginary parts we get
Q.71. If show that
Solution:
Now
Then
But
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Hence
Q.72. Find the principle value of show that it is purely real if
is multiple of
Solution:
Now
If is entirely real then
i.e. multiple of
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Q.73. If prove that the general value of x is given by
Where
Solution:
If
Taking log general of both sides we get
Comparing both side we get
Then gives
Also gives