Ekeeda – Chemical Engineering 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
irreducibilis).
test if
This
the
cubic
conundrum
led
is
irreducible Italian
(the
so-called casus
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 re-expressed by the following wellknown formula which bears his name, “DeMoivre's formula”
Definition
Ekeeda – Chemical Engineering 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 y=0 then
is purely real.
If x=0 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 i.e. For
and Or
,
If
and
have no meaning. 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
Ekeeda – Chemical Engineering 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.
Ekeeda – Chemical Engineering 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.
:-
Ekeeda – Chemical Engineering 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 given by
i.e. 19. 20. 21.
22. 23. 24. 25. 26.
is denoted with
capital by
and
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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 Argument of
[Modulus of z = 1,
]
5. If 6. If
.
, show that and
.
are any two complex numbers, prove that .
7. If
and
are two complex numbers such that
difference of their amplitudes is
.(or prove that arg
, prove the .)
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Part II : DeMoivre’s Theorem 1. If n is a positive integer, prove that
2. Prove that where
is equal to -1, if
and 2, if
is an integer.
3. Show that (4n)th 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
6. If
are the roots of
7. If
are the roots of the equation
, prove that , find the equation whose roots are
, prove that
Ekeeda – Chemical Engineering 8. If
then show that
the general value of 9. If
is 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
16. Expand
where
in a series of cosines of multiples of
17. If
.
, prove that .
18. Using De Moivre’s Theorem prove that,
Ekeeda – Chemical Engineering 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
28. Show that the roots of 29. Solve the equation
and show that their continued product is
are given by 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 3. Express
4. If
.
, find tan hx. in terms of hyperbolic sines of multiples of x.
,
, show that
(i) (ii)
5. Prove that 6. If
. , prove that (i) (iii)
(ii) (iv)
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7. If
, prove that
8. If
and
.
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
and
. express x and y in terms of
. Hence show that
16. If
are the roots of the equation
, prove that
,
and n is an integer, prove that 17. Prove that
.
.
. Further, if .
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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 (i) r = 1
. , show that
(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 6. If 7. Prove that
, find
,
.
, prove that , where
, where
. .
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8. If
,
prove
that
,
when
. 9. Considering only the principal value, if value is
, is real, prove that its
.
10. If
, prove that the general value of x is given by where
and
.
11. If
, prove that
12. Prove that
where n is any positive integer. .
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.
6. Prove that
.
7. If
, find
(i)
(ii)
(iv)
(v)
8. If z = x + iy, prove that
9. If
, prove that
Part II : DeMoivre’s Theorem 1. Show that (i) (ii) (iii) 2.
[Ans : z = 2]
(iii) (vi)
.
.
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3. Hint: 4. Prove that
5. If
, Prove that
6. If
then show that
Hint:
Also 7. If
and where
in
the
Argand’s
diagram
if
then prove that
Hint: 8. If
are three complex numbers with modulus .
Prove that (i)
(ii)
9. If Prove that (i) (ii) 10. Using De Moivre’s Theorem, prove the following. (i)
each and
Ekeeda – Chemical Engineering (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 đ?œƒ .
(iii) Expand
as a series of sines of multiples of đ?œƒ .
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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 Îą, y+1/y = 2 cos β, z+1/z = 2 cos đ?›ž Show that xyz + √đ?‘Ľđ?‘Śđ?‘§ +
1 √đ?‘Ľđ?‘Śđ?‘§
= 2 cos (
) đ?‘š
21. If x -1/x = 2i sin đ?œƒ , y -1/y = 2i sin đ?›&#x; show that
22. If
√đ?‘Ľ
đ?‘›
√đ?‘Ś
+
đ?‘›
√đ?‘Ś = √đ?‘Ľ
đ?‘š
show that
23. If then by using De Moivre’s theorem simplify 24. If n is the + ve integer, show that
25. If Îą, β are the roots of quadratic equation x2- 2x+ 4 = 0, then đ?‘›đ?œ‹
(i)
Prove that ιn+ βn = 2n+1 cos ( 3 )
(ii)
Find the value of ι15+ β15Ans : -216
đ?œƒ
đ?œ™
2cos (đ?‘š − đ?‘› )
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26. Find all the values of where
, k = 0,1,2,3]
27. Solve :
(i) (ii) 28. (i) x7+ x4+ x3 + 1 = 0 (ii) x10+ 11x5+ 10 = 0 (iii) x9- x5+ x4 - 1 = 0. 3đ?œ‹ 5
[Ans: -1, 1/2 Âą đ?œ‹
1√3 2
,
1 √2
đ?œ‹
Âą
1 √2
−1
,
√2
3đ?œ‹
1
Âą
]
√2
3đ?œ‹
[Ans: (-10)1/5, -1, cos 5 Âąi sin 5 , cos ( 5 ) Âąi sin ( 5 )] [Ans:Âą -1, Âąi, cos
đ?œ‹ 5
đ?œ‹
Âąi sin 5 , cos
3đ?œ‹ 5
Âąi sin
] đ?œ‹
(iv) x14+ 127x7- 128 = 0
[Ans: 2 [ cos (2k+ 1) 7 + i sin (2k+ 1)
đ?œ‹
]
7
k = o to 6] (v) x7+ x4+ i (x3+1) = 0 [Ans: -1, 1/2 Âą i
√3 , 2
Âą( cos
đ?œ‹ 8
- i sin
đ?œ‹ 8
), Âą (cos
3đ?œ‹ 8
3đ?œ‹
+ i sin
8
),] 29. Solve (i) x4- x2+ 1 = 0 [M96]
[Ans:Âą
√3 2
,
đ?‘–
Âą .] 2
(ii) x4- x3+ x2- x+ 1 = 0.
[Ans: cos
đ?œ‹
đ?œ‹
Âąi sin 5 ,cos 5
3đ?œ‹ 5
Âąi sin
3đ?œ‹ 5
]
30. Find the continued product of all the values of (i) [1+ i]2/3 (ii) [1+ i]1/5 (iii) (1+ i√3 )1/4 31. Show that the nth roots of unity are given by
[Ans: 2i ] [Ans: 1+i] [Ans: - ( 1+ i√3 )] where đ?œ† =
cos 2đ?œ‹/đ?‘› + i sin 2đ?œ‹/đ?‘›. Show that continued products of the all these nth roots is (-1)n+1
Ekeeda – Chemical Engineering 32. Prove that nth roots of unity are in geometric progression. Also find sum of nth 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:
]
.State true of false.
[Ans: True]
where Solve
where k = 0,1, 2, 3, 4.
36. If
37. If arg (z+ 1) =
, then
and arg (z- 1) =
38. Find z if amp (z+ 2i) =
find z.
, 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 42. (a) Express
where in the form
. Find value of a & b in terms of
x and y.
(b) If 43. If
, prove that , Prove that
44. Prove that Hint: Let
45. If 46. If
where
and
, prove that , prove that
47. Prove that 48. If number.
. Prove that z lies on imaginary axis where z is a complex
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Part III : Exponential form of Complex Number 2
1. If z = x+ iy and � � = a+ ib. Find the a and b. 2
2
Hint : a + ib = � � = � (�+��) = � � Ans: a = � �
2 −đ?‘Ś 2
cos 2xy, b = đ?‘’ đ?‘Ľ
2. If
2 −đ?‘Ś 2 +12đ?‘Ľđ?‘Ś
2 −đ?‘Ś 2
sin 2xy
find R and đ?›&#x;.
Ans: 3. If p = a + ib, q = a - ib where a and b are real then prove that pep + qeq is real. 4. Prove that (1 - đ?‘’ đ?‘–đ?œƒ )-1/2 + (1 - đ?‘’ đ?‘–đ?œƒ )-1/2 = ( 1 + cosec đ?œƒ/2)1/2.
5. Prove that ( 1 - sec đ?œƒ/2 )1/2 = ( 1 + đ?‘’ đ?‘–đ?œƒ )-1/2 - (1 + đ?‘’ đ?‘–đ?œƒ )-1/2 6. Show that
đ?‘ đ?‘–đ?‘›đ?œƒ 2
+
đ?‘ đ?‘–đ?‘›2đ?œƒ 22
+
đ?‘ đ?‘–đ?‘›3đ?œƒ 23
+ ‌‌‌‌..=
2đ?‘ đ?‘–đ?‘›đ?œƒ 5−4đ?‘?đ?‘œđ?‘ đ?œƒ
7. Solve the equation 7 cosh x + 8 sinh x = 1 for real values of x. 8. If tanh x = 1/2, find sinh 2x and cosh 2x
[Ans: -log 3]
[Ans: 4/3, 5/3]
9. If x = tanh-1 (0.5). show that sinh 2x = 4/3 Hint: sinh 2x = 2 tanh x/ 1tanh2 x 10. Prove that tanh (log√3 ) = 1/2. Hint: use definition of tanhx. 11. Prove that 16 sinh5 x = sinh 5 x – 5 sinh 3x + 10 sinh x. 12. Prove that 32 cosh6x- 10 = cosh 6x+ 6 cosh 4x+ 15 cosh 3x. 13. If cosh6x= a cosh 6x + b cosh 4x + c cosh 2x + d, prove that 5a+ 5b+ 3c- 4d = 0
14. Prove that 1+ ������ ℎ�
15. Prove that (i) [1−đ?‘Ąđ?‘Žđ?‘›â„Žđ?‘Ąđ?‘Žđ?‘›â„Ž đ?‘Ľ ]n = cosh2nx + sinh2nx (ii) (cos hx – sin hx)n = cosh nx – sinh nx đ?‘?đ?‘œđ?‘ â„Žđ?‘?đ?‘œđ?‘ â„Ž đ?‘Ľ+đ?‘ đ?‘–đ?‘›â„Žđ?‘ đ?‘–đ?‘›â„Ž đ?‘Ľ
16. Prove that [ đ?‘?đ?‘œđ?‘ â„Žđ?‘?đ?‘œđ?‘ â„Ž đ?‘Ľâˆ’đ?‘ đ?‘–đ?‘›â„Žđ?‘ đ?‘–đ?‘›â„Ž đ?‘Ľ ]n = cosh 2nx + sinh 2nx 17. If log ( tan x) = y, prove that (i) sinh ny = 1/2 (tannx – cotnx)
Ekeeda – Chemical Engineering (ii) 2 cosh ny cosec 2x = cosh (n+ 1) y + cosh (n- 1) y 18. If sin (đ?œƒ + iđ?›&#x;) = đ?‘’ đ?‘–đ?›ź , prove that sin đ?›ź = Âą cos2đ?œƒ = Âąsinh2đ?›&#x; 19. If cosh (đ?œƒ + iđ?›&#x;) = đ?‘’ đ?‘–đ?›ź , prove that sin2đ?›ź = sin4đ?›&#x; = sinh4đ?œƒ 20. If
prove that
21. If cos (x+iy) = eiπ/6, Prove that (i) 3sin2x-cos2x = 4sin2x.cos2x (ii)3sinh2y + cosh2y = 4sinh2y.cosh2y 22. If log [cos(x-iy)] = ι + iβ, prove that ι =
log
and find β.
23. If sin-1(ι+iβ) = Ν + iΟ. Prove that sin2 Ν and cosh2 Ο are the roots of the equations
x2 – (1+ ι2 + β2)x + ι2 = 0
24. Let P(z) where z = sin(ι+iβ). If ι is variable, show that the locus of the P(z) is
an ellipse
. Also show that x2 cosec2ι – y2sec2 ι = 1 if β is
variable. 25. If sinh (x+ iy) = eiĎ€/3,prove that (i) 3cos2y – sin2y = 4sin2y cos2y (ii)3sinh2 x + cosh2 x = 4sinh2x.cosh2 x 26. If u+ i v = cosh ( đ?›ź + i đ?œ‹/4 ).Find the value of u2 – v2
[Ans : ½]
27. If x+ iy = 2 cosh (đ?›ź+ i đ?œ‹/3), prove that 3x2- y2 = 3 28. If x = 2 sin đ?›ź cosh β, y = 2 cos đ?›źsinh β , Show that (i) cosec(đ?›ź − đ?‘– β ) + cosec (đ?›ź + đ?‘– β ) = (ii)cosec(đ?›ź − đ?‘– β ) - cosec (đ?›ź + đ?‘– β ) =
4đ?‘Ľ đ?‘Ľ 2 + đ?‘Ś2 4đ?‘–đ?‘Ś đ?‘Ľ2+ đ?‘Ś2
đ?œ‹
29. If tan( 6 + đ?‘–đ?›ź) = x+ iy, prove that x2+ y2 + 2x/√3 = 1. đ?œ‹
30. If cot ( 6 + đ?‘–đ?›ź) = x+ iy, prove that x2+ y2 - 2x/√3 = 1 31. Show that tan
=
32. If tan h (� +i β ) = x+ iy, prove that x2+ y2- 2x cot 2 �= 1, x2+ y2+ 2y coth
Ekeeda – Chemical Engineering 2 β + 1 = 0. đ?œ‹
33. If cot (� +i β ) = i. Prove that β = 4 , � = 0 34. If � +i β = tan h ( x + i
đ?œ‹ 4
), prove that �2 + � 2 = 1
35. If tan h (a+ ib )= x+ iy, Prove that x2+ y2- 2x coth 2� + 1 = 0&x2+ y2+ 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 + đ?‘Ľ 2 ) (ii) tanh-1 (đ?›&#x;) = sinh-1 (
đ?œ™
√1−đ?œ™2
)
(iii) Prove that tanh-1 (sinđ?œƒ ) = cosh-1 (secđ?œƒ ). 40. Show that sech-1 (sinđ?œƒ ) = log (cot đ?œƒ/2) 41. Show that sinh-1 (tan x) = log [ tan ( 42. Prove that cosech-1 z = log (
1+√1+đ?‘§ 2 đ?‘§
đ?œ‹ 4
đ?‘Ľ
+ 2) ]
).Is defined for all values of z?
43. Show that cos-1 z = - i log ( zÂąâˆšđ?‘§2 − 1 ) 44. If cosh-1 a + cosh-1 b = cosh-1 x, then prove that a √đ?‘? 2 − 1 + b √đ?‘Ž2 − 1= √đ?‘Ľ 2 − 1. 45. If cosh-1 (x+ iy) + cosh-1 (x- iy) = cosh-1 a, prove that 2(a- 1) x2+ 2(a+ 1) y2 = a2- 1. 46. If A+ iB = C tan (x+ iy), prove that tan 2x =
2đ??śđ??´ đ??ś 2 −đ??´2 −đ??ľ2
47. Separate tan-1 (cosđ?œƒ + i sinđ?œƒ ) into real and imaginary parts 48. If tan (đ?œƒ + iđ?›&#x;) = cos đ?›ź + i sin đ?›ź, show that đ?œƒ =
đ?‘›đ?œ‹ 2
+
đ?œ‹ 4
1+đ?‘ đ?‘–đ?‘›đ?›ź
, đ?›&#x; = Âź log (1−đ?‘ đ?‘–đ?‘›đ?›ź)
Ekeeda – Chemical Engineering 49. If tan (đ?œƒ + i đ?›&#x; ) = đ?‘’ đ?‘–đ?›ź show that đ?œƒ = ( n+ 1/2) đ?œ‹/2 and đ?›&#x; = 1/2 log tan (đ?œ‹/4 + đ?›ź/2 ) 50. Separate into real and imaginary parts : tan-1(a+ iy)or Prove that tan-1 (a+ iy) = 1/2 tan-1 ( 2a/1- a2- y2 ) + i/4 log |
(1+đ?‘Ś)2 +đ?‘Ž2 (1−đ?‘Ś)2 +đ?‘Ž2
|
51. Prove that one value of tan-1 (x+ iy/x- iy) is đ?œ‹/4 + đ?‘–/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: tan(x- iy) – I, then tan 2x=tan[ (x+ iy)+(x- iy)] & tan 2iy = tan [(x+ iy)-(x-iy)] 53. If tan (u+ iv) = x+ iy then prove that curves u = constant and v = constant are families of circles.
Ekeeda – Chemical Engineering
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 14. Prove that the general values of
is is Hence find the principal value.
15. If
, show that
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EXTRA SOLVED PROBLEMS Q.1. If Z1 , Z2 are non-zero 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 &
Ekeeda – Chemical Engineering
Also
Dividing (i) by (ii) we get,
which is purely imaginary.
Q.2. If
prove that
(i) Solution: Now (i)
Then
(ii)
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[For G.P., Sum= (ii) Also
Q.3. If Solution: (i) L.H.S.
prove that
and
]
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(ii)
L.H.S.
[Multiplying Numerator& Denominator by i]
Q.4. If
Prove that (i) Solution: Now
(ii)
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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
Ekeeda – Chemical Engineering 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 Solution:
Now
Hence
power of
is
where n is a positive integer.
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Q.10. Find the roots common to
and
.
Solution: We have [General polar form]
Putting k=0,1,2,3,4,5, we get the roots as,
Also [General polar form]
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Putting k=0,1,2,3, 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 (x-1)]
where k=0,1,2,3,4. When k=0,
Root
When k=1,
Root
(say)
When k=2,
Root
(say)
When k=3,
Root
(say)
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When k=4,
Root
Since
are the roots of
Putting x=1,
we get,
Note: Hence
Q.12. Prove that Solution: Consider
When k= 0, k = 1,
k=2,
k=3,
(say) , we have
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k= 4,
Also
From(I) and (II) we have
Q.13. If Solution: (i) Now
Then
then
prove
that
(i)
(ii)
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(ii) Now
Comparing both sides we get,
Then
Also
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Q.14. If Solution:
then show that
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,
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)
Ekeeda – Chemical Engineering (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 Solution: Let Now
& Then
Also i tanh
Prove that
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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
Solution:
Now
=
=
= =
= =
and
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= = =
But
………(i)
= =
[Subs. from (i)]
= Comparing both sides we get,
Q.19. Find the value of log [ sin(x+ iy) ] Solution: sin (x+ iy) = sin x.cos iy + cos x. sin iy = sin x. cosh y + i cos x. sinh y [sin(x+ iy)] = log (sin x. cosh y + i cos x. sinh y) = =
= = =
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Q.20. If Solution:
, find
and
Now
…… (I) But
Subs in (I) we get,
(say)
Hence
,
Q.21. Prove that Solution: Let a – b = x, a + b = y
Then
where
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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. from(i) ]
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)
&
………(ii)
Also
Hence from (i) & (ii), &
Q.25. If
Show that
Solution: Let
But
…..(given)
=1
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∴
Q.26. If
Show that (i)
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(Putting n = p – q) (ii) Solution: Let Where &
p=1,2,3…..n
Let Where & Now,
(given)
(i) Comparing amplitude we get,
i.e. (ii) Comparing modulii we get,
∴ ∴
Q.27. Prove that Solution:
(Squaring both thet sides)
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Then &
Hence Q.28. If
then prove that
Solution: Let
,
Comparing both the sides
Then
Q.29. If Solution: Now
and
then prove that
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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 Q.31. If
and
&
Or , find z
&
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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) ………(ii) Then
using (i) and (ii) Q.35. If Show that
and
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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 Solution:
then show that
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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
&
.
Q.42. If
are the root of the equation
Hence, deduce that Solution: Now are its roots we have
Let
Hence
, prove that
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Similarly Hence Putting Q.43. If
,
We get
are the roots of
Solution:
Now
Adding we get,
Q.44. Find the continued product of Solution: Now
When When When
. Prove that
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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 k=0,
Root
When k=1,
Root
When k=2,
Root
When k=3,
Root
When k=4, Discarded,
Root (as we have taken it in the equation)
Also & Hence Required Roots are
,
Ekeeda – Chemical Engineering Q.47. Given that
is one root of the equation
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
.
. Find
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Q.48. Show that all the roots of
are given by
Solution:
[By Componendo-Dividendo]
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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 by-i]
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 multiple of 3. Solution: Now
Similarly Hence If n is not a multiple then, Let Where
where k is an integer
is not a
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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 Q.54. Show that Solution: Let
Comparing both sides, we get From (i), Also
Hence
Q.55. Prove that Solution: (i)Let
From (i) & (ii) we have, (ii) Now
&
is constant)
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From (i) & (ii) we have, Q.56. Prove that Solution: Let
From (i) & (ii) we have, Q.57. Prove that
Solution: Let
hence deduce that
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[By Componendo - Dividendo]
From (i) & (ii) we get, Putting
Q.58. If Solution: Now Let
Then R.H.S.
and
resp. in (i) and then adding we get,
then prove that
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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.
Q.60. If (i) (ii)
(iii) Solution:
[Dividing by
Prove that
]
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(ii)
(iii) Now,
Q.61. If (ii) Solution: (i) Now
Prove that (i) (iii)
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(ii)
[from (i)]
Where
Hence (iii) Now
[By componendo-Dividendo] 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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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 Solution: Now Then
But
show that
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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
Also
gives,
gives,