Ekeeda - Chemical Engineering- Applied Mathematics - Complex Number

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Chemical engineering is the type of engineering, which influences vast areas of technology. In simpler terms, chemical engineers conceive and design the process to produce, convert, and transport materials. Ekeeda offers Online Chemical Engineering Courses for all the Subjects as per the Syllabus.

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Ekeeda – Chemical Engineering 1 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 so-called 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 re-expressed by the following well- known formula which bears his name “DeMoivres formula” Definition

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Ekeeda – Chemical Engineering 2 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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Ekeeda – Chemical Engineering 3 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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Ekeeda – Chemical Engineering 4 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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Ekeeda – Chemical Engineering 5 Circular Functions of a Complex Number Hyperbolic functions 1. 2. 3. 4. 5. 6. 7. 8. 9.

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Ekeeda – Chemical Engineering 6 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

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Ekeeda – Chemical Engineering 7 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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Ekeeda – Chemical Engineering 8 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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Ekeeda – Chemical Engineering 9 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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Ekeeda – Chemical Engineering 10 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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Ekeeda – Chemical Engineering 11 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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Ekeeda – Chemical Engineering 12 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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Ekeeda – Chemical Engineering 13 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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Ekeeda – Chemical Engineering 14 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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Ekeeda – Chemical Engineering 15 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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Ekeeda – Chemical Engineering 16 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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Ekeeda – Chemical Engineering 17 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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Ekeeda – Chemical Engineering 18 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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Ekeeda – Chemical Engineering 19 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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Ekeeda – Chemical Engineering 20 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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Ekeeda – Chemical Engineering 21 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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Ekeeda – Chemical Engineering 22 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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Ekeeda – Chemical Engineering 23 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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Ekeeda – Chemical Engineering 24 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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Ekeeda – Chemical Engineering 25 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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Ekeeda – Chemical Engineering 26 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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Ekeeda – Chemical Engineering 27 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 x-cos 2 x 4sin 2 x.cos 2 x ii 3sinh 2 y + cosh 2 y 4sinh 2 y.cosh 2 y 22. If log cosx-iy 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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Ekeeda – Chemical Engineering 28 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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Ekeeda – Chemical Engineering 29 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+ iy-x-iy 53. If tan u+ iv x+ iy then prove that curves u constant and v constant are families of circles.

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Ekeeda – Chemical Engineering 30 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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Ekeeda – Chemical Engineering 31 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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Ekeeda – Chemical Engineering 32 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

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Ekeeda – Chemical Engineering 33 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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Ekeeda – Chemical Engineering 34 For G.P. Sum ii Also Q.3. If prove that and Solution: i L.H.S.

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Ekeeda – Chemical Engineering 35 ii L.H.S. Multiplying Numerator Denominator by i Q.4. If Prove that i ii Solution: Now

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Ekeeda – Chemical Engineering 36 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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Ekeeda – Chemical Engineering 37 Solution: Now Similarly As above Subtracting ii from i we get

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Ekeeda – Chemical Engineering 38 Q.6. Prove that Solution: Let Then Multiplying N D by -i From i ii we get

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Ekeeda – Chemical Engineering 39 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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Ekeeda – Chemical Engineering 40 Q.8. If prove that Solution: Let Also Then Hence Comparing this with the given equation we get

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Ekeeda – Chemical Engineering 41 Q.9. Show that the power of is where n is a positive integer. Solution: Now Hence

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Ekeeda – Chemical Engineering 42 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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Ekeeda – Chemical Engineering 43 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 x-1 where k01234. When k0 Root When k1 Root say When k2 Root say When k3 Root say

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Ekeeda – Chemical Engineering 44 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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Ekeeda – Chemical Engineering 45 k 4 Also FromI and II we have Q.13. If then prove that i ii Solution: i Now Then

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Ekeeda – Chemical Engineering 46 ii Now Comparing both sides we get Then Also

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Ekeeda – Chemical Engineering 47 From I and II we have Q.14. If then show that Solution:

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Ekeeda – Chemical Engineering 48 Now Adding two equations we get ……………………i Similarly subtracting we get as above ……………………ii From i ii we get Alternately

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Ekeeda – Chemical Engineering 49 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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Ekeeda – Chemical Engineering 50 ii Solution: Now log i sinh ny ii Alternately But

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Ekeeda – Chemical Engineering 51 Subs .in I we get

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Ekeeda – Chemical Engineering 52 Q.16. If Prove that Solution: Let Now Then Also i tanh

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Ekeeda – Chemical Engineering 53 Subs. in I we get z Q.17. Find the sum of the series Solution: Let ……… ..... ……… for a Geom. Series

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Ekeeda – Chemical Engineering 54 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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Ekeeda – Chemical Engineering 55 ………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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Ekeeda – Chemical Engineering 56 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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Ekeeda – Chemical Engineering 57 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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Ekeeda – Chemical Engineering 58 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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Ekeeda – Chemical Engineering 59 ∴ Q.26. If Show that i

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Ekeeda – Chemical Engineering 60 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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Ekeeda – Chemical Engineering 61 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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Ekeeda – Chemical Engineering 62 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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Ekeeda – Chemical Engineering 63 Solution: Let Now Also From i ii we get Then

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Ekeeda – Chemical Engineering 64 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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Ekeeda – Chemical Engineering 65 Q.33. If then prove that Solution: Now Hence Then

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Ekeeda – Chemical Engineering 66 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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Ekeeda – Chemical Engineering 67 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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Ekeeda – Chemical Engineering 68 Q.38. Prove that Solution: L.H.S.

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Ekeeda – Chemical Engineering 69 Q.39. If prove that Solution: Now Using componendo - dividendo

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Ekeeda – Chemical Engineering 70 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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Ekeeda – Chemical Engineering 71 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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Ekeeda – Chemical Engineering 72 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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Ekeeda – Chemical Engineering 73 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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Ekeeda – Chemical Engineering 74 When When When Q.46. Solve Solution: Now Multiply by x+1 on both the side

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Ekeeda – Chemical Engineering 75 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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Ekeeda – Chemical Engineering 76 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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Ekeeda – Chemical Engineering 77 Q.48. Show that all the roots of are given by Solution: By Componendo-Dividendo

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Ekeeda – Chemical Engineering 78 Multiplying N D by - 1 When not defined Hence discard Hence the solution are given by where

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Ekeeda – Chemical Engineering 79 Q.49. Show that the points representing the roots of the equation on Argand’s diagram are collinear. Solution: Now

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Ekeeda – Chemical Engineering 80 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 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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Ekeeda – Chemical Engineering 81 Value of expression When Values of expression Subs in i we get Q.51. Show that Solution:

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Ekeeda – Chemical Engineering 82 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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Ekeeda – Chemical Engineering 83 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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Ekeeda – Chemical Engineering 84 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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Ekeeda – Chemical Engineering 85 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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Ekeeda – Chemical Engineering 86 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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Ekeeda – Chemical Engineering 87 i Now ii iii Now Q.61. If Prove that i ii iii Solution: i Now

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Ekeeda – Chemical Engineering 88 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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Ekeeda – Chemical Engineering 89 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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Ekeeda – Chemical Engineering 90 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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Ekeeda – Chemical Engineering 91 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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Ekeeda – Chemical Engineering 92 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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Ekeeda – Chemical Engineering 93 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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Ekeeda – Chemical Engineering 94 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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Ekeeda – Chemical Engineering 95 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

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