Complex Number de Moives Theorem-5

7/26/2019 Complex Number de Moives Theorem-5

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Complex number

7on7.in 1

1. (a) Show that any non-real n-th root of unity satisfies the equation

1 +z+z2+ . . . +z

n1= 0(2)

(b) Letz1denote the non-real n-th root of unity which has the smallest positive argument.

(i) Derive an expression forz1in the form cos+ isin.

(3)

(ii) By substitutingz =z1in the equation in part (a), find the sum of each of the series.

n

n

nn

)1(2cos...

4cos

2cos

and .)1

(2

sin...4sin

2sin

n

n

nn

(5)

(Total 10 marks)

2. (a) (i) Show that 52 i

ew

is one of the fifth roots of unity(1)

(ii) Show that the other fifth roots of unity are 1, w2, w

3and w

4.

(3)

(b) Letp= w+w4and q= w

2+ w

3, where 5

2 i

ew

.

(i) Show that

p+ q=1 and pq =1.(6)

(ii) Write down a quadratic equation, with integer coefficients, whose roots arepandq.

(1)


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(iii) Express p and q as integer multiples of cos5

2and cos

5

4, respectively.

(4)

(iv) Hence obtain the values of cos5

2and cos

5

4in surd form.

(2)

(Total 17 marks)

3. (a) Convert 2 + 23i to modulus-argument form. Hence find both square roots of 2 + 23iGive the answers in Cartesian form.

(4)

(b) An alternative method of finding the square roots of 2 + 23i starts as follows:

Let a + bi be one of the square roots.

Then a+bi2= 2 + 23i.

Continue this working to obtain two equations in aand band use the equations to

evaluate the square roots of 2 + 23i. Show clearly how the answers are obtained. Markswill notbe awarded for substituting the answers into an equation.

(8)

(Total 12 marks)

4. (a) Show that

3

tani

1

tani1

= cosp+ i sinp,

wherepis an integer to be determined(5)


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(b) Hence find the general solution of the equation

i

2

1

2

1

tani

1

tani1 3

(3)

(Total 8 marks)

5. (a) Express the complex number 2 + 2i in the form rei, where r> 0 and


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Complex number

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6. (a) (i) Express 2

2

i

e

i

e in terms of sin2

.

(1)

(ii) Hence, or otherwise, show that

1e,2cot2

i

2

1

1e

1 ii

(3)

(b) Derive expressions, in the form whereei , for the four non-real roots of the

equationz6= 1.

(4)

(c) The equation

1

1 6

w

w

has one real root and four non-real roots.

(i) Explain why the equation has only five roots in all.(1)

(ii) Find the real root.(2)

(iii) Show that the non-real roots are

,1

1,1

1,1

1,1

1

4321 zzzz

wherez1,z2,z3andz4are the non-real roots of the equation z6= 1

(3)

(iv) Deduce that the points in an Argand diagram which represent the roots of equation

(*) lie on a straight line.(3)

(Total 17 marks)


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7. (a) (i) Use de Moivres theorem to show that if z= cos + i sin , then

zn

+n

z

1

= 2 cosn

(3)

(ii) Write down the corresponding result forznn

z

1.

(1)

(b) (i) Show that

2

2

6

6

331

1

1

1

zzB

zzAzzzz

whereAandBare numbers to be determined.(4)

(ii) By substitutingz= cos + i sin in the above identity, deduce that

cos3sin

3=

32

1 (3sin2sin6).

(4)

(c) By using the identity in part (b)(ii), show that the general term in the expansion of

cos 3 sin 3 in ascending powers of is

16

3 (2

2r6

2r)

,1!12

1 2

rr

r r= 0, 1, 2, ...

(3)

(Total 15 marks)


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8. In this question, give all answers in exact form with the arguments in radians.

(a) Write in modulus-argument form:

(i) 1 + j3 , (ii) (1 + 3j)2, (iii)j31

1

(6)

(b) Solve the equationz4=

4,16

, leaving the solutions in modulus-argument form.

(4)

(Total 10 marks)

9. (a) Express the complex number

i31

in the form re1

, where ris real and< .(2)

(b) (i) Verify that is a root of the equation

z4= 8 (1 + 3 i).

(2)

(ii) Find the other three roots of the above equation giving your answers in the form

re1

, whereris real and< .(3)

(Total 7 marks)

10. (a) Write down the expansion of (a+ b)5.(1)

(b) Use de Moivres theorem to show that

cos 516 cos520 cos3+ 5 cos(5)


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(c) (i) By considering the equation 05cos , show that

16

5

10

9cos

10

7cos

10

3cos

10

cos

(3)

(ii) Hence find the exact value of

10

3cos10

cos .

(3)

(Total 12 marks)

11. The complex number is defined by

=2i310i2

(a) Show that = 22i.(2)

(b) Express in the form rei, where r is real and< .(2)

(c) Hence

(i) show that 4

is real,(2)

(ii) solve the equation

z3 = ,

giving your answers in the form rei

, where r is real and< .(4)

(Total 10 marks)


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12. (a) Use de Moivres theorem to show that ifz = cos + i sin , then

zn

+n

z

1= 2 cos n .

(3)

(b) (i) Write down the expansion of (z z

1 )

4in terms ofz.

(2)


8 sin4= cos 44 cos 2+ 3.

(5)

(c) Solve the equation

8 sin4 = cos 4+ 1

in the interval< , giving your answers in terms of .(3)

(Total 13 marks)

13. (a) Express 3 + i in the form r(cos + i sin ), where r0 and(3)

(b) Obtain a similar expression for 3 i.

(2)

(c) Hence find the set of integer values of n for which

( 3 + i)n( 3 i)n= 0.

(6)

(Total 11 marks)


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14. (a) (i) Use de Moivres theorem to show that

(cos + i sin )4+ (cos i sin )

4= 2 cos 4.

(2)

(ii) Deduce that

(cot + i)4+ (cot i)

4=

4sin

4cos2, r.

(1)

(b) Verify that cot81 is a root of

(z+ i)4+ (z1)

4= 0

and find the three other roots of this equation giving each answer in the form

+ cot orcot , where 0 < 2

.

(4)

(c) Express the equation in part (b) in the form

z4+ bz

2+ c= 0,

where band care real numbers to be determined,(2)

(d) Hence, or otherwise, find in surd form the value of cot28

.

(3)(Total 12 marks)


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Complex number

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15. It is given that

w=

2

1(1 + i).

(a) (i) Show that |w| = 1.

(ii) Express win the form eiwhere< .(3)

(b) Solvez3= w, giving your answer in the form e

i, where< .(4)

(c) (i) Show that

(1w)(1w*) = 2 + 2 ,

where w* is the complex conjugate of w.(3)

(ii) The sum of the geometric series

11

0r

r

w is S.

Show that

S=w1

2

and hence express Sin the form 1 +pi, wherepis real.(5)

(Total 15 marks)

16. Use de Moivres Theorem to show that

i3sini

3cos

6sini

6cos

57

.

(Total 6 marks)


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Complex number

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17. (a) (i) Verify thatz=i4

1

e2

is a root of the equationz4=16.(1)

(ii) Find the other three roots of this equation, giving each root in the form rei

, where

ris real and.

(3)

(iii) Illustrate the four roots of the equation by points on an Argand diagram.(2)

(b) (i) Show that

422e2e2 2ii4

1

4

1

zzzz

.

(3)

(ii) Expressz4+ 16 as the product of two quadratic factors with real coefficients.(3)

(Total 12 marks)

18. (a) Find the constantsAandBin the identity

6

6

2

2

3

2

2 1

1

1

z

zB

z

zA

z

z .

(3)

(b) (i) Use the result

zn=

n

z

1= 2i sin n,

where

z = cos+ i sin,

to show that

sin32=

4

3sin 2

4

1sin 6.

(4)


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Complex number

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4

1

0

3sin 2d=

3

1.

(3)

(Total 10 marks)

19. (a) Express 1 + i in the form rei

, wherer> 0 and < .(2)

(b) Show that

(1 + i)21

(1 i)21= ki,

where kis an integer to be found.(5)

(Total 7 marks)

20. (a) Express the complex numbers 3 + i and 22i in the form rei, where r > 0 and

< .(4)

(b) Solve the equation

(22i)z3= 3 + i,

giving each answer in the form rei

, where r >0 and


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Complex number

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(b) The diagram shows the curve with equationy= f(x), where f(x) =x33x

2+ 3.

p q r

y

xO

The curve crosses thex-axis at the points with coordinates (p, 0), (q, 0) and (r, 0), wherep 0 and

< . Determine the exact value of r and give in radians to four decimal places.(2)


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(b) The roots of the equationz3+ 21.6 = 16.2i are , and .

(i) Find these roots, giving your answers in polar form.

(5)

(ii) The complex numbers , and are represented in the complex plane bythe pointsA, Band C. Describe the geometrical relationship betweenA, B, and Cand their position relative to the origin.

(2)

(Total 9 marks)

23. (a) (i) Given the complex numberz= 1 + 2i, write down z in the form re

i

, where r> 0and< .

(2)

(ii) Hence determine the modulus ofzin an exact surd form, and the argument ofz5in

an exact trigonometric form in the interval between and .(3)

(b) Use the binomial theorem to expresszin the form a+ bi (a, breal).

Deduce the exact value of cos(5 tan1

2).(5)

(Total 10 marks)

24. A complex number,z, is said to be in polar form when it is written as [ r, ], where ris the

modulus ofzand is the argument ofzwith< .

(a) Express the complex number 512i in polar form.(2)

(b) Hence determine, in polar form, the nine complex roots of the equation

z9512i = 0.

(3)

(c) On a sketch of the complex plane draw the nine points which represent these roots.Explain why these nine points lie at the vertices of a regular nonagon. Find the area of

this plane figure, giving your answer in the form psin , stating the values ofpand .(5)

(Total 10 marks)


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Complex number

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25. (a) Use one of the double-angle formulae for cos 2, with a suitable value of , to show that

.224

1

8sin2

(2)

(b) The complex numberzis such thatz= sin + i(1cos ), where 0 .

(i) Show that 2

1sin2z and find the argument of z in as simplified a form as

possible.(5)

(ii) In the case when =4

, expressz

8in the polar form [r, ], where r> 0 is the

modulus ofz8and its argument (betweenand ).

(2)

(iii) Using the result of part (a), or otherwise, determine the exact value ofz8as a real

number in the form a+ b 2 for integers aand bto be determined.(3)

(Total 12 marks)

26. (a) Express the complex numberz= (2 + i)(5 + i)(8 + i) in the form a+ bi, where aand bare

real numbers.(3)

(b) Deduce the exact value of tan1

21 + tan

151 + tan

181 .

(2)

(Total 5 marks)

27. (a) Given thatz= cos + i sin , use de Moivres theorem to prove that

cos n = nn zz 2

1 ,

and state a similar result for sin n.(3)


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(b) Use the binomial theorem to show that

(z+z1

)4+ (zz

1)4= 2z

4+ 2z

4+ 12.

Hence show that cos4+ sin4 p+ qcos4for rational numberspand qwhose valuesshould be determined.

(5)

(c) Deduce the exact value of

0

44 sincos d.

(3)

(Total 11 marks)

28. (a) Find the exact values of the solutions of the equation

052016 24 xx .(3)

(b) By consideringz = cos + i sin and using De Moivres theorem, show that

5sin20sin16sin5sin 24 .(4)

(c) By considering the equation sin 5 = 0, show that

,8

55

5sin

and find a similar expression for .

5

2sin

(4)

(Total 11 marks)


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Complex number

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29. (a) Given that z =2 + 23 i, show thatz2+ 4z is real.(3)

(b) Find the modulus and argument of the complex number2 + 23 i, giving the argumentin terms of .(4)

(Total 7 marks)

30. (a) Find the modulus and argument of 1 + i 3

(2)

(b) Hence solve the equation

z2= 1 + i 3 ,

giving your answers in the form rei

where r> 0 and< .(3)

(Total 5 marks)

31. (a) A complex number has polar form

3

2,4 , Find its real part.

(2)

(b) (i) Show that the real part of the complex number 1isi2

i31

.

(3)

(ii) Find the modulus and argument of the complex numberi2

i31

, giving the

argument in terms of (3)

(Total 8 marks)


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32. (a) Given thatz= cos + i sin , use de Moivres theoremto show that

zn+z

n= 2 cos n

(3)

(b) By considering the expansion of

51

zz , find the values of the constantsp,qand r for

which 32 cos5cos5+qcos 3+r cos

(5)

(Total 8 marks)

33. The complexnumberzisi43i105 .

(a) Expresszin the form a= bi, where aand bare integers.(3)

(b) Hence:

(i) find the modulus ofz;(1)

(ii) find the argument ofzin radians.(2)

(Total 6 marks)

34. (a) Use de Moivres Theorem to prove the identity

cos 34 cos3 3 cos (4)

(b) Hence determine the solutions of the equation

8x36x1 = 0

giving your answers in exact trigonometric form.(6)


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(c) Evaluate, in an exact numerical form,

3

6

d)cos3cos(4 24

(6)

(Total 16 marks)

35. (a) State the modulus and argument of the complex number 8i.(2)

(b) Determine the three complex roots of the equationz3= 8i , giving each answer in the

form rei

, where r> 0 and< .(4)

(c) On a diagram of the complex plane (the Argand diagram), mark the three points whichrepresent these roots.

(2)

(d) Find the exact value of each of these roots in the form a+ ib.(3)

(Total 11 marks)


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36. The complex number = 30 31 +194 i.

(a) Express in the form rei

, where r> 0 and is to be given to four decimal places in

radians betweenand .(2)

(b) The complex numberz1is the complex number with least positive argument such that

z18= . Determine z1in an exact form, and find arg(z1) to four decimal places.

(2)

(c) On a diagram of the complex plane, mark the position ofz1, and describe the positions of

the remaining seven complex roots of the equation z8= .

(3)

(Total 7 marks)

37. (a) Use mathematical induction to prove that (cos + i sin )n= cos n+ i sin nfor all

positive integers n.(4)

(b) (i) Express ( 3 + i)nin the form 2

n(cos n+ i sin n), giving the value of in terms

of .(3)

(ii) Hence find the least positive integer value of nfor which ( 3 + i)nis a positive

real number.(2)

(Total 9 marks)

38. (a) Determine the real part of (1 + i tan )3.(2)

(b) Deduce the identity

1 3 tan2

3cos

3cos

(3)

(Total 5 marks)


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Complex number

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39. (a) (i) Show that sinicos

1

= cos i sin .

(1)

(ii) Given thatz= cos4 + i sin

4 , evaluate

121

zz

(4)

(b) (i) Use de Moivres Theorem to simplify the expression

[r(cos + i sin )]n(1)

(ii) Using this result, or otherwise, find the integer m for which

m

12

i2121

(5)

(Total 11 marks)

40. The complex numbersz1andz2are given by

z1=i1

i1 and z2=

2

3

2

1 i

(a) Show thatz1= i.

(2)

(b) Show that |z1 | = |z2|.

(2)

(c) Express bothz1andz2in the form rei

, where r> 0 and < .

(3)

(d) Draw an Argand diagram to show the points representingz1,z2andz1+z2.

(2)


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(e) Use your Argand diagram to show that

tan12

5 = 2 + 3

(3)

(Total 12 marks)

41. It is given thatz= ei

.

(a) (i) Show that

z+z

1 = 2 cos

(2)

(ii) Find a similar expression for

z2+2

1

z

(2)

(iii) Hence show that

z2z+ 2

z

1 +

2

1

z

= 4 cos22 cos

(3)

(b) Hence solve the quartic equation

z4z

3+ 2z

2z + 1 = 0

giving the roots in the form a+ ib.(5)

(Total 12 marks)


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42. (a) Find the six roots of the equationz6= 1, giving your answers in the form e

i,

where < .

(3)

(b) It is given that w = ei, where n.

(i) Show thatw

w 12 = 2i sin .

(2)

(ii) Show thatsin2i

12

ww .

(2)

(iii) Show that 1i22w

cot i.

(3)

(iv) Given thatz= cot i, show thatz + 2i =zw2.(2)

(c) (i) Explain why the equation

(z+ 2i)6=z6

has five roots.(1)

(ii) Find the five roots of the equation

(z+ 2i)6=z6

giving your answers in the form a + ib.(4)

(Total 17 marks)

43. (a) Find the three roots ofz3= 1, giving the non-real roots in the form e

i, where < .

(2)

(b) Given that is one of the non-real roots ofz3= 1, show that

1 + + 2= 0

(2)


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Complex number

(c) By using the result in part (b), or otherwise, show that:

(i) ;1

1

(2)

(ii) ;1

2

2

(1)

(iii) ,cos2)1(11 3

2

2

2

kkkk

where kis an integer.

(5)

(Total 12 marks)

44. Use De Moivres Theorem to find the smallest positive angle for which

(cos + i sin )15

= i(Total 5 marks)

45. (a) (i) Given that z64z3+ 8 = 0, show thatz3= 2 2i.(2)

(ii) Hence solve the equation

z64z3+ 8 = 0

giving your answers in the form rei, where r> 0 and < .(6)

(b) Show that, for any real values of kand ,

22ii cos2)e()e( kkzzkzkz (2)

(c) Expressz64z3+ 8 as the product of three quadratic factors with real coefficients.

(3)

(Total 13 marks)

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Complex Number de Moives Theorem-5

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