question bank complex numberp1.rtf

8/17/2019 question bank complex numberp1.rtf

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1. Given that 2+ z z

= 2 – i, z ∈ , find z in the form a + ib.(Total 4 marks)

2. (a) Write down the expansion of (cos θ + i sin θ )3 in the form a + ib, where a and b

are in terms of sin θ and cos θ .(2)

(b) Hence show that cos 3θ = 4 cos3 θ – 3 cos θ .

(3)

(c) Similarly show that cos θ = !" cos θ – 20 cos3 θ + cos θ .

(3)

(d) Hence sol#e the e$%ation cos θ + cos 3θ + cos θ = &, where θ

−∈

2

π,

2

π

.(6)

(e) 'y considerin the sol%tions of the e$%ation cos θ = &, show that

8

55

10

πcos

+=

and state the #al%e of 10

π7cos

.(8)

(Total 22 marks)

3. he complex n%mbers z ! = * – 2i and z * = ! – 3i are represented by the points and' respecti#ely on an rand diaram. i#en that - is the oriin,

(a) find ', i#in yo%r answer in the form 3−ba , where a, b ∈ +(3)

IB Questionbank Mathematics Higher Level 3rd edition 1


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(b) calc%late BÔA in terms of π.(3)

(Total 6 marks)

4. (a) /actori0e z 3 + ! into a linear and $%adratic factor.

(2)

1et γ = 2

3i1+

.

(b) (i) Show that γ is one of the c%be roots of –1.

(ii) Show that γ* = γ – 1.

(iii) ence !ind the va"#e o! (1 – γ)".(9)

he matrix A is defined by A =

γ

γ 1

0

1

.

(c) Show that A* – A + I = 0, where 0 is the 0ero matrix.

(4)

(d) 2ed%ce that

(i) A3 = – I

(ii) A –1 = I – A.(5)

(Total 20 marks)

5. i#en that z = (b + i)*, where b is real and positi#e, find the #al%e of b when ar z =

"&$.(Total 6 marks)



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6. he roots of the e$%ation z * + *z + 4 = & are denoted by α and β

(a) /ind α and β in the form r eiθ .

(6)

(b) i#en that α lies in the second $%adrant of the rand diaram, mar α and β onan rand diaram.

(2)

(c) 5se the principle of mathematical ind%ction to pro#e 2e 6oi#re%s theo&e', hichstates that cos nθ + i sin nθ = (cos θ + i sin θ )n for n ∈ +.

(8)

(d) 5sin 2e 6oi#re%s theo&e' !ind2

3

β

α

in the form a + ib.(4)

(e) 5sin 2e 6oi#re%s theo&e' o& othe&ise, sho that α3 = β 3.(3)

(f) /ind the exact #al%e of αβ + βα where α7 is the con8%ate of α and β is thecon8%ate of β .

(5)

() /ind the set of #al%es of n for which αn is real.

(3)(Total 31 marks)

7. 9onsider the complex eometric series eiθ +

θ θ 3i2i e

1e

2

1+

+ ....

(a) /ind an expression for z , the common ratio of this series.(2)



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(b) Show that *z * + 1.(2)

(c) Write down an expression for the s%m to infinity of this series.(2)

(d) (i) :xpress yo%r answer to part (c) in terms of sin θ and cos θ .

(ii) Hence show that

cos θ + 2

1

cos *θ + )

1

cos 3θ + ... = θ

θ

cos5

2cos

−

−

.(10)

(Total 16 marks)

8. Sol#e the sim%ltaneo%s e$%ations

iz ! + *z * = 3

z ! + (! – i)z * = 4i#in z ! and z * in the form ! + i" , where ! and " are real.

(Total 9 marks)

9. /ind b wherei

10

107

i1i2

+=−

+

b

b

.(Total 6 marks)

10. (a) Show that sin * n! = sin((*n + !) ! ) cos ! – cos((2n + !) ! ) sin ! .(2)

IB Questionbank Mathematics Higher Level 3rd edition #


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(b) Hence pro#e, by ind%ction, that

cos ! + cos 3 ! + cos ! + ... + cos((*n – 1) ! ) = xnx

sin2

2sin

,

for all n ∈ +, sin ! - 0.

(12)

(c) Sol#e the e$%ation cos ! + cos 3 ! = 2

1

, & ; ! ; π.(6)

(Total 20 marks)

11. /ind the three c%be roots of the complex n%mber


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(b) i#en that b is a root of the e$%ation z –1 = 0 hich does not "ie on the &ea" ais in

the A&and dia&a', sho that 1 b + b* + b3 + b4 = &.(3)

(c) f u = b + b4 and v = b

* + b

3 show that

(i) u + v = uv = –1

(ii) u – v = 5 , i#en that u – v > &.(8)

(Total 13 marks)

16. (a) 5se de 6oi#re%s theo&e' to !ind the &oots o! the e4#ation z 4 = ! – i.(6)

(b) 2raw these roots on an rand diaram.(2)

(c) f z ! is the root in the first $%adrant and z * is the root in the second $%adrant, find

1

2

z

z

in the form a + ib.(4)

(Total 12 marks)

17. 1et % = cos .5

2

sini5

2 π+

π

(a) Show that % is a root of the e$%ation z 1 = 0.

(3)

(b) Show that (% 1) (% 4 + % 3 + % * + % + !) = % 1 and ded#ce that%

4 + %

3 + %

* + % + ! = &.

(3)

IB Questionbank Mathematics Higher Level 3rd edition &


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(c) Hence show that cos.

2

1

5

cos

5

2−=

π+

π

(6)(Total 12 marks)

18. (a) /ind the s%m of the infinite eometric se$%ence *?, , 3, 1, ... .(3)

(b) 5se mathematical ind%ction to pro#e that for n∈+,

a + ar + ar * + ... + ar

n –1 =

( ) .1

1

r

r a n

−−

(7)(Total 10 marks)

19. :xpress

( )33i11

− in the form

b

a

where a, b∈

. (Total 5 marks)

20. /ind the #al%es of n s%ch that (! + 3 i)n is a real n%mber.(Total 5 marks)

21. (a) 1et z = ! + i" be any non@0ero complex n%mber.

(i) :xpress z

1

in the form u + iv .

(ii) f∈=+ k k

z z ,

1

, show that either " = & or ! * + "

* = !.

(iii) Show that if ! * + "

* = ! then *k * 6 2.

(8)

IB Questionbank Mathematics Higher Level 3rd edition '


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(b) 1et % = cos θ + i sin θ .

(i) Show that % n + %

– n = *cos nθ , n ∈ .

(ii) Sol#e the e$%ation 3%

*

– % + * – % –1

+ 3%

–2

= &, i#in the roots in theform ! + i" .(14)

(Total 22 marks)

22. he complex n%mber z is defined as z = cos θ + i sin θ .

(a) State de 6oi#re%s theo&e'.(1)

(b) Show that z n

n z

1−

= *i sin (nθ ).(3)

(c) 5se the binomial theorem to expand

5

1

− z

z

i#in yo%r answer in simplifiedform.(3)

(d) Hence show that !" sin θ = sin θ –5 sin 3θ + !& sin θ .

(4)

(e) 9hec that yo%r res%lt in part (d) is tr%e for θ =

π

.(4)

(f) /indθ θ dsin2

π

0

5∫ .(4)

IB Questionbank Mathematics Higher Level 3rd edition (


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(c) Hence show that the property L(z !z *) = L(z !) + L(z *) does not hold for all #al%es

of z ! and z *.

(2)(Total 9 marks)

25. 9onsider the complex n%mbers z = ! + *i and % = * +ai, where a ∈ .

/ind a when

(a) *% * = 2*z *(3)

(b) Ae (z% ) = * m(z% ).

(3)(Total 6 marks)

26. 9onsider the complex n%mber ω = 2

i

++

z

z

, where z = ! + i" and i = 1− .

(a) f ω = i, determine z in the form z = r cis θ .

(6)

(b) Bro#e that ω =22

22

)2(

)22i()2(

y x

y x y y x x

++++++++

.(3)

(c) Hence show that when Ae(ω) = ! the points ( ! , " ) lie on a straiht line, l !, and

write down its radient. (4)

(d) i#en ar (z ) = ar(ω) =

π

, find *z *.(6)

(Total 19 marks)



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27. 9onsider ω =

+

3

2πsini

3

π2cos

.

(a) Show that

(i) ω3 = !

(ii) ! + ω + ω* = &.

(5)

(b) (i) 2ed%ce that eiθ +

+

+

+ 3π

i3

π2i

eeθ θ

= &.

(ii) ll%strate this res%lt for θ = 2

π

on an rand diaram.

(4)

(c) (i) :xpand and simplify , (z ) = (z – 1)(z – ω)(z – ω*) where z is a complexn%mber.

(ii) Sol#e , (z ) = ?, i#in yo%r answers in terms of ω.(7)

(Total 16 marks)


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question bank complex numberp1.rtf