Ch.3 Further Complex Numbers - FP2

7/23/2019 Ch.3 Further Complex Numbers - FP2

1/26

PhysicsAndMathsTutor.com

Edexcel Maths FP2

Topic Questions from Papers

Complex Numbers


2/26

4

*M35144A0428*

Leave

blank

2. Solve the equation

z3= 42 42i,

giving your answers in the form r(cos + i sin ), where < .

(6)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009


3/26

Leave

blank

16

*M35144A01628*

6. A transformation Tfrom thez-plane to the w-plane is given by

w z

zz=

+

ii,

The circle with equation |z | = 3 is mapped by Tonto the curve C.

(a) Show that Cis a circle and find its centre and radius.

(8)

The region |z | < 3 in thez-plane is mapped by Tonto the regionRin the w-plane.

(b) Shade the regionRon an Argand diagram.

(2)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________



4/26

Leave

blank

17

Turn over*M35144A01728*

Question 6 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________



5/26

Leave

blank

8

*N35388A0824*

4. z 8 8 3( )i

(a) Find the modulus of zand the argument of z.

(3)

Using de Moivres theorem,

(b) find 3z ,

(2)

(c) find the values of wsuch that 4w z= , giving your answers in the form a+ ib, where

,a b .

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



6/26

Leave

blank

14

*N35388A01424*

6. A complex number zis represented by the point Pin the Argand diagram.

(a) Given that 6z z = , sketch the locus of P.

(2)

(b) Find the complex numbers zwhich satisfy both 6z z = and 3 4i 5z = .

(3)

The transformation Tfrom the z-plane to the w-plane is given by30

wz

= .

(c) Show that Tmaps 6z z = onto a circle in the w-plane and give the cartesian

equation of this circle.

(5)



7/26

Leave

blank

15

Turn over*N35388A01524*


___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



8/26

Leave

blank

12

*P35413A01228*

5. The point Prepresents the complex number zon an Argand diagram, where

The locus of Pas zvaries is the curve C.

(a) Find a cartesian equation of C.(2)

(b) Sketch the curve C.

(2)

A transformation Tfrom the z-plane to the w-plane is given by

The point Qis mapped by Tonto the point R. Given that Rlies on the real axis,

(c) show that Qlies on C.

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



9/26

Leave

blank

13

*P35413A01328* Turn over


___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



10/26

Leave

blank

20

*P35413A02028*

7. (a) Use de Moivres theorem to show that

(5)

Hence, given also that

(b) find all the solutions of

in the interval Give your answers to 3 decimal places.

(6)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



11/26

Leave

blank

21



___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



12/26

Leave

blank

6

*P40104A0628*

3. (a) Express the complex number in the form .

(3)

(b) Solve the equation

giving the roots in the form .

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



13/26

Leave

blank

24

*P40104A02428*

8. The point Prepresents a complex number zon an Argand diagram such that

z z = 6 2 3i

(a) Show that, as zvaries, the locus of Pis a circle, stating the radius and the coordinates

of the centre of this circle.

(6)

The point Qrepresents a complex number zon an Argand diagram such that

arg ( )z = 63

4

(b) Sketch, on the same Argand diagram, the locus of Pand the locus of Qas zvaries.

(4)

(c) Find the complex number for which both z z = 6 2 3i and arg ( )z = 63

4

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



14/26


___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

28

*P40104A02828*

Q8

(Total 14 marks)

Leave

blank



15/26

Leaveblank

2

*P42955A0232*

1. A transformation Tfrom thez-plane to the w-plane is given by

wz

z=

+2ii

z

The transformation maps points on the real axis in thez-plane onto a line in the w-plane.

Find an equation of this line.(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com June 2013 (R)


16/26

Leaveblank

20

*P42955A02032*

6. The complex numberz= ei, where is real.

(a) Use de Moivres theorem to show that

12cos

n

nz n

z+ =

where n is a positive integer.(2)

(b) Show that

cos5 1

16(cos5+ 5cos3+ 10cos)

(5)

(c) Hence find all the solutions of

cos5+ 5cos3+ 12cos= 0

in the interval 0


17/26

Leaveblank

21



___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com June 2013 (R)


18/26

Leaveblank

4

*P43149A0428*

2. z= 53 5i

Find

(a) |z|,(1)

(b) arg(z), in terms of .(2)

2 cos i sin4 4

w = +

Find

(c)w

z,

(1)

(d) arg ,w

z

in terms of .(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________



19/26

Leaveblank

8

*P43149A0828*

4. (a) Given thatz= r(cos + i sin ), r

prove, by induction, thatzn= rn (cos + i sin ), n+

(5)

3 33 cos i sin4 4w

= +

(b) Find the exact value of w5, giving your answer in the form a+ ib, where a, b .(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________



20/26

Leaveblank

9



___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

______________________________________________________________________________________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



21/26

Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 Issue 1 September 2009 9

Further Pure Mathematics FP2

Candidates sitting FP2 may also require those formulae listed under Further Pure

Mathematics FP1 and Core Mathematics C1C4.

Area of a sector

A= d

2

1 2r (polar coordinates)

Complex numbers

sinicose i +=

)sini(cos)}sini(cos{ nnrr nn +=+

The roots of 1=nz are given by nk

zi2

e

= , for 1,,2,1,0 = nk

Maclaurins and Taylors Series

)0(f!

)0(f!2

)0(f)0f()f( )(

2

+++++= rr

r

xxxx

)(f!

)()(f

!2

)()(f)()f()f( )(

2

+

++

++= ar

axa

axaaxax

r

r

)(f!

)(f!2

)(f)f()f( )(

2

+++++=+ ar

xa

xaxaxa

r

r

xr

xxxx

r

x

allfor!!21)exp(e

2

+++++==

)11()1(32

)1(ln 132


22/26

8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 Issue 1 September 2009

Further Pure Mathematics FP1

Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1

and C2.

Summations

)12)(1(61

1

2 ++==

nnnrn

r

22

41

1

3 )1( +==

nnrn

r

Numerical solution of equations

The Newton-Raphson iteration for solving 0)f( =x : )(f

)f(1

n

n

nnx

x

xx =+

Conics

ParabolaRectangular

Hyperbola

Standard

Formaxy 42 = xy= c2

Parametric

Form(at2, 2at)

t

cct,

Foci )0,(a Not required

Directrices ax = Not required

Matrix transformations

Anticlockwise rotation through aboutO:

cossin

sincos

Reflection in the line xy )(tan= :

2cos2sin

2sin2cos

In FP1, will be a multiple of 45.


23/26

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 Issue 1 September 2009 7

Core Mathematics C4

Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2

and C3.

Integration (+constant)

f(x)

xx d)f(

sec2 kxk

1tan kx

xtan xsecln

xcot xsinln

xcosec )tan(ln,cotcosecln 21

xxx +xsec )tan(ln,tansecln

41

21 ++ xxx

= x

x

uvuvx

x

vu d

d

dd

d

d


24/26

6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 Issue 1 September 2009

Core Mathematics C3

Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and

C2.

Logarithms and exponentials

xaxa=lne

Trigonometric identities

BABABA sincoscossin)(sin =

BABABA sinsincoscos)(cos =

))((tantan1

tantan)(tan

21 +

= kBA

BA

BABA

2cos2sin2sinsin

BABA

BA

+

=+

2sin

2cos2sinsin

BABABA

+=

2cos

2cos2coscos

BABABA

+=+

2sin

2sin2coscos BABA

BA +

=

Differentiation

f(x) f(x)

tankx ksec2

kx

secx secx tanx

cotx cosec2x

cosecx cosecxcotx

)g(

)f(

x

x

))(g(

)(g)f()g()(f2x

xxxx


25/26

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 Issue 1 September 2009 5

Core Mathematics C2

Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.

Cosine rule

a2= b2+ c2 2bccosA

Binomial series

21)( 221 nrrnnnnn bba

r

nba

nba

naba ++

++

+

+=+ (n )

where)!(!

!C

rnr

n

r

nr

n

==


26/26

Core Mathematics C1

Mensuration

Surface area of sphere = 4r2

Area of curved surface of cone = rslant height

Arithmetic series

un= a+ (n 1)d

Sn=2

1n(a+ l) =

2

1n[2a+ (n1)d]

Documents

Ch.3 Further Complex Numbers - FP2