95 to 06 Ext 2 Complex Numbers

HSC by Topic 1995 to 2006 Complex Numbers

Page 1

Mathematics Extension 2 HSC Examination Topic: Complex Numbers 06

2a

z2

1

(ii)

zw

1

(iii)

w z

1

(i)

Express

(i)

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06

Let z = 3 + i and w = 2 – 5i. Find, in the form x + iy,

2b

(ii)

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(iii) 06

3 - i in modulus-argument form. 7

Express ( 3 - i) in modulus-argument form. 7

Hence express ( 3 - i) in the form x + iy. 3

2c

Find, in modulus-argument form, all solutions of z = –1.

2d

The equation |z – 1 – 3i| + |z – 9 – 3i|corresponds to an ellipse in the Argand

1 1 2 3

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06

diagram.

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(i)

Write down the complex number corresponding to the centre of the ellipse.

1

(ii)

Sketch the ellipse, and state the lengths of the major and minor axes.

3

(iii)

Write down the range of values of arg(z) for complex numbers z corresponding

1

to points on the ellipse. 05

2a

HSC

Let z = 3 + i and w = 1 – i. Find, in the form x + iy , (i)

2z + iw

1

(ii)

zw 6 w

1

(iii) 05

2b

Let β = 1 - i 3 (i)

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(ii) (iii) 05

2c

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Express β in modulus-argument form.

2

5

Express β in modulus-argument form. 5

Hence express β in the form x + iy.

2d

2 1

Sketch the region on the Argand diagram where the inequalities |z - z | < 2 and |z − 1| ≥1 hold simultaneously.

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05

1

3

Let l be the line in the complex plane that passes through the origin and makes an angle α with the positive real axis, where 0 < arg(z1) < α. The point P represents the complex number z1, where 0 < arg(z1) < α. The point P is reflected in the line l to produce the point Q, which represents the complex number z2. Hence |z1| = |z2|. (i) (ii)

Explain why arg(z1) + arg(z2) = 2α . 2

2

Deduce that z1z2 = |z1| (cos 2α + i sin 2α)

1

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HSC by Topic 1995 to 2006 Complex Numbers

(iii)

Let α =

π 4

Page 2

and let R be the point that represents the complex number z1z2.

1

Describe the locus of R as z1 varies. 04

2a

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Let z = 1 + 2i and w = 3 − i. Find, in the form x + iy, (i) zw (ii)

04

2b

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1

 10     z 

1

Let α = 1 + i 3 and β = 1 + i.

α , in the form x + iy. β

1

(i)

Find

(ii)

Express α in modulus-argument form.

(iii)

Given that β has the modulus-argument form β =

2

2(cos

π 4

+ i sin

π 4

) , find the

1

α modulus-argument form of . β (iv) 04

2c

π 12

.

1

Sketch the region in the complex plane where the inequalities

3

|z + z | ≤ 1 and |z – i| ≤ 1 hold simultaneously.

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04

Hence find the exact value of sin

2d

The diagram shows two distinct points A and B that represent

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the

complex

numbers

z

and

w

respectively. The points A and B lie on the circle of radius r centred at O. The point C representing the complex number z + w also lies on this circle. Copy the diagram into your writing booklet.

04 HSC

7b

2

(i)

Using the fact that C lies on the circle, 2π show geometrically that ∠ AOB = . 3

(ii)

Hence show that z3 = w3.

(iii)

Show that z2 + w2 + zw = 0.

2 1

Let α be a real number and suppose that z is a complex number such that 1 z+ = 2 cos α. z (i)

(ii)

(iii)

By reducing the above equation to a quadratic equation in z, solve for z and 1 use de Moivre’s theorem to show that zn + n = 2 cos nα. z 1 Let w = z + . z 1 1 1 Prove that w3 + w2 – 2w – 2 = (z + ) + (z2 + 2 )+ (z3 + 3 ). z z z

3

Hence, or otherwise, find all solutions of cos α + cos 2α + cos 3α = 0, in the

3

2

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HSC by Topic 1995 to 2006 Complex Numbers

Page 3

range 0 ≤ α ≤ 2π. 03

2a

Let z = 2 + i and w = 1 − i. Find, in the form x + iy, (i)

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(ii) 03

2c

2e

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02

Sketch the region in the complex plane where the inequalities

π 4

3

hold simultaneously.

Suppose that the complex number z lies on the unit circle, and 0 ≤ arg (z) ≤

2a

π 2

.

2

Let z = 1 + 2i and w = 1 + i. Find, in the form x + iy, (i) (ii)

2b

zw 1 w

1 1

On an Argand diagram, shade in the region where the inequalities

3

0 ≤ Re z ≤ 2 and |z – 1 + i| ≤ 2 both hold.

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02

1

Prove that 2 arg (z + 1) = arg (z).

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02

1

|z – 1 – i| < 2 and 0 < arg(z – 1 - i) <

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03

zw 4 z

2d

Prove by induction that, for all integers n ≥ 1, (cos θ − i sin θ)n = cos nθ − i sin nθ.

2e

Let z = 2(cos θ + i sin θ ).

3

HSC

02

(i)

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(ii) (iii) 02

7b

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Find 1 − z .

1

1 1 − 2 cos θ Show that the real part of is . 1− z 5 − 4 cos θ 1 Express the imaginary part of in terms of θ. 1− z

Suppose 0 < α, β <

π 2

2

1

and define complex numbers zn by

zn = cos(α + nβ ) + i sin(α + nβ ) for n = 0, 1, 2, 3, 4. The points P0, P1, P2 and P3 are the points in the Argand diagram that correspond to the complex numbers z0, z0 + z1 , z0 + z1 + z2 and z0 + z1 + z2 + z3 respectively. The angles

θ 0, θ 1 and θ 2 are the external angles at P0, P1 and P2 as shown in the diagram below. (i)

Using vector addition, explain why θ 0 = θ 1 = θ 2 = β .

2

(ii)

Show that ∠P0 O P1 = ∠P0 P2 P1, and explain why

2

O P0 P1 P2 is a cyclic quadrilateral. (iii)

Show that P0 P1 P2 P3 is a cyclic quadrilateral, and explain why the points O, P0,

2

P1, P2 and P3 are concyclic. (iv) 01

2a

Suppose that z0 + z1 + z2 + z3 + z4 = 0. Show that β =

Let z = 2 + 3i and w = 1 + i.

2π 5

2 2

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HSC by Topic 1995 to 2006 Complex Numbers

Find zw and

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01

2b

(ii)

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01

2c

2d

Express 1 + 3 i in modulus-argument form. Hence evaluate (1 + 3 i)

10

in the form x + iy.

Sketch the region in the complex plane where the inequalities

π 3

≤ arg z ≤

π 4

1 1 3

both hold.

Find all solutions of the equation z4 = –1.

3

Give your answers in modulus-argument form.

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01

1 in the form x + iy. w

|z + 1 – 2i| ≤ 3 and -

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01

(i)

Page 4

2e

In the diagram the vertices of a triangle ABC are represented by the complex numbers z1, z2 and z3,

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respectively. The triangle is isosceles and rightangled at B.

2 2

2

(i)

Explain why (z1 – z2) = –(z3– z2) .

(ii)

Suppose D is the point such that ABCD is a

1

square. Find the complex number, expressed in 01

7a

HSC

terms of z1, z2 and z3, that represents D. 1 Suppose that z = (cos θ + i sin θ) where θ is real. 2 (i)

Find |z|.

1

(ii)

Show that the imaginary part of the geometric series 1 2 sin θ 1 + z + z2 + z3 + … = is . 1− z 5 − 4 cos θ 1 1 1 Find an expression for 1 + cos θ + cos 2θ + cos 3θ + … in terms of 2 2 2 23

3

(iii)

2

cos θ . 00

2a

Find all pairs of integers x and y that satisfy (x + iy)2 = 24 + 10i.

2c

(i)

Let z = cos

(ii)

Plot, on the Argand diagram, all complex numbers that are solutions of z6 = –1.

3

HSC

00 HSC

00 HSC

2d

π 6

+ i sin

π 6

. Find z6.

4

Sketch the region in the Argand diagram that satisfies the inequality

3

z z + 2(z+ z ) ≤ 0

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HSC by Topic 1995 to 2006 Complex Numbers

00

2e

Page 5

In the Argand diagram, OABC is a rectangle,

3

where OC = 2OA. The vertex A corresponds to

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the complex number ω. (i)

What complex number corresponds to the vertex C?

(ii)

What complex number corresponds to the point of intersection D of the diagonals OB and AC?

99

2a

2b

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99

2c

Let α = 1 + i 3 .

4

(i)

Find the exact value of |α| and arg α.

(ii)

Find the exact value of α

11

in the form a + ib, where a and b are real numbers.

Sketch the region in the Argand diagram where the two inequalities |z – i| ≤ 2 and 0 ≤ arg (z + 1) ≤

HSC

99

3

Express the following in the form a + ib, where a and b are real numbers: 2 (i) zw (ii) iw

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99

Let z = 3 + 2i and w = –1 + i.

2e

4

to

complex

numbers

z1

and

2

both hold.

The points A and B in the complex plane correspond

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π

4

z2

respectively. Both triangles OAP and OBQ are right-angled isosceles triangles. (i)

Explain why P corresponds to the complex number (1 + i) z1.

(ii)

Let M be the midpoint of PQ. What

complex 99

8a

HSC

number corresponds to M? 2π 2π Let ρ = cos + i sin . The complex number α = ρ + ρ2 + ρ4 is a root of the 7 7

8

quadratic equation x2 + ax + b = 0, where a and b are real. (i)

Prove that 1 + ρ + ρ2 +… + ρ6 = 0.

(ii)

The second root of the quadratic equation is β. Express β in terms of positive powers of ρ. Justify your answer.

(iii) (iv) 98

Find the values of the coefficients a and b. 7 π 2π 3π + sin + sin = . Deduce that –sin 7 7 7 2

2a

Evaluate i1998.

2b

Let z =

2

HSC

98 HSC

(i)

18 + 4i 3−i

5

Simplify (18 + 4i) ( 3 − i ) http://members.optuszoo.com.au/hscsupport/index.htm

HSC by Topic 1995 to 2006 Complex Numbers

98

2c

(ii)

Express z in the form a + ib, where a and b are real numbers.

(iii)

Hence, or otherwise, find |z| and arg (z).

Sketch the region in the complex plane where the inequalities

2

|z – 2 + i| ≤ 2 and Im(z) ≥ 0 both hold.

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98

Page 6

2d

The points P and Q in the complex plane

1

correspond to the complex numbers z and w

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respectively. The triangle OPQ is isosceles and

∠ POQ is a right angle. Show that z2 + w2 = 0. 98

7a

HSC

(i)

97

2a

HSC

97

2b

(ii)

Find one of the roots of P(z) = 0 in exact form.

(iii)

Hence find all the roots of P(z) = 0.

(i)

Express

3 - i in modulus–argument form.

(ii)

Hence evaluate ( 3 - i)

(i)

Simplify (-2i)3 .

(ii)

HSC

5 4 z + 1. The complex number w is a root of P(z) = 0. 2 1 Show that iw and are also roots of P(z) = 0. w

Let P(z) = z8 -

6

4

6

4 3

Hence find all complex numbers z such that z = 8i. Express your answers in the form x+iy.

97

2c

3

hold.

HSC

97

Sketch the region where the inequalities |z – 3 + i| ≤ 5 and |z + 1| ≤ |z – 1| both

2d

Let w =

3 + 4i 5 + 12i and z = , so that |w| = |z| = 1 5 13

4

HSC

(i)

Find wz and w z in the form x+iy.

(ii)

Hence find two distinct ways of writing 652 as the sum a2 + b2, where a and b are integers and 0 < a < b.

97

8b

The diagram shows points O, R, S, T, and U in

5

the complex plane. These points correspond to

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the

complex

numbers

0,

r,

s,

t,

and

u

respectively. The triangles ORS and OTU are equilateral. Let ω = cos

π 3

+ i sin

π 3

(i)

Explain why u = ωt.

(ii)

Find the complex number r in terms of s.

(iii)

Using complex numbers, show that the lengths of RT and SU are equal.

96

2a

Suppose that c is a real number, and that z = c - i.

2

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HSC by Topic 1995 to 2006 Complex Numbers

Express the following in the form x+iy, where x and y are real numbers:

HSC

96

2b

(i)

(iz )

(ii)

1 z

On an Argand diagram, shade the region specified by both the conditions

2c

(i)

Prove by induction that (cos θ + i sin θ)n = cos(nθ) + i sin(nθ) for all

7

integers n ≥ 1.

HSC

96

2

Re(z) ≤ 4 and |z – 4 + 5i| ≤ 3

HSC

96

Page 7

2d

3 – i in modulus–argument form.

(ii)

Express w=

(iii)

Hence express w5 in the form x + iy, where x and y are real numbers.

The diagram shows the locus of points z in the

4

complex plane such that

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arg(z – 3) – arg(z + 1) =

π 3

.

This locus is part of a circle. The angle between the lines from –1 to z and from 3 to z is θ, as shown. Copy this diagram into your Writing Booklet.

95

π

(i)

Explain why θ =

(ii)

Find the centre of the circle.

3

2a

Let w1 = 8 – 2i and w2 = -5 + 3i. Find w1+ w 2 .

2b

(i)

Show that (1 – 2i)2 = -3 – 4i.

(ii)

Hence solve the equation z2 – 5z + (7 + i) = 0.

2

HSC

95 HSC

95

2c

HSC

95 HSC

2d

Sketch the locus of z satisfying: 3π (i) arg(z – 4) = 4

2

5 (ii)

Im z = |z|

The diagram shows a complex plane with origin O. The

6

points P and Q represent arbitrary non-zero complex numbers z and w respectively. Thus the length of PQ is |z – w|. (i)

Copy the diagram into your Writing Booklet, and use it to show that |z – w | ≤ |z| + |w|.

(ii)

Construct the point R representing z + w. What can be said about the quadrilateral OPRQ?

(iii)

If |z – w| = |z + w|, what can be said about the complex number

w ? z

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HSC by Topic 1995 to 2006 Complex Numbers

A HSC

2006 2a.(i)8+6i (ii)1-17i (iii) cis(

π 3

),-1, cis(-

π 3

Page 8

π 1 17 5π i 2b.(i)2cis(- ) (ii) 128cis( )(iii) -64 3 +64i 10 10 6 6

) 2d.(i)5+3i (ii)major 10, minor 9 (iii)0 ≤ arg(z) ≤

(ii)2-4i (iii)3+3i 2b.(i) 2cis(2a.(i)5+5i (ii)2+4i 2b.(i)

π 3

) (ii) 32cis(

3 +1 3 −1 +i 2 2

π 3

π

2005 2a.(i)7+3i

2

) (iii)16+i16 3 2d.(iii)+ve y-axis

(ii)2cis

π 3

(iii) 2 cis

π 12

2c.

(iv)

3 −1

2004 7b.(iii)xx

2 3

1 i 1 − 2 cos θ - 2e.(i)(1-2cosθ)+2isinθ (ii) 2 2 5 − 4 cos θ 2 sin θ 1−i π π 3π (iii) 2001 2a.-1+5i, 2b.(i)2cis (ii)-512[1+i 3 ] 2d.cis , cis , 2 4 4 5 − 4 cos θ 3 −π −3π 2(2 − cos θ ) 1 cis , cis 2e.(ii) z1+ z2 –z3 7a.(i) (iii) 2000 2a.x=5 y=1 or x=-5 y=-1 4 4 2 5 − 4 cos θ π 3π 5π −π −3π −5π w 2c.(i)-1 (ii) cis , cis , cis , cis , cis , cis 2e.(i)2iw (ii) +iw 1999 2 6 6 6 6 6 6 (1 + i )z1 + (1 − i )z 2 π 2a.(i)-5+i (ii)-1+i 2b.(i)2, (ii)210 – i. 210 3 2e.(ii) 8a.(iii)xx 1998 2 3 1 1 2a.-1 2b.(i)50+30i (ii)5+3i (iii) 34 , 0.54 rad 7a.(ii) 4 2 (iii) ± 4 2 , ± 4 2 i, ± , ± i 4 4 2 2 π π −33 66i 63 16i 1997 2a.(i)2(cos -isin ) (ii)-26 2b.(i)8i (ii) 3 +i, - 3 +i, -2i 2d.(i) + 6 6 65 65 65 65 c i π (ii)332+652 or 632+162 8b.(ii) r=s w 1996 2a.(i)1-ic (ii) 2 + 2 2c.(ii)2(cos 6 c +1 c +1 π 2 isin ) (iii)-16 3 -16i 2d.(ii)(1, ) 1995 2a.3-5i 2b.(i)-3-4i (ii)3-i,2+i 6 3 2003 2a.(i)1+3i (ii)

8 4i 5 5

2d.(ii)parallelogram (iii)

2002 2a.(i)3+i (ii)

w is imaginary z

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