95 to 06 Ext 2 Complex Numbers

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Mathematics Extension 2 HSC Examination Topic: Complex Numbers

06 HSC

2a Let z = 3 + i and w = 2 – 5i. Find, in the form x + iy, (i) z2 (ii) z w (iii) z w 1 1 1 06 HSC

2b (i) Express 3 - i in modulus-argument form. (ii) Express ( 3 - i)7 in modulus-argument form. (iii) Hence express ( 3 - i)7 in the form x + iy.

1 1 2 06

HSC

2c Find, in modulus-argument form, all solutions of z3 = –1. 3

06 HSC

2d The equation |z – 1 – 3i| + |z – 9 – 3i|corresponds to an ellipse in the Argand diagram.

(i) Write down the complex number corresponding to the centre of the ellipse. (ii) Sketch the ellipse, and state the lengths of the major and minor axes.

(iii) Write down the range of values of arg(z) for complex numbers z corresponding to points on the ellipse.

1 3 1

05 HSC

2a Let z = 3 + i and w = 1 – i. Find, in the form x + iy , (i) 2z + iw (ii) z w (iii) w 6 1 1 1 05 HSC 2b Let β = 1 - i 3

(i) Express β in modulus-argument form. (ii) Express β5 in modulus-argument form. (iii) Hence express β5 in the form x + iy.

2 2 1 05

HSC

2c Sketch the region on the Argand diagram where the inequalities

|z - z | < 2 and |z − 1| ≥1 hold simultaneously. 3

05 HSC

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

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(iii) Let α = 4 π

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

Describe the locus of R as z1 varies.

1

04 HSC

2a Let z = 1 + 2i and w = 3 − i. Find, in the form x + iy,

(i) zw (ii)       z 10 1 1 04 HSC 2b Let α = 1 + i 3 and β = 1 + i. (i) Find β α

, in the form x + iy.

(ii) Express α in modulus-argument form.

(iii) Given that β has the modulus-argument form β = ) 4 sin 4 (cos 2 π +i π , find the modulus-argument form of β α . (iv) Hence find the exact value of sin

12 π . 1 2 1 1 04 HSC

2c Sketch the region in the complex plane where the inequalities |z + z | ≤ 1 and |z – i|≤ 1 hold simultaneously.

3

04 HSC

2d The diagram shows two distinct points A and B that represent 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.

(i) Using the fact that C lies on the circle, show geometrically that ∠AOB =

3 2π

. (ii) Hence show that z3 = w3.

(iii) Show that z2 + w2 + zw = 0.

2

2 1 04

HSC

7b Let

α

be a real number and suppose that z is a complex number such that z +

z 1

= 2cos

α

.

(i) By reducing the above equation to a quadratic equation in z, solve for z and use de Moivre’s theorem to show that zn +

n z 1 = 2cosn

α

. (ii) Let w = z + z 1 . Prove that w3 + w2 – 2w – 2 = (z + z 1 ) + (z2 + 2 1 z )+ (z 3 + 3 1 z ).

(iii) Hence, or otherwise, find all solutions of cos

α

+ cos2

α

+ cos3

α

= 0, in the 3

2

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range 0 ≤

α

≤ 2

π

. 03

HSC

2a Let z = 2 + i and w = 1 − i. Find, in the form x + iy, (i) zw (ii) z 4 1 1 03 HSC

2c Sketch the region in the complex plane where the inequalities |z – 1 – i| < 2 and 0 < arg(z – 1 - i) < 4 π hold simultaneously. 3 03 HSC

2e Suppose that the complex number z lies on the unit circle, and 0 ≤ arg (z) ≤ 2 π

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

2

02 HSC

2a Let z = 1 + 2i and w = 1 + i. Find, in the form x + iy, (i) zw (ii) w 1 1 1 02 HSC

2b On an Argand diagram, shade in the region where the inequalities 0 ≤ Re z ≤ 2 and |z – 1 + i| ≤ 2 both hold.

3

02 HSC

2d Prove by induction that, for all integers n ≥ 1, (cos θ − i sin θ)n = cos nθ − i sin nθ. 3

02 HSC

2e Let z = 2(cos θ + i sin θ ). (i) Find 1− z.

(ii) Show that the real part of z − 1 1 is θ θ cos 4 5 cos 2 1 − − . (iii) Express the imaginary part of

z − 1 1 in terms of θ. 1 2 1 02 HSC 7b Suppose 0 <

α

,

β

< 2 π

and define complex numbers zn by

zn = cos(

α

+ n

β

) + isin(

α

+ n

β

) 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

.

(ii) Show that ∠P0 OP1 = ∠P0 P2 P1, and explain why

OP0 P1 P2 is a cyclic quadrilateral.

2 2

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

P1, P2 and P3 are concyclic.

(iv) Suppose that z0 + z1 + z2 + z3 + z4 = 0. Show that

β

=

5 2π

2

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HSC Find zw and

w 1

in the form x + iy. 01

HSC

2b (i) Express 1 + 3i in modulus-argument form. (ii) Hence evaluate (1 + 3i)10 in the form x + iy.

1 1 01

HSC

2c Sketch the region in the complex plane where the inequalities |z + 1 – 2i| ≤ 3 and - 3 π ≤ arg z ≤ 4 π both hold. 3 01 HSC

2d Find all solutions of the equation z4 = –1. Give your answers in modulus-argument form.

3

01 HSC

2e In the diagram the vertices of a triangle ABC are represented by the complex numbers z1, z2 and z3,

respectively. The triangle is isosceles and right-angled

at B.

(i) Explain why (z1 – z2) 2

= –(z3– z2) 2

.

(ii) Suppose D is the point such that ABCD is a square. Find the complex number, expressed in terms of z1, z2 and z3, that represents D.

2 1 01 HSC 7a Suppose that z = 2 1

(cos θ + i sin θ) where θ is real. (i) Find |z|.

(ii) Show that the imaginary part of the geometric series 1 + z + z2 + z3 + … = z − 1 1 is θ θ cos 4 5 sin 2 − .

(iii) Find an expression for 1 + 2 1 cos θ + 2 2 1 cos 2θ + 3 2 1 cos 3θ + … in terms of cos θ . 1 3 2 00 HSC

2a Find all pairs of integers x and y that satisfy (x + iy)2 = 24 + 10i. 3

00 HSC

2c (i) Let z = cos 6 π + i sin 6 π . Find z6_.

(ii) Plot, on the Argand diagram, all complex numbers that are solutions of z6 = –1. 4

00 HSC

2d Sketch the region in the Argand diagram that satisfies the inequality zz + 2(z+z) ≤ 0

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00 HSC

2e In the Argand diagram, OABC is a rectangle, where OC = 2OA. The vertex A corresponds to 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?

3

99 HSC

2a Let z = 3 + 2i and w = –1 + i.

Express the following in the form a + ib, where a and b are real numbers:

(i) zw (ii) iw 2 3 99 HSC 2b Let

α

= 1 + i 3.

(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. 4

99 HSC

2c Sketch the region in the Argand diagram where the two inequalities |z – i| ≤ 2 and 0 ≤ arg (z + 1) ≤ 4 π both hold. 2 99 HSC

2e The points A and B in the complex plane correspond to complex numbers z1 and 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 number corresponds to M?

4 99 HSC 8a Let ρ = cos 7 2π + i sin 7 2π

. The complex number

α

= ρ + ρ2 + ρ4 is a root of the 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) Find the values of the coefficients a and b. (iv) Deduce that –sin

7 π + sin 7 2π + sin 7 3π = 2 7 . 8 98 HSC 2a Evaluate i1998. 2 98 HSC 2b Let z = i i − + 3 4 18

(i) Simplify (18 + 4i) (3−i)

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(ii) Express z in the form a + ib, where a and b are real numbers. (iii) Hence, or otherwise, find |z| and arg (z).

98 HSC

2c Sketch the region in the complex plane where the inequalities |z – 2 + i| ≤ 2 and Im(z) ≥ 0 both hold.

2

98 HSC

2d The points P and Q in the complex plane correspond to the complex numbers z and w respectively. The triangle OPQ is isosceles and

∠ POQ is a right angle. Show that z2_{+ w}2_{= 0.} 1 98 HSC 7a Let P(z) = z8 - 2 5

z4 + 1. The complex number w is a root of P(z) = 0.

(i) Show that iw and w

1

are also roots of P(z) = 0. (ii) Find one of the roots of P(z) = 0 in exact form. (iii) Hence find all the roots of P(z) = 0.

6

97 HSC

2a (i) Express 3 - i in modulus–argument form. (ii) Hence evaluate ( 3 - i)6

4

97 HSC

2b (i) Simplify (-2i)3 .

(ii) Hence find all complex numbers z such that z3 = 8i. Express your answers in the form x+iy.

4

97 HSC

2c Sketch the region where the inequalities |z – 3 + i| ≤ 5 and |z + 1| ≤ |z – 1| both hold. 3 97 HSC 2d Let w = 5 4 3+ i and z = 13 12 5+ i , so that |w| = |z| = 1 (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.

4

97 HSC

8b The diagram shows points O, R, S, T, and U in the complex plane. These points correspond to the complex numbers 0, r, s, t, and u respectively. The

triangles ORS and OTU are equilateral. Let

ω

= cos 3 π + i sin 3 π (i) Explain why u =

ω

t.

(ii) Find the complex number r in terms of s. (iii) Using complex numbers, show that the

lengths of RT and SU are equal.

5

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HSC Express the following in the form x+iy, where x and y are real numbers: (i) (iz) (ii) z 1 96 HSC

2b On an Argand diagram, shade the region specified by both the conditions Re(z) ≤ 4 and |z – 4 + 5i| ≤ 3

2

96 HSC

2c (i) Prove by induction that (cos θ + i sin θ)n = cos(nθ) + i sin(nθ) for all integers n ≥ 1.

(ii) Express w= 3 – i in modulus–argument form.

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

7

96 HSC

2d The diagram shows the locus of points z in the complex plane such that

arg(z – 3) – arg(z + 1) = 3 π

.

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. (i) Explain why θ =

3 π

(ii) Find the centre of the circle.

4

95 HSC

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

95 HSC

2b (i) Show that (1 – 2i)2 = -3 – 4i.

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

2

95 HSC

2c Sketch the locus of z satisfying: (i) arg(z – 4) = 4 3π (ii) Im z = |z| 5 95 HSC

2d The diagram shows a complex plane with origin O. The 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|.

6

(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 z w

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A HSC

2006 2a.(i)8+6i (ii)1-17i (iii) 10 1 -10 17 i 2b.(i)2cis(-6 π _{) (ii) 128cis(} 6 5π _{)(iii) -64} ₃_{+64i 2c.} cis( 3 π ),-1, cis(-3 π

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

2005 2a.(i)7+3i (ii)2-4i (iii)3+3i 2b.(i)

2cis(-3 π

) (ii) 32cis( 3 π

) (iii)16+i16 3 2d.(iii)+ve y-axis 2004 2a.(i)5+5i (ii)2+4i 2b.(i)

2 1 3 + +i 2 1 3 − (ii)2cis 3 π (iii) 2 cis 12 π (iv) 3 2 1 3 − 7b.(iii)xx 2003 2a.(i)1+3i (ii) 5 8 -5 4i 2002 2a.(i)3+i (ii) 2 1 -2 i

2e.(i)(1-2cosθ)+2isinθ (ii)

θ θ cos 4 5 cos 2 1 − − (iii) θ θ cos 4 5 sin 2 − 2001 2a.-1+5i, 2 1−i 2b.(i)2cis 3 π (ii)-512[1+i 3] 2d.cis 4 π , cis 4 3π , cis 4 π − , cis 4 3π − 2e.(ii) z1+ z2 –z3 7a.(i) 2 1 (iii) θ θ cos 4 5 ) cos 2 ( 2 − −

2000 2a.x=5 y=1 or x=-5 y=-1 2c.(i)-1 (ii) cis

6 π , cis 6 3π , cis 6 5π , cis 6 π − , cis 6 3π − , cis 6 5π − 2e.(i)2iw (ii) 2 w +iw 1999 2a.(i)-5+i (ii)-1+i 2b.(i)2,

3 π (ii)210 – i. 210 3 2e.(ii) 2 ) 1 ( ) 1 ( +i z1 + −i z2 _{8a.(iii)xx 1998}

2a.-1 2b.(i)50+30i (ii)5+3i (iii) 34, 0.54 rad 7a.(ii)42 (iii)± 42 , ± 42 i, 4₂ 1 ± , 4₂ 1 ± i 1997 2a.(i)2(cos 6 π -isin 6 π

) (ii)-26 2b.(i)8i (ii) 3+i, - 3+i, -2i 2d.(i) 65 33 − + 65 66i 65 63 -65 16i

(ii)332+652 or 632+162 8b.(ii) r=s w 1996 2a.(i)1-ic (ii) 1 2 + c c + 1 2 + c i 2c.(ii)2(cos 6 π -isin 6 π

) (iii)-16 3-16i 2d.(ii)(1, 3 2

) 1995 2a.3-5i 2b.(i)-3-4i (ii)3-i,2+i 2d.(ii)parallelogram (iii)

z w