Complex Numbers
1a. Express
[369 marks]
−3 + √3i in the form reiθ , where r > 0 and −π < θ ⩽ π.
3
[5 marks]
Let the roots of the equation z 3
= −3 + √3i be u, v and w.
1b. Find u, v and w expressing your answers in the form
and −π < θ ⩽ π.
reiθ , where r > 0
[5 marks]
On an Argand diagram,
respectively.
u, v and w are represented by the points U, V and W
1c. Find the area of triangle UVW.
[4 marks]
1d. By considering the sum of the roots u,
cos 518π + cos 718π + cos 1718π = 0.
v and w, show that
[4 marks]
2. The complex numbers
w
z
z∗
[7 marks]
w and z satisfy the equations
= 2i
− 3w = 5 + 5i.
Find w and z in the form a + bi where a , b ∈ Z.
4
3a. Use the binomial theorem to expand (cos θ + i sin θ)4 . Give your answer[3 marks]
in the form a + bi where a and b are expressed in terms of sin θ and cos θ.
3b. Use de Moivre’s theorem and the result from part (a) to show that
cot 4θ =
cot4 θ −6cot2 θ +1
.
4cot3 θ −4cot θ
[5 marks]
3c. Use the identity from part (b) to show that the quadratic equation
x2 − 6x + 1 = 0 has roots cot2 π8 and cot2 38π .
23
[5 marks]
3d. Hence find the exact value of cot2 3π .
8
[4 marks]
3e. Deduce a quadratic equation with integer coefficients, having roots
cosec2 π8 and cosec2 38π .
[3 marks]
This question will explore connections between complex numbers and regular
polygons.
The diagram below shows a sector of a circle of radius 1, with the angle
,0<
<
π
subtended at the centre O being
point P to intersect the x-axis at
-axis at R.
α, 0 < α < π2 . A perpendicular is drawn from
Q. The tangent to the circle at P intersects the x
4a. By considering the area of two triangles and the area of the sector show [5 marks]
sin α
that cos α sin α < α < cos
.
α
lim
4b.
lim
Hence show that α→0 sinα
α
n
= 1,
∈ C,
∈ N,
= 1.
⩾5
[2 marks]
4c. Let z n = 1, z ∈ C, n ∈ N, n ⩾ 5. Working in modulus/argument form
find the n solutions to this equation.
[8 marks]
4d. Represent these n solutions on an Argand diagram. Let their positions be [1 mark]
denoted by P0 , P1 , P2 , … Pn−1 placed in order in an anticlockwise
direction round the circle, starting on the positive x-axis. Show the positions of
P0 , P1 , P2 and Pn−1 .
2 sin π
4e. Show that the length of the line segment P0 P1 is
2 sin πn .
[4 marks]
4f. Hence, write down the total length of the perimeter of the regular n sided [1 mark]
polygon P0 P1 P2 … Pn−1 P0 .
→∞
4g. Using part (b) find the limit of this perimeter as n
→ ∞.
[2 marks]
[3 marks]
4h. Find the total area of this n sided polygon.
→∞
4i. Using part (b) find the limit of this area as n
4
3
→ ∞.
2
[2 marks]
C
+ 4z 3 + 8z 2 + 80z + 400 = 0, z ∈ C. [8 marks]
Two of the roots of this equation are a + bi and b + ai , where a, b ∈ Z.
Find the possible values of a .
5. Consider the quartic equation z 4
π
π
2(cos π5 + i sin π5 ) and
+ i sin 2kπ
), where k ∈ Z+ .
5
Consider the complex numbers z =
w = 8(cos
2kπ
5
6a. Find the modulus of zw.
6b. Find the argument of zw in terms of k.
∈Z
[1 mark]
[2 marks]
Suppose that
zw ∈ Z.
[3 marks]
6c. Find the minimum value of k.
6d. For the value of
k found in part (i), find the value of zw.
= cos + i sin
∈ C, ≠ 1
[1 mark]
7. Consider z =
cos θ + i sin θ where z ∈ C, z ≠ 1.
Show that Re( 1−z )= 0.
1+
z
[5 marks]
This question asks you to investigate and prove a geometric property
involving the roots of the equation z n = 1 where z ∈ C for integers n,
where n ≥ 2.
= 1 where z ∈ C are 1, ω, ω2 , … , ωn−1 , where
2π i
ω = e n . Each root can be represented by a point P0 , P1 , P2 , … , Pn−1 ,
The roots of the equation z n
respectively, on an Argand diagram.
For example, the roots of the equation z 2 = 1 where z ∈ C are 1 and ω. On an
Argand diagram, the root 1 can be represented by a point P 0 and the root ω can
be represented by a point P1 .
= 3.
The roots of the equation z 3 = 1 where z ∈ C are 1, ω and ω2 . On the following
Argand diagram, the points P0 , P 1 and P 2 lie on a circle of radius 1 unit with
centre O(0, 0).
Consider the case where n
8a. Show that (ω − 1)(ω2
2
+ ω + 1) = ω3 − 1.
[2 marks]
8b. Hence, deduce that ω2
[P P ]
+ ω + 1 = 0.
[P P ]
[2 marks]
Line segments [P0 P1 ] and [P0 P2 ] are added to the Argand diagram in part (a)
and are shown on the following Argand diagram.
P0 P1 is the length of [P0 P1 ] and P0 P2 is the length of [P0 P2 ].
8c. Show that P0 P 1
× P 0 P 2 = 3.
=4
[3 marks]
= 4.
The roots of the equation z 4 = 1 where z ∈ C are 1, ω, ω2 and ω3 .
Consider the case where n
8d. By factorizing z 4
− 1, or otherwise, deduce that ω3 + ω2 + ω + 1 = 0.
[2 marks]
On the following Argand diagram, the points P 0 , P 1 , P 2 and P 3 lie on a circle of
radius 1 unit with centre O(0, 0). [P 0 P 1 ] , [P 0 P 2 ] and [P 0 P 3 ] are line segments.
8e. Show that P0 P 1
× P 0 P 2 × P 0 P 3 = 4.
[4 marks]
5
C
2
3
For the case where n
and ω4 .
= 5, the equation z 5 = 1 where z ∈ C has roots 1, ω, ω2 , ω3
It can be shown that P0 P1
× P 0 P 2 × P 0 P 3 × P 0 P 4 = 5.
Now consider the general case for integer values of n, where n ≥ 2.
The roots of the equation z n = 1 where z ∈ C are 1, ω, ω2 , … , ωn−1 . On an
Argand diagram, these roots can be represented by the points
P0 , P1 , P2 , … , Pn−1 respectively where [P0 P1 ], [P0 P2 ], … , [P0 Pn−1 ] are line
segments. The roots lie on a circle of radius 1 unit with centre O(0, 0).
8f. Suggest a value for
P0 P1 × P0 P2 × … × P0 Pn−1 .
[1 mark]
P0 P1 can be expressed as |1 − ω|.
8g. Write down expressions for P0 P2 and
P0 P3 in terms of ω.
8h. Hence, write down an expression for P 0 P n−1 in terms of
−1
−2
[2 marks]
[1 mark]
n and ω.
C
Consider z n − 1
8i. Express z n−1
the set C.
= (z − 1)(z n−1 + z n−2 + … + z + 1)where z ∈ C.
+ z n−2 + … + z + 1 as a product of linear factors over
[3 marks]
8j. Hence, using the part (g)(i) and part (f) results, or otherwise, prove your [4 marks]
suggested result to part (e).
2
R
9. Consider the equation
2z
3−z*
= i, where z = x + iy and x, y ∈ R.
[5 marks]
Find the value of x and the value of y.
9
10.
Find the term independent of x in the expansion of 13 ( 1 2
x
12
3x
9
− x2 ) .
[6 marks]
At a gathering of 12 teachers, seven are male and five are female. A group of five
of these teachers go out for a meal together. Determine the possible number of
groups in each of the following situations:
11a. There are more males than females in the group.
[4 marks]
11b. Two of the teachers, Gary and Gerwyn, refuse to go out for a meal
together.
[3 marks]
[5 marks]
12. Three planes have equations:
2x − y + z = 5
x + 3y − z = 4 , where a, b ∈ R.
3x − 5y + az = b
Find the set of values of a and b such that the three planes have no points of
intersection.
4
C
Consider the equation z 4
= −4, where z ∈ C.
13a. Solve the equation, giving the solutions in the form
a, b ∈ R.
a + ib, where
[5 marks]
13b. The solutions form the vertices of a polygon in the complex plane. Find [2 marks]
the area of the polygon.
A random variable
X has probability density function
3a
⎧
⎪
f (x) = ⎨ a (x − 5) (1 − x)
⎩
⎪
0
14a. Find, in terms of
,
,
,
0⩽x<2
2⩽x⩽b
otherwise
a,b ∈ R+ ,3 < b ⩽ 5.
a, the probability that X lies between 1 and 3.
=5
[4 marks]
Consider the case where b
= 5.
14b. Sketch the graph of f . State the coordinates of the end points and any [4 marks]
local maximum or minimum points, giving your answers in terms of a .
Find the value of
14c. a .
[4 marks]
14d. E (X).
[3 marks]
[4 marks]
14e. the median of X .
(2 + )n
⩾3
∈Z
15. Consider the expansion of (2 + x)n , where
The coefficient of
3
n ⩾ 3 and n ∈ Z.
[6 marks]
x3 is four times the coefficient of x2 . Find the value of n.
2
C
Z
16. Let
P (z) = az 3 − 37z 2 + 66z − 10, where z ∈ C and a ∈ Z.
One of the roots of P
(z) = 0 is 3 + i. Find the value of a.
[6 marks]
Eight boys and two girls sit on a bench. Determine the number of possible
arrangements, given that
17a. the girls do not sit together.
[3 marks]
17b. the girls do not sit on either end.
[2 marks]
17c. the girls do not sit on either end and do not sit together.
5
4
3
2
[3 marks]
R
Consider the equation x5
.
− 3x4 + mx3 + nx2 + px + q = 0, where m, n, p, q ∈ R
The equation has three distinct real roots which can be written as
and log2 c.
log2 a, log2 b
The equation also has two imaginary roots, one of which is di where d ∈ R.
18a. Show that abc
= 8.
[5 marks]
The values
a, b, and c are consecutive terms in a geometric sequence.
18b. Show that one of the real roots is equal to 1.
2
[3 marks]
18c. Given that q
= 8d 2 , find the other two real roots.
πi
[9 marks]
Solve z 2
= 4e 2 i , giving your answers in the form
π
19a. reiθ where r,
θ ∈ R, r > 0.
[3 marks]
b ∈ R.
[2 marks]
19b. a + ib where a ,
=
+ i
b ∈ R+
arg =
Let
z = a + bi, a, b ∈ R+ and let arg z = θ.
20a. Show the points represented by
diagram.
z and z − 2a on the following Argand
20b. Find an expression in terms of θ for
arg (z − 2a).
20c.
z ).
arg ( z−2
a
Find an expression in terms of θ for
[1 mark]
[1 mark]
[2 marks]
(
)
20d.
Hence or otherwise find the value of θ for which
21a. Find the roots of the equation w3
Cartesian form.
z ) = 0.
Re ( z−2
a
= 8i, w ∈ C. Give your answers in
Re (
)=0
[3 marks]
[4 marks]
21b. One of the roots
Given that
w1 =
w1 satisfies the condition Re (w1 ) = 0.
z , express
z −i
[3 marks]
z in the form a + bi, where a, b ∈ Q.
4
3
2
C
Consider the polynomial P
(z) ≡ z 4 − 6z 3 − 2z 2 + 58z − 51, z ∈ C.
22a. Sketch the graph of y = x4 − 6x3 − 2x2 + 58x − 51, stating clearly the [6 marks]
coordinates of any maximum and minimum points and intersections with axes.
∈R
22b. Hence, or otherwise, state the condition on
the equation P (z) = k are real.
23a. Find the roots of z 24
k ∈ R such that all roots of [2 marks]
= 1 which satisfy the condition 0 < arg (z) <
, expressing your answers in the form reiθ , where r, θ ∈ R+ .
π
2
[5 marks]
Let S be the sum of the roots found in part (a).
23b. Show that Re S = Im S.
[4 marks]
23c.
π
π
By writing 12
as ( π4 − π6 ), find the value of cos 12
in the form
, where a , b and c are integers to be determined.
(
)(
)
√ a+√b
c
[3 marks]
23d.
Hence, or otherwise, show that S = 12 (1 + √2) (1 + √3) (1 + i).
Consider the following system of equations where a
2x + 4y − z = 10
x + 2y + az = 5
5x + 12y = 2a.
[4 marks]
∈ R.
24a. Find the value of a for which the system of equations does not have a
unique solution.
=2
[2 marks]
24b. Find the solution of the system of equations when a
4
3
2
= 2.
[5 marks]
R
25. Consider the equation z 4
and z ∈ C.
+ az 3 + bz 2 + cz + d = 0, where a, b, c, d ∈ R [7 marks]
Two of the roots of the equation are log26 and i√3 and the sum of all the roots is
3 + log23.
Show that 6a + d + 12 = 0.
π
π
Consider w
= 2 (cos π3 + i sin π3 )
26a. Express w2 and w3 in modulus-argument form.
[3 marks]
26b. Sketch on an Argand diagram the points represented by w0 , w1 , w2
and w3.
[2 marks]
These four points form the vertices of a quadrilateral, Q .
26c.
Show that the area of the quadrilateral Q is
π
π
Z+
21√3
2 .
[3 marks]
[6 marks]
z = 2 (cos πn + i sin πn ) , n ∈ Z+ . The points represented on an
Argand diagram by z 0 , z 1 , z 2 , … , z n form the vertices of a polygon Pn .
Show that the area of the polygon Pn can be expressed in the form
a (bn − 1) sin πn , where a, b ∈ R.
26d. Let
2+7i
2+7i
Consider the complex number z = 6+2i .
27a. Express
z in the form a + ib, where a, b ∈ Q.
[2 marks]
27b. Find the exact value of the modulus of z.
[2 marks]
27c. Find the argument of z, giving your answer to 4 decimal places.
[2 marks]
28. Determine the roots of the equation (z + 2i)3 = 216i ,
answers in the form z = a√3 + bi where a, b ∈ Z.
z ∈ C, giving the [7 marks]
[4 marks]
29. Boxes of mixed fruit are on sale at a local supermarket.
Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.
Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.
Box C contains 5 bananas and 4 kiwifruit and costs $3.00.
Find the cost of each type of fruit.
z
Consider the complex numbers z1
= 1 + √3i,z2 = 1 + i and w =
z1
.
z2
30a. By expressing z1 and z2 in modulus-argument form write down the
modulus of w;
[3 marks]
30b. By expressing z1 and z2 in modulus-argument form write down the
argument of w.
[1 mark]
30c. Find the smallest positive integer value of n, such that wn is a real
number.
[2 marks]
2 sin( + 60∘ ) = cos( + 30∘ ), 0∘ ⩽
⩽ 180∘
31a. Solve
2 sin(x + 60∘ ) = cos(x + 30∘ ), 0∘ ⩽ x ⩽ 180∘ .
31b. Show that sin 105∘
+ cos 105∘ =
1
.
√2
= 1 − cos 2 − i sin 2 , ∈ C, 0 ⩽ ⩽
[5 marks]
[3 marks]
Let
z = 1 − cos 2θ − i sin 2θ, z ∈ C, 0 ⩽ θ ⩽ π.
31c. Find the modulus and argument of
in its simplest form.
z in terms of θ. Express each answer [9 marks]
31d. Hence find the cube roots of z in modulus-argument form.
[5 marks]
32. In the following Argand diagram the point A represents the complex
[4 marks]
number −1 + 4i and the point B represents the complex number
−3 + 0i. The shape of ABCD is a square. Determine the complex numbers
represented by the points C and D.
3
Let
ω be one of the non-real solutions of the equation z 3 = 1.
[4 marks]
33a. Determine the value of
(i)
(ii)
1 + ω + ω2 ;
1 + ω* + (ω*)2 .
2
2
33b. Show that (ω − 3ω2 )(ω2
[4 marks]
− 3ω) = 13.
Consider the complex numbers p
= 1 − 3i and q = x + (2x + 1)i, where x ∈ R.
33c. Find the values of x that satisfy the equation |p|
= |q|.
[5 marks]
33d. Solve the inequality
Re(pq) + 8 < (Im(pq))2 .
34a. Use de Moivre’s theorem to find the value of (cos( π ) + i sin( π ))3 .
3
3
[6 marks]
[2 marks]
34b. Use mathematical induction to prove that
[6 marks]
(cos θ − i sin θ)n = cos nθ − i sin nθ for n ∈ Z+ .
Let
z = cos θ + i sin θ.
34c. Find an expression in terms of
the complex conjugate of z.
*=1
θ for (z)n + (z*)n , n ∈ Z+ where z* is
[2 marks]
34d. (i)
Show that zz*
= 1.
[5 marks]
(ii)
Write down the binomial expansion of (z + z*)3 in terms of
(iii)
Hence show that cos 3θ =
34e. Hence solve
z and z*.
4 cos3 θ − 3 cos θ.
4 cos3 θ − 2 cos2 θ − 3 cos θ + 1 = 0 for 0 ⩽ θ < π.
[6 marks]
35. The following system of equations represents three planes in space.
[6 marks]
x + 3y + z = −1
x + 2y − 2z = 15
2x + y − z = 6
Find the coordinates of the point of intersection of the three planes.
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