Quadratics

Question 1 NC

By completing the square, show that the solution to the equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 is

−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎

Question 2 NC

Determine the coordinates of all points which are 10 units from both (−2, 4) and (6, 0).

Page 1
Question 3 NC

ℎ(𝑥) = 2𝑥 2 + (𝑘 + 4)𝑥 + 𝑘, where 𝑘 is a real constant.

a. Find the discriminant of ℎ(𝑥) in terms of 𝑘.

b. Hence or otherwise, prove that ℎ(𝑥) has two distinct real roots for all values of 𝑘.

Question 4 NC

The equation 3𝑝𝑥 2 + 2𝑝𝑥 + 1 = 𝑝 has two real, distinct roots.

a. Find the possible values for 𝑝.

b. Consider the case when 𝑝 = 4. The roots of the equation can be expressed in the form
𝑎±√13
𝑥= 6
, where 𝑎 ∈ ℤ. Find the value of 𝑎.

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Question 5 NC

In the quadratic equation 7𝑥 2 − 8𝑥 + 𝑝 = 0, 𝑝 ∈ ℚ, one root is three times the other root. Find the
value of 𝑝.

Question 6 NC

The roots 𝛼 and 𝛽 of the quadratic equation 𝑥 2 − 𝑘𝑥 + (𝑘 + 1) = 0 are such that 𝛼2 + 𝛽 2 = 13. Find
the possible values of the real number 𝑘.

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Question 7 C

a. Prove the identity (𝑝 + 𝑞)3 − 3𝑝𝑞(𝑝 + 𝑞) = 𝑝3 + 𝑞 3 .

The equation 2𝑥 2 − 5𝑥 + 1 = 0 has two real roots, 𝛼 and 𝛽.
1 1
Consider the equation 𝑥 2 + 𝑚𝑥 + 𝑛 = 0, where 𝑚, 𝑛 ∈ ℤ and which has roots 𝛼3 and 𝛽3 .

b. Without solving 2𝑥 2 − 5𝑥 + 1 = 0, determine the values of 𝑚 and 𝑛.

Question 8 NC

The parabola 𝑦 = 𝑎𝑥 2 + 4𝑥 − 1, where 𝑎 ∈ ℤ, has an axis of symmetry of 𝑥 = 1.

a. Find the value of 𝑎.

b. Find the coordinates of
i. the vertex
ii. the 𝑥-intercepts.

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Question 9 NC
Let 𝑓(𝑥) = 𝑥 2 − 4𝑎𝑥 + 𝑎 + 3 where 𝑎 ∈ ℝ. Consider the graph of 𝑦 = 𝑓(𝑥).
a. Determine the coordinates of the vertex.
The parabola is tangential to the 𝑥-axis.
b. Calculate the possible values of 𝑎.

Question 10 NC

Find the range of values of 𝑚 such that for all 𝑥, 𝑚(𝑥 + 1) ≤ 𝑥 2 .

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Question 11 NC

Consider two different quadratic functions of the form 𝑓(𝑥) = 4𝑥 2 − 𝑞𝑥 + 25. The graph of each
function has its vertex on the 𝑥-axis.

a. Find both values of 𝑞.

b. For the greater value of 𝑞, solve 𝑓(𝑥) = 0.
c. Find the coordinates of the point of intersection of the two graphs.

Question 12 NC

A circle has equation 𝑥 2 + (𝑦 − 2)2 = 1. The line with equation 𝑦 = 𝑘𝑥, where 𝑘 ∈ ℝ, is a tangent to
the circle. Find all possible values of 𝑘.

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Question 13 NC

Show that the graphs of 𝑦 = 2𝑥 2 + 𝑘𝑥 − 3 and 𝑦 = 𝑘𝑥 2 − 𝑥 − 1 where 𝑘 ∈ ℝ will always intersect

twice.

Question 14 NC

The following diagram contains one square and two similar rectangles.

Determine the exact value of 𝑥.

Page 7
Question 15 C

A spear is thrown over level ground from the top of a tower. The height, in metres, of the spear above
the ground after 𝑡 seconds is modelled by the function: ℎ(𝑡) = 12.25 + 14.7𝑡 − 4.9𝑡 2 , 𝑡 ≥ 0

a. Interpret the meaning of the constant term 12.25 in the model.

b. After how many seconds does the spear hit the ground?
c. Write ℎ(𝑡) in the form 𝐴 − 𝐵(𝑡 − 𝐶)2 , where 𝐴, 𝐵 and 𝐶 are constants to be found.
d. Using your answer to part c or otherwise, find the maximum height of the spear above the
ground, and the time at which this maximum height is reached.

Question 16 NC

A room is to be built beneath the roof of a

house. The shaded area in the diagram below
represents the cross-section of the room.

a. Determine the equation of

i. the left side of the roof.
ii. the right side of the roof.
b. Point 𝑃 has coordinates (𝑥, 𝑦). Find the shaded area in terms of 𝑥.
c. Determine the values of 𝑥 and 𝑦 that maximise the area of the shaded region.

Page 8
Question 17 NC

Find the shortest distance from the origin to the line 𝑦 = 2𝑥 + 1.

Question 18 NC

𝑥2
Consider the graphs of 𝑦 = 𝑥−3 and 𝑦 = 𝑚(𝑥 + 3), 𝑚 ∈ ℝ. Find the set of values for 𝑚 such that the

two graphs have no intersection points.

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Question 19 NC

The parabolas 𝑦 = 𝑘𝑥 2 − 2𝑥 + 𝑘 and 𝑦 = 𝑥 2 − 𝑘𝑥 + 1 do not intersect. Determine the restrictions on

the values of 𝑘.

Question 20 NC

Let 𝑓(𝑥) = 𝑥 2 + 𝑏𝑥 − 1 and 𝑔(𝑥) = −𝑥 2 + 2𝑥 + 𝑏 where 𝑏 ∈ ℝ . Consider the parabolas 𝑦 = 𝑓(𝑥)

and 𝑦 = 𝑔(𝑥).

a. Show that the 𝑥-coordinates of the points of intersection of the two parabolas are
2 − 𝑏 ± √𝑏 2 + 4𝑏 + 12
𝑥=
4
b. Prove that the parabolas will always intersect twice.

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 Basic Complex Numbers

Question 1 NC

One root of the equation 𝑥 2 + 𝑎𝑥 + 𝑏 = 0 is 2 + 3𝑖 where 𝑎, 𝑏 ∈ ℝ. Find the value of 𝑎 and the value
of 𝑏.

Question 2 NC

𝑎+𝑖
A complex number 𝑧 is given by 𝑧 = 𝑎−𝑖, 𝑎 ∈ ℝ.

a. Determine the set of values of a such that

ⅰ. 𝑧 is real;

ⅱ. 𝑧 is purely imaginary.

b. Show that |𝑧| is constant for all values of 𝑎.

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Question 3 NC

Given that z is the complex number 𝑥 + 𝑖𝑦 and that |𝑧| + 𝑧 = 6 − 2𝑖, find the value of 𝑥 and the value
of 𝑦.

Question 4 C

Given that (4 − 5𝑖)𝑚 + 4𝑛 = 16 + 15𝑖, where 𝑖 2 = −1, find 𝑚 and 𝑛 if

a. 𝑚 and 𝑛 are real numbers;

b. 𝑚 and 𝑛 are conjugate complex numbers.

Page 12
Question 5 NC

Consider complex numbers 𝑢 = 2 + 3𝑖 and 𝑣 = 3 + 2𝑖.

1 1 10
Given that + = , express w in the form 𝑎 + 𝑏𝑖, 𝑎, 𝑏 ∈ ℝ.
𝑢 𝑣 𝑤

Question 6 NC

Consider the distinct complex numbers 𝑧 = 𝑎 + 𝑖𝑏, 𝑤 = 𝑐 + 𝑖𝑑, where 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ.

𝑧+𝑤
a. Find the real part of 𝑧−𝑤.

𝑧+𝑤
b. Find the value of the real part of 𝑧−𝑤 when |𝑧| = |𝑤|.

Page 13
Question 7 NC

Consider the complex numbers 𝑧 = 1 + 2𝑖 and 𝑤 = 2 + 𝑎𝑖, where 𝑎 ∈ ℝ.

Find 𝑎 when

a. |𝑤| = 2|𝑧|

b. Re(𝑧𝑤) = 2Im(𝑧𝑤).

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Question 8 NC

Given that 𝑧1 = 2 and 𝑧2 = 1 + √3𝑖 are roots of the cubic equation 𝑧 3 + 𝑏𝑧 2 + 𝑐𝑧 + 𝑑 = 0 where
𝑏, 𝑐, 𝑑 ∈ ℝ.

a. Write down the third root, 𝑧3 , of the equation.

b. Find the values of 𝑏, 𝑐 and 𝑑.

Question 9 NC

Consider the complex number 𝑝 = 1 − 3𝑖 and 𝑞 = 𝑥 + (2𝑥 + 1)𝑖, where 𝑥 ∈ ℝ.

a. Find the values of 𝑥 that satisfy the equation |𝑝| = |𝑞|.

b. Solve the inequality Re(𝑝𝑞) + 8 < (Im(𝑝𝑞))2 .

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Question 10 NC

Given the complex numbers 𝑧1 = 1 + 3𝑖 and 𝑧2 = −1 − 𝑖 , find the minimum value of |𝑧1 + 𝛼𝑧2 | ,
where 𝛼 ∈ ℝ.

Question 11 NC

The complex numbers 𝑤 and 𝑧 satisfy the equations

𝑤
= 2𝑖
𝑧

𝑧 ∗ − 3𝑤 = 5 + 5𝑖

Find 𝑤 and 𝑧 in the form 𝑎 + 𝑏𝑖 where 𝑎, 𝑏 ∈ ℤ.

Page 16
Question 12 NC

Consider the quartic equation 𝑧 4 + 4𝑧 3 + 8𝑧 2 + 80𝑧 + 400 = 0, 𝑧 ∈ ℂ.

Two of the roots of this equation are 𝑎 + 𝑏𝑖 and 𝑏 + 𝑎𝑖, where 𝑎, 𝑏 ∈ ℤ.

Find the possible values of 𝑎.

Page 17
 Counting

Question 1 C

Find the number of ways in which twelve different baseball cards can be given to Emily, Harry, John
and Olivia, if Emily is to receive 5 cards, Harry is to receive 3 cards, John is to receive 3 cards and
Olivia is to receive 1 card.

Question 2 C

A police department has 4 male and 7 female officers. A special group of 5 officers is to be assembled
for an undercover operation.

a. Determine how many possible groups can be chosen.

b. Determine how many groups can be formed consisting of 2 males and 3 females.

c. Determine how many groups can be formed consisting of at least one male.

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Question 3 C

Eight runners compete in a race where there are no tied finishes. Andrea and Jack are two of the eight
competitors in this race.

Find the total number of possible ways in which the eight runners can finish if Jack finishes

a. in the position immediately after Andrea;

b. in any position after Andrea.

Question 4 C

A set of positive integers {1,2,3,4,5,6,7,8,9} is used to form a pack of nine cards.

Each card displays one positive integer without repetition from this set. Grace wishes to select four
cards at random from this pack of nine cards.

a. Find the number of selections Grace could make if the largest integer drawn among the four
cards is either a 5, a 6 or a 7.

b. Find the number of selection Grace could make if at least two of the four integers drawn are
even.

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Question 5 C

Three boys and three girls are to sit on a bench for a photograph.

a. Find the number of ways this can be done if the three girls must sit together.

b. Find the number of ways this can be done if the three girls must all sit apart.

Question 6 C

Ten students are to be arranged in a new chemistry lab. The chemistry lab is set out in two rows of five
desks as shown in the diagram.

a. Find the number of ways the ten students may be arranged in the lab.

Two of the students, Hugo and Leo, were noticed to talk to each other during previous lab sessions.

b. Find the number of ways the students may be arranged if Hugo and Leo must sit so that one
is directly behind the other. For example, Desk 1 and Desk 6.
c. Find the number of ways the students may be arranged if Hugo and Leo must not sit next to
each other in the same row.

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Question 7 C

A farmer has six sheep pens, arranged in a grid with

three rows and two columns as shown in the following
diagram.

Five sheep called Amber, Brownie, Curly, Daisy and Eden are to be placed in the pens. Each pen is
large enough to hold all of the sheep. Amber and Brownie are known to fight.

Find the number of ways of placing the sheep in the pens in each of the following cases:

a. Each pen is large enough to contain five sheep. Amber and Brownie must not be placed in the
same pen.

b. Each pen may only contain one sheep. Amber and Brownie must not be placed in pens which
share a boundary.

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Question 8 C

There are 11 players on a football team who are asked to line up in one straight line for a team photo.
Three of the team members named Adam, Brad and Chris refuse to stand next to each other. There is
no restriction on the order in which the other team members position themselves.

Find the number of different orders in which the 11 team members can be positioned for the photo.

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Question 9 C

Chloe and Selena play a game where each have four cards
showing capital letters 𝐴, 𝐵, 𝐶 and 𝐷.

Chloe lays her cards faced up on the table in order 𝐴, 𝐵, 𝐶, 𝐷 as

shown in the diagram.

Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to
see if her card matches with Chloe’s card directly above.

Chloe wins if no matches occur; otherwise Selena wins.

3
Show that the probability that Chloe wins the game is .
8

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 Binomial Theorem

Question 1 C
1 10
Find the term independent of 𝑥 in the binomial expansion of (2𝑥 2 + ) .
2𝑥 3

Question 2 C
2 4
Determine the constant term in the expansion (2𝑥 2 + 1) (𝑥 − 𝑥) .

Page 24
Question 3 C

Consider the expansion of (3 + 𝑥 2 )𝑛+1 , where 𝑛 ∈ ℤ+ .

Given that the coefficient of 𝑥 4 is 20412, find the value of 𝑛.

Question 4 C

𝑝 8
Consider the expansion of (3𝑥 + 𝑥) , where 𝑝 > 0. The coefficient of the term in 𝑥 4 is equal to the

coefficient of the term in 𝑥 6 . Find 𝑝.

Page 25
Question 5 NC

In the expansion of (2𝑥 + 1)𝑛 , the coefficient of the term in 𝑥 2 is 40𝑛, where 𝑛 ∈ ℤ+ . Find 𝑛.

Question 6 NC

Given that (1 + 𝑥)3 (1 + 𝑝𝑥)4 = 1 + 𝑞𝑥 + 93𝑥 2 + ⋯ + 𝑝4 𝑥 7, find the possible values of 𝑝 and 𝑞.

Page 26
Question 7 C

1 1 5
Find the coefficient of the term in 𝑥 in the expansion of (2𝑥 + 3𝑥) (𝑥 + 1)4 .

Page 27
Question 8 NC

1
Consider the expression − √1 − 𝑥 where 𝑎 ∈ ℚ, 𝑎 ≠ 0.
√1+𝑎𝑥

The binomial expansion of this expression, in ascending powers of 𝑥, as far as the term in 𝑥 2 is 4𝑏𝑥 +
𝑏𝑥 2 , where 𝑏 ∈ ℚ.

a. Find the value of 𝑎 and the value of 𝑏.

b. State the restriction which must be placed on 𝑥 for this expansion to be valid.

Page 28
Question 9 NC

𝑥
Show that (1+𝑥)2 ≈ 𝑥 − 2𝑥 2 + 3𝑥 3 , |𝑥| < 1.

Page 29
Question 10 NC

a. Simplify the difference of binomial coefficients

𝑛 2𝑛
( ) − ( ), where 𝑛 ≥ 3
3 2

b. Hence, solve the inequality

𝑛 2𝑛
( ) − ( ) > 32𝑛, where 𝑛 ≥ 3
3 2

Question 11 C

a. Write down the quadratic expression 𝑥 2 + 2𝑥 − 3 as the product of two linear factors.

b. Hence, or otherwise, find the coefficient of 𝑥 in the expansion of (2𝑥 2 + 𝑥 − 3)8 .

Page 30
 Sequences and Series

Question 1 NC

Consider an arithmetic sequence where 𝑢8 = 𝑆8 = 8. Find the value of the first term, 𝑢1 , and the value
of the common difference, 𝑑.

Question 2 C

On 1st January 2020, Laurie invests $𝑃 in an account that pays a nominal annual interest rate of 5.5%
compounded quarterly.

The amount of money in Laurie’s account at the end of each year follows a geometric sequence with
common ratio 𝑟.

a. Find the value of 𝑟, giving your answer to four significant figures.

Laurie makes no further deposits to or withdrawals from the account.

b. Find the year in which the amount of money in Laurie’s account will become double the
amount she invested.

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Question 3 C

The arithmetic sequence {𝑢𝑛 : 𝑛 ∈ ℤ+ } has first term 𝑢1 = 1.6 and common difference 𝑑 = 1.5. The
geometric sequence {𝑣𝑛 : 𝑛 ∈ ℤ+ } has first term 𝑣1 = 3 and common ratio 𝑟 = 1.2.

a. Find an expression for 𝑢𝑛 − 𝑣𝑛 in terms of 𝑛.

b. Determine the set of values of 𝑛 for which 𝑢𝑛 > 𝑣𝑛 .
c. Determine the greatest value of 𝑢𝑛 − 𝑣𝑛 . Give your answer correct to four significant figures.

Question 4 NC

6
The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is 13. If each

term of this sequence is positive, and the product of the first term and the third term is 32, find the sum
of the first 100 terms of this sequence.

Page 32
Question 5 NC

The sum of the first 𝑛 terms of an arithmetic sequence is 𝑆𝑛 = 3𝑛2 − 2𝑛. Find the 𝑛 th term 𝑢𝑛 .

Question 6 C

81
A geometric sequence 𝑢1 , 𝑢2 , 𝑢3 … has 𝑢1 = 27 and a sum to infinity of 2
.

a. Find the common ratio of the geometric sequence.

An arithmetic sequence 𝑣1 , 𝑣2 , 𝑣3 … is such that 𝑣2 = 𝑢2 and 𝑣4 = 𝑢4 .

b. Find the greatest value of 𝑁 such that ∑𝑁

𝑛=1 𝑣𝑛 > 0.

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Question 7 C

The seventh, third and first terms of an arithmetic sequence form the first three terms of a geometric
sequence. The arithmetic sequence has first term 𝑎 and non-zero common difference 𝑑.

𝑎
a. Show that 𝑑 = 2 .

b. The seventh term of the arithmetic sequence is 3. The sum of the first 𝑛 terms in the arithmetic
sequence exceeds the sum of the first 𝑛 terms in the geometric sequence by at least 200.
Find the least value of 𝑛 for which this occurs.

Question 8 NC

The sum of the first two terms of a geometric series is 10 and the sum of the first four terms is 30.

a. Show that the common ratio 𝑟 satisfies 𝑟 2 = 2.

b. Given 𝑟 = √2,
i. Find the first term.
ii. Find the sum of the first ten terms.

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Question 9 C

Grant wants to save $40000 over 5 years to help his son pay for his college tuition. He deposits $20000
into a savings account that has an interest rate of 6% per annum compounded monthly for 5 years.

a. Show that Grant will not be able to reach his target.

b. Find the minimum amount, to the nearest dollar, that Grant would need to deposit initially for
him to reach his target.
Grant only has $20000 to invest, so he asks his sister, Caroline, to help him accelerate the saving process.
Caroline is happy to help and offers to contribute part of her income each year. Her annual income is
$37500 per year. She starts by contributing one fifth of her annual income, and then decreases her
contributions by half each year until the target is reached. Caroline’s contributions do not yield any
interest.

c. Show that Grant and Caroline together can reach the target in 5 years.
Grant and Caroline agree that Caroline should stop contributing once she contributes enough to
complement the deficit of Grant’s investment.

d. Find the whole number of years after which Caroline will stop contributing.

Page 35
Question 10 C

A geometric sequence has first term 𝑎, common ratio 𝑟 and sum to infinity 76. A second geometric
sequence has first term 𝑎, common ratio 𝑟 3 and sum to infinity 36.

Find 𝑟.

Question 11 NC

1 𝑟
Find the value of 𝑘 if ∑∞
𝑟=1 𝑘 (3) = 7.

Page 36
Question 12 C

An infinite geometric series is given by ∑∞ 𝑘

𝑘=1 2(4 − 3𝑥) .

a. Find the values of 𝑥 for which the series has a finite sum.
b. When 𝑥 = 1.2, find the minimum number of terms needed to give a sum which is greater than
1.328.

Question 13 C

1
An infinite geometric series has 𝑢1 = 𝑘 and 𝑢2 = 2 𝑘 2 − 2𝑘, where 𝑘 > 0.

a. Find an expression for the common ratio, 𝑟, in terms of 𝑘.

b. Find the values of 𝑘 for which the sum to infinity of the series exists.

c. Find the value of 𝑘 when the sum of the infinite sequence is 𝑆∞ = 46.

Page 37
Question 14 NC

a. The following diagram shows [𝑃𝑄], with length 4𝑐𝑚. The line is divided into an infinite
number of line segments. The diagram shows the first four segments.

The length of the line segments are 𝑚 𝑐𝑚, 𝑚2 𝑐𝑚, 𝑚3 𝑐𝑚, …, where 0 < 𝑚 < 1.
4
Show that 𝑚 = .
5

b. The following diagram shows [𝑅𝑆], with length 𝑙 𝑐𝑚, where 𝑙 > 1. Squares with side lengths
𝑛 𝑐𝑚, 𝑛2 𝑐𝑚, 𝑛3 𝑐𝑚, …, where 0 < 𝑛 < 1, are drawn along [𝑅𝑆]. This process is carried on
indefinitely. The diagram shows the first four squares.

25
The total sum of the areas of all the squares is 11. Find the value of 𝑙.

Page 38
Question 15 C

2 7 𝑟
The sum of the first 𝑛 terms of a geometric sequence is given by 𝑆𝑛 = ∑𝑛𝑟=1 3 (8) .

a. Find the first term of the sequence, 𝑢1 .

b. Find 𝑆∞ .
c. Find the least value of 𝑛 such that 𝑆∞ − 𝑆𝑛 < 0.001.

Question 16 NC

7𝑛 −𝑎 𝑛
The sum, 𝑆𝑛 , of the first 𝑛 terms of a geometric sequence, whose 𝑛𝑡ℎ term is 𝑢𝑛 , is given by 𝑆𝑛 = ,
7𝑛

where 𝑎 > 0.

a. Find an expression for 𝑢𝑛 .

b. Find the first term and the common ratio of the sequence.
c. Consider the sum to infinity of the sequence.
i. Determine the values of a such that the sum to infinity exists.
ii. Find the sum to infinity when it exists.

Page 39
Question 17 NC

Let {𝑢𝑛 }, 𝑛 ∈ ℤ+ , be an arithmetic sequence with first term equal to 𝑎 and common difference of 𝑑,
where 𝑑 ≠ 0. Let another sequence {𝑣𝑛 }, 𝑛 ∈ ℤ+ , be defined by 𝑣𝑛 = 2𝑢𝑛 .

𝑣𝑛+1
a. i. Show that is a constant.
𝑣𝑛

ii. Write down the first term of the sequence {𝑣𝑛 }.

iii. Write down a formula for 𝑣𝑛 in terms of 𝑎, 𝑑 and 𝑛.

b. Let 𝑆𝑛 be the sum of the first 𝑛 term of the sequence {𝑣𝑛 }.

i. Find 𝑆𝑛 , in term of 𝑎, 𝑑 and 𝑛.

ii. Find the values of 𝑑 for which ∑∞
𝑖=1 𝑣𝑖 exists.

You are now told that ∑∞

𝑖=1 𝑣𝑖 does exist and is denoted by 𝑆∞ .

ⅲ. Write down 𝑆∞ in terms of 𝑎 and 𝑑.

iv. Given that 𝑆∞ = 2𝑎+1, find the value of 𝑑.

Page 40
Question 18 C

The diagram shows a square ABCD of side 4cm. The midpoints P, Q, R,

S of the sides are joined to form a second square.

a. i. Show that PQ = 2√2 cm.

ii. Find the area of PQRS.
The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a
third square as shown:

b. i. Write down the area of the third square, WXYZ.

ii. Show that the areas of ABCD, PQRS, WXYZ form a
geometric sequence, and find the common ratio of this sequence.
The process of forming smaller and smaller squares (by joining the
midpoints) is continued indefinitely.

c. i. Find the area of the 11th square.

ii. Calculate the sum of the areas of all the squares.

Page 41
Question 19 C

Phil takes out a bank loan of $150000 to buy a house, at an annual interest rate of 3.5%. The interest is
calculated at the end of each year and added to the amount outstanding.

a. Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest
dollar.
To pay off the loan, Phil makes annual deposits of $𝑃 at the end of every year in a savings account,
paying an annual interest rate of 2%. He makes his first deposit at the end of the first year after taking
out the loan.

(1.0220 −1)𝑃
b. Show that the total value of Phil’s savings after 20 years is 1.02−1
.

c. Given that Phil’s aim is to own the house after 20 years, find the value for 𝑃 to the nearest
dollar.
David visits a different bank and makes a single deposit of $𝑄, the annual interest rate being 2.8%.

d. i. David wishes to withdraw $5000 at the end of each year for a period of 𝑛 years. Show that
5000 5000 5000
an expression for the minimum value of 𝑄 is + + ⋯+ .
1.028 1.0282 1.028𝑛

ii. Hence or otherwise, find the minimum value of 𝑄 that would permit David to withdraw
annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.

Page 42
Question 20 NC

A geometric sequence {𝑢𝑛 }, with complex terms, is defined by 𝑢𝑛+1 = (1 + 𝑖)𝑢𝑛 and 𝑢1 = 3.

a. Find the fourth term of the sequence, giving your answer in the form 𝑥 + 𝑦𝑖, 𝑥, 𝑦 ∈ ℝ.

b. Find the sum of the first 20 terms of {𝑢𝑛 }, giving your answer in the form 𝑎 × (1 + 2𝑚 ) where
𝑎 ∈ ℂ and 𝑚 ∈ ℤ are to be determined.

A second sequence {𝑣𝑛 } is defined by 𝑣𝑛 = 𝑢𝑛 𝑢𝑛+𝑘 , 𝑘 ∈ ℕ.

c. ⅰ. Show that {𝑣𝑛 } is a geometric sequence.

ⅱ. State the first term.

ⅲ. Show that the common ratio is independent of 𝑘.

A third sequence {𝑤𝑛 } is defined by 𝑤𝑛 = |𝑢𝑛 − 𝑢𝑛+1 |.

d. Show that {𝑤𝑛 } is a geometric sequence.

Page 43
Question 21 NC

The cubic equation 𝑥 3 + 𝑝𝑥 2 + 𝑞𝑥 + 𝑐 = 0, has roots α, β, γ. By expanding (𝑥 − 𝛼)(𝑥 − 𝛽)(𝑥 − 𝛾)

show that

a.

i. 𝑝 = −(𝛼 + 𝛽 + 𝛾);

ii. 𝑞 = 𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼;

iii. 𝑐 = −𝛼𝛽𝛾.

b. It is now given that 𝑝 = −6 and 𝑞 = 18 for part b. and c. below.

i. In the case that the three roots 𝛼, 𝛽, 𝛾 form an arithmetic sequence, show that one of
the roots is 2.

ii. Hence determine the value of 𝑐.

c. In another case the three roots 𝛼, 𝛽, 𝛾 form a geometric sequence. Determine the value of 𝑐.

Page 44
 Functions
Question 1 NC

𝑥+4 𝑥−2
Let 𝑓(𝑥) = 𝑥+1 , 𝑥 ≠ −1 and 𝑔(𝑥) = 𝑥−4 , 𝑥 ≠ 4. Find the set of values of 𝑥 such that 𝑓(𝑥) ≤ 𝑔(𝑥).

Question 2 NC

Let 𝑓(𝑥) = 𝑥 2 + (𝑎 + 𝑏)𝑥 + 𝑎𝑏 where 𝑎, 𝑏 ∈ ℝ and 𝑎 < 𝑏. Consider the graph of 𝑦 = 𝑓(𝑥).

a. Find the coordinates of the 𝑥-intercepts.

𝑎+𝑏 (𝑎−𝑏)2
b. Show that the coordinates of the vertex are (− 2
, − 4
).
1
Let 𝑔(𝑥) =
√𝑓(𝑥)

c. Find
i. the domain of 𝑔(𝑥)
ii. the range of 𝑔(𝑥)

Page 45
Question 3 NC

4
Determine the domain and range of the function 𝑓(𝑥) = .
√𝑥 2 +6𝑥+8

Question 4 NC

The functions 𝑓 and 𝑔 are defined for 𝑥 ∈ ℝ by 𝑓(𝑥) = 6𝑥 2 − 12𝑥 + 1 and 𝑔(𝑥) = −𝑥 + 𝑐 , where
𝑐 ∈ ℝ.

a. Find the range of 𝑓.

b. Given that (𝑔 ∘ 𝑓)(𝑥) ≤ 0 for all 𝑥 ∈ ℝ, determine the set of possible values for 𝑐.

Page 46
Question 5 NC

The function 𝑓 is defined by 𝑓(𝑥) = (𝑥 + 2)2 − 3.

The function 𝑔 is defined by 𝑔(𝑥) = 𝑎𝑥 + 𝑏, where 𝑎 and 𝑏 are constants.

a. Find the value of 𝑎, 𝑎 > 0 and the corresponding value of 𝑏, such that 𝑓(𝑔(𝑥)) = 4𝑥 2 +
3
6𝑥 − 4.

b. The functions ℎ and 𝑘 are defined by ℎ(𝑥) = 5𝑥 + 2 and 𝑘(𝑥) = 𝑐𝑥 2 − 𝑥 + 2 respectively.

Find the value of 𝑐 such that ℎ(𝑘(𝑥)) = 0 has equal roots.

Question 6 NC

2𝑥+4
The function 𝑓 is defined by 𝑓(𝑥) = , where 𝑥 ∈ ℝ, 𝑥 ≠ 3.
3−𝑥

a. Write down the equation of asymptotes of the graph of 𝑓.

b. Find the coordinates where the graph of 𝑓 crosses the axes.
c. Sketch the graph of 𝑓.
𝑎𝑥+4
The function 𝑔 is defined by 𝑔(𝑥) = 3−𝑥
, where 𝑥 ∈ ℝ, 𝑥 ≠ 3 and 𝑎 ∈ ℝ.

d. Given that 𝑔(𝑥) = 𝑔−1 (𝑥), determine the value of 𝑎.

Page 47
Question 7 NC

𝑥−4
a. Sketch the graph of 𝑦 = 2𝑥−5, starting the equations of any asymptotes and the coordinates of

any points of intersection with the axes.

𝑥−4
b. Consider the function 𝑓: 𝑥 → √2𝑥−5. Write down:

i. The largest possible domain of 𝑓.

ii. The corresponding range of 𝑓.

Page 48
Question 8 NC

(2−𝑎)𝑥−4
Let 𝑓(𝑥) = 𝑥−2𝑎
where 𝑎 ∈ [0, 2]. The diagram below shows

the graph of 𝑦 = 𝑓(𝑥).

The region completely bound by the axes and the asymptotes is

shaded.

a. Determine the equation of

i. the vertical asymptote
ii. the horizontal asymptote
b. Hence show that the area 𝐴 of the shaded region is given by 𝐴 = −2𝑎2 + 4𝑎.
c. Determine the value of 𝑎 which maximises the value of 𝐴.
For the value of 𝑎 found in part c., let 𝑔(𝑥) = 𝑓(𝑥 − 3) + 2. The area completely bound by the axes
and the asymptotes on the graph of 𝑦 = 𝑔(𝑥) is shaded.
d. Determine the area of this shaded region.

Page 49
Question 9 C

a. Sketch the graphs of 𝑦 = sin3 𝑥 +ln 𝑥 and 𝑦 = 1 + cos 𝑥 for 0 < 𝑥 ≤ 9.

b. Hence solve sin3 𝑥 + ln 𝑥 − cos 𝑥 − 1 < 0 in the range 0 < 𝑥 ≤ 9.

Question 10 NC

𝑎𝑥+𝑏 𝑑
The function 𝑓 is defined by 𝑓(𝑥) = 𝑐𝑥+𝑑 , for 𝑥 ∈ ℝ, 𝑥 ≠ − 𝑐 .

a. Find the inverse function 𝑓 −1 , stating its domain.

2𝑥−3
The function 𝑔 is defined by 𝑔(𝑥) = 𝑥−2
, 𝑥 ∈ ℝ, 𝑥 ≠ 2.

b. Sketch the graph of 𝑦 = 𝑔(𝑥). State the equations of any asymptotes and the coordinates of
any intercepts with the axes.
c. The function ℎ is defined by ℎ(𝑥) = √𝑥, for 𝑥 ≥ 0. State the domain and range of ℎ ∘ 𝑔.

Page 50
Question 11 NC

The following graph represents a function 𝑦 = 𝑓(𝑥),

where −3 ≤ 𝑥 ≤ 5. The function has a maximum at
(3, 1) and a minimum at (−1, −1).

a. The functions 𝑢 and 𝑣 are defined as 𝑢(𝑥) =

𝑥 − 3, 𝑣(𝑥) = 2𝑥 where 𝑥 ∈ ℝ.

i. State the range of the function 𝑢 ∘ 𝑓.

ii. State the range of the function 𝑢 ∘ 𝑣 ∘ 𝑓.
iii. Find the largest possible domain of the function 𝑓 ∘ 𝑣 ∘ 𝑢.
b.
i. Explain why 𝑓 does not have an inverse.
ii. The domain of 𝑓 is restricted to define a function 𝑔 so that it has an inverse 𝑔−1 . State
the largest possible domain of 𝑔.
iii. Sketch a graph of 𝑦 = 𝑔−1 (𝑥), showing clearly the 𝑦-intercept and stating the
coordinates of the endpoints.
2𝑥−5
c. Consider the function defined by ℎ(𝑥) = 𝑥+𝑑
,𝑥 ≠ −𝑑 and 𝑑 ∈ ℝ.

i. Find an expression for the inverse function ℎ−1 (𝑥).

ii. Find the value of 𝑑 such that ℎ is a self-inverse function.
2𝑥
For this value of 𝑑, there is a function 𝑘 such that ℎ ∘ 𝑘(𝑥) = 𝑥+1 , 𝑥 ≠ −1.

iii. Find 𝑘(𝑥).

Page 51
Question 12 NC

The function 𝑓(𝑥) is one-to-one and defined such that 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 13, 𝑥 ≥ 𝑘, 𝑥 ∈ ℝ, 𝑘 ∈ ℝ.

a. Find the least possible values for 𝑘.

b. Find an expression for the inverse function 𝑓 −1 (𝑥).
c. State the domain and the range of 𝑓 −1 (𝑥).

Page 52
Question 13 NC

1 4
Let 𝑓(𝑥) = 𝑥 − 𝑥+1 . Consider the graph of 𝑦 = 𝑓(𝑥).

−𝑦−3±√𝑦 2 +10𝑦+9
a. By rearranging 𝑦 = 𝑓(𝑥), show that 𝑥 = 2𝑦
.

b. Hence find the range of 𝑓(𝑥).

Question 14 NC

The diagram shows the graph of 𝑦 = 𝑓(𝑥), −3 ≤ 𝑥 ≤ 5.

a. Find the value of (𝑓 ∘ 𝑓)(1).

b. Given that 𝑓 −1 (𝑎) = 3, determine the value of 𝑎.
c. Given that 𝑔(𝑥) = 2𝑓(𝑥 − 1), find the domain and
range of 𝑔.

Page 53
Question 15 NC

The quadratic function 𝑓 is defined by 𝑓(𝑥) = 3𝑥 2 − 12𝑥 + 11.

a. Write 𝑓 in the form 𝑓(𝑥) = 3(𝑥 − ℎ)2 − 𝑘.

b. The graph of 𝑓 is translated 3 units in the positive 𝑥-direction and 5 units in the positive 𝑦-
direction. Find the function 𝑔 for the translated graph, giving your answer in the form 𝑔(𝑥) =
3(𝑥 − 𝑝)2 + 𝑞.

Question 16 NC

1
Find the rational function when 𝑦 = is transformed by a vertical stretch of 2, then stretched
𝑥
1 −2
horizontally by a factor of 3, followed by a translation of ( ). Find the domain and range of the new
3
function.

Page 54
Question 17 NC

1 14−4𝑥
List the transformations, in order, that transform the graph of 𝑓(𝑥) = 𝑥 to the graph of 𝑔(𝑥) = 3−𝑥
.

Page 55
Question 18 NC

Let 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 8 and 𝑔(𝑥) = −2𝑥 2 + 4𝑥 + 1 . The

following diagram shows the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥).

a. Write 𝑓(𝑥) and 𝑔(𝑥) in the form 𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞 where

𝑎, 𝑝, 𝑞 ∈ ℤ
b. Describe a series of transformations that maps the graph of
𝑦 = 𝑓(𝑥) onto the graph of 𝑦 = 𝑔(𝑥).
The graph of 𝑦 = 𝑔(𝑥) is translated 𝑘 units upwards so that it intersects with the graph of 𝑦 = 𝑓(𝑥) at
least once.

20
c. Show that 𝑘 ≥ .
3

d. If the translated graph intersects with the graph of 𝑦 = 𝑓(𝑥) exactly once, find the 𝑥-coordinate
of the point of intersection.

Page 56
Question 19 NC

1
The function 𝑓 is defined by 𝑓(𝑥) = 4𝑥 2 −4𝑥+5.

a. Express 4𝑥 2 − 4𝑥 + 5 in the form 𝑎(𝑥 − ℎ)2 + 𝑘 where 𝑎, ℎ, 𝑘 ∈ ℚ.

b. The graph of 𝑦 = 𝑥 2 is transformed onto the graph of 𝑦 = 4𝑥 2 − 4𝑥 + 5. Describe a
sequence of transformations that does this, making the order of transformations clear.
c. Sketch the graph of 𝑦 = 𝑓(𝑥).
d. Find the range of 𝑓.

Page 57
Question 20 NC

Let 𝑓(𝑥) = 𝑥 4 + 𝑥 3 − 3. The graph of 𝑦 = 𝑓(𝑥) is translated 𝑎 units to the right to produce the graph
of 𝑦 = 𝑔(𝑥). The coefficient of 𝑥 2 in the function 𝑔(𝑥) is equal to 18. Find the value of 𝑎.

Page 58
 Exponentials and Logarithms

Question 1 NC

Find the exact solution to the equation 7𝑥−2 = 2 × 5𝑥+1.

ln 𝑎
Write your answer in the form ln 𝑏−ln 𝑐 where 𝑎, 𝑏, 𝑐 ∈ ℤ+ .

Question 2 NC

3
Solve the equation √4𝑥−2 = 8𝑥+3 .

Page 59
Question 3 C

All living plants contain an isotope of carbon called carbon-14. When a plant dies, the isotope decays
so that the amount of carbon-14 present in the remains of the plant decreases. The time since the death
of a plant can be determined by measuring the amount of carbon-14 still present in the remains.

The amount, 𝐴, of carbon-14 present in a plant 𝑡 years after its death can be modelled by 𝐴 = 𝐴0 𝑒 −𝑘𝑡 ,
where 𝑡 ≥ 0 and 𝐴0 , 𝑘 are positive constants.

At the time of death, a plant is defined to have 100 units of carbon-14.

a. Show that 𝐴0 = 100.

The time taken for half the original amount of carbon-14 to decay is known to be 5730 years.

ln 2
b. Show that 𝑘 = .
5730

c. Find, correct to the nearest 10 years, the time taken after the plant’s death for 25% of the carbon-
14 to decay.

Question 4 NC

1
Solve the equation log 3 √𝑥 = 2 log + log 3 (4𝑥 3 ), where 𝑥 > 0.
23

Page 60
Question 5 NC

Solve the equation 4𝑥+1 + 2𝑥+5 − 17 = 0 for 𝑥 ∈ ℝ.

Question 6 NC

Find the integer values of 𝑚 and 𝑛 for which

𝑚 − 𝑛 log 3 2 = 10 log 9 6

Page 61
Question 7 NC

Consider the equation 𝑒 0.4𝑥 = 2(𝑒 0.2𝑥 + 1).

a. Show that 𝑒 0.2𝑥 = 1 + √3.

b. Find the exact solution to the equation in the form 𝑥 = ln(𝑎 + 𝑏√3) where 𝑎, 𝑏 ∈ ℤ.

Question 8 NC

Let 𝑔(𝑥) = log 5 |2 log 3 𝑥|. Find the product of the zeros of 𝑔.

Page 62
Question 9 NC

Solve (ln 𝑥)2 − (ln 2)(ln 𝑥) < 2(ln 2)2 .

Question 10 NC

3𝑥 +1
The function 𝑓 is given by 𝑓(𝑥) = 3𝑥 −3−𝑥 , for 𝑥 > 0.

a. Show that 𝑓(𝑥) > 1 for all 𝑥 > 0.

b. Solve the equation 𝑓(𝑥) = 4.

Page 63
Question 11 NC

1 1 1 1
The first terms of an arithmetic sequence are log 𝑥, log 𝑥, log , log ,…
2 8 32 𝑥 128 𝑥

Find 𝑥 if the sum of the first 20 terms of the sequence is equal to 100.

Question 12 NC

The first three terms of geometric sequences are ln 𝑥 9 , ln 𝑥 3 , ln 𝑥, for 𝑥 > 0.

a. Find the common ratio.

b. Solve ∑∞
𝑘=1 3
3−𝑘
ln 𝑥 = 27.

Page 64
Question 13 NC

The function 𝑓 is defined by 𝑓(𝑥) = 𝑒 2𝑥 − 6𝑒 𝑥 + 5, 𝑥 ∈ ℝ, 𝑥 ≤ 𝑎 . The

graph of 𝑦 = 𝑓(𝑥) is shown in the following diagram.

a. Find the largest value of 𝑎 such that 𝑓 has an inverse function.

b. For this value of 𝑎 , find an expression for 𝑓 −1 (𝑥) , stating its
domain.

Question 14 NC

The graph of 𝑦 = ln (5𝑥 + 10) is obtained from the graph of 𝑦 = ln 𝑥 by a translation of 𝑎 units in the
direction of the 𝑥-axis followed by a translation of 𝑏 units in the direction of the 𝑦-axis.

Find the value of 𝑎 and the value of 𝑏.

Page 65
Question 15 C

1
Consider 𝑓(𝑥) = 2 − ln(√𝑥 2 − 4).

a. Find the largest possible domain 𝐷 for 𝑓 to be a function.

1
The function 𝑓 is defined by 𝑓(𝑥) = 2 − ln(√𝑥 2 − 4), for 𝑥 ∈ 𝐷.

b. Sketch the graph of 𝑦 = 𝑓(𝑥), showing clearly the equations of asymptotes and the
coordinates of any intercepts with the axes.
c. Explain why the inverse function 𝑓 −1 does not exist.
1
The function 𝑔 is defined by 𝑔(𝑥) = − ln(√𝑥 2 − 4), for 𝑥 ∈ (2, ∞).
2

d. Find the inverse function 𝑔−1 and state its domain.

Page 66
Question 16 NC
1
A function is defined by ℎ(𝑥) = 2𝑒 𝑥 − 𝑒 𝑥 , 𝑥 ∈ ℝ. Find an expression for ℎ−1 (𝑥).

Question 17 NC

The geometric sequence 𝑢1 , 𝑢2 , 𝑢3 , … has common ratio 𝑟.

Consider the sequence 𝐴 = {𝑎𝑛 = log 2 |𝑢𝑛 | : 𝑛 ∈ ℤ+ }.

a. Show that 𝐴 is an arithmetic sequence, stating its common difference 𝑑 in terms of 𝑟.

b. A particular geometric sequence has 𝑢1 = 3 and a sum to infinity of 4.
Find the value of 𝑑.

Page 67
Question 18 NC

1
a. Prove that log 𝑎 𝑏 = log for 𝑎, 𝑏 > 0.
𝑏𝑎

Let 𝑓(𝑥) = 4𝑥 . The value of 𝑐 is such that 𝑓(𝑐) = 7.

b. Find the value of 𝑐.

c. For the graph of 𝑦 = 𝑓(𝑥), find the coordinates of the 𝑦-intercept and the equation of the
asymptote.
The graph of 𝑦 = 𝑓(𝑥) is translated by 𝑐 units to the left resulting in the graph of 𝑦 = 𝑔(𝑥).

d. Write down the function 𝑔(𝑥).

e. Find a different transformation that can be applied to the graph of 𝑦 = 𝑓(𝑥) to produce the
graph of 𝑦 = 𝑔(𝑥).
A stretch by a factor of log 𝑎 𝑏 is applied to the graph of 𝑦 = 𝑓(𝑥) to produce the graph of 𝑦 = 7𝑥 .

f. Find the direction of the stretch and the values of 𝑎 and 𝑏.

Page 68
Question 19 NC

Let {𝑢𝑛 }, 𝑛 ∈ ℤ+ , be an arithmetic sequence with first term equal to a and common difference of 𝑑,
where 𝑑 ≠ 0. Let another sequence {𝑣𝑛 }, 𝑛 ∈ ℤ+ , be defined by 𝑣𝑛 = 2𝑢𝑛 .

a.

𝑣𝑛+1
i. Show that is a constant.
𝑣𝑛

ii. Write down the first term of the sequence {𝑣𝑛 }.

iii. Write down a formula for 𝑣𝑛 in terms of 𝑎, 𝑑 and 𝑛.

b. Let 𝑆𝑛 be the sum of the first 𝑛 term of the sequence {𝑣𝑛 }.

i. Find 𝑆𝑛 , in term of 𝑎, 𝑑 and 𝑛.

ii. Find the values of 𝑑 for which ∑∞

𝑖=1 𝑣𝑖 exists.

You are now told that∑∞

𝑖=1 𝑣𝑖 does exist and is denoted by 𝑆∞ .

iii. Write down 𝑆∞ in terms of 𝑎 and 𝑑.

iv. Given that 𝑆∞ = 2𝑎+1, find the value of 𝑑.

c. Let {𝑤𝑛 }, 𝑛 ∈ ℤ+ , be a geometric sequence with first term equal to 𝑝 and common ratio 𝑞,
where 𝑝 and 𝑞 are both greater than zero. Let another sequence {𝑧𝑛 } be defined by 𝑧𝑛 = ln 𝑤𝑛 .

Find ∑𝑛𝑖=1 𝑧𝑖 giving your answer in the form ln 𝑘 with 𝑘 in terms of 𝑛, 𝑝 and 𝑞.

Page 69
Question 20 NC

1
Consider the series ln 𝑥 + 𝑝 ln 𝑥 + 3 ln 𝑥 + ⋯, where 𝑥 ∈ ℝ, 𝑥 > 1 and 𝑝 ∈ ℝ, 𝑝 ≠ 0.

a. Consider the case where the series is geometric.

1
i. Show that 𝑝 = ± .
√3

ii. Hence or otherwise, show that the series is convergent.

iii. Given that 𝑝 > 0 and 𝑆∞ = 3 + √3, find the value of 𝑥.
b. Now consider the case where the series is arithmetic with common difference 𝑑.
2
i. Show that 𝑝 = 3 .

ii. Write down 𝑑 in the form 𝑘 ln 𝑥, where 𝑘 ∈ ℚ.

1
iii. The sum of the first 𝑛 terms of the series is ln (𝑥 3 ). Find the value of 𝑛.

Page 70
 Further Functions

Question 1 NC

The functions 𝑓 and 𝑔 are defined by 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, 𝑥 ∈ 𝑅 and 𝑔(𝑥) = 𝑝 sin 𝑥 +𝑞𝑥 + 𝑟, 𝑥 ∈
ℝ where 𝑎, 𝑏, 𝑐, 𝑝, 𝑞, 𝑟 are real constants.

a. Given that 𝑓 is an even function, show that 𝑏 = 0.

b. Given that 𝑔 is an odd function, find the value of 𝑟.
c. The function ℎ is both odd and even, with domain ℝ.
Find ℎ(𝑥).

Question 2 NC

Determine algebraically whether the function is odd, even or neither.

𝑥 2 +1
a. 𝑦= |𝑥|
;𝑥 ≠0
𝑥
b. 𝑦=
𝑥 2 +1

√𝑥
c. 𝑦= 𝑥
;𝑥 ≠0

Page 71
Question 3 C

a. A function 𝑓 is defined by 𝑓(𝑥) = (𝑥 + 1)(𝑥 − 1)(𝑥 − 5), 𝑥 ∈ ℝ. Find the values of 𝑥 for
which 𝑓(𝑥) < |𝑓(𝑥)|.
b. A function 𝑔 is defined by 𝑔(𝑥) = 𝑥 2 + 𝑥 − 6, 𝑥 ∈ ℝ. Find the values of 𝑥 for which 𝑔(𝑥) <
1
.
𝑔(𝑥)

Question 4 NC

The following diagram shows the graph of 𝑦 =

𝑓(𝑥). The graph has a horizontal asymptote at 𝑦 =
−1 . The graph crosses the 𝑥 -axis at 𝑥 = −1 and
𝑥 = 1, and the 𝑦-axis at 𝑦 = 2.

Sketch the graph of 𝑦 = [𝑓(𝑥)]2 + 1 , clearly

showing any asymptotes with their equations and
the coordinates of any local maxima or minima.

Page 72
Question 5 NC

3𝑥
The function 𝑓 is defined by 𝑓(𝑥) = 𝑥−2 , 𝑥 ∈ ℝ, 𝑥 ≠ 2.

a. Sketch the graph of 𝑦 = 𝑓(𝑥), indicating clearly any asymptotes and points of intersection with
the 𝑥 and 𝑦 axes.
b. Find an expression for 𝑓 −1 (𝑥).
c. Find all values of 𝑥 for which 𝑓(𝑥) = 𝑓 −1 (𝑥).
3
d. Solve the inequality |𝑓(𝑥)| < 2.
3
e. Solve the inequality 𝑓(|𝑥|) < 2.

Page 73
Question 6 NC

𝑥 2 −𝑥−12 15
Consider the function 𝑓(𝑥) = 2𝑥−15
,𝑥 ∈ ℝ, 𝑥 ≠ 2
.

a. Find the coordinates where the graph of 𝑓 crosses the

i. 𝑥-axis
ii. 𝑦-axis
b. Write down the equation of the vertical asymptotes of the graph of 𝑓.
c. The oblique asymptote of the graph 𝑓 can be written as 𝑦 = 𝑎𝑥 + 𝑏 where 𝑎, 𝑏 ∈ ℚ.
Find the value of 𝑎 and the value of 𝑏.
d. Sketch the graph of 𝑓 for −30 ≤ 𝑥 ≤ 30, clearly indicating the points of intersection with
each axis and any asymptotes.

Page 74
Question 7 NC

4𝑥−3
Consider the function defined by 𝑓(𝑥) = 9𝑥 2 −4 for 𝑥 ∈ ℝ, 𝑥 ≠ 𝑝, 𝑥 ≠ 𝑞, 𝑝 < 𝑞.

a. Find the value of 𝑝 and the value of 𝑞.

b. Sketch the graph of 𝑦 = 𝑓(𝑥) for −3 ≤ 𝑥 ≤ 3, showing the values of axes intercepts and
asymptotes.

Question 8 NC

1−3𝑥
a. Sketch the graph of 𝑦 = 𝑥−2
, indicating clearly any asymptotes and points of intersection with

the 𝑥 and 𝑦 axes.

1−3𝑥
b. Hence or otherwise, solve the inequality | 𝑥−2 | < 2.

Page 75
Question 9 NC
Consider the graphs of 𝑦 = |𝑥| and 𝑦 = −|𝑥| + 𝑏, where 𝑏 ∈ ℤ+ .

a. Sketch the graph on the same axes.

b. Given that the graphs enclose a region of area 18 square units, find the value of 𝑏.

Question 10 NC
9−12𝑥 20
Let 𝑓(𝑥) = , for 𝑥 ≠ , where 𝑐 ≠ 0.
𝑐𝑥−20 𝑐

a. The line 𝑥 = 5 is a vertical asymptote to the graph of 𝑦 = 𝑓(𝑥).

a. Find the value of 𝑐.
b. Write down the equation of the horizontal asymptote to the graph of 𝑦 = 𝑓(𝑥).
b. The line 𝑦 = ℎ, where ℎ ∈ ℝ, intersects the graph of 𝑦 = |𝑓(𝑥)| at exactly one point. Find the
possible values of ℎ.

Page 76
Question 11 NC

9
a. Sketch the curve 𝑦 = |𝑥+5| and line 𝑦 = 5 − 𝑥 on the same axes, clearly indicating any 𝑥 and

𝑦 intercepts and any asymptotes.

9
b. Find the exact solutions to the equation 5 − 𝑥 = |𝑥+5| .

Page 77
Question 12 NC

The diagram below shows the graph of 𝑦 = 𝑓(𝑥) where 𝑓(𝑥) = 𝑥 2 +

𝑏𝑥 + 𝑐 for 𝑏, 𝑐 ∈ ℝ.

a. Find the values of 𝑏 and 𝑐.

The graph of 𝑦 = 𝑓(𝑥) is translated 𝑝 units to the right to produce the

graph of 𝑦 = 𝑔(𝑥).

b. Find the function 𝑔(𝑥) in expanded form.

The graph of 𝑦 = 𝑔(|𝑥|) intersects four times with the line 𝑦 = 𝑘 where 𝑘 ∈ ℝ.

c. Determine any restrictions on the value of 𝑝 and 𝑘.

Page 78
Question 13 NC

𝑥 2 +𝑏𝑥+4
The asymptotes of the graph of 𝑦 = 𝑥−3
intersect at the point (𝑘, 5) where 𝑏, 𝑘 ∈ ℝ.

a. Write down the value of 𝑘.

b. Find the value of 𝑏.

Question 14 NC
Find the conditions on the value of 𝑘 if the equation |𝑥 2 − 𝑘𝑥 − 1| = 𝑘 has four unique solutions.

Page 79
Question 15 NC

1
The graph below shows the function 𝑦 = 𝑓(𝑥). Sketch the graph of 𝑦 = 𝑓(𝑥).

Question 16 NC

The graph shows the function 𝑦 = 𝑓(𝑥).

a. Sketch the graph of 𝑦 = |𝑓(𝑥)|.

b. Sketch the graph of 𝑦 = 𝑓(|𝑥|).

Page 80
 Unit Circle and Radian Measure

Question 1 NC

The logo, for a company that makes chocolate, is a sector of a circle of radius
2𝑐𝑚, shown as shaded in the diagram. The area of the logo is 3𝜋 𝑐𝑚2 .

a. Find, in radians, the value of the angle 𝜃, as indicated on the diagram.

b. Find the total length of the perimeter of the logo.

Question 2 C

A system of equations is given by

cos 𝑥 + cos 𝑦 = 1.2

sin 𝑥 + sin 𝑦 = 1.4

a. For each equation, express 𝑦 in terms of 𝑥.

b. Hence solve the system for 0 < 𝑥 < 𝜋, 0 < 𝑦 < 𝜋.

Page 81
Question 3 NC

A sector of a circle with radius 𝑟 cm, where 𝑟 > 0 , is shown on the

following diagram. The sector has an angle of 1 radian at the centre.

Let the area of the sector be 𝐴 cm2 and the perimeter be 𝑃 cm. Given that
𝐴 = 𝑃 , find the value of 𝑟.

Question 4 C

The diagram shows a metallic pendant made out of four equal

sectors of a larger circle of radius 𝑂𝐵 = 9 𝑐𝑚 and four equal
sectors of a smaller circle of radius 𝑂𝐴 = 3 𝑐𝑚 . The angle
𝐵𝑂𝐶 = 20. Find the area of the pendant.

Page 82
Question 5 C

Farmer Bill owns a rectangular field, 10 m by 4 m, Bill attaches a rope to a wooden post at one corner
of his field and attaches the other end to his goat Gruff.

a. Given that the rope is 5 m long, calculate the percentage of Bill’s field that Gruff is able to
graze. Give your answer correct to the nearest integer.
b. Bill replaces Gruff’s rope with another, this time of length 𝑎, 4 < 𝑎 < 10, so that Gruff can
now graze exactly one half of Bill’s field.
Show that 𝑎 satisfies the equation

4
𝑎2 arcsin ( ) + 4√𝑎2 − 16 = 40.
𝑎

c. Fine the value of 𝑎.

Page 83
Question 6 C

A rectangle is drawn around a sector of a circle as shown. If the angle

of the sector is 1 radian and the area of the sector is 7 cm 2, find the
dimensions of the rectangle, giving your answers to the nearest
millimetre.

Question 7 NC

The diagram shows two concentric circles with centre O. The radius of
the smaller circle is 8 cm and the radius of the larger circle is 10 cm.
Points A, B, and C are on the circumference of the larger circle such that
𝜋
AOB is 3 radians.

a. Find the length of the arc ACB.

b. Find the area of the shaded region.

Page 84
Question 8 NC

The diagram below shows a triangle and two arcs of circles. The
triangle ABC is a right-angles isosceles triangle, with 𝐴𝐵 = 𝐴𝐶 =
2. The point 𝑃 is the midpoint of [BC]. The arc BDC is part of a
circle with centre 𝐴. The arc BEC is part of a circle with centre 𝑃.

a. Calculate the area of the segment BDCP.

b. Calculate the area of the shaded region BECD.

Question 9 NC

The diagram shows a tangent, (TP), to the circle with

centre O and radius r. The size of 𝑃𝑂̂ 𝐴 is 𝜃 radians.

a. Find the area of triangle AOP in terms of r

and 𝜃.
b. Find the area of triangle POT in terms of r
and 𝜃.
c. Using your results from part a. and part b., show that sin 𝜃 < 𝜃 < tan 𝜃.

Page 85
Question 10 NC

The diagram below shows two straight lines intersecting at O and two circles,
each with centre O. The outer circle has radius R and the inner circle has radius
r. Consider the shaded regions with areas A and B. Given that 𝐴: 𝐵 = 2: 1, find
the exact value of the ratio 𝑅: 𝑟.

Question 11 C

The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas of these
sectors form a geometric sequence with common ratio k. the angle of the first sector is 𝜃 radians.

a. Show that 𝜃 = 2𝜋(1 − 𝑘).

b. The perimeter of the third sector is half the perimeter of the first sector. Find the value of 𝑘
and of 𝜃.

Page 86
Question 12 C

Two non-intersecting circles C1, containing points M and S, and C2, containing points N and R, have
centres P and Q where 𝑃𝑄 = 50. The line segments [MN] and [SR] are common tangents to the circles.
The size of the reflex angle MPS is 𝛼, the size of the obtuse angle NQR is 𝛽, and the size of the angle
MPQ is 𝜃. The arc length MS is l1 and the arc length NR is l2. This information is represented in the
diagram below. The radius of C1 is 𝑥, where 𝑥 ≥ 10 and the radius of C2 is 10.

a. Explain why 𝑥 < 40.

𝑥−10
b. Show that cos 𝜃 = .
50

c. i. Find an expression for MN in terms of 𝑥.

ii. Find the value of 𝑥 that maximizes MN.

d. Find an expression in terms of 𝑥 for

i. 𝛼

ii. 𝛽

e. The length of the perimeter is given by l1 + l2 + MN+ SR.

i. Find an expression, 𝑏(𝑥), for the length of the perimeter in terms of 𝑥.

ii. Find the maximum value of the length of the perimeter.

iii. find the value of 𝑥 that gives a perimeter of length 200.

Page 87
Page 88
Question 13 C

The diagram shows a fenced triangular enclosure in the middle of a large

grassy field. The points 𝐴 and 𝐶 are 3 𝑚 apart. A goat 𝐺 is tied by 5 𝑚
length of rope at point 𝐴 on the outside edge of the enclosure.

Given that the corner of the enclosure at 𝐶 forms an angle of 𝜃 radians and
the area of field that can be reached by the goat is 44 𝑚2, find the value of
𝜃.

Page 89
 Non-Right-Angled Trigonometry

Question 1 C

The following diagram shows a part of a circle with centre 𝑂 and

radius 4𝑐𝑚.

Chord 𝐴𝐵 has a length of 5𝑐𝑚 and 𝐴𝑂̂𝐵 = 𝜃.

a. Find the value of 𝜃, giving your answer in radians.

b. Find the area of the shaded region.

Question 2 NC

The following diagram shows triangle 𝐴𝐵𝐶, with 𝐴𝐵 = 10,

𝐵𝐶 = 𝑥, and 𝐴𝐶 = 2𝑥.

3
Given that cos 𝐶̂ = 4, find the area of the triangle.

𝑝√𝑞
Give your answer in the form 2
where 𝑝, 𝑞 ∈ ℤ+ .

Page 90
Question 3 C

The following diagram shows a semicircle with centre 𝑂

and radius 𝑟. Points 𝑃, 𝑄 and 𝑅 lie on the circumference of
the circle, such that 𝑃𝑄 = 2𝑟 and 𝑅𝑂̂𝑄 = 𝜃 , where 0 <
𝜃 < 𝜋.

a. Given that the areas of the two shaded regions are equal, show that 𝜃 = 2 sin 𝜃.
b. Hence determine the value of 𝜃.

Question 4 NC

The lengths of two of the sides in a triangle are 4 cm and 5 cm. Let 𝜃 be the angle between the two
5√15
given sides. The triangle has an area of 2
𝑐𝑚2 .

√15
a. Show that sin 𝜃 = 4
.

b. Find the two possible values for the length of the third side.

Page 91
Question 5 C

Consider the following diagram. The sides of the equilateral

triangle ABC have lengths 1m. The midpoint of [AB] is denoted
by P. The circular arc AB has centre, M, the midpoint of [CP].

a. i. Find AM.
̂ 𝑃 in radians.
ii. Find 𝐴𝑀

b. Find the area of the shaded region.

Page 92
Question 6 C

Boat A is situated 10 km away from boat B, and each boat has a marine radio transmitter on board. The
range of the transmitter on boat A is 7 km, and the range of the transmitter on boat B is 5 km. The region
in which both transmitters can be detected is represented by the shaded region in the following diagram.
Find the area of this region.

Page 93
Question 7 NC

a. Find the set of values of k that satisfy the inequality 𝑘 2 − 𝑘 −

12 < 0.
b. The triangle ABC is shown in the following diagram. Given that
1
cos 𝐵 < 4, find the range of possible values for AB.

Question 8 C

Barry is at the top of a cliff, standing 80 m above sea level, and observes
two yachts in the sea.

“Seaview” (S) is at an angle of depression of 25°.

“Nauti Buoy” (N) is at an angle of depression of 35°.

The following three-dimensional diagram shows Barry and the two

yachts at S and N. 𝑋 lies at the foot of the cliff and angle SXN = 70°. Find, to 3 significant figures, the
distance between the two yachts.

Page 94
Question 9 C

The diagram below shows a semi-circle of diameter 20 cm, centre O and two points A and B such that
𝐴𝑂̂𝐵 = 𝜃, where 𝜃 is in radians.

a. Show that the shaded area can be expressed as 50𝜃 − 50 sin 𝜃.

b. Find the value of 𝜃 for which the shaded area is equal to half that of the unshaded area, giving
your answer correct to four significant figures.

Page 95
Question 10 C

A triangle ABC has 𝐴̂ = 50°, 𝐴𝐵 = 7 𝑐𝑚 and 𝐵𝐶 = 6 𝑐𝑚. Find the area of the triangle given that it
is smaller than 10 cm2.

Question 11 NC

The triangle ABC is equilateral of side 3 cm. The point D lies on [BC] such that 𝐵𝐷 = 1 𝑐𝑚. Find
cos 𝐷𝐴̂𝐶.

Page 96
Question 12 C

The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the
smaller circle lies on the circumference of the bigger circle. 𝑂 is the centre of the bigger circle and the
two circles intersect at points 𝐴 and 𝐵.

Find:

a. 𝐵𝑂̂𝐶
b. The area of the shaded region.

Page 97
Question 13 C

An electricity station is on the edge of a straight coastline. A

lighthouse is located in the sea 200 m from the electricity
station. The angle between the coastline and the line joining the
lighthouse with the electricity station is 60°. A cable needs to
be laid connecting the lighthouse to the electricity station. It is
decided to lay the cable in a straight line to the coast and then
along the coast to the electricity station. The length of cable laid
along the coastline is 𝑥 metres. This information is illustrated
in the diagram below.

The cost of laying the cable along the seabed is US $80 per
metre, and the cost of laying it on land is US $20 per metre.

a. Find, in terms of 𝑥, an expression for the cost of laying the cable.

b. Find the value of 𝑥, to the nearest metre, such that this cost is minimized.

Question 14 NC

From a vertex of an equilateral triangle of side 2𝑥, a circular arc is drawn

to divide the triangle into two regions, as shown in the diagram below.
Given that the area of the two regions are equal, find the radius of the arc
in terms of 𝑥.

Page 98
Question 15 C

The diagram below shows a circle with centre O and radius

8 cm. The points A, B, C, D, E and F are on the circle and
[AF] is a diameter. The length of arc ABC is 6 cm.

a. Find the size of angle AOC

b. Hence find the area of the shaded region
The area of sector OCDE is 45 cm2.

c. Find the size of angle COE.

d. Find EF.

Page 99
Question 16 C

The following diagram shows a circle with centre 𝑂 and radius 4 cm. The points A, B, and C lie on the
circle. The point D is outside the circle, on
(OC). Angle 𝐴𝐷𝐶 = 0.3 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 and
angle 𝐴𝑂𝐶 = 0.8 𝑟𝑎𝑑𝑖𝑎𝑛𝑠.

a. Find AD
b. Find OD
c. Find the area of sector OABC
d. Find the area of region ABCD

Page 100
Question 17 C

The following diagram shows two semi-circles. The larger

one has centre 𝑂 and radius 4 cm. The smaller one has centre
𝑃, radius 3 cm, and passes through O. The line (𝑂𝑃) meets
the larger semi-circle at 𝑆. The semi-circles intersect at 𝑄.

a. i. Explain why OPQ is an isosceles triangle.

1
ii. Use the cosine rule to show that cos 𝑂𝑃̂𝑄 = 9.
√80
iii. Hence show that sin 𝑂𝑃̂𝑄 = .
9

iv. Find the area of the triangle OPQ.

b. Consider the smaller semi-circle, with centre P.
i. Write down the size of 𝑂𝑃̂𝑄.
ii. Calculate the area of the sector OPQ.
c. Consider the larger semi-circle, with centre O. Calculate the area of the sector QOS.
d. Hence calculate the area of the shaded region.

Page 101
Question 18 C

The following diagram shows a circle centre 𝑂 , radius r. The angle

𝐴𝑂̂𝐵 at the centre of the circle is 𝜃 radians. The chord AB divides the
circle into a minor segment (the shaded region) and a major segment.

1
a. Show that the area of the minor segment is 2 𝑟 2 (𝜃 − sin 𝜃).

b. Find the area of the major segment.

c. Given that the ratio of the areas of the two segments is 2 ∶ 3,
4𝜋
show that sin 𝜃 = 𝜃 − 5
.

d. Hence find the value of 𝜃.

Page 102
Question 19 NC

a. Given that cos 75° = 𝑞, show that cos 105° = −𝑞.

In the following diagram, the points 𝐴 , 𝐵 , 𝐶 and 𝐷 are on the
circumference of a circle with centre O and radius 𝑟 . [𝐴𝐶 ] is a
diameter of the circle. 𝐵𝐶 = 𝑟 , 𝐴𝐷 = 𝐶𝐷 and 𝐴𝐵̂𝐶 = 𝐴𝐷
̂𝐶 =
90°.
b. Show that 𝐵𝐴̂𝐷 = 75°.
c. i. By considering triangle 𝐴𝐵𝐷, show that 𝐵𝐷 2 = 5𝑟 2 − 2𝑟 2 𝑞√6.
ii. By considering triangle 𝐶𝐵𝐷, find another expression for 𝐵𝐷 2 in terms of 𝑟 and 𝑞.
1
d. Use your answers to part c. to show that cos 75° = .
√ √2
6+

Page 103
 Trigonometric Functions

Question 1 C

A function is defined by 𝑓(𝑥) = 𝐴 sin 𝐵𝑥 + 𝐶, −𝜋 ≤ 𝑥 ≤

𝜋, where 𝐴, 𝐵, 𝐶 ∈ ℤ. The following diagram represents the
graph of 𝑦 = 𝑓(𝑥).
a. Find the value of 𝐴, 𝐵, and 𝐶.
b. Solve 𝑓(𝑥) = 3 for 0 ≤ 𝑥 ≤ 𝜋.

Question 2 NC

A Ferris wheel with diameter 110 metres rotates at a constant

speed. The lowest point on the wheel is 10 metres above the
ground, as shown on the following diagram. 𝑃 is a point on the
wheel. The wheel starts moving with 𝑃 at the lowest point and
completes one revolution in 20 minutes.

The height, ℎ metres, of 𝑃 above the ground after 𝑡 minutes is

given by ℎ(𝑡) = 𝑎 cos(𝑏𝑡) + 𝑐, where 𝑎, 𝑏, 𝑐 ∈ ℝ.

Find the values of 𝑎, 𝑏 and 𝑐.

Page 104
Question 3 NC

The following diagram represents a large Ferris wheel, with a diameter of 100 meters. Let P be a point
on the wheel. The wheel starts with P at the lowest point, at ground level. The wheel rotates at a constant
rate, in an anticlockwise direction. One revolution takes 20 minutes.

a. Write down the height of P above ground level after

i. 10 minutes;
ii. 15 minutes;

Let ℎ(𝑡) metres be the height of P above ground level after 𝑡 minutes. Some values of ℎ(𝑡) are given in
the table. 𝑡 ℎ(𝑡)
𝑓(𝑥)

b. i. Show that ℎ(8) = 90.5. 0 0.0

ii. Find ℎ(21). 1 2.4

c. Sketch the graph of ℎ, for 0 ≤ 𝑡 ≤ 40. 2 9.5

d. Given that ℎ can be expressed in the form ℎ(𝑡) = 𝑎 cos 𝑏𝑡 + 𝑐, find 3 20.6
𝑎, 𝑏 and c. 4 34.5
5 50.0

Page 105
Question 4 NC

Consider 𝑔(𝑥) = 3sin 2𝑥.

a. Write down the period of 𝑔.

b. On the diagram below, sketch the curve of 𝑔, for 0 ≤ 𝑥 ≤ 2𝜋.
c. Write down the number of solutions to the equation 𝑔(𝑥) = 2, for 0 ≤ 𝑥 ≤ 2𝜋

Question 5 C

Let 𝑓(𝑥) = 6 sin 𝜋𝑥, and 𝑔(𝑥) = 6𝑒 −𝑥 − 3, for 0 ≤ 𝑥 ≤ 2. The graph of 𝑓 is shown on the diagram
below. There is a maximum value at B (0.5, 𝑏).

a. Write down the value of 𝑏.

b. On the same diagram, sketch the graph of 𝑔.
c. Solve 𝑓(𝑥) = 𝑔(𝑥), 0.5 ≤ 𝑥 ≤ 1.5.

Page 106
Question 6 C

a. Consider the equation 4𝑥 2 + 𝑘𝑥 + 1 = 0. For what values of k does this equation have two
equal roots?
Let 𝑓 be the function 𝑓(𝜃) = 2cos 2𝜃 + 4 cos 𝜃 + 3, for −360° ≤ 𝜃 ≤ 360°.

b. Show that this function may be written as 𝑓(𝜃) = 4cos2 𝜃 + 4 cos 𝜃 + 1.

c. Consider the equation 𝑓(𝜃) = 0, for −360° ≤ 𝜃 ≤ 360°.
i. How many distinct values of cos 𝜃 satisfy this equation?
ii. Find all values of 𝜃 which satisfy this equation.
d. Given that 𝑓(𝜃) = 𝑐 is satisfied by only three values of 𝜃, find the value of 𝑐.

Page 107
Question 7 C

The height of water, in metres, in Dungeness harbour is modelled by the function

𝐻(𝑡) = 𝑎 sin(𝑏(𝑡 − 𝑐)) + 𝑑 , where 𝑡 is the number of hours after midnight, and 𝑎, 𝑏, 𝑐 and 𝑑 are
constants, where 𝑎 > 0, 𝑏 > 0 and 𝑐 > 0.

The graph shows the height of the water for 13 hours, starting
at midnight.

The first high tide occurs at 04:30 and the next high tide occurs
12 hours later. Throughout the day, the height of the water
fluctuates between 2.2𝑚 and 6.8𝑚.

All heights are given correct to one decimal place.

𝜋
a. Show that 𝑏 = .
6

b. Find the value of 𝑎.

c. Find the value of 𝑑.
d. Find the smallest possible value of 𝑐.
e. Find the height of the water at 12:00.
f. Determine the number of hours, over a 24-hour period, for which the tide is higher than 5
metres.
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal
pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur 50 minutes
earlier than at Dungeness.

g. Find a suitable equation that may be used to model the tidal height of water at Folkestone
harbour.

Page 108
Question 8 NC (calculator only for part c)

The following graph shows the depth of water, 𝑦 meters, at a point P, during one day. The time 𝑡 is given
in hours, from midnight to noon.

a. Use the graph to write down an estimate of the value of 𝑡 when

i. the depth of water is minimum;
ii. the depth of water is maximum;
iii. the depth of water is increasing most rapidly.
b. The depth of water can be modeled by the function 𝑦 = A cos(𝐵(𝑡 − 1)) + 𝐶.
i. Show that 𝐴 = 8 .
ii. Write down the value of 𝐶.
iii. Find the value of 𝐵.
c. A sailor knows that he cannot sail past P when the depth of the water is less than 12 m.
Calculate the values of 𝑡 between which he cannot sail past P.

23a. proof

Page 109
Question 9 C

A formula for the depth 𝑑 meters of water in a harbour at a time 𝑡 hours after midnight is
𝜋
𝑑 = 𝑃 + 𝑄 cos ( 𝑡) , 0 ≤ 𝑡 ≤ 24,
6
, where 𝑃 and 𝑄 are positive constants. In the following graph the point (6, 8.2) is a minimum point
and the point (12, 14.6) is a maximum point.

a. Find the value of

i. 𝑄;
ii. 𝑃.
b. Find the first time the 24-hour period when the depth of the water is 10 meters.
c. i. Use the symmetry of the graph to find the next time when the depth of the water is 10
meters.
ii. Hence find the time intervals in the 24-hours period during which the water is less than
10 meters deep.

23a. proof
1 3
23b. 𝑓(𝑥) ∈ [− , ]
4 4

23c?

Page 110
 Further Trigonometry

Question 1 C

𝜋
In triangle ABC, BC = √3 cm, 𝐴𝐵̂𝐶 = 𝜃 𝑎𝑛𝑑 𝐵𝐶̂ 𝐴 = 3 .

3
a. Show that length AB = .
√3 cos 𝜃+sin 𝜃

b. Given that AB has a minimum value, determine the value of 𝜃 for which this occurs.

Question 2 NC

𝜋 3
Given that 2 < 𝛼 < 𝜋 and cos 𝛼 = − 4, find the value of sin 2α.

Page 111
Question 3 C

Solve the equation 3 cos 𝑥 = 5 sin 𝑥, for 𝑥 in the interval 0° ≤ 𝑥 ≤ 360°, by making tan 𝑥 the subject,
giving your answers to the nearest degree.

Question 4 NC

Solve the equation sin 2𝑥 − cos 2𝑥 = 1 + sin 𝑥 − cos 𝑥 for 𝑥 ∈ [−𝜋, 𝜋].

Page 112
Question 5 NC

Consider the equation 3 cos 2𝑥 + sin 𝑥 = 1.

a. Write this equation in the form 𝑓(𝑥) = 0, where 𝑓(𝑥) = 𝑝sin2 𝑥 + 𝑞 sin 𝑥 + 𝑟, and 𝑝, 𝑞, 𝑟 ∈
ℤ.
b. Factorize 𝑓(𝑥).
c. Write down the number of solutions of 𝑓(𝑥) = 0, for 0 ≤ 𝑥 ≤ 2π.

Question 6 NC

Find all solutions to the equation tan 𝑥 + tan 2𝑥 = 0 where 0° ≤ 𝑥 < 360° .

Page 113
Question 7 NC

2 tan 𝜃 𝜋
Use the double angle identity tan 2𝜃 = 1−tan2 𝜃 to show that tan 8 = √2 − 1.

Question 8 NC

1 1 1
a. Given that arctan (5) + arctan (8) = arctan (𝑝), where 𝑝 ∈ ℤ+ , find 𝑝.
1 1 1
b. Hence find the value of arctan (2) + arctan (5) + arctan (8).

Page 114
Question 9 NC

2
Given that sin 𝑥 + cos 𝑥 = 3, find cos 4𝑥.

Question 10 NC

6 2𝑥 2 −5𝑥−3
a. Show that 2𝑥 − 3 − 𝑥−1 = 𝑥−1
, 𝑥 ∈ ℝ, 𝑥 ≠ 1.
6 𝜋
b. Hence or otherwise, solve the equation 2 sin 2𝜃 − 3 − sin 2𝜃−1 = 0 for 0 ≤ 𝜃 ≤ 𝜋, 𝜃 ≠ 4
.

Page 115
Question 11 NC

3 𝜋 3𝜋
It is given that cosec 𝜃 = 2 , where 2 < 𝜃 < 2
. Find the exact value of cot 𝜃.

Question 12 NC

cos A+sin A
Show that cos A−sin A = sec 2A + tan 2A.

Page 116
Question 13 C

a. Solve the equation 3cos2 𝑥 − 8 cos 𝑥 + 4 = 0 , where 0 ≤ 𝑥 ≤ 180° , expressing your

answer(s) to the nearest degree.
b. Find the exact values of sec 𝑥 satisfying the equation 3sec 4 𝑥 − 8sec 2 𝑥 + 4 = 0.

Question 14 NC

3
The straight line with equation 𝑦 = 𝑥 makes an acute angle 𝜃 with the 𝑥-axis.
4

a. Write down the value of tan 𝜃.

b. Find the value of
i. sin 2𝜃
ii. cos 2𝜃.

Page 117
Question 15 NC

a. Show that 4 − cos 2𝜃 + 5 sin 𝜃 = 2sin2 𝜃 + 5 sin 𝜃 + 3.

b. Hence, solve the equation 4 − cos 2𝜃 + 5 sin 𝜃 = 0 for 0 ≤ 𝜃 ≤ 2𝜋.

Question 16 NC

a. Show that log 9 (cos 2𝑥 + 2) = log 3 √cos 2𝑥 + 2.

𝜋
b. Hence or otherwise solve log 3 (2 sin 𝑥) = log 9 (cos 2𝑥 + 2) for 0 < 𝑥 < 2
.

Page 118
Question 17 NC

𝜋 𝜋
The first three terms of a geometric sequence are sin 𝑥, sin 2𝑥 and 4 sin 𝑥 cos 2 𝑥, − 2 < 𝑥 < 2 .

a. Find the common ratio 𝑟.

b. Find the set of values of 𝑥 for which the geometric series sin 𝑥 + sin 2𝑥 + 4 sin 𝑥 cos 2 𝑥 + ⋯
converges.

1
Consider 𝑥 = arccos (4), 𝑥 > 0.

√15
c. Show that the sum to infinity of this series is 2
.

Page 119
Question 18 NC

1−tan2 𝜃
a. Show that cot 2𝜃 = 2 tan 𝜃
.

b. Verify that 𝑥 = tan 𝜃 and 𝑥 = − cot 𝜃 satisfy the equation 𝑥 2 + (2 cot 2𝜃)𝑥 − 1 = 0.
𝜋
c. Hence, or otherwise, show that the exact value of tan 12 = 2 − √3.
𝜋 𝜋
d. Using the results from parts b. and c. find the exact value of tan 24 − cot 24.

Give your answer in the form 𝑎 + 𝑏√3, where 𝑎, 𝑏 ∈ ℤ.

Page 120