lOMoARcPSD|31096826

Mathematics 2U HSC Questions

SYLLABUS CONTENT

• Basic arithmetic and algebra ----------------------------------------- Page 2

• Real functions ----------------------------------------------------------- Page 5

• Trigonometric ratios --------------------------------------------------- Page 9

• Linear functions --------------------------------------------------------- Page 14

• The quadratic polynomial and the parabola ---------------------- Page 20

• Plane geometry --------------------------------------------------------- Page 24

• Tangent to a curve and derivative of a function ------------------ Page 29

• Geometrical applications of differentiation ----------------------- Page 34

• Integration --------------------------------------------------------------- Page 40

• Trigonometric functions ---------------------------------------------- Page 54

• Logarithmic and exponential functions ---------------------------- Page 62

• Applications of calculus to the physical world -------------------- Page 66

• Probability --------------------------------------------------------------- Page 74

• Series and series applications ---------------------------------------- Page 79

1|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

BASIC ARITHMETIC AND ALGEBRA

Question 1 (2015 HSC Q1)
What is 0.005 233 59 written in scientific notation, correct to 4 significant figures?

A. 5.2336 × 10−2

B. 5.234 × 10−2

C. 5.2336 × 10−3

D. 5.234 × 10−3

Question 2 (2015 HSC Q11a) – 1 mark

Simplify 4𝑥 − (8 − 6𝑥).

Question 3 (2015 HSC Q11b) – 2 marks

Factorise fully 3𝑥 2 − 27.

Question 4 (2015 HSC Q11c) – 2 marks

8
Express 2+ with a rational denominator.
√7

Question 5 (2014 HSC Q1)

𝜋2
What is the value of , correct to 3 significant figures?
6

A. 1.64

B. 1.65

C. 1.644

D. 1.645

2|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 6 (2014 HSC Q6)

Which expression is a factorisation of 8𝑥 3 + 27?

A. (2𝑥 − 3)(4𝑥 2 + 12𝑥 − 9)

B. (2𝑥 + 3)(4𝑥 2 − 12𝑥 + 9)

C. (2𝑥 − 3)(4𝑥 2 + 6𝑥 − 9)

D. (2𝑥 + 3)(4𝑥 2 − 6𝑥 + 9)

Question 7 (2014 HSC Q11a) – 2 marks

1
Rationalise the denominator of .
√5−2

Question 8 (2014 HSC Q11b) – 2 marks

Factorise 3𝑥 2 + 𝑥 − 2.

Question 9 (2013 HSC Q11a) – 1 mark

Evalute ln 3 correct to three significant figures.

Question 10 (2012 HSC Q1)

What is 4.097 84 correct to three significant figures?
A. 4.09

B. 4.10

C. 4.097

D. 4.098

3|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 11 (2012 HSC Q2)

1
Which of the following is equal to 2√5− 3?
√

2√5−√3
A.
7

2√5+√3
B. 7

2√5−√3
C. 17

2√5+√3
D. 17

Question 12 (2012 HSC Q11a) – 2 marks

Factorise 2𝑥 2 − 7𝑥 + 3.

Question 13 (2011 HSC Q1a) – 2 marks

3 651
Evaluate √ correct to four significant figures.
4𝜋

Question 14 (2011 HSC Q1b) – 1 mark

𝑛2 −25
Simplify 𝑛−5
.

Question 15 (2011 HSC Q1c) – 2 marks

Solve 22𝑥+1 = 32.

Question 16 (2011 HSC Q1f) – 2 marks

4
Rationalise the denominator of .
√5−√3

Give your answer in the simplest form.

Question 17 (2010 HSC Q1a) – 2 marks

Solve 𝑥 2 = 4𝑥.

Question 18 (2010 HSC Q1b) – 2 marks

1
Find integers 𝑎 and 𝑏 such that = 𝑎 + 𝑏√5.
√5−2

4|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

REAL FUNCTIONS
Question 1 (2016 HSC Q4)
Which diagram shows the graph of an odd function?

Question 2 (2016 HSC Q11a) – 2 marks

Sketch the graph of (𝑥 − 3)2 + (𝑦 + 2)2 = 4

Question 3 (2016 HSC Q11c) – 2 marks

Solve |𝑥 − 2| ≤ 3.

Question 4 (2015 HSC Q13b)

i) Find the domain and range for the function 𝑓(𝑥) = √9 − 𝑥 2 . (2 marks)

ii) On a number plane, shade the region where the points (𝑥, 𝑦) satisfy both of the
inequalities 𝑦 ≤ √9 − 𝑥 2 and 𝑦 ≥ 𝑥. (2 marks)

5|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 5 (2013 HSC Q3)

1
Which inequality defines the domain of the function 𝑓(𝑥) = ?
√𝑥+3

A. 𝑥 > −3

B. 𝑥 ≥ −3

C. 𝑥 < −3

D. 𝑥 ≤ −3

Question 6 (2013 HSC Q11b) – 2 marks

𝑥 3 −8
Evaluate lim 𝑥 2 −4.
𝑥→2

Question 7 (2013 HSC Q11g) – 3 marks

Sketch the region defined by (𝑥 − 2)2 + (𝑦 − 3)2 ≥ 4.

Question 8 (2013 HSC Q15c)

i) Sketch the graph 𝑦 = |2𝑥 − 3|. (1 mark)

ii) Using the graph from part (i), or otherwise, find all values of 𝑚 for which the
equation |2𝑥 − 3| = 𝑚𝑥 + 1 has exactly one solution. (2 marks)

6|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 9 (2012 HSC Q8)

The diagram shows the region enclosed by 𝑦 = 𝑥 − 2 and 𝑦 2 = 4 − 𝑥.

Which of the following pairs of inequalities describes the shaded region in the diagram?

A. 𝑦 2 ≤ 4 − 𝑥 and 𝑦 ≤ 𝑥 − 2

B. 𝑦 2 ≤ 4 − 𝑥 and 𝑦 ≥ 𝑥 − 2

C. 𝑦 2 ≥ 4 − 𝑥 and 𝑦 ≤ 𝑥 − 2

D. 𝑦 2 ≥ 4 − 𝑥 and 𝑦 ≥ 𝑥 − 2

Question 10 (2012 HSC Q11b) – 2 marks

Solve |3𝑥 − 1| < 2.

Question 11 (2011 HSC Q1e) – 2 marks

Solve 2 − 3𝑥 ≤ 8.

7|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 12 (2011 HSC Q4e) – 2 marks

The diagram shows the graphs 𝑦 = |𝑥| − 2 and 𝑦 = 4 − 𝑥 2 .

Write down inequalities that together describe the shaded region.

Question 13 (2010 HSC Q1c) – 1 mark

Write down the equation of the circle with centre (−1, 2) and radius 5.

Question 14 (2010 HSC Q1d) – 2 marks

Solve |2𝑥 + 3| = 9.

Question 15 (2010 HSC Q1g) – 1 mark

Let 𝑓(𝑥) = √𝑥 − 8. What is the domain of 𝑓(𝑥)?

Question 16 (2010 HSC Q2b) – 2 marks

Solve the inequality 𝑥 2 − 𝑥 − 12 < 0.

Question 17 (2010 HSC Q4d) – 2 marks

Let 𝑓(𝑥) = 1 + 𝑒 𝑥 .
Show that 𝑓(𝑥) × 𝑓(−𝑥) = 𝑓(𝑥) + 𝑓(−𝑥).

8|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

TRIGONOMETRIC RATIOS
Question 1 (2016 HSC Q1)
7 24
For the angle 𝜃, sin 𝜃 = and cos 𝜃 = − .
25 25

Which diagram best shows the angle 𝜃?

Question 2 (2016 HSC Q8)

How many solutions does the equation |cos(2𝑥)| = 1 have for 0 ≤ 𝑥 ≤ 2𝜋?
A. 1

B. 3

C. 4

D. 5

9|Page

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 3 (2016 HSC Q12c) – 3 marks

Square tiles of side length 20 cm are being used to tile a bathroom.
The tiler needs to drill a hole in one of the tiles at a point 𝑃 which is 8 cm from one corner and
15 cm from an adjacent corner.
To locate the point 𝑃, the tiler needs to know the size of the angle 𝜃 shown in the diagram.

Find the size of the angle 𝜃 to the nearest degree.

Question 4 (2015 HSC Q12a) – 2 marks

Find the solutions of 2 sin 𝜃 = 1 for 0 ≤ 𝜃 ≤ 2𝜋.

Question 5 (2015 HSC Q13a)

The diagram shows ∆𝐴𝐵𝐶 with sides 𝐴𝐵 = 6 cm, 𝐵𝐶 = 4 cm and 𝐴𝐶 = 8 cm.

7
i) Show that cos 𝐴 = 8. (1 mark)

ii) By finding the exact value of sin 𝐴, determine the exact value of the area of ∆𝐴𝐵𝐶.
(2 marks)

10 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 6 (2014 HSC Q13d)

Chris leaves island 𝐴 in a boat and sails 142 km on a bearing of 078° to island 𝐵. Chris then sails
on a bearing of 191° for 220 km to island 𝐶, as shown in the diagram.

i) Show that the distance from island 𝐶 to island 𝐴 is approximately 210 km. (2 marks)

ii) Chris wants to sail from island 𝐶 directly to island 𝐴. On what bearing should Chris
sail? Give your answer correct to the nearest degree. (3 marks)

Question 7 (2014 HSC Q15a) – 3 marks

Find all solutions of 2 sin2 𝑥 + cos 𝑥 − 2 = 0, where 0 ≤ 𝑥 ≤ 2𝜋.

11 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 8 (2013 HSC Q14c) – 3 marks

The right-angled triangle 𝐴𝐵𝐶 has hypotenuse 𝐴𝐵 = 13. The point 𝐷 is on 𝐴𝐶 such that 𝐷𝐶 = 4,
𝜋
∠𝐷𝐵𝐶 = and ∠𝐴𝐵𝐷 = 𝑥.
6

Using the sine rule, or otherwise, find the exact value of sin 𝑥.

Question 9 (2012 HSC Q6)

What are the solutions of √3 tan 𝑥 = −1 for 0 ≤ 𝑥 ≤ 2𝜋?

2𝜋 4𝜋
A. 3
and 3

2𝜋 5𝜋
B. and
3 3

5𝜋 7𝜋
C. 6
and 6

5𝜋 11𝜋
D. 6
and 6

Question 10 (2011 HSC Q2b) – 2 marks

Find the exact values of 𝑥 such that 2 sin 𝑥 = −√3, where 0 ≤ 𝑥 ≤ 2𝜋.

12 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 11 (2011 HSC Q8a)

In the diagram, the shop at 𝑆 is 20 kilometres across the bay from the post office at 𝑃. The
distance from the shop to the lighthouse at 𝐿 is 22 kilometres and ∠𝑆𝑃𝐿 is 60°.
Let the distance 𝑃𝐿 be 𝑥 kilometres.

i) Use the cosine rule to show that 𝑥 2 − 20𝑥 − 84 = 0. (1 mark)

ii) Hence, find the distance from the post office to the lighthouse. Give your answer
correct to the nearest kilometre. (2 marks)

13 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

LINEAR FUNCTIONS
Question 1 (2016 HSC Q11e) – 3 marks

Find the points of intersection of 𝑦 = −5 − 4𝑥 and 𝑦 = 3 − 2𝑥 − 𝑥 2 .

Question 2 (2016 HSC Q12a)

The diagram shows points 𝐴(1, 0), 𝐵(2, 4) and 𝐶(6, 1). The point 𝐷 lies on 𝐵𝐶 such that 𝐴𝐷 ⊥
𝐵𝐶.

i) Show that the equation of 𝐵𝐶 is 3𝑥 + 4𝑦 − 22 = 0. (2 marks)

ii) Find the length of 𝐴𝐷. (2 marks)

iii) Hence, or otherwise, find the area of ∆𝐴𝐵𝐶. (2 marks)

Question 3 (2015 HSC Q2)

What is the slope of the line with equation 2𝑥 − 4𝑦 + 3 = 0?
A. −2

1
B. −
2

1
C. 2

D. 2

14 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 4 (2015 HSC Q12b)

The diagram shows the rhombus 𝑂𝐴𝐵𝐶.
The diagonal from the point 𝐴(7, 11) to the point 𝐶 lies on the line ℓ1 .

The other diagonal, from the origin 𝑂 to the point 𝐵, lies on the line ℓ2 which has equation
𝑥
𝑦=− .
3

i) Show that the equation of the line ℓ1 is 𝑦 = 3𝑥 − 10. (2 marks)

ii) The lines ℓ1 and ℓ2 intersect at the point 𝐷.

Find the coordinates of 𝐷. (2 marks)

Question 5 (2014 HSC Q5)

Which equation represents the line perpendicular to 2𝑥 − 3𝑦 = 8, passing through the point
(2, 0)?

A. 3𝑥 + 2𝑦 = 4

B. 3𝑥 + 2𝑦 = 6

C. 3𝑥 − 2𝑦 = −4

D. 3𝑥 − 2𝑦 = 6

15 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 6 (2014 HSC Q12b)

The points 𝐴(0, 4), 𝐵(3, 0) and 𝐶(6, 1) form a triangle, as shown in the diagram.

i) Show that the equation of 𝐴𝐶 is 𝑥 + 2𝑦 − 8 = 0. (2 marks)

ii) Find the perpendicular distance from 𝐵 to 𝐴𝐶. (2 marks)

iii) Hence, or otherwise, find the area of Δ𝐴𝐵𝐶. (2 marks)

Question 7 (2013 HSC Q2)

The diagram shows the line ℓ.

What is the slope of the line ℓ?

A. √3

B. −√3

1
C.
√3

1
D. −
√3

16 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 8 (2013 HSC Q12b)

The points 𝐴(−2, −1), 𝐵(−2, 24), 𝐶(22, 42) and 𝐷(22, 17) form a parallelogram as shown. The
point 𝐸(18, 39) lies on 𝐵𝐶. The point 𝐹 is the midpoint of 𝐴𝐷.

i) Show that the equation of the line through 𝐴 and 𝐷 is 3𝑥 − 4𝑦 + 2 = 0. (2 marks)

ii) Show that the perpendicular distance from 𝐵 to the line through 𝐴 and 𝐷 is 20 units.
(1 mark)

iii) Find the length of 𝐸𝐶. (1 mark)

iv) Find the area of the trapezium 𝐸𝐹𝐷𝐶. (2 marks)

Question 9 (2012 HSC Q5)

What is the perpendicular distance of the point (2, −1) from the line 𝑦 = 3𝑥 + 1?
6
A.
√10

6
B.
√5

8
C.
√10

8
D.
√5

Question 10 (2012 HSC Q13a)

The diagram shows a triangle 𝐴𝐵𝐶. The line 2𝑥 + 𝑦 = 8 meets the 𝑥 and 𝑦 axes at the points 𝐴
and 𝐵 respectively. The point 𝐶 has coordinates (7, 4).

i) Calculate the distance 𝐴𝐵. (2 marks)

ii) It is known that 𝐴𝐶 = 5 and 𝐵𝐶 = √65. (Do NOT prove this.)

Calculate the size of ∠𝐴𝐵𝐶 to the nearest degree. (2 marks)

iii) The point 𝑁 lies on 𝐴𝐵 such that 𝐶𝑁 is perpendicular to 𝐴𝐵.

Find the coordinates of 𝑁. (3 marks)

17 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 11 (2011 HSC Q3c)

The diagram shows a line ℓ1 , with equation 3𝑥 + 4𝑦 − 12 = 0, which intersects the 𝑦-axis at 𝐵.
A second line ℓ2 , with equation 4𝑥 − 3𝑦 = 0, passes through the origin 𝑂 and intersects ℓ1 at 𝐸.

i) Show that the coordinates of 𝐵 are (0, 3). (1 mark)

ii) Show that ℓ1 is perpendicular to ℓ2 . (2 marks)

12
iii) Show that the perpendicular distance from 𝑂 to ℓ1 is 5
. (1 mark)

iv) Using Pythagoras’ theorem, or otherwise, find the length of the interval 𝐵𝐸. (1 mark)

v) Hence, or otherwise, find the area of Δ𝐵𝑂𝐸. (1 mark)

18 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 12 (2010 HSC Q3a)

In the diagram, 𝐴, 𝐵 and 𝐶 are the points (−2, −4), (12, 6) and (6, 8) respectively. The point
𝑁(2, 2) is the midpoint of 𝐴𝐶. The point 𝑀 is the midpoint of 𝐴𝐵.

i) Find the coordinates of 𝑀. (1 mark)

ii) Find the gradient of 𝐵𝐶. (1 mark)

iii) Prove that ∆𝐴𝐵𝐶 is similar to ∆𝐴𝑀𝑁. (2 marks)

iv) Find the equation of 𝑀𝑁. (2 marks)

v) Find the exact length of 𝐵𝐶. (1 mark)

vi) Given that the area of ∆𝐴𝐵𝐶 is 44 square units, find the perpendicular distance from
𝐴 to 𝐵𝐶. (1 mark)

19 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

THE QUADRATIC POLYNOMIAL AND THE PARABOLA

Question 1 (2016 HSC Q3)

Which diagram best shows the graph of the parabola 𝑦 = 3 − (𝑥 − 2)2 ?

Question 2 (2016 HSC Q13b)

Consider the parabola 𝑥 2 − 4𝑥 = 12𝑦 + 8.

i) By completing the square, or otherwise, find the focal length of the parabola.
(2 marks)

ii) Find the coordinates of the focus. (1 mark)

Question 3 (2015 HSC Q12d) – 2 marks

For what values of 𝑘 does the quadratic equation 𝑥 2 − 8𝑥 + 𝑘 = 0 have real roots?

20 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 4 (2015 HSC Q12e)

𝑥2 1
The diagram shows the parabola 𝑦 = 2
with focus 𝑆(0, 2). A tangent to the parabola is drawn at
1
𝑃(1, 2).

i) Find the equation of the tangent at the point 𝑃. (2 marks)

ii) What is the equation of the directrix of the parabola? (1 mark)

iii) The tangent and directrix intersect at 𝑄.

Show that 𝑄 lies on the 𝑦-axis. (1 mark)

iv) Show that ∆𝑃𝑄𝑆 is isosceles. (1 mark)

Question 5 (2014 HSC Q2)

Which graph best represents 𝑦 = (𝑥 − 1)2 ?

21 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 6 (2014 HSC Q14b)

The roots of the quadratic equation 2𝑥 2 + 8𝑥 + 𝑘 = 0 are 𝛼 and 𝛽.

i) Find the value of 𝛼 + 𝛽. (1 mark)

ii) Given that 𝛼 2 𝛽 + 𝛼𝛽 2 = 6, find the value of 𝑘. (2 marks)

Question 7 (2013 HSC Q1)

What are the solutions of 2𝑥 2 − 5𝑥 − 1 = 0?
−5±√17
A. 𝑥 =
4

5±√17
B. 𝑥 = 4

−5±√33
C. 𝑥 =
4

5±√33
D. 𝑥 = 4

Question 8 (2013 HSC Q7)

A parabola has focus (5, 0) and directrix 𝑥 = 1.
What is the equation of the parabola?

A. 𝑦 2 = 16(𝑥 − 5)

B. 𝑦 2 = 8(𝑥 − 3)

C. 𝑦 2 = −16(𝑥 − 5)

D. 𝑦 2 = −8(𝑥 − 3)

Question 9 (2012 HSC Q3)

The quadratic equation 𝑥 2 + 3𝑥 − 1 = 0 has roots 𝛼 and 𝛽.

What is the value of 𝛼𝛽 + (𝛼 + 𝛽)?

A. 4

B. 2

C. −4

D. −2

22 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 10 (2012 HSC Q11e) – 2 marks

Find the coordinates of the focus of the parabola 𝑥 2 = 16(𝑦 − 2).

Question 11 (2012 HSC Q16c)

The circle 𝑥 2 + (𝑦 − 𝑐)2 = 𝑟 2 , where 𝑐 > 0 and 𝑟 > 0, lies inside the parabola 𝑦 = 𝑥 2 . The circle
touches the parabola at exactly two points located symmetrically on opposite sides of the 𝑦-axis,
as shown in the diagram.

i) Show that 4𝑐 = 1 + 4𝑟 2. (2 marks)

1
ii) Deduce that 𝑐 > 2. (1 mark)

Question 12 (2011 HSC Q2a)

The quadratic equation 𝑥 2 − 6𝑥 + 2 = 0 has roots 𝛼 and 𝛽.

i) Find 𝛼 + 𝛽. (1 mark)

ii) Find 𝛼𝛽. (1 mark)

1 1
iii) Find 𝛼 + . (1 mark)
𝛽

Question 13 (2011 HSC Q3b) – 2 marks

A parabola has focus (3, 2) and directrix 𝑦 = −4. Find the coordinates of the vertex.

Question 14 (2011 HSC Q6b) – 3 marks

A point 𝑃(𝑥, 𝑦) moves so that the sum of the squares of its distance from each of the points
𝐴(−1, 0) and 𝐵(3, 0) is equal to 40.
Show that the locus of 𝑃(𝑥, 𝑦) is a circle, and state its radius and centre.

23 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

PLANE GEOMETRY
Question 1 (2016 HSC Q12b)
The diagram shows a semicircle with centre 𝑂. It is given that 𝐴𝐵 = 𝑂𝐵, ∠𝐶𝑂𝐷 = 87° and
∠𝐵𝐴𝑂 = 𝑥°.

i) Show that ∠𝐶𝐵𝑂 = 2𝑥°, giving reasons. (1 mark)

ii) Find the value of 𝑥, giving reasons. (2 marks)

Question 2 (2016 HSC Q15c)

Maryam wishes to estimate the height, ℎ metres, of a tower, 𝑆𝑇, using a square, 𝐴𝐵𝐶𝐷, with side
length 1 metre.
She places the point 𝐴 on the horizontal ground and ensures that the point 𝐷 lies on the line
joining 𝐴 to the top of the tower 𝑇. The point 𝐹 is the intersection of the line joining 𝐵 and 𝑇 and
the side 𝐶𝐷. The point 𝐸 is the foot of the perpendicular from 𝐵 to the ground. Let 𝐶𝐹 has length
𝑥 metres and 𝐴𝐸 have length 𝑦 metres.

Copy or trace the diagram into your writing booklet.

i) Show that ∆𝐹𝐶𝐵 and ∆𝐵𝐴𝑇 are similar. (2 marks)

ii) Show that ∆𝑇𝑆𝐴 and ∆𝐴𝐸𝐵 are similar. (2 marks)

iii) Find ℎ in terms of 𝑥 and 𝑦. (2 marks)

24 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 3 (2015 HSC Q15b)

The diagram shows ∆𝐴𝐵𝐶 which has a right angle at 𝐶. The point 𝐷 is the midpoint of the side
𝐴𝐶. The point 𝐸 is chosen on 𝐴𝐵 such that 𝐴𝐸 = 𝐸𝐷. The line segment 𝐸𝐷 is produced to meet
the line 𝐵𝐶 at 𝐹.

Copy or trace the diagram into your writing booklet.

i) Prove that ∆𝐴𝐶𝐵 is similar to ∆𝐷𝐶𝐹. (2 marks)

ii) Explain why ∆𝐸𝐹𝐵 is isosceles. (1 mark)

iii) Show that 𝐸𝐵 = 3𝐴𝐸. (2 marks)

Question 4 (2014 HSC Q15b)

In Δ𝐷𝐸𝐹, a point 𝑆 is chosen on the side 𝐷𝐸. The length of 𝐷𝑆 is 𝑥, and the length of 𝐸𝑆 is 𝑦. The
line through 𝑆 parallel to 𝐷𝐹 meets 𝐸𝐹 at 𝑄. The line through 𝑆 parallel to 𝐸𝐹 meets 𝐷𝐹 at 𝑅.

The area of Δ𝐷𝐸𝐹 is 𝐴. The areas of Δ𝐷𝑆𝑅 and Δ𝑆𝐸𝑄 are 𝐴1 and 𝐴2 respectively.
i) Show that Δ𝐷𝐸𝐹 is similar to Δ𝐷𝑆𝑅. (2 marks)

𝐷𝑅 𝑥
ii) Explain why 𝐷𝐹 = . (1 mark)
𝑥+𝑦

25 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

𝐴1 𝑥
iii) Show that √ = . (2 marks)
𝐴 𝑥+𝑦

𝐴
iv) Using the result from part (iii) and a similar expression for √ 𝐴2 , deduce that √𝐴 =

√𝐴1 + √𝐴2 . (2 marks)

Question 5 (2013 HSC Q16c)

The diagram shows triangles 𝐴𝐵𝐶 and 𝐴𝐵𝐷 with 𝐴𝐷 parallel to 𝐵𝐶. The sides 𝐴𝐶 and 𝐵𝐷
intersect at 𝑌. The point 𝑋 lies on 𝐴𝐵 such that 𝑋𝑌 is parallel to 𝐴𝐷 and 𝐵𝐶.

i) Prove that Δ𝐴𝐵𝐶 is similar to Δ𝐴𝑋𝑌. (2 marks)

1 1 1
ii) Hence, or otherwise, prove that
𝑋𝑌
= + . (2 marks)
𝐴𝐷 𝐵𝐶

Question 6 (2012 HSC Q16a)

The diagram shows a triangle 𝐴𝐵𝐶 with sides 𝐵𝐶 = 𝑎 and 𝐴𝐶 = 𝑏. The points 𝐷, 𝐸 and 𝐹 lie on
the sides 𝐴𝐶, 𝐴𝐵 and 𝐵𝐶, respectively, so that 𝐶𝐷𝐸𝐹 is a rhombus with sides of length 𝑥.

i) Prove that Δ𝐸𝐵𝐹 is similar to Δ𝐴𝐸𝐷. (2 marks)

ii) Find an expression for 𝑥 in terms of 𝑎 and 𝑏.

26 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 7 (2011 HSC Q6a)

The diagram shows a regular pentagon 𝐴𝐵𝐶𝐷𝐸. Sides 𝐸𝐷 and 𝐵𝐶 are produced to meet at 𝑃.

Copy or trace the diagram into your writing booklet.

i) Find the size of ∠𝐶𝐷𝐸. (1 mark)

ii) Hence, show that Δ𝐸𝑃𝐶 is isosceles. (2 marks)

Question 8 (2011 HSC Q9a)

The diagram shows ∆𝐴𝐷𝐸, where 𝐵 is the midpoint of 𝐴𝐷 and 𝐶 is the midpoint of 𝐴𝐸. The
intervals 𝐵𝐸 and 𝐶𝐷 meet at 𝐹.

i) Explain why ∆𝐴𝐵𝐶 is similar to ∆𝐴𝐷𝐸. (1 mark)

ii) Hence, or otherwise, prove that the ratio 𝐵𝐹: 𝐹𝐸 = 1: 2. (2 marks)

27 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 9 (2010 HSC Q10a)

In the diagram 𝐴𝐵𝐶 is an isosceles triangle with 𝐴𝐶 = 𝐵𝐶 = 𝑥. The point 𝐷 on the interval 𝐴𝐵 is
chosen so that 𝐴𝐷 = 𝐶𝐷. Let 𝐴𝐷 = 𝑎, 𝐷𝐵 = 𝑦 and ∠𝐴𝐷𝐶 = 𝜃.

i) Show that ∆𝐴𝐵𝐶 is similar to ∆𝐴𝐶𝐷. (2 marks)

ii) Show that 𝑥 2 = 𝑎2 + 𝑎𝑦. (1 mark)

iii) Show that 𝑦 = 𝑎(1 − 2 cos 𝜃). (2 marks)

iv) Deduce that 𝑦 ≤ 3𝑎. (1 mark)

28 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

TANGENT TO A CURVE AND DERIVATIVE OF A FUNCTION

Question 1 (2016 HSC Q11b) – 2 marks
𝑥+2
Differentiate .
3𝑥−4

Question 2 (2014 HSC Q11c) – 2 marks

𝑥3
Differentiate .
𝑥+1

Question 3 (2014 HSC Q11f) – 2 marks

The gradient function of a curve 𝑦 = 𝑓(𝑥) is given by 𝑓 ′ (𝑥) = 4𝑥 − 5. The curve passes through
the point (2, 3).
Find the equation of the curve.

Question 4 (2014 HSC Q14e) – 3 marks

The diagram shows the graph of a function 𝑓(𝑥).
The graph has a horizontal turning point of inflexion at 𝐴, a point of inflexion at 𝐵 and a
maximum turning point at 𝐶.

Sketch the graph of the derivative 𝑓 ′ (𝑥).

29 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 5 (2013 HSC Q8)

The diagram shows the points 𝐴, 𝐵, 𝐶 and 𝐷 on the graph 𝑦 = 𝑓(𝑥).

At which point is 𝑓 ′ (𝑥) > 0 and 𝑓 ′′ (𝑥) = 0?

A. 𝐴

B. 𝐵

C. 𝐶

D. 𝐷

Question 6 (2013 HSC Q16a) – 3 marks

The derivative of a function 𝑓(𝑥) is 𝑓’(𝑥) = 4𝑥 − 3. The line 𝑦 = 5𝑥 − 7 is tangent to the graph
of 𝑓(𝑥).
Find the function 𝑓(𝑥).

30 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 7 (2012 HSC Q4)

The diagram shows the graph of 𝑦 = 𝑓(𝑥).

Which of the following statements is true?

A. 𝑓’(𝑎) > 0 and 𝑓’’(𝑎) < 0

B. 𝑓’(𝑎) > 0 and 𝑓’’(𝑎) > 0

C. 𝑓’(𝑎) < 0 and 𝑓’’(𝑎) < 0

D. 𝑓’(𝑎) < 0 and 𝑓’’(𝑎) > 0

Question 8 (2012 HSC Q11c) – 2 marks

Find the equation of the tangent to the curve 𝑦 = 𝑥 2 at the point where 𝑥 = 3.

Question 9 (2012 HSC Q11d) – 2 marks

Differentiate (3 + 𝑒 2𝑥 )5.

Question 10 (2011 HSC Q2c) – 3 marks

Find the equation of the tangent to the curve 𝑦 = (2𝑥 + 1)4 at the point where 𝑥 = −1.

Question 11 (2011 HSC Q4c) – 2 marks

𝑑𝑦
The gradient of a curve is given by 𝑑𝑥 = 6𝑥 − 2. The curve passes through the point (−1, 4).

What is the equation of the curve?

31 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 12 (2011 HSC Q9c) – 3 marks

The graph 𝑦 = 𝑓(𝑥) in the diagram has a stationary point when 𝑥 = 1, a point of inflexion when
𝑥 = 3 and a horizontal asymptote at 𝑦 = −2.

Sketch the graph 𝑦 = 𝑓’(𝑥), clearly indicating its features at 𝑥 = 1 and at 𝑥 = 3, and the shape of
the graph as 𝑥 → ∞.

Question 13 (2010 HSC Q7b)

The parabola shown in the diagram is the graph 𝑦 = 𝑥 2 . The points 𝐴(−1, 1) and 𝐵(2, 4) are on
the parabola.

i) Find the equation of the tangent to the parabola at 𝐴. (2 marks)

ii) Let 𝑀 be the midpoint of 𝐴𝐵.

There is a point 𝐶 on the parabola such that the tangent at 𝐶 is parallel to 𝐴𝐵.

Show that the line 𝑀𝐶 is vertical. (2 marks)

iii) The tangent at 𝐴 meets the line 𝑀𝐶 at 𝑇.

Show that the line 𝐵𝑇 is a tangent to the parabola. (2 marks)

32 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 14 (2010 HSC Q8d) – 2 marks

Let 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 𝑘𝑥 + 8, where 𝑘 is a constant.

Find the values of 𝑘 for which 𝑓(𝑥) is an increasing function.

Question 15 (2010 HSC Q9b)

Let 𝑦 = 𝑓(𝑥) be a function defined for 0 ≤ 𝑥 ≤ 6, with 𝑓(0) = 0.
The diagram shows the graph of the derivative of 𝑓, 𝑦 = 𝑓′(𝑥).

The shaded region 𝐴1 has area 4 square units. The shaded region 𝐴2 has area 4 square units.
i) For what values of 𝑥 is 𝑓(𝑥) increasing? (1 mark)

ii) What is the maximum value of 𝑓(𝑥)? (1 mark)

iii) Find the value of 𝑓(6). (1 mark)

iv) Draw a graph of 𝑦 = 𝑓(𝑥) for 0 ≤ 𝑥 ≤ 6. (2 marks)

33 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

GEOMETRICAL APPLICATIONS OF DIFFERENTIATION

Question 1 (2016 HSC Q13a)

Consider the function 𝑦 = 4𝑥 3 − 𝑥 4 .

i) Find the two stationary points and determine their nature. (4 marks)

ii) Sketch the graph of the function, clearly showing the stationary points and the 𝑥 and
𝑦 intercepts. (2 marks)

Question 2 (2016 HSC Q14c)

A farmer wishes to make a rectangular enclosure of area 720 m2. She uses an existing straight
boundary as one side of the enclosure. She uses wire fencing for the remaining three sides and
also to divide the enclosure into four equal rectangular areas of width 𝑥 m as shown.

i) Show that the total length, ℓ m, of the wire fencing is given by

(1 mark)
720
ℓ = 5𝑥 +
𝑥

ii) Find the minimum length of wire fencing required, showing why this is the
minimum length. (3 marks)

Question 3 (2015 HSC Q13c)

Consider the curve 𝑦 = 𝑥 3 − 𝑥 2 − 𝑥 + 3.

i) Find the stationary points and determine their nature. (4 marks)

1 70
ii) Given that the point 𝑃(3 , 27) lies on the curve, prove that there is a point of inflexion
at 𝑃. (2 marks)

iii) Sketch the curve, labelling the stationary points, point of inflexion and 𝑦-intercept.
(2 marks)

34 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 4 (2015 HSC Q16c)

The diagram shows a cylinder of radius 𝑥 and height 𝑦 inscribed in a cone of radius 𝑅 and height
𝐻, where 𝑅 and 𝐻 are constants.

1
The volume of a cone of radius 𝑟 and height ℎ is 3 𝜋𝑟 2 ℎ.

The volume of a cylinder of radius 𝑟 and height ℎ is 𝜋𝑟 2 ℎ.

𝐻
i) Show that the volume, 𝑉, of the cylinder can be written as 𝑉 = 𝑅
𝜋𝑥 2 (𝑅 − 𝑥).
(3 marks)

ii) By considering the inscribed cylinder of maximum volume, show that the volume of
4
any inscribed cylinder does not exceed 9 of the volume of the cone. (4 marks)

Question 5 (2014 HSC Q14c)

The diagram shows a window consisting of two sections. The top section is a semicircle of
diameter 𝑥 m. The bottom section is a rectangle of width 𝑥 m and height 𝑦 m.
The entire frame of the window, including the piece that separates the two sections, is made
using 10 m of thin metal.

The semicircular section is made of coloured glass and the rectangular section is made of clear
glass.
Under test conditions, the amount of light coming through one square metre of the coloured
glass is 1 unit and the amount of light coming through one square metre of the clear glass is 3
units.
The total amount of light coming through the window under test conditions is 𝐿 units.

35 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

𝜋
i) Show that 𝑦 = 5 − 𝑥(1 + 4 ). (2 marks)

5𝜋
ii) Show that 𝐿 = 15𝑥 − 𝑥 2 (3 + 8
). (2 marks)

iii) Find the values of 𝑥 and 𝑦 that maximise the amount of light coming through the
window under test conditions. (3 marks)

Question 6 (2013 HSC Q12a) – 2 marks

The cubic 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 has a point of inflexion at 𝑥 = 𝑝.

𝑏
Show that 𝑝 = − 3𝑎.

Question 7 (2013 HSC Q14b)

Two straight roads meet at 𝑅 at an angle of 60°. At time 𝑡 = 0 car 𝐴 leaves 𝑅 on one road, and
car 𝐵 is 100 km from 𝑅 on the other road. Car 𝐴 travels away from 𝑅 at a speed of 80 km/h, and
car 𝐵 travels towards 𝑅 at a speed of 50 km/h.

The distance between the cars at time t hours is r km.

i) Show that 𝑟 2 = 12 900𝑡 2 − 18 000𝑡 + 10 000. (2 marks)

ii) Find the minimum distance between the cars. (3 marks)

36 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 8 (2012 HSC Q14a)

A function is given by 𝑓(𝑥) = 3𝑥 4 + 4𝑥 3 − 12𝑥 2.

i) Find the coordinates of the stationary points of 𝑓(𝑥) and determine their nature.
(3 marks)

ii) Hence, sketch the graph 𝑦 = 𝑓(𝑥) showing the stationary points. (2 marks)

iii) For what values of 𝑥 is the function increasing? (1 mark)

iv) For what values of 𝑘 will 3𝑥 4 + 4𝑥 3 − 12𝑥 2 + 𝑘 = 0 have no solution? (1 mark)

Question 9 (2012 HSC Q16b)

The diagram shows a point 𝑇 on the unit circle 𝑥 2 + 𝑦 2 = 1 at angle 𝜃 from the positive 𝑥-axis,
𝜋
where 0 < 𝜃 < 2 .

The tangent to the circle at 𝑇 is perpendicular to 𝑂𝑇, and intersects the 𝑥-axis at 𝑃, and the line
𝑦 = 1 at 𝑄. The line 𝑦 = 1 intersects the 𝑦-axis at 𝐵.

i) Show that the equation of the line 𝑃𝑇 is

(2 marks)
𝑥 cos 𝜃 + 𝑦 sin 𝜃 = 1

ii) Find the length of 𝐵𝑄 in terms of 𝜃. (1 mark)

iii) Show that the area, 𝐴, of the trapezium 𝑂𝑃𝑄𝐵 is given by

(2 marks)
2 − sin 𝜃
𝐴=
2 cos 𝜃

iv) Find the angle 𝜃 that gives the minimum area of the trapezium. (3 marks)

37 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 10 (2011 HSC Q7a)

Let 𝑓(𝑥) = 𝑥 3 − 3𝑥 + 2.
i) Find the coordinates of the stationary points of 𝑦 = 𝑓(𝑥), and determine their
nature. (3 marks)

ii) Hence, sketch the graph 𝑦 = 𝑓(𝑥) showing all stationary points and the 𝑦-intercept.
(2 marks)

Question 11 (2011 HSC Q16b)

A farmer is fencing a paddock using 𝑃 metres of fencing. The paddock is to be in the shape of a
sector of a circle with radius 𝑟 and sector angle 𝜃 in radians, as shown in the diagram.

i) Show that the length of fencing required to fence the perimeter of the paddock is
𝑃 = 𝑟(𝜃 + 2). (1 mark)

1
ii) Show that the area of the sector is 𝐴 = 2 𝑃𝑟 − 𝑟 2 . (1 mark)

iii) Find the radius of the sector, in terms of 𝑃, that will maximise the area of the
paddock. (2 marks)

iv) Find the angle 𝜃 that gives the maximum area of the paddock. (1 mark)

v) Explain why it is only possible to construct a paddock in the shape of a sector if

𝑃 𝑃
< 𝑟 < . (2 marks)
2(𝜋+1) 2

38 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 12 (2010 HSC Q5a)

A rainwater tank is to be designed in the shape of a cylinder with radius 𝑟 metres and height ℎ
metres.

The volume of the tank is to be 10 cubic metres. Let 𝐴 be the surface area of the tank, including
its top and base, in square metres.
20
i) Given that 𝐴 = 2𝜋𝑟 2 + 2𝜋𝑟ℎ, show that 𝐴 = 2𝜋𝑟 2 + . (2 marks)
𝑟

ii) Show that 𝐴 has a minimum value and find the value of 𝑟 for which the minimum
occurs. (3 marks)

Question 13 (2010 HSC Q6a)

Let 𝑓(𝑥) = (𝑥 + 2)(𝑥 2 + 4).

i) Show that the graph 𝑦 = 𝑓(𝑥) has no stationary points. (2 marks)

ii) Find the values of 𝑥 for which the graph 𝑦 = 𝑓(𝑥) is concave down, and the values
for which it is concave up. (2 marks)

iii) Sketch the graph 𝑦 = 𝑓(𝑥), indicating the values of the 𝑥 and 𝑦 intercepts. (2 marks)

39 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

INTEGRATION
Question 1 (2016 HSC Q9)
2
What is the value of ∫−3|𝑥 + 1|𝑑𝑥 ?
5
A.
2

11
B. 2

13
C.
2

17
D. 2

Question 2 (2016 HSC Q11d) – 2 marks

1
Evaluate ∫0 (2𝑥 + 1)3 𝑑𝑥.

Question 3 (2016 HSC Q12d)

i) Differentiate 𝑦 = 𝑥𝑒 3𝑥 . (1 mark)

2
ii) Hence find the exact value of ∫0 𝑒 3𝑥 (3 + 9𝑥)𝑑𝑥. (2 marks)

Question 4 (2016 HSC Q14a) – 3 marks

The diagram shows the cross-section of a tunnel and a proposed enlargement.

The heights, in metres, of the existing section at 1 metre intervals are shown in Table 𝐴.

40 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

The heights, in metres, of the proposed enlargement are shown in Table 𝐵.

Use Simpson’s rule with the measurements given to calculate the approximate increase in area.

Question 5 (2016 HSC Q15) – 4 marks

The diagram shows two curves 𝐶1 and 𝐶2 . The curve 𝐶1 is the semicircle 𝑥 2 + 𝑦 2 = 4,
𝑥2 𝑦2
−2 ≤ 𝑥 ≤ 0. The curve 𝐶2 has equation + = 1, 0 ≤ 𝑥 ≤ 3.
9 4

An egg is modelled by rotating the curves about the x-axis to form a solid of revolution.
Find the exact volume of the solid of revolution.

Question 6 (2015 HSC Q5)

Using the trapezoidal rule with 4 subintervals, which expression gives the approximate area
under the curve 𝑦 = 𝑥𝑒 𝑥 between 𝑥 = 1 and 𝑥 = 3?
1
A. 4
(𝑒 1 + 6𝑒 1.5 + 4𝑒 2 + 10𝑒 2.5 + 3𝑒 3 )

1
B. 4
(𝑒 1 + 3𝑒 1.5 + 4𝑒 2 + 5𝑒 2.5 + 3𝑒 3 )

1
C. (𝑒 1 + 6𝑒 1.5 + 4𝑒 2 + 10𝑒 2.5 + 3𝑒 3 )
2

1
D. (𝑒 1 + 3𝑒 1.5 + 4𝑒 2 + 5𝑒 2.5 + 3𝑒 3 )
2

41 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 7 (2015 HSC Q6)

The diagram shows the parabola 𝑦 = 4𝑥 − 𝑥 2 meeting the line 𝑦 = 2𝑥 at (0, 0) and (2, 4).

Which expression gives the area of the shaded region bound by the parabola and the line?
2
A. ∫0 𝑥 2 − 2𝑥 𝑑𝑥

2
B. ∫0 2𝑥 − 𝑥 2 𝑑𝑥

4
C. ∫0 𝑥 2 − 2𝑥 𝑑𝑥

4
D. ∫0 2𝑥 − 𝑥 2 𝑑𝑥

Question 8 (2015 HSC Q10)

2
The diagram shows the area under the curve from 𝑦 = 𝑥 from 𝑥 = 1 to 𝑥 = 𝑑.

What value of 𝑑 makes the shaded area equal to 2?

A. 𝑒

B. 𝑒 + 1

C. 2𝑒

D. 𝑒 2

42 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 9 (2015 HSC Q11h) – 2 marks

𝑥
Find ∫ 𝑑𝑥.
𝑥 2 −3

Question 10 (2015 HSC Q12c) – 2 marks

𝑥 2 +3
Find 𝑓′(𝑥), where 𝑓(𝑥) = 𝑥−1
.

Question 11 (2015 HSC Q16a)

The diagram shows the curve with equation 𝑦 = 𝑥 2 − 7𝑥 + 10. The curve intersects the 𝑥-axis at
points 𝐴 and 𝐵. The point 𝐶 on the curve has the same 𝑦-coordinate as the 𝑦-intercept of the
curve.

i) Find the 𝑥-coordinates of points 𝐴 and 𝐵. (1 mark)

ii) Write down the coordinates of 𝐶. (1 mark)

2
iii) Evaluate ∫0 (𝑥 2 − 7𝑥 + 10)𝑑𝑥 . (1 mark)

iv) Hence, or otherwise, find the area of the shaded region. (2 marks)

43 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 12 (2015 HSC Q16b) – 3 marks

A bowl is formed by rotating the curve 𝑦 = 8 log 𝑒 (𝑥 − 1) about the 𝑦-axis for 0 ≤ 𝑦 ≤ 6.

Find the volume of the bowl. Give your answer correct to 1 decimal place.

Question 13 (2014 HSC Q11d) – 2 marks

1
Find ∫ (𝑥+3)2 𝑑𝑥.

Question 14 (2014 HSC Q12d)

The parabola 𝑦 = −2𝑥 2 + 8𝑥 and the line 𝑦 = 2𝑥 intersect at the origin and at the point 𝐴.

i) Find the 𝑥-coordinate of the point 𝐴. (1 mark)

ii) Calculate the area enclosed by the parabola and the line. (3 marks)

Question 15 (2014 HSC Q13a)

i) Differentiate 3 + sin 2𝑥. (1 mark)

cos 2𝑥
ii) Hence, or otherwise, find ∫ 3+sin 2𝑥 𝑑𝑥. (2 marks)

44 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 16 (2014 HSC Q14c) – 3 marks

The region bounded by the curve 𝑦 = 1 + √𝑥 and the 𝑥-axis between 𝑥 = 0 and 𝑥 = 4 is rotated
about the 𝑥-axis to form a solid.

Find the volume of the solid.

Question 17 (2013 HSC Q13b)

The diagram shows the graphs of the functions 𝑓(𝑥) = 4𝑥 3 − 4𝑥 2 + 3𝑥 and 𝑔(𝑥) = 2𝑥. The
graphs meet at 𝑂 and at 𝑇.

i) Find the 𝑥-coordinate of 𝑇. (1 mark)

ii) Find the area of the shaded region between the graphs of the functions 𝑓(𝑥) and
𝑔(𝑥). (3 marks)

45 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 18 (2013 HSC Q14d) – 1 mark

The diagram shows the graph 𝑦 = 𝑓(𝑥).

𝑎
What is the value of 𝑎, where 𝑎 > 0, so that ∫−𝑎 𝑓(𝑥) 𝑑𝑥 = 0?

Question 19 (2013 HSC Q15a)

The diagram shows the front of a tent supported by three vertical poles. The poles are 1.2 m
apart. The height of each outer pole is 1.5 m and the height of the middle pole is 1.8 m. The roof
hangs between the poles.

The front of the tent has area 𝐴 m2 .

i) Use the trapezoidal rule to estimate 𝐴. (1 mark)

ii) Use Simpson’s rule to estimate 𝐴. (1 mark)

iii) Explain why the trapezoidal rule gives the better estimate of 𝐴. (1 mark)

46 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 20 (2013 HSC Q15b) – 4 marks

The region bound by the 𝑥-axis, the 𝑦-axis and the parabola 𝑦 = (𝑥 − 2)2 is rotated about the
𝑦-axis to form a solid.

Find the volume of the solid.

Question 21 (2012 HSC Q10)

The graph of 𝑦 = 𝑓(𝑥) has been drawn to scale for 0 ≤ 𝑥 ≤ 8.

Which of the following integrals has the greatest value?

1
A. ∫0 𝑓(𝑥) 𝑑𝑥

2
B. ∫0 𝑓(𝑥) 𝑑𝑥

7
C. ∫0 𝑓(𝑥) 𝑑𝑥

8
D. ∫0 𝑓(𝑥) 𝑑𝑥

47 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 22 (2012 HSC Q12d)

At a certain location, a river is 12 metres wide. At this location, the depth of the river, in metres,
has been measured at 3 metre intervals. The cross-section is shown below.

i) Use Simpson’s rule with the five depth measurements to calculate the approximate
area of the cross-section. (3 marks)

ii) The river flows at 0.4 metres per second.

Calculate the approximate volume of water flowing through the cross-section in 10

seconds. (1 mark)

Question 23 (2012 HSC Q13b)

The diagram shows the parabolas 𝑦 = 5𝑥 − 𝑥 2 and 𝑦 = 𝑥 2 − 3𝑥. The parabolas intersect at the
origin 𝑂 and the point 𝐴. The region between the two parabolas is shaded.

i) Find the 𝑥-coordinate of the point 𝐴. (1 mark)

ii) Find the area of the shaded region. (3 marks)

48 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 24 (2012 HSC Q14b) – 3 marks

3
The diagram shows the region bounded by 𝑦 = (𝑥+2)2, the 𝑥-axis, the 𝑦-axis and the line 𝑥 = 1.

The region is rotated about the 𝑥-axis to form a solid.

Find the volume of the solid.

Question 25 (2011 HSC Q2e) – 2 marks

1
Find ∫ 3𝑥 2 𝑑𝑥.

Question 26 (2011 HSC Q4d)

i) Differentiate 𝑦 = √9 − 𝑥 2 with respect to 𝑥. (2 marks)

6𝑥
ii) Hence, or otherwise, find ∫ 𝑑𝑥. (2 marks)
√9−𝑥 2

Question 27 (2011 HSC Q5c) – 3 marks

The table gives the speed 𝑣 of a jogger at time 𝑡 in minutes over a 20-minute period. The speed 𝑣
is measured in metres per minute, in intervals of 5 minutes.

The distance covered by the jogger over the 20-minute period is given by
20
∫ 𝑣 𝑑𝑡
0

Use Simpson’s rule and the speed at each of the five time values to find the approximate
distance the jogger covers in the 20-minute period.

49 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 28 (2011 HSC Q8b)

The diagram shows the region enclosed by the parabola 𝑦 = 𝑥 2, the 𝑦-axis and the line 𝑦 = ℎ,
where ℎ > 0. This region is rotated about the 𝑦-axis to form a solid called a paraboloid. The
point 𝐶 is the intersection of 𝑦 = 𝑥 2 and 𝑦 = ℎ. The point 𝐻 has coordinates (0, ℎ).

i) Find the exact volume of the paraboloid in terms of ℎ. (2 marks)

ii) A cylinder has radius 𝐻𝐶 and height 𝐻.

What is the ratio of the volume of the paraboloid to the volume of the cylinder?
(1 mark)

Question 29 (2010 HSC Q2d)

i) Find ∫ √5𝑥 + 1 𝑑𝑥. (2 marks)

𝑥
ii) Find ∫ 4+𝑥2 𝑑𝑥. (2 marks)

Question 30 (2010 HSC Q2e) – 2 marks

6
Given that ∫0 (𝑥 + 𝑘) 𝑑𝑥 = 30, and 𝑘 is a constant, find the value of 𝑘.

50 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 31 (2010 HSC Q3b)

i) Sketch the curve 𝑦 = ln 𝑥. (1 mark)

ii) Use the trapezoidal rule with three function values to find an approximation to
(2 marks)
3
∫ ln 𝑥 𝑑𝑥 .
1

iii) State whether the approximation found in part (ii) is greater than or less than the
3
exact value of ∫1 ln 𝑥 𝑑𝑥 . Justify your answer. (1 mark)

Question 32 (2010 HSC Q4b) – 3 marks

The curves 𝑦 = 𝑒 2𝑥 and 𝑦 = 𝑒 −𝑥 intersect at the point (0, 1) as shown in the diagram.

Find the exact area enclosed by the curves and the line 𝑥 = 2.

51 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 33 (2010 HSC Q5c) – 3 marks

1
The diagram shows the curve 𝑦 = 𝑥, for 𝑥 > 0.

The area under the curve between 𝑥 = 𝑎 and 𝑥 = 1 is 𝐴1 . The area under the curve between
𝑥 = 1 and 𝑥 = 𝑏 is 𝐴2 .

The areas 𝐴1 and 𝐴2 are each equal to 1 square unit.

Find the values of 𝑎 and 𝑏.

Question 34 (2010 HSC Q10b)

The circle 𝑥 2 + 𝑦 2 = 𝑟 2 has radius r and centre O. The circle meets the positive x-axis at B. The
point A is on the interval OB. A vertical line through A meets the circle at P. Let 𝜃 = ∠𝑂𝑃𝐴.

i) The shaded region bounded by the arc PB and the intervals AB and AP is rotated
about the x-axis. Show that the volume, V, formed is given by
(3 marks)
𝜋𝑟 3
𝑉= (2 − 3 sin 𝜃 + sin3 𝜃)
3

52 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

ii) A container is in the shape of a hemisphere of radius 𝑟 metres. The container is

initially horizontal and full of water. The container is then tilted at an angle of 𝜃 to
the horizontal so that some water spills out.

1. Find 𝜃 so that the depth of water remaining is one half of the original depth.
(1 mark)

2. What fraction of the original volume is left in the container? (2 marks)

53 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

TRIGONOMETRIC FUNCTIONS
Question 1 (2016 HSC Q5)
What is the derivative of ln(cos 𝑥)?
A. − sec 𝑥

B. − tan 𝑥

C. sec 𝑥

D. tan 𝑥

Question 2 (2016 HSC Q6)

What is the period of the function 𝑓(𝑥) = tan(3𝑥)?
𝜋
A. 3

2𝜋
B. 3

C. 3𝜋

D. 6𝜋

Question 3 (2016 HSC Q7)

The circle centred at 𝑂 has radius 5. Arc 𝐴𝐵 has length 7 as shown in the diagram.

What is the area of the shaded sector 𝑂𝐴𝐵?

35
A.
2

35
B. 2
𝜋

125
C. 14

125
D. 𝜋
14

54 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 4 (2016 HSC Q11f) – 2 marks

𝜋
Find the gradient of the tangent to the curve 𝑦 = tan 𝑥 at the point where 𝑥 = .
8

Give your answer correct to 3 significant figures.

Question 5 (2016 HSC Q11g) – 2 marks

𝑥 1
Solve sin(2) = 2 for 0 ≤ 𝑥 ≤ 2𝜋.

Question 6 (2016 HSC Q13d) – 3 marks

𝜋
The curve 𝑦 = √2 cos( 𝑥) meets the line 𝑦 = 𝑥 at 𝑃(1, 1), as shown in the diagram.
4

Find the exact area of the shaded area.

Question 7 (2015 HSC Q11g) – 2 marks

𝜋
Evaluate ∫04 cos 2𝑥 𝑑𝑥 .

Question 8 (2015 HSC Q7)

How many solutions of the equation (sin 𝑥 − 1)(tan 𝑥 + 2) = 0 lie between 0 and 2𝜋?

A. 1

B. 2

C. 3

D. 4

55 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 9 (2014 HSC Q11e) – 3 marks

𝜋
𝑥
Evaluate ∫02 sin 2 𝑑𝑥 .

Question 10 (2014 HSC Q11g) – 2 marks

𝜋
The angle of a sector in a circle of radius 8 cm is 7 radians, as shown in the diagram.

Find the exact value of the perimeter of the sector.

Question 11 (2014 HSC Q16a) – 3 marks

Use Simpson’s rule with five function values to show that
𝜋
3 𝜋 8
∫ sec 𝑥 𝑑𝑥 ≈ (3 + )
−
𝜋 9 √3
3

Question 12 (2013 HSC Q4)

𝑥
What is the derivative of cos 𝑥?
cos 𝑥+𝑥 sin 𝑥
A. cos2 𝑥

cos 𝑥−𝑥 sin 𝑥

B. cos2 𝑥

𝑥 sin 𝑥−cos 𝑥
C. cos2 𝑥

−𝑥 sin 𝑥−cos 𝑥
D. cos2 𝑥

56 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 13 (2013 HSC Q6)

𝜋
Which diagram shows the graph 𝑦 = sin(2𝑥 + )?
3

57 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 14 (2013 HSC Q11c) – 2 marks

Differentiate (sin 𝑥 − 1)8 .

Question 15 (2013 HSC Q13a)

The population of a herd of wild horses is given by
𝜋
𝑃(𝑡) = 400 + 50 cos ( 𝑡)
6
where 𝑡 is time in months.
i) Find all times during the first 12 months when the population equals 375 horses.
(2 marks)

ii) Sketch the graph of 𝑃(𝑡) for 0 ≤ 𝑡 ≤ 12. (2 marks)

Question 15 (2013 HSC Q13c) – 2 marks

The region 𝐴𝐵𝐶 is a sector of a circle with radius 30 cm, centred at 𝐶. The angle of the sector is
𝜃. The arc 𝐷𝐸 lies on a circle also centred at 𝐶, as shown in the diagram.

The arc 𝐷𝐸 divides the sector 𝐴𝐵𝐶 into two regions of equal area.
Find the exact length of the interval 𝐶𝐷.

Question 16 (2012 HSC Q11f) – 2 marks

The area of a sector of a circle of radius 6 cm is 50 cm2.
Find the length of the arc of the sector.

Question 17 (2012 HSC Q11g) – 3 marks

𝜋
𝑥
Find ∫02 sec 2 2 𝑑𝑥.

58 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 18 (2012 HSC Q12a) – 2 marks

cos 𝑥
Differentiate .
𝑥2

Question 19 (2011 HSC Q4a) – 2 marks

𝑥
Differentiate sin 𝑥 with respect to 𝑥.

Question 20 (2011 HSC Q6c)

The diagram shows the graph 𝑦 = 2 cos 𝑥.

i) State the coordinates of 𝑃. (1 mark)

𝜋
ii) Evaluate the integral ∫02 2 cos 𝑥 𝑑𝑥 . (2 marks)

iii) Indicate which area in the diagram, 𝐴, 𝐵, 𝐶 or 𝐷, is represented by the integral

2𝜋
∫3𝜋 2 cos 𝑥 𝑑𝑥. (1 mark)
2

iv) Using parts (ii) and (iii), or otherwise, find the area of the region bounded by the
curve 𝑦 = 2 cos 𝑥 and the 𝑥-axis, between 𝑥 = 0 and 𝑥 = 2𝜋. (1 mark)

2𝜋
v) Using the parts above, write down the value of ∫𝜋 2 cos 𝑥 𝑑𝑥 . (1 mark)
2

Question 21 (2010 HSC Q1e) – 2 marks

Differentiate 𝑥 2 tan 𝑥 with respect to 𝑥.

Question 22 (2010 HSC Q2a) – 2 marks

cos 𝑥
Differentiate 𝑥
with respect to 𝑥.

59 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 23 (2010 HSC Q5b)

i) Prove that
(1 mark)
1 + sin 𝑥
sec 2 𝑥 + sec 𝑥 tan 𝑥 =
cos 2 𝑥

ii) Hence prove that

(1 marks)
1
sec 2 𝑥 + sec 𝑥 tan 𝑥 =
1 − sin 𝑥

iii) Hence use the table of standard integrals to find the exact value of
(2 marks)
𝜋
4 1
∫ 𝑑𝑥
0 1 − sin 𝑥

Question 24 (2010 HSC Q6b)

The diagram shows a circle with centre 𝑂 and radius 5 cm.
The length of the arc 𝑃𝑄 is 9 cm. Lines drawn perpendicular to 𝑂𝑃 and 𝑂𝑄 at 𝑃 and 𝑄
respectively meet at 𝑇.

i) Find ∠𝑃𝑂𝑄 in radians. (1 mark)

ii) Prove that ∆𝑂𝑃𝑇 is congruent to ∆𝑂𝑄𝑇. (2 marks)

iii) Find the length of 𝑃𝑇. (1 mark)

iv) Find the area of the shaded region. (2 marks)

60 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 25 (2010 HSC Q8c)

The graph shown is 𝑦 = 𝐴 sin 𝑏𝑥.

i) Write down the value of 𝐴. (1 mark)

ii) Find the value of 𝑏. (1 mark)

iii) Copy or trace the graph into your writing booklet.

On the same set of axes, draw the graph 𝑦 = 3 sin 𝑥 + 1, for 0 ≤ 𝑥 ≤ 𝜋. (2 marks)

61 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

LOGARITHMIC AND EXPONENTIAL FUNCTIONS

Question 1 (2016 HSC Q10)
Which expression is equivalent to 4 + log 2 𝑥?
A. log 2 (2𝑥)

B. log 2 (16 + 𝑥)

C. 4 log 2 (2𝑥)

D. log 2 (16𝑥)

Question 2 (2016 HSC Q14e) – 2 marks

Write log 2 + log 4 + log 8 + ⋯ + log 512 in the form where 𝑎 log 𝑏 where 𝑎 and 𝑏 are integers
greater than 1.

Question 3 (2015 HSC Q8)

The diagram shows the graph of 𝑦 = 𝑒 𝑥 (1 + 𝑥).

How many solutions are there to the equation 𝑒 𝑥 (1 + 𝑥) = 1 − 𝑥 2 ?

A. 0

B. 1

C. 2

D. 3

Question 4 (2015 HSC Q11e) – 2 marks

Differentiate (𝑒 𝑥 + 𝑥)5 .

62 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 5 (2015 HSC Q11f) – 2 marks

Differentiate 𝑦 = (𝑥 + 4) ln 𝑥.

Question 6 (2014 HSC Q3)

What is the solution to the equation log 2(𝑥 − 1) = 8?
A. 4

B. 17

C. 65

D. 257

Question 7 (2014 HSC Q4)

Which expression is equal to ∫ 𝑒 2𝑥 𝑑𝑥?

A. 𝑒 2𝑥 + 𝑐

B. 2𝑒 2𝑥 + 𝑐

𝑒 2𝑥
C. 2
+𝑐

𝑒 2𝑥+1
D. 2𝑥+1
+𝑐

Question 8 (2014 HSC Q14a) – 3 marks

Find the coordinates of the stationary point on the graph 𝑦 = 𝑒 𝑥 − 𝑒𝑥, and determine its nature.

Question 9 (2014 HSC Q15c)

The line 𝑦 = 𝑚𝑥 is a tangent to the curve 𝑦 = 𝑒 2𝑥 at a point 𝑃.
i) Sketch the line and the curve on one diagram. (1 mark)

ii) Find the coordinates of 𝑃. (3 marks)

iii) Find the value of 𝑚. (1 mark)

63 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 10 (2013 HSC Q9)

What is the solution of 5𝑥 = 4?
log𝑒 4
A. 𝑥 = 5

4
B. 𝑥 = log
𝑒5

log 4
C. 𝑥 = log𝑒 5
𝑒

4
D. 𝑥 = log 𝑒 (5)

Question 11 (2013 HSC Q11d) – 2 marks

Differentiate 𝑥 2 𝑒 𝑥 .

Question 12 (2013 HSC Q11e) – 2 marks

Find ∫ 𝑒 4𝑥+1 𝑑𝑥.

Question 13 (2013 HSC Q11f) – 3 marks

1 𝑥2
Evaluate ∫0 𝑑𝑥.
𝑥 3 +1

Question 14 (2012 HSC Q7)

Let 𝑎 = 𝑒 𝑥 .
Which expression is equal to log 𝑒 (𝑎2 ) ?

A. 𝑒 2𝑥

2
B. 𝑒 𝑥

C. 2𝑥

D. 𝑥 2

64 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 15 (2012 HSC Q9)

4 1
What is the value of ∫1 3𝑥
𝑑𝑥?
1
A. ln 3
3

1
B. ln 4
3

C. ln 9

D. ln 12

Question 16 (2012 HSC Q12a) – 2 marks

Differentiate (𝑥 − 1) log 𝑒 𝑥.

Question 17 (2012 HSC Q12b) – 2 marks

4𝑥
Find ∫ 𝑥 2 +6 𝑑𝑥.

Question 18 (2011 HSC Q1d) – 2 marks

Differentiate ln(5𝑥 + 2) with respect to 𝑥.

Question 19 (2011 HSC Q2d) – 2 marks

Find the derivative of 𝑦 = 𝑥 2 𝑒 𝑥 with respect to 𝑥.

Question 20 (2011 HSC Q4b) – 2 marks

𝑒3 5
Evaluate ∫𝑒 𝑑𝑥.
𝑥

Question 21 (2010 HSC Q2c) – 2 marks

Find the gradient of the tangent to the curve 𝑦 = ln(3𝑥) at the point where 𝑥 = 2.

65 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

APPLICATIONS OF CALCULUS TO THE PHYSICAL WORLD

Question 1 (2016 HSC Q13c)
A radioactive isotope of Curium has a half-life of 163 days. Initially there are 10 mg of Curium in
a container.
The mass 𝑀(𝑡) in milligrams of Curium, after 𝑡 days, is given by

𝑀(𝑡) = 𝐴𝑒 −𝑘𝑡 ,
where 𝐴 and 𝑘 are constants.
i) State the value of 𝐴. (1 mark)

ii) Given that after 163 days only 5 mg of Curium remain, find the value of 𝑘. (2 marks)

Question 2 (2016 HSC Q16a)

A particle moves in a straight line. Its velocity 𝑣 m s −1 at time 𝑡 seconds is given by
4
𝑣 =2−
𝑡+1
i) Find the initial velocity. (1 mark)

ii) Find the acceleration of the particle when the particle is stationary. (2 marks)

iii) By considering the behaviour of 𝑣 for large 𝑡, sketch a graph of 𝑣 against 𝑡 for 𝑡 ≥ 0,
showing any intercepts. (2 marks)

iv) Find the exact distance travelled by the particle in the first 7 seconds. (3 marks)

66 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 3 (2016 HSC Q16b)

Some yabbies are introduced into a small dam. The size of the population, 𝑦, of yabbies can be
modelled by the function
200
𝑦=
1 + 19𝑒 −0.5𝑡
where 𝑡 is the time in months after the yabbies are introduced into the dam.
i) Show that the rate of growth of the size of the population is
(2 marks)
1900𝑒 −0.5𝑡
(1 + 19𝑒 −0.5𝑡 )2

ii) Find the range of the function 𝑦, justifying your answer. (2 marks)

iii) Show that the rate of growth of the size of the population can be rewritten as
(1 mark)
𝑦
(200 − 𝑦)
400

iv) Hence, find the size of the population when it is growing at its fastest rate. (2 marks)

Question 4 (2015 HSC Q9)

A particle is moving along the 𝑥-axis. The graph shows its velocity 𝑣 metres per second at time 𝑡
seconds.

When 𝑡 = 0 the displacement 𝑥 is equal to 2 metres.

What is the maximum value of the displacement 𝑥?
A. 8 m

B. 14 m

C. 16 m

D. 18 m

67 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 5 (2015 HSC Q14a)

In a theme park ride, a chair is released from a height of 110 metres and falls vertically.
Magnetic brakes are applied when the velocity of the chair reaches −37 metres per second.

The height of the chair at time 𝑡 seconds is 𝑥 metres. The acceleration of the chair is given by
𝑥̈ = −10. At the release point, 𝑡 = 0, 𝑥 = 110 and 𝑥̇ = 0.

i) Using calculus, show that 𝑥 = −5𝑡 2 + 110. (2 marks)

ii) How far has the chair fallen when the magnetic brakes are applied? (2 marks)

Question 6 (2015 HSC Q15a)

The amount of caffeine, 𝐶, in the human body decreases according to the equation
𝑑𝐶
= −0.14𝐶
𝑑𝑡
where 𝐶 is measured in mg and 𝑡 is the time in hours.
𝑑𝐶
i) Show that 𝐶 = 𝐴𝑒 −0.14𝑡 is a solution to 𝑑𝑡 = −0.14𝐶, where 𝐴 is a constant. (1 mark)

When 𝑡 = 0, there are 130 mg of caffeine in Lee’s body.

ii) Find the value of 𝐴. (1 mark)

iii) What is the amount of caffeine in Lee’s body after 7 hours? (1 mark)

iv) What is the time taken for the amount of caffeine in Lee’s body to halve? (2 marks)

68 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 7 (2015 HSC Q15c)

Water is flowing in and out of a rock pool. The volume of water in the pool at time 𝑡 hours if 𝑉
litres. The rate of change of the volume is given by
𝑑𝑉
= 80 sin(0.5𝑡)
𝑑𝑡
At time 𝑡 = 0, the volume of water in the pool is 1200 litres and is increasing.
i) After what time does the volume of water first start to decrease? (2 marks)

ii) Find the volume of water in the pool when 𝑡 = 3. (2 marks)

iii) What is the greatest volume of water in the pool? (1 mark)

Question 8 (2014 HSC Q9)

The graph shows the displacement 𝑥 of a particle moving along a straight line as a function of
time 𝑡.

Which statement describes the motion of the particle at the point 𝑃?

A. The velocity is negative and the acceleration is positive.

B. The velocity is negative and the acceleration is negative.

C. The velocity is positive and the acceleration is positive.

D. The velocity is positive and the acceleration is negative.

Question 9 (2014 HSC Q13b)

A quantity of radioactive material decays according to the equation
𝑑𝑀
= −𝑘𝑀
𝑑𝑡
where 𝑀 is the mass of the material in kg, 𝑡 is the time in years and 𝑘 is a constant.

i) Show that 𝑀 = 𝐴𝑒 −𝑘𝑡 is a solution to the equation, where 𝐴 is a constant. (1 mark)

ii) The time for half of the material to decay is 300 years. If the initial amount of
material is 20 kg, find the amount remaining after 1000 years. (3 marks)

69 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 10 (2014 HSC Q13c)

The displacement of a particle moving along the 𝑥-axis is given by
1
𝑥=𝑡−
1+𝑡
where 𝑥 is the displacement from the origin in metres, 𝑡 is the time in seconds, and 𝑡 ≥ 0.
i) Show that the acceleration of the particle is always negative. (2 marks)

ii) What value does the velocity approach as 𝑡 increases indefinitely? (1 mark)

Question 11 (2013 HSC Q10)

A particle is moving along the 𝑥-axis. The displacement of the particle at time 𝑡 seconds is 𝑥
metres.

At a certain time, 𝑥̇ = −3 m s −1 and 𝑥̈ = 2 m s −2 .

Which statement describes the motion of the particle at that time?
A. The particle is moving to the right with increasing speed.

B. The particle is moving to the left with increasing speed.

C. The particle is moving to the right with decreasing speed.

D. The particle is moving to the left with decreasing speed.

Question 12 (2013 HSC Q14a)

The velocity of a particle moving along the 𝑥-axis is given by 𝑥̇ = 10 − 2𝑡, where 𝑥 is the
displacement from the origin in metres and 𝑡 is the time in seconds. Initially the particle is 5
metres to the right of the origin.
i) Show that the acceleration of the particle is constant. (1 mark)

ii) Find the time when the particle is at rest. (1 mark)

iii) Show that the position of the particle after 7 seconds is 26 metres to the right of the
origin. (2 marks)

iv) Find the distance travelled by the particle during the first 7 seconds. (2 marks)

70 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 13 (2013 HSC Q16b)

Trout and carp are types of fish. A lake contains a number of trout. At a certain time, 10 carp are
introduced into the lake and start eating the trout. As a consequence, the number of trout, 𝑁,
decreases according to

𝑁 = 375 − 𝑒 0.04𝑡
where 𝑡 is the time in months after the carp are introduced.
The population of carp, 𝑃, increases according to
𝑑𝑃
= 0.02𝑃
𝑑𝑡
i) How many trout were in the lake when the carp were introduced? (1 mark)

ii) When will the population of trout be zero? (1 mark)

iii) Sketch the number of trout as a function of time. (1 mark)

iv) When is the rate of increase of carp equal to the rate of decrease of trout? (3 marks)

v) When is the number of carp equal to the number of trout? (2 marks)

Question 14 (2012 HSC Q14c)

Professor Smith has a colony of bacteria. Initially there are 1000 bacteria. The number of
bacteria, 𝑁(𝑡), after 𝑡 minutes is given by

𝑁(𝑡) = 1000𝑒 𝑘𝑡
i) After 20 minutes, there are 2000 bacteria.

Show that 𝑘 = 0.0347 correct to four decimal places. (1 mark)

ii) How many bacteria are there when 𝑡 = 120? (1 mark)

iii) What is the rate of change of the number of bacteria per minute, when 𝑡 = 120?
(1 mark)

iv) How long does it take for the number of bacteria to increase from 1000 to 100 000?
(2 marks)

71 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 15 (2012 HSC Q14b)

The velocity of a particle is given by
𝑥̇ = 1 − 2 cos 𝑡
where 𝑥 is the displacement in metres and 𝑡 is the time in seconds. Initially the particle is 3 m to
the right of the origin.
i) Find the initial velocity of the particle. (1 mark)

ii) Find the maximum velocity of the particle. (1 mark)

iii) Find the displacement, 𝑥, of the particle in terms of 𝑡. (2 marks)

iv) Find the position of the particle when it is at rest for the first time. (2 marks)

Question 16 (2011 HSC Q7b)

The velocity of a particle moving along the 𝑥-axis is given by

𝑥̇ = 8 − 8𝑒 −2𝑡
where 𝑡 is the time in seconds and 𝑥 is the displacement in metres.
i) Show that the particle is initially at rest. (1 mark)

ii) Show that the acceleration of the particle is always positive. (1 mark)

iii) Explain why the particle is moving in the positive direction for all 𝑡 > 0. (2 marks)

iv) As 𝑡 → ∞, the velocity of the particle approaches a constant.

Find the value of this constant. (1 mark)

v) Sketch the graph of the particle’s velocity as a function of time. (2 marks)

Question 17 (2011 HSC Q9b)

𝑡2
A tap released liquid 𝐴 into a tank at the rate of (2 + 𝑡+1) litres per minute, where 𝑡 is time in
1
minutes. A second tap released liquid 𝐵 into the same tank at the rate of (1 + 𝑡+1) litres per
minute. The taps are opened at the same time and release the liquids into an empty tank.
i) Show that the rate of flow of liquid 𝐴 is greater than the rate of flow of liquid 𝐵 by 𝑡
litres per minute. (1 mark)

ii) The taps are closed after 4 minutes. By how many litres is the volume of liquid 𝐴
greater than the volume of liquid 𝐵 in the tank when the taps are closed? (2 marks)

72 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 18 (2011 HSC Q10a)

The intensity I, measured in watt/m2, of a sound is given by

𝐼 = 10−12 × 𝑒 0.1𝐿
where L is the loudness of the sound in decibels.
i) If the loudness of a sound at a concert is 110 decibels, find the intensity of the sound.
Give your answer in scientific notation. (1 mark)

ii) Ear damage occurs if the intensity of a sound is greater than 8.1 × 10−9 watt/m2.

What is the maximum loudness of a sound so that no ear damage occurs? (2 marks)

iii) By how much will the loudness of a sound have increased if its intensity has
doubled? (2 marks)

Question 19 (2010 HSC Q7a)

The acceleration of a particle is given by
𝑥̈ = 4 cos 2𝑡
where 𝑥 is displacement in metres and 𝑡 is time in seconds.
Initially the particle is at the origin with a velocity of 1 m s −1.

i) Show that the velocity of the particle is given by

(2 marks)
𝑥̇ = 2 sin 2𝑡 + 1

ii) Find the time when the particle first comes to rest. (2 marks)

iii) Find the displacement, 𝑥, of the particle in terms of 𝑡. (2 marks)

Question 20 (2010 HSC Q8a) – 4 marks

Assume that the population, 𝑃, of cane toads in Australia has been growing at a rate
𝑑𝑃
proportional to 𝑃. That is, 𝑑𝑡 = 𝑘𝑃 where 𝑘 is a positive constant.

There were 102 cane toads brought to Australia from Hawaii in 1935.
Seventy-five years later, in 2010, it is estimated that there are 200 million cane toads in
Australia.
If the population continues to grow at this rate, how many cane toads will there be in Australia
in 2035?

73 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

PROBABILITY
Question 1 (2016 HSC Q2)
In a raffle, 30 tickets are sold and there is one prize to be won.
What is the probability that someone buying 6 tickets wins the prize?
1
A. 30

1
B.
6

1
C.
5

1
D. 4

Question 2 (2016 HSC Q15b)

An eight-sided die is marked with numbers 1, 2, …, 8. A game is played by rolling the die until an
8 appears on the uppermost face. At this point the game ends.
i) Using a tree diagram, or otherwise, explain why the probability of the game ending
before the fourth roll is
(2 marks)
1 7 1 7 2 1
+ × +( ) ×
8 8 8 8 8

ii) What is the smallest value for 𝑛 for which the probability of the game ending before
3
the 𝑛th roll is more than 4? (3 marks)

Question 3 (2015 HSC Q4)

5
The probability that Mel’s soccer team wins this weekend is .
7
2
The probability that Mel’s rugby team wins this weekend is 3.

What is the probability that neither team wins this weekend?

2
A.
21

10
B. 21

13
C. 21

19
D. 21

74 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 4 (2015 HSC Q14b)

Weather records for a town suggest that:
5
• If a particular day is wet (W), the probability of the next day being dry is 6

1
• If a particular day is dry (D), the probability of the next day being dry is 2.

In a specific week Thursday is dry. The tree diagram shows the possible outcomes for the next
three days: Friday, Saturday and Sunday.

2
i) Show that the probability of Saturday being dry is 3. (1 mark)

ii) What is the probability of both Saturday and Sunday being wet? (2 marks)

iii) What is the probability of at least one of Saturday and Sunday being dry? (1 mark)

75 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 5 (2014 HSC Q10)

Three runners compete in a race. The probabilities that the three runners finish the race in
1 1 2
under 10 seconds are , and respectively.
4 6 5

What is the probability that at least one of the three runners will finish the race in under 10
seconds?
1
A. 60

37
B. 60

3
C.
8

5
D. 8

Question 6 (2014 HSC Q12c)

A packet of lollies contains 5 red lollies and 14 green lollies. Two lollies are selected at random
without replacement.
i) Draw a tree diagram to show the possible outcomes. Include the probability on each
branch. (2 marks)

ii) What is the probability that the two lollies are of different colours? (1 mark)

Question 7 (2013 HSC Q5)

A bag contains 4 red marbles and 6 blue marbles. Three marbles are selected at random without
replacement.
What is the probability that at least one of the marbles selected is red?
1
A.
6

1
B.
2

5
C. 6

29
D.
30

76 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 8 (2013 HSC Q15d)

Pat and Chandra are playing a game. They take turns throwing two dice. The game is won by the
first player to throw a double six. Pat starts the game.
i) Find the probability that Pat wins the game on the first throw. (1 mark)

ii) What is the probability that Pat wins the game on the first or on the second throw?
(2 marks)

iii) Find the probability that Pat eventually wins the game. (2 marks)

Question 9 (2012 HSC Q13c)

Two buckets each contain red marbles and white marbles. Bucket 𝐴 contains 3 red and 2 white
marbles. Bucket 𝐵 contains 3 red and 4 white marbles.
Chris randomly chooses one marble from each bucket.
i) What is the probability that both marbles are red? (1 mark)

ii) What is the probability that at least one of the marbles is white? (1 mark)

iii) What is the probability that both marbles are the same colour? (2 marks)

Question 10 (2011 HSC Q1g) – 1 mark

A batch of 800 items is examined. The probability that an item from this batch is defective is
0.02.
How many items from this batch are defective?

Question 11 (2011 HSC Q5b)

Kim has three red shirts and two yellow shirts. On each of the three days, Monday, Tuesday and
Wednesday, she selects one shirt at random to wear. Kim wears each shirt that she selects only
once.
i) What is the probability that Kim wears a red shirt on Monday? (1 mark)

ii) What is the probability that Kim wears a shirt of the same colour on all three days?
(1 mark)

iii) What is the probability that Kim does not wear a shirt of the same colour on
consecutive days? (2 marks)

77 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 12 (2010 HSC Q4c)

There are twelve chocolates in a box. Four of the chocolates have mint centres, four have
caramel centres and four have strawberry centres. Ali randomly selects two chocolates and eats
them.
i) What is the probability that the two chocolates have mint centres? (1 mark)

ii) What is the probability that the two chocolates have the same centre? (1 mark)

iii) What is the probability that the two chocolates have different centres? (1 mark)

Question 13 (2010 HSC Q8b) – 2 marks

Two identical biased coins are tossed together, and the outcome is recorded. After a large
number of trials it is observed that the probability that both coins land showing heads is 0.36.
What is the probability that both coins land showing tails?

78 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

SERIES AND APPLICATIONS

Question 1 (2016 HSC Q14b)
A gardener develops an eco-friendly spray that will kill harmful insects on fruit trees without
contaminating the fruit. A trial is to be conducted with 100 000 insects. The gardener expects
the spray to kill 35% of the insects each day and that exactly 5000 new insects will be produced
each day.
The number of insects expected at the end of the 𝑛th day of the trial is 𝐴𝑛 .
i) Show that 𝐴2 = 0.65(0.65 × 100 000 + 5000) + 5000. (2 marks)

(1−0.65𝑛 )
ii) Show that 𝐴𝑛 = 0.65𝑛 × 100 000 + 5000 0.35
. (1 mark)

iii) Find the expected insect population at the end of the fourteenth day, correct to the
nearest 100. (1 mark)

Question 2 (2016 HSC Q14d) – 2 marks

𝑥 5 −1
By summing the geometric series 1 + 𝑥 + 𝑥 2 + 𝑥 3 + 𝑥 4 , or otherwise, find lim .
𝑥→1 𝑥−1

Question 3 (2015 HSC Q3)

The first three terms of an arithmetic series are 3, 7 and 11.
What is the 15th term of this series?
A. 59

B. 63

C. 465

D. 495

Question 4 (2015 HSC Q11d) – 2 marks

1 1 1
Find the limiting sum of the geometric series 1 − + − + ⋯.
4 16 64

79 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 5 (2015 HSC Q14c)

Sam borrows $100 000 to be repaid at a reducible interest rate of 0.6% per month. Let $𝐴𝑛 be
the amount owing at the end of 𝑛 months and $𝑀 be the monthly repayment.
i) Show that 𝐴2 = 100 000(1.006)2 − 𝑀(1 + 1.006). (1 mark)

(1.006)𝑛 −1
ii) Show that 𝐴𝑛 = 100 000(1.006)𝑛 − 𝑀( 0.006
). (2 marks)

iii) Sam makes monthly repayments of $780.

Show that after making 120 monthly repayments, the amount owing is $68 500 to
the nearest $100. (1 mark)

iv) Immediately after making the 120th repayment, Sam makes a one-off payment,
reducing the amount owing to $48 500. The interest rate and monthly repayment
remain unchanged.
After how many more months will the amount owing be completely repaid?
(3 marks)

Question 6 (2014 HSC Q8)

Which expression is a term of the geometric series 3𝑥 − 6𝑥 2 + 12𝑥 3 − ⋯?

A. 3072𝑥 10

B. −3072𝑥10

C. 3072𝑥 11

D. −3072𝑥11

Question 7 (2014 HSC Q12a) – 2 marks

Evaluate the arithmetic series 2 + 5 + 8 + 11 + ⋯ + 1094.

Question 8 (2014 HSC Q14d)

At the beginning of every 8-hour period, a patient is given 10 mL of a particular drug.
1
During each of these 8-hour periods, the patient’s body partially breaks down the drug. Only 3 of
the total amount of the drug present in the patient’s body at the beginning of each 8-hour period
remains at the end of that period.
i) How much of the drug is in the patient’s body immediately after the second dose is
given? (1 mark)

ii) Show that the total amount of the drug in the patient’s body never exceeds 15 mL.
(2 marks)

80 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 9 (2014 HSC Q16b)

At the start of a month, Jo opens a bank account and makes a deposit of $500. At the start of
each subsequent month, Jo makes a deposit which is 1% more than the previous deposit.
At the end of every month, the bank pays interest of 0.3% (per month) on the balance of the
account.
i) Explain why the balance of the account at the end of the second month is
(2 marks)
$500(1.003)2 + $500(1.01)(1.003)

ii) Find the balance of the account at the end of the 60th month, correct to the nearest
dollar (3 marks)

Question 10 (2013 HSC Q12c)

Kim and Alex start jobs at the beginning of the same year. Kim’s annual salary in the first year is
$30 000, and increases by 5% at the beginning of each subsequent year. Alex’s annual salary in
the first year is $33 000, and increases by $1500 at the beginning of each subsequent year.
i) Show that in the 10th year, Kim’s annual salary is higher than Alex’s annual salary.
(2 marks)

ii) In the first 10 years how much, in total, does Kim earn? (2 marks)

1
iii) Every year, Alex saves 3 of her annual salary. How many years does it take her to
save $87 500? (3 marks)

Question 11 (2013 HSC Q13d)

A family borrows $500 000 to buy a house. The loan is to be repaid in equal monthly
instalments. The interest, which is charged at 6% per annum, is reducible and calculated
monthly. The amount owing after 𝑛 months, $𝐴𝑛 , is given by

𝐴𝑛 = 𝑃𝑟 𝑛 − 𝑀(1 + 𝑟 + 𝑟 2 + ⋯ + 𝑟 𝑛−1 )
(Do NOT prove this)
where $𝑃 is the amount borrowed, 𝑟 = 1.005 and $𝑀 is the monthly repayment.
i) The loan is to be repaid over 30 years. Show that the monthly repayment is $2998 to
the nearest dollar. (2 marks)

ii) Show that the balance owing after 20 years is $270 000 to the nearest thousand
dollars. (1 mark)

iii) After 20 years the family borrows an extra amount, so that the family then owes a
total of $370 000. The monthly repayment remains $2998, and the interest rate
remains the same.
How long will it take to repay the $370 000? (2 marks)

81 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 12 (2012 HSC Q12c)

Jay is making a pattern using triangular tiles. The pattern has 3 tiles in the first row, 5 tiles in the
second row, and each successive row has 2 more tiles than the previous row.

i) How many tiles would Jay use in row 20? (2 marks)

ii) How many tiles would Jay use altogether to make the first 20 rows? (1 mark)

iii) Jay has only 200 tiles.

How many complete rows of the pattern can Jay make? (2 marks)

Question 13 (2012 HSC Q15a)

Rectangles of the same height are cut from a strip and arranged in a row. The first rectangle has
width 10 cm. The width of each subsequent rectangle is 96% of the width of the previous
rectangle.

i) Find the length of the strip required to make the first ten rectangles. (2 marks)

ii) Explain why a strip of length 3 m is sufficient to make any number of rectangles.
(1 mark)

Question 14 (2012 HSC Q15c)

Ari takes out a loan of $360 000. The loan is to be repaid in equal monthly repayments, $𝑀, at
the end of each month, over 25 years (300 months). Reducible interest is charged at 6% per
annum, calculated monthly.
Let $𝐴𝑛 be the amount owing after the 𝑛th repayment.
i) Write down an expression for the amount owing after two months, $𝐴2 . (1 mark)

ii) Show that the monthly repayment is approximately $2319.50. (2 marks)

iii) After how many months will the amount owing, $𝐴𝑛 , become less than $180 000?
(3 marks)

82 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 15 (2011 HSC Q3a)

A skyscraper of 110 floors is to be built. The first floor to be built will cost $3 million. The cost of
building each subsequent floor will be $0.5 million more than the floor immediately below.
i) What will be the cost of building the 25th floor? (2 marks)

ii) What will be the cost of building all 110 floors of the skyscraper? (2 marks)

Question 16 (2011 HSC Q5a)

The number of members of a new social networking site doubles every day. On Day 1 there
were 27 members and on Day 2 there were 54 members.
i) How many members were there on Day 12? (1 mark)

ii) On which day was the number of members first greater than 10 million? (2 marks)

iii) The site of 0.5 cents per member per day. How much money did the site earn in the
first 12 days? Give your answer to the nearest dollar. (2 marks)

Question 17 (2011 HSC Q8c)

When Jules started working she began paying $100 at the beginning of each month into a
superannuation fund.
The contributions are compounded monthly at an interest rate of 6% per annum.
She intends to retire after having worked for 35 years.
i) Let $𝑃 be the final value of Jules’s superannuation when she retires after 35 years
(420 months).

Show that $𝑃 = $143 183 to the nearest dollar. (2 marks)

ii) Fifteen years after she started working, Jules read a magazine article about
retirement, and realised that she would need $800 000 in her fund when she retires.
At the time of reading the magazine article, she had $29 227 in her fund. For the
remaining 20 years she intends to work, she decides to pay a total of $𝑀 into her
fund at the beginning of each month. The contributions continue to attract the same
interest rate of 6% per annum, compounded monthly.

At the end of 𝑛 months after starting the new contributions, the amount in the fund
is $𝐴𝑛 .

1. Show that 𝐴2 = 29 227 × 1.0052 + 𝑀(1.005 + 1.0052 ). (1 mark)

2. Find the value of 𝑀 so that Jules will have $800 000 in her fund after the
remaining 20 years (240 months). (3 marks)

83 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)

 lOMoARcPSD|31096826

Question 18 (2011 HSC Q9d)

i) Rationalise the denominator in the expression

1
√𝑛 + √𝑛 + 1

where 𝑛 is an integer and 𝑛 ≥ 1. (1 mark)

ii) Using your result from part (i), or otherwise, find the value of the sum
(2 marks)
1 1 1 1
+ + + ⋯+
√1 + √2 √2 + √3 √3 + √4 √99 + √100

Question 19 (2010 HSC Q1f) – 2 marks

1 1 1
Find the limiting sum of the geometric series 1 − 3 + − + ⋯.
9 27

Question 20 (2010 HSC Q4a)

Susannah is training for a fun run by running every week for 26 weeks. She runs 1 km in the
first week and each week after than she runs 750 m more than the previous week, until she
reaches 10 km in a week. She then continues to run 10 km each week.
i) How far does Susannah run in the 9th week? (1 mark)

ii) In which week does she first run 10 km? (1 mark)

iii) What is the total distance that Susannah runs in 26 weeks? (2 marks)

Question 21 (2010 HSC Q9a)

i) When Chris started a new job, $500 was deposited into his superannuation fund at
the beginning of each month. The money was invested at 0.5% per month,
compounded monthly.
Let $𝑃 be the value of the investment after 240 months, when Chris retires.

Show that 𝑃 = 232 175.55. (2 marks)

ii) After retirement, Chris withdraws $2000 from the account at the end of each month,
without making any further deposits. The account continues to earn interest at 0.5%
per month.
Let $𝐴𝑛 be the amount left in the account 𝑛 months after Chris’s retirement.

1. Show that 𝐴𝑛 = (𝑃 − 400 000) × 1.005𝑛 + 400 000. (3 marks)

2. For how many months after retirement will there be money left in the account?
(2 marks)

84 | P a g e

Downloaded by Danielle Lynch (daniellelynch2007@gmail.com)