**Practice HSC Exam**

*Reading time:* 5 minutes
*Working time:* Two and a half hours

*You may use the list of formulas printed at the start of Chapter 11 (pages 413 and 414).*

**SECTION 1**

*22 multiple choice questions:* 1 mark each (Total 22 marks)
*Allow about 30 minutes for this section.*

*Choose the correct answer:* A, B, C or D.

**1.** On this prize wheel, what is the probability of
*not* spinning the dinner or holiday?

A. B.

C. D.

**2.** The local time in India is 5 hours ahead of GMT while the time in Fiji is 12 hours ahead
of GMT. What is the time in India when it is 11 am in Fiji?

A. 4:30 am B. 4:30 pm C. 5:30 am D. 5:30 pm

**3.** The distance between the Earth and the Sun is 1.49 × 1011_{ m. What is this distance in }

kilometres?

A. 1 490 000 km B. 149 000 000 km

C. 149 000 000 000 km D. 14 900 000 km

**4.** Which one of these is an example of continuous data?

A. the home suburb of a car driver B. the number of bedrooms in a house C. the mass of a truck D. the age a soccer player turns this year

**5.** This table shows the distribution of students in
each year at Westvale High School.

A stratified sample of 80 students is to be taken from the school population for a survey on the new school uniform. On a proportional basis, how many Year 12 students should be surveyed?

A. 11 B. 7

C. 13 D. 12

Dinner Holiday

Cash CD

Show 3

10

--- 7

10

---1 2

--- 3

5

---1 2

**---Year** **No. of students**

7 188

8 195

9 176

10 179

11 161

12 138

Total 1037

**6.** In the diagram, Keira’s home is due east of the school.
What is the bearing of the pool from Keira’s home?

A. 235° B. 215°

C. 325° D. 305°

**7.** Use the cosine rule to find the size of θ
to the nearest degree.

A. 25° B. 47°

C. 76° D. 58°

**8.** This tin can has a height of 16 cm and a base
with diameter 10 cm. Calculate its surface area
to the nearest square centimetre.

A. 581 cm2 _{B. 660 cm}2

C. 1257 cm2 _{D. 1634 cm}2

**9.** Which one of these could be the graph of *y* = *x*3_{?}

**10.** Simplify 3*x*2× _{.}

A. B. 6*xy* C. 12*x*2_{y}_{D. 6}* _{x}*2

_{y}**11.** This backyard swimming pool is in the
shape of a trapezoidal prism. Calculate its
volume in kilolitres, given that 1 m3=_{ 1 kL.}

A. 68.5 kL B. 15.8 kL

C. 50.6 kL D. 101.2 kL

School Keira’s

home Pool

49° 55°

16 m 14 m 12 m

θ

10 cm

16 cm

1 2

---A. B.

C. * y* D.

**x**

**y**

**x****y**

**x****y**

**x**

*4y*
*2x*

*---7xy*
2

---2.1 m

1.0 m 10.2 m

3.2 m

**12.** The kinetic energy of a moving object is proportional to the square of its speed. A toy
car travelling at a speed of 2 m/s has a kinetic energy of 0.8 J (joules). Calculate the
kinetic energy of the toy car when it is travelling at 1.5 m/s.

A. 0.3 J B. 0.45 J C. 0.71 J D. 1.07 J

**13.** Make *h* the subject of the formula *V* = π*r*2_{h}_{.}

A. *h* = B. *h* = C. *h* = D. *h* =

**14.** How many possible outcomes are there when a coin is tossed 3 times?

A. 6 B. 4 C. 8 D. 9

**15.** Solve − = 6.

A. *w* = 2 B. *w* = −18 C. *w* = 12 D. *w* = 20

**16.** Sonya plays a game in which she rolls a die. If she rolls a 6 she wins $1.05; if she rolls
an odd number she loses 5c. What is Sonya’s financial expectation from each game?

A. 15c B. 90c C. 55c D. 18c

**17.** The value of a 1950s record collection appreciates by 3.5% each year. If its value is
$17 500 this year, use the compound interest formula to calculate its value in 25 years.

A. $32 813 B. $41 357 C. $153 125 D. $31 724

**18.** What is the length of the shadow of a flagpole
14 m high when the angle of elevation of the
sun is 63°?

A. 7.1 m B. 12.5 m

C. 27.5 m D. 15.7 m

**19.** Caroline has 10 different-coloured highlighter pens in her bag. She takes out 3 of them
at random. How many different combinations of colours are possible?

A. 720 B. 167 C. 120 D. 30

**20.** To calculate the width of a river, Sam placed
2 pegs *A* and *B* on one bank and peg *C* on the other
bank. She took some measurements and made a
scale drawing as shown. Use the scale drawing to
find *w*, the width of the river.

A. 8 m B. 48 m

C. 32 m D. 16 m

**21.** What type of correlation is illustrated by
this diagram?

A. low positive correlation B. low negative correlation C. high positive correlation D. high negative correlation

**22.** Beijing, China (40°N, 116°E) and Perth, WA (32°S, 116°E) lie on the same meridian of
longitude. Given that 1° on a great circle subtends 60 M (nautical miles), what is the
shortest distance between Beijing and Perth?

A. 1200 M B. 480 M C. 894 M D. 4320 M

1 3

*---3V*

π*r*

--- *3V*

π*r*2

--- *V*π*r*

2

3

--- *3V*

π*r*

*---w*
2
---- *w*

5

----63° 14 m

*C*

*A*

*w*

*B*

80 m

33°

47°

**SECTION 2**

*Questions 23 to 28*

*6 questions:* 13 marks each (Total 78 marks)
*Allow about 2 hours for this section.*
*Answer each question on a separate sheet of paper.*

**Question 23**

(a) A balloon has the shape of a sphere with diameter 27 cm.

(i) Calculate its volume to the nearest cubic centimetre.

(ii) The balloon is inflated until its volume
is 73 600 cm3_{. Calculate its new }

diameter to the nearest metre.

(b) Two buildings are situated on level ground 70 m apart. Ivory Tower is 220 m high. The angle of depression from the top of Ivory Tower to the top of the NCM Building is 48°.

(i) Calculate the length of *CB*
correct to 1 decimal place.
(ii) Show that *CD*≈ 77.7 m.
(iii) Hence write down the height

of the NCM Building.
(iv) Explain why ∠*CAB* = 138°.

(v) Use the sine rule to calculate
the size of ∠*ACB* to the nearest
degree.

(c) On a hot summer’s night, crickets chirp more frequently. There is a linear relationship between the temperature of the air and the number of times a cricket chirps per minute.

(i) Find a formula for *C* in terms of *T*.

(ii) Use your formula to calculate the crickets’ chirping rate at temperature 24°C.

(iii) Why is this linear model probably inaccurate for low values of *T*?
(iv) Rewrite the formula in (i) with *T* as the subject.

**Question 24**

(a) A biased coin shows heads 2 times out of 5 when tossed. This coin is tossed twice. Calculate the probability that:

(i) a head comes up both times (ii) a head comes up exactly once (b) The graphs on the next page illustrate two types of income tax rates:

n the current progressive tax rate n a proposed flat tax rate

**Temperature, ****T**** (°C)** 12 15 19 22 28

**Chirping rate, *** C (chirps/min)* 72 96 128 152 200

27 cm

[4 marks]

[5 marks]

70 m 48°

NCM Building

Ivory Tower

220 m

*B*
*A*

*C*

*D*

*E*
*F*

[4 marks]

[4 marks]

[6 marks]

.

(i) Alex has a taxable income of $30 000. Use the graph to estimate his income tax under the current progressive tax rate.

(ii) Calculate Alex’s income tax under the current tax rate if it is $2380 plus 30 cents for each $1 over $20 000.

(iii) Describe the type of income earner who would suffer most under the proposed flat rate system. Give a reason for your answer. (iv) Kim pays $11 250 in income tax under the current system. Use the

graph to find how much tax she would pay under the proposed flat rate system.

(v) Express the proposed flat tax rate as a percentage (of taxable income). (c) An offset survey of a park was taken, and the notebook diagram shown

at right was recorded.

*(i) Draw a neat sketch of the park DEFG.*

*(ii) Calculate the length of EF to the nearest metre.*
*(iii) Calculate the area of the park DEFG.*

**Income tax payable, **

**T**

** ($’000)**

**Taxable income, I ($‘000)**

5 10 15 20 25 30 35 40 45

7

6

5

4

3

2

1

50 0

12

11

10

9

8 15

14

13

55 60
**Flat tax rate**
**Progressive **
**tax rate**

**Income tax rates**

*D*

*G 32*
120

74 62 0

*58 E*

*F*

[3 marks]

**Question 25**

(a) The price of a football is $26.95 after a GST of 10% has been added. What was its original price?

(b) A large bucket holds 5 L of water when full. It is placed under a ceiling that leaks at a rate of 65 drops per minute. Each drop of water is 0.2 mL. How long will it take to fill the bucket? Answer in hours and minutes. (c) The maximum temperature in Emu Springs for each day in April is

illustrated in the dot plot.

(i) Describe the shape of this distribution.

(ii) Make a five-number summary of this data set and hence construct a box-and-whisker plot.

(iii) Calculate the mean and standard deviation of this data set correct to 1 decimal place.

**Question 26**

(a) Calculate the surface area of this winners’ platform.

(b) A competition draw is organised so that 6 hockey teams (A, B, C, D, E and F) play each other twice: once at their home ground and once (away) at the other team’s home ground. This table lists all 30 games in the draw. The home team is listed first. For example, DB means team D plays team B at team D’s home ground.

(i) How many games does team D play over the competition? (ii) Why are 6 cells in this grid marked ‘—’?

(iii) If an extra team were added to this draw to make 7 teams, how many games would be required in the competition?

*(iv) If there were n teams in the competition, write an algebraic *
expression for the number of games required in the competition.

**Away team**

A B C D E F

A — AB AC AD AE AF

B BA — BC BD BE BF

**Home**
**team**

C CA CB — CD CE CF

D DA DB DC — DE DF

E EA EB EC ED — EF

F FA FB FC FD FE —

[2 marks]

[3 marks]

[8 marks]

17 18 19 20 21 22

**Maximum temperature (°C)**

[3 marks]

1.8 m 0.8 m

0.6 m

0.6 m

0.6 m 0.8 m

0.4 m

0.8 m

[4 marks]

(c) From each of two gyms, 20 members were randomly selected and their heart rates measured (in beats/minute). The data is listed in the double stem-and-leaf plot.

(i) Calculate the median heart rate for the Best Fit Gym sample. (ii) One entry (represented by ) is missing for the Jim’s Gym sample.

Write a possible heart rate that could be represented here. (iii) Calculate the mean and standard deviation of the Best Fit Gym

sample, correct to 1 decimal place.

(iv) The mean and standard deviation of the Best Fit Gym sample are both greater than the mean and standard deviation of the Jim’s Gym sample. Describe how the shape and other features of the stem-and-leaf plot illustrate this fact.

**Question 27**

(a) $1 is paid into an account each year and the investment is compounded annually at a rate of 5% p.a. The spreadsheet below shows the amount in the account at the end of each year.

(i) Calculate the missing values in cells D6 and E6.

(ii) Write an appropriate spreadsheet formula for calculating the value in cell C8.

(iii) Basam makes annual payments of $1500 into an account with an interest rate of 5% p.a. Use the table to calculate the amount in his account at the end of 5 years.

(b) This table lists some statistics for the performances of Nancy and David in their mathematics tests over a semester.

Which student, Nancy or David, had the more consistent test results? Give a reason for your answer.

**Best Fit Gym** **Jim’s Gym**

6 9 8 5 3 2 0 9 9 6 4 4 4 3 1 0 8 3 1 2

5 6 7 8 9

4 6 8

4 6 6 7 9 9

0 0 1 2 2 4 8 0 1 5

**A** **B** **C** **D** **E**

**1** **Annuity** Annual payment $1.00

**2** Compounding rate of interest 5%

**3** **Year** **Payment** **Start of year** **Interest** **End of year**

**4** 1 $1.00 $1.00 $0.05 $1.05

**5** 2 $1.00 $2.05 $0.10 $2.15

**6** 3 $1.00 $3.15

**7** 4 $1.00 $4.31 $0.22 $4.53

**8** 5 $1.00 $0.28 $5.81

**Student** **Mean** **Standard deviation** **Range**

Nancy 70 3.8 11

David 64 2.5 11

[6 marks]

[4 marks]

[2 marks]

(c) Dave claims that a quick rule-of-thumb for converting Celsius temperatures to Fahrenheit is to ‘double and add 30’.

(i) Graph Dave’s rule and the actual conversion formula on the same
*axes for values of C from 0 to 100.*

Dave’s rule: *F = 2C + 30*

*Actual conversion formula: F = C + 32*

*where C represents the temperature in °C and F represents the *
temperature in °F.

*(ii) According to your graph, for what values of C does Dave’s rule *
give answers that are too high?

(iii) Use the two formulas to calculate the error resulting from Dave’s rule when it is used to convert 16°C to Fahrenheit.

(iv) What are the limitations of Dave’s rule?

**Question 28**

(a) There are 8 horses in a race. A quinella is a bet on the first 2 horses in the race, in any order. How many quinella bets are possible from a race with 8 horses?

(b) This diagram illustrates a compass
*radial survey of a field PQRS.*

(i) Find the size of ∠*POQ.*
(ii) Use the cosine rule to calculate

*the length of PQ to the nearest *
metre.

(c) A machine is set to make bolts with a mean diameter of 18.0 mm and standard deviation 0.5 mm. The diameters of these bolts are normally distributed.

(i) Between which two lengths should 99.7% of the manufactured bolts lie?

(ii) What percentage of bolts should have lengths between 17.0 mm and 18.0 mm?

(iii) A bolt produced by the machine is selected at random. It has a diameter of 16.3 mm. Why does the machine operator then think that this machine has been set incorrectly?

*(d) The petrol consumption, P L/100 km, of a car is related to its speed, *
*s km/h, by the quadratic function*

*P = 0.01s*2−_{ s + 33}

(i) Calculate the petrol consumption of the car when it is travelling at a speed of 65 km/h.

*(ii) Graph the quadratic function for values of s from 20 to 80.*
(iii) At what speed does minimum petrol consumption occur?
(iv) This quadratic function is a good model only for speeds between

*20 km/h and 80 km/h. Why would this model not be useful for s = 0?*

**END OF EXAM**

[7 marks]

9 5

---[2 marks]

[3 marks]
*P*

*Q*

*R*

*S* *O*

45 m

70 m

66 m 51 m

332°

082°

156° 264°

[3 marks]

[5 marks]