LEVEL 2 MATHEMATICS - Maths Centre

NCEA

LEVEL 2

MATHEMATICS QUESTIONS AND ANSWERS P J Kane

Published by Mahobe Resources (NZ) Ltd

The NZ Centre of Mathematics

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ACKNOWLEDGEMENTS

NCEA Level 2 Mathematics Questions & Answers P. J. Kane This eBook was published in 2010. Mahobe Resources (NZ) Ltd P.O. Box 109-760 Newmarket, Auckland New Zealand www.mahobe.co.nz © Mahobe Resources (NZ) Ltd ISBN 9781877216824 This eBook has been provided by Mahobe Resources (NZ) Ltd to The New Zealand Centre of Mathematics. Schoolteachers, University lecturers, and their students are able to freely download this book from The New Zealand Centre of Mathematics website www.mathscentre.co.nz. Electronic copies of the complete eBook may not be copied or distributed. Students have permission to print one copy for their personal use. Any photocopying by teachers must be for training or educational purposes and must be recorded and carried out in accordance with Copyright Licensing Ltd guidelines. The content presented within the book represents the views of the publisher and his contributors as at the date of publication. Because of the rate with which conditions change, the publisher and his contributors reserve the right to alter and update the contents of the book at any time based on the new conditions. This eBook is for informational purposes only and the publisher and his contributors do not accept any responsibilities for any liabilities resulting from the use of the information within. While every attempt has been made to verify the content provided, neither the publisher nor his contributors and partners assume any responsibility for errors, inaccuracies oromissions. All rights reserved. All the views expressed in this book are those of the author. The questions and suggested answers are the responsibility of the author and have not been moderated for use in NCEA examinations. Thats all the legal stuff over. We hope that the book is helpful!

3

CONTENTS 2.1

Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2

Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3

Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4

Co-ordinate Geometry . . . . . . . . . . . . . . . . . . 55

2.5

Sample Statistics . . . . . . . . . . . . . . . . . . . . . 73

2.6

Probability & Normal Distribution . . . . . . . . . . 87

2.7

Sequences . . . . . . . . . . . . . . . . . . . . . . . . 105

2.8

Trigonometry Problems . . . . . . . . . . . . . . . . 117

2.9

Trigonometric Equations . . . . . . . . . . . . . . . 130

The Answers . . . . . . . . . . . . . . . . . . . . . . . 144

Areas Under the Normal Curve . . . . . . . . . . . 171

Formulae Sheet . . . . . . . . . . . . . . . . . . . . . 172

Pages for Extra Notes . . . . . . . . . . . . . . . . . 173

YEAR 12 MATHEMATICS

4

STUDYING NCEA LEVEL 2 MATHS Ø This book has been written for you to practise NCEA Level Two- type assessments. Nine chapters have been designed to match the nine achievement standards at this level.

Ù Each chapter begins with a schedule of the requirements for that achievement standard. As you read down each schedule, you will see that the challenges become more complex.

Ú In most chapters a preliminary set of exercises has been provided to set in motion the set of skills required for the achievement objectives. Once you think that you have mastered the skill set progress onto the first model assessment. For external achievement standards allow 45 - 60 minutes. For internal achievement standards allow 4-5 hours as these are more project orientated. Check the solutions, and if yours do not quite match these, rework your calculations, or check with friends or teachers until you are satisfied.

Û Attempt the second model assessment 1-2 weeks later to see if the themes you covered still ‘click’. Again, check the solutions at the back of the book with yours. Remember, you can still learn from your mistakes .... this side of the final exams.

Ü It is worth recognising that in Year 12 your mathematics may appear to have begun at a roundabout. It introduces new themes which seem to go down different roads. One of the strengths of this subject, however, is that these themes or roads are connected, though this may not be evident just yet. Therefore as you are being assessed in discrete themes or standards, try to develop an eye for the bigger picture. As always mathematics is about solving problems and finding patterns and reasons. Hopefully your experiences this year will provide you with confidence and judgement for future challenges.

YEAR 12 MATHEMATICS

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Algebra

MATHEMATICS 2.1 ACHIEVEMENT STANDARD 90284 Manipulate algebraic expressions and solve equations

Below are the requirements of this Achievement Standard.

C

expand brackets (up to 3 bracket sets)

C

factorise expressions including quadratics

C

simplify and use fractional exponents

C

simplify and use integer exponents

C

interchange between exponential and

From the straightforward

logarithmic statements (less steps) C

solve linear equations or inequations using at least 2 steps

C

cases

solve quadratic equations which can be factorised

C

solve simple logarithmic equations

C

form then solve pairs of linear simultaneous equations

to

situations involving

C

solve quadratics using the quadratic formula

C

solve a pair of simultaneous equations, with one being linear and one being non-linear

C

solve exponential equations, which may involve logarithmic methods

C

complete algebraic challenges such as proving an algebraic statement

C

explore the nature of the roots of a quadratic equation.

YEAR 12 MATHEMATICS

more depth, more steps and growing complexities with sensible interpretations of the solutions(s).

5

6

Algebra

ALGEBRA - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. Write your answers on the opposite page. 1.

ASSUMED KNOWLEDGE Rearrange (change the subject of) these formulae: a.

x, in y = 3x - 7

b.

r, in A = 4πr2

Solve in factored form:

2.

3.

4.

5.

c.

(2x - 5)(x + 11) = 0

d.

5x2(1 - 3x)2(x + 8) = 0

EXPAND BRACKETS AND SIMPLIFY a. 3x(2x - 3) - 6(x2 - 2) b.

2x(3x + 1)(3x - 5)

c.

(x + 3)(2x - 7)2

FACTORISE ALGEBRAIC EXPRESSIONS Factorise completely: a.

x2 - 19x + 84

b.

10x2 - x - 21

c.

a2 - av + aw - vw

FRACTIONAL AND NEGATIVE INDICES a.

Write

b.

Simplify

c.

Simplify

with positive indices. . .

ELEMENTARY PROPERTIES OF LOGARITHMS a. Write log3 243 = 5 in exponential form. b.

Express as the log of a single number: i.

log 15 - log 3

ii.

3 log 2 + 2 log 3

iii. 2 log 6 c.

log 36 + log 5

Simplify

YEAR 12 MATHEMATICS

Algebra

YEAR 12 MATHEMATICS

7

8

6.

Algebra

SIMPLIFY RATIONAL EXPRESSIONS a.

b.

c.

d.

7.

SOLVE LINEAR EQUATIONS AND INEQUATIONS a.

8.

×

x-3=x+7

b.

2(n + 3) = 5(n - 1) - 7(2n - 3)

c.

3(2x + 1) < 2x + 9

d.

2(x + 3) >

SOLVE QUADRATIC EQUATIONS By factorising a. x2 - x - 42 = 0 b.

3x2 - 7x + 2 = 0

c.

5x = 2x3 + 3x2

By quadratic formula (to 2dp) d. 4x2 - 2x - 3 = 0

9.

e.

x2 + 4x - 2 = 0

f.

8 - x - x2 = 0

SOLVE LOGARITHMIC AND EXPONENTIAL EQUATIONS Find the value of x in: a. log3 x = 7 b.

logx 343 = 3

c.

7x-1 = 26

YEAR 12 MATHEMATICS

Algebra

YEAR 12 MATHEMATICS

9

10

10.

Algebra

SOLVE SIMULTANEOUS EQUATIONS Linear pairs a. x + 3y = 5

b.

2x + y = -5 Line and curve c. y = x2 - 3x y = 2x - 6 11.

y=x+5 3y + 4x = 1

d.

xy = 2 y-x=1

QUADRATIC ROOTS a. Find the nature of the roots of:

b.

i.

4x2 - 13x + 7 = 0

ii.

25x2 - 30x + 9 = 0

Use the discriminant of 2x2 - 2nx + 5 = 0 to find the values of n for which there will be no real roots (i.e. imaginary).

YEAR 12 MATHEMATICS

Algebra

ALGEBRA - PRACTICE TEST 1 QUESTION ONE

1.

Simplify:

2.

Simplify fully:

3.

Write as the log of a single number: log 112 - log 14

4.

Solve the following equations:

a.

=

b. logx 44 = 5

c. 3x2 - x = 4

YEAR 12 MATHEMATICS

11

Algebra

12

QUESTION TWO A suburb in a major city has been infected by a foreign moth which could have devastating effects on neighbouring farms and forests. An aerial spray campaign is launched where an aeroplane flies over the area and spreads an insecticide, which though fatal to the moth is harmless to humans and other creatures. The formula M = M0(0.85)t gives the number of moths (M) in the spray zone t days after the plane has sprayed. M0 is the initial number of moths that the Ministry officials believe were in the zone. If they believe that 800 moths were present in the zone, how many days after spraying would it take the population to fall to 500 moths?

QUESTION THREE A circular traffic island in the middle of an intersection is planned. The circle is represented by x2 + y2 = 36. Also in the plan is a path of an electrical cable which runs underneath the traffic island. The cable path may be shown by y = 2x + 6. a.

Find the x ordinates of the points where the cable meets the perimeter of the traffic island.

b.

Find the y ordinates and hence write the points of intersection.

YEAR 12 MATHEMATICS

Algebra

13

QUESTION FOUR A team of netballers and their supporters are fundraising in order to attend a Golden Oldies tournament in the Cook Islands. One of their activities is a social at a local hall. The team has two options for pricing tickets to this event. Option Price A PA =

Option Price B PB =

Where x is the number of tickets sold

Where x is the number of tickets sold

and the 9 best ticket sellers get free

and the 12 best ticket sellers get free

tickets to the social.

tickets to the social.

Solve PA = PB and find the minimum number of tickets which need to be sold so that the price of Option B tickets would be less than the price of Option A tickets.

YEAR 12 MATHEMATICS

14

Algebra

QUESTION FIVE The quadratic equation x2 - (k + 1)x + 4k = 0 has 2 roots. If the difference between the roots is 1, find the value of k.

YEAR 12 MATHEMATICS

Algebra

15

ALGEBRA - PRACTICE TEST 2 QUESTION ONE 1.

Expand and simplify: (x + 8)(3x - 1)(4x + 3)

2.

Write this expression in positive index form:

3.

Write as a single number:

4.

Solve the equations: a. logx 243 = 5

5.

b. 3x2 - 8x = -4

c. 6x = 31

At the movies during the weekend, Moira served ice creams to a group of children from a birthday party. Of the 9 she served, 7 wanted chocolate dipped while the other 2 wanted plain. If it cost a total of $19.25 with a chocolate dipped ice cream being 50 cents more than a plain one, calculate the cost of a plain ice cream.

YEAR 12 MATHEMATICS

16

Algebra

QUESTION TWO Show that there is only one point of intersection between: x2 + y2 + 2x - 7 = 0 and y = x - 3.

QUESTION THREE The height of a door is 1 metre longer than its width. The area of the door is 1.7 m2. What are the dimensions of the door? (Give your answer to 1 dp.)

YEAR 12 MATHEMATICS

Algebra

17

QUESTION FIVE After t hours of use, the value (V) of a certain brand of jetski (which was purchased new for $19 995) may be estimated by: V = P(0.993)t where P is the retail price. After how many hours of use would the jetski be worth

YEAR 12 MATHEMATICS

of its original retail price?

18

Algebra

QUESTION SIX A certain aeroplane can cover a distance of 5000 km travelling over a time, t hours, at a velocity v =

.

If the same aeroplane flew the 5000 km again, this time increasing its speed by 250 km/h (i.e. v + 250), and cutting the travelling time by an hour (i.e. t - 1), what would its speed have been in both instances?

YEAR 12 MATHEMATICS

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Graphs

19

MATHEMATICS 2.2 ACHIEVEMENT STANDARD 90285 Draw straightforward non-linear graphs


C

C C

draw a quadratic graph whose equation may be factorised, y = (cx - d)(x + e) or expressed as y = ±(x - a)2 + b draw a polynomial graph whose equation could be factorised (leading term (±1)xn ) draw a rectangular hyperbola from an equation such as y =

C C C

C

or

From more straightforward relations and their

draw a circle from a given equation whose centre is at (0, 0) draw an exponential function from a given equation, y = ax draw a logarithmic function from a given equation y = loga x show relevant features including: intercepts, maxima or minima, asymptotes, symmetry

graphs to

graphs with more complexity

C C C C C C

YEAR 12 MATHEMATICS

draw rectangular hyperbola y = 2

and more features. 2

2

draw circles such as (x - a) + (y - b) = r draw exponential functions y = ax - b + c draw log functions y = loga(x - b) + c write equations and interpret features of any graphs of the above model any of the above (or combinations of) to describe a situation, find points of intersection and to solve related problems

Relations given may have coefficients constraints and exponents other than ± 1.

20

Graphs

GRAPHS - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. 1.

NON LINEAR GRAPHS Draw graphs for each of the following. Insure that you include any intercepts, asymptotes, symmetry and maxima or minima for quadratic curves. a. i.

y = x2 + 3x - 4

ii.

y = (x + 3)2 - 4

iii.

(x + 1)(x - 2)(x + 3)

b. i.

y=

ii. y =

Draw each of the following pairs of graphs on the same axes. c. i.

x2 + y2 = 49

ii. (x + 2)2 + (y - 1)2 = 49

d. i.

y = 5x

ii. y = 5x - 3

e. i.

y = log8 x

ii. y = log8 *x - 3*

YEAR 12 MATHEMATICS

Graphs

YEAR 12 MATHEMATICS

21

22

Graphs

YEAR 12 MATHEMATICS

Graphs

2.

For each of the graphs sketched below, write the equations. a. b.

c.

YEAR 12 MATHEMATICS

d.

23

24

3.

Graphs

Draw the curve xy = -1 and y = x - 2 on the same axes. Label any points of intersection with the correct co-ordinates.

YEAR 12 MATHEMATICS

Graphs

GRAPHS - PRACTICE TEST 1 QUESTION ONE Draw the graphs of the three equations below. a.

y = x2 - 4x -5

YEAR 12 MATHEMATICS

b. y =

c. y = 4x

25

26

Graphs

QUESTION TWO 1.

Identify THREE features of the graphs y = x2 - 3x + 5 and y + x = 4.

2.

Millie bought a car for $11995 some years ago. She knows that the current value of her car may be modelled by the equation: V = $11995(0.82)t, where V is the current value of her car and t is the number of years since she bought her car. a. Plot the graph of this equation of the car’s current value over the six years since she bought it. b. During which year did the value of the car fall under half of the purchase price?

1.

2. V

(Value in $)

t(years)

YEAR 12 MATHEMATICS

Graphs

QUESTION THREE Draw graphs of these equations: a.

(x - 2)2 + (y + 3)2 = 9

b.

y=

YEAR 12 MATHEMATICS

= 3+

27

28

Graphs

QUESTION FOUR Another car depreciation model that Millie discovered is given as: V =

.

V is the value of the car (V) in dollars over t years. The graph of the equation for the current value of the car is shown below. a.

What does the graph tell us about the rate at which the value of the car decreased?

b.

What does the graph indicate about the value of the car after many years?

c.

What does the y intercept tell us about the purchase price of the car? V (Value in $)

t(years)

YEAR 12 MATHEMATICS

Graphs

29

QUESTION FIVE For each of the graphs, write the equation.

y

y

a.

b.

x

c.

YEAR 12 MATHEMATICS

x

30

Graphs

QUESTION SIX Millie’s geology class has been studying volcanic crater lakes of the central North Island. One crater lake that she studied had suddenly filled then burst one of its walls sending a torrent of water, mud and rock down the mountain side. The data from the seismic monitoring station at the lake gave these figures: Time

Lake Depth

Number of hours later

4pm, 10 Feb

6.6m (initial)

0

7am, 11 Feb

15.2m (burst)

15

10am, 13 Feb

8.0m

66

Millie models this situation with two hyperbolae (see graph below).

After the first 15 hours, the depth of the lake could be modelled by this hyperbole: D is the depth of the crater lake (in metres) and t is the time (in hours) since the lake began to fill. a.

Write the equation for the (second) hyperbola which models the lakes depth after 15 hours.

b.

Use your model equation above to estimate the time and the date when the crater lake returns to its initial depth of 6.6 metres.

YEAR 12 MATHEMATICS

Graphs

GRAPHS - PRACTICE TEST 2 QUESTION ONE a.

Draw the graph of y = x(x - 1)(x + 3), showing all intercepts.

b.

Draw the graph of y = 4 - (x + 1)2 showing key features.

YEAR 12 MATHEMATICS

31

32

Graphs

c. Draw the graph of y = log10 x. d. Write three features of the circular graph illustrated below.

y

x

YEAR 12 MATHEMATICS

Graphs

QUESTION TWO Write the equation of each of the following graphs.

a.

c.

YEAR 12 MATHEMATICS

b.

33

34

Graphs

QUESTION THREE Draw graphs of EACH of the following: a.

y = 2x2 - 3x - 5

b.

y = -x3 + 1

YEAR 12 MATHEMATICS

Graphs

QUESTION FOUR

Draw the graph of y = 4

on the axes below for -5 # x #4.

y

x

YEAR 12 MATHEMATICS

35

36

Graphs

QUESTION FIVE Helen and Don invest a sum of money into an education fund which compounds at 8% annually. The amount in the account after t years may be given by the equation y = 45(1.08)t, where y the amount of money is in hundreds of dollars. Below, a graph is given for the first 11 years.

a. What sum did Helen and Don invest initially? b. If interest is calculated and added on at the end of every year, during which year would you expect their original sum to have doubled? c. 12 years after the fund began, Helen and Don need to withdraw $3000 for a family emergency. How would this be represented on the graph?

YEAR 12 MATHEMATICS

Graphs

YEAR 12 MATHEMATICS

37

38

Graphs

QUESTION SIX In the year of a general election a certain government department has been ordered to trim its spending (S) according to this model equation: S = A - B log10 (x + 0.5), where S is in dollars, and x is the number of weeks since the order was given.

By the end of Week 1, the Department has spent $107 449 for that week, but by the end of the tenth week, their weekly spending was $64 011. a.

Find A and B (to the nearest $10), then rewrite the model equation with these values.

b.

If the election was held seven months (30 weeks) after the order was given to the government department, use your model equation to estimate how much had been spent by them in that election week.

YEAR 12 MATHEMATICS

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Calculus

39

MATHEMATICS 2.3 ACHIEVEMENT STANDARD 90286 Find and use straightforward derivatives and integrals


C C C C C C

C

find derivatives of polynomial expressions such as 5x9 - 7x3 + 4 find integrals of polynomial expressions relate the derived function to the gradient of a curve relate the integral to the area under a curve use the derivative to find the gradient at a point and locate the point given a gradient value use the integral to find a straight forward area under a curve, and to extract an equation given the gradient function use differentiation techniques to locate turning points where f!(x) = 0, then determine their nature(s), find the equation of a tangent to a curve and solve rate of change problems such as kinematics

C

using integration techniques to find areas (including compound) under polynomials

C

use various calculus techniques to form equations, to interpret results, to optimise situations, to solve rates of change cases (including kinematics) and to find relevant areas

YEAR 12 MATHEMATICS

From more straightforward uses of calculus techniques and applications and familiarity with , f!(x) and Idx notations

to wider ranging applications and contexts involving those techniques requiring interpretation of the solutions.

40

Calculus

CALCULUS - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. 1.

Differentiate these functions with respect to x: a. y = 3x9 - 5x2 + 7 b. f(x) = (3x - 5)(2x2 + 7)

2.

(Hint: expand, then find f !(x).)

Find the indefinite integrals for: a. f ! (x) = 3x2 + 8x - 11 b.

3.

= 18x5 +

+1

a. Find f!(2) if f(x) = x4 - 5x2 + x. b. Find the gradient of (the tangent to) the curve, y = x2 - 2x, at x = -1.

YEAR 12 MATHEMATICS

Calculus

4.

For the curve y = x2 + 5x: a. Find the equation of the tangent to this curve at (1, 6). b. Find the equation of the tangent to this curve at x = 0.

5.

6.

Find the point on the curve y = x2 - 3x + 2 where the gradient is 1.

a. Evaluate the definite integral

3x2 . dx.

b. Find the area between the curve, y = 2x - x2 and the x axis between x = 0 and x = 2. c. Find the total area between y = x(x - 1)(x + 3) and the x axis.

YEAR 12 MATHEMATICS

41

42

7.

Calculus

Consider the curve, y = x3 - 3x = x(x2 - 3). a. Find an expression for the gradient function

.

b. Determine any turning points on y = x3 - 3x. c. Along which values of x is the curve increasing and decreasing?

YEAR 12 MATHEMATICS

Calculus

8.

43

A large model rocket is fired vertically into the air with an initial velocity of 245 m/s. After t seconds the height of the rocket (h metres) is given by: h = 245t - 4.9t2. a. Find an expression for the instantaneous velocity, v, of the rocket after t seconds. b. What is the velocity of the rocket after 5 seconds? c. What is the height of the rocket at the same time? d. Show that the acceleration of the rocket is constant. e. When does the rocket reach its maximum height above the ground, and what is this height?

YEAR 12 MATHEMATICS

44

9.

Calculus

Optimisation situations require the use of calculus to find the maximum or minimum solution. For example, in a new subdivision the developers wish to create rectangular sections, each having a total boundary (or perimeter) of 108 m. What are the dimensions of such a rectangle, so that its area could be a maximum?

YEAR 12 MATHEMATICS

Calculus

CALCULUS - PRACTICE TEST 1 Show ALL working. QUESTION ONE Find the gradient of the curve y = x3 - 6x - 5 at the point where x = 5.

QUESTION TWO The graph shown below has the equation y = 3x2 + 1. Calculate the shaded area.

YEAR 12 MATHEMATICS

45

46

Calculus

QUESTION THREE The gradient function of a curve is f!(x) = 6x2 - 4x + 5. The curve passes through the point (2, 11). Find the equation of the curve.

QUESTION FOUR Find the x co-ordinates of the two points on the graph of y = 2x3 - 6x + 8 where the gradient is parallel to the x-axis.

YEAR 12 MATHEMATICS

Calculus

47

QUESTION FIVE Find the equation of the tangent to the curve y = x3 - 3x2 - 7x + 1 at the point (-1, 4).

QUESTION SIX Graeme returns to his car at the end of work and realises that he left the lights on, draining the battery. Fortunately he parked on a slight slope earlier in the day, so he can roll the manual geared vehicle to push start it. As the vehicle slowly rolls forward, its velocity is given by v = 0.75t (m/s) where v = velocity in metres per second and t = time in seconds from when the car begins to roll. How far has the car rolled over the first 8 seconds?

YEAR 12 MATHEMATICS

48

Calculus

QUESTION SEVEN Graeme designs rest areas along the edges of major highways. One of his more recent designs was the computer designed area (part of which is shown as the shaded region on the graph below). As edges for the area, he used these three equations: y = 12 - 3x2 y = -36 and

x = 1,

where x and y are in metres.

Calculate the shaded (rest area) region .

YEAR 12 MATHEMATICS

Calculus

49

QUESTION EIGHT To prevent flooding near a rural school the local council asked Graeme to design a drain along the rear boundary. Graeme saves costs by designing a concrete structure to fit into an existing ditch. Together the floor (width) and the heights of the two walls have a total length of 5.6m. Find the width of the floor which will allow the greatest flow of stormwater through the cross-sectional area shown. Also, give this maximum cross-sectional area.

YEAR 12 MATHEMATICS

50

Calculus

CALCULUS - PRACTICE TEST 2 Show ALL working. QUESTION ONE a.

Find the gradient of the curve y = x4 - 3x2 + 5 at the point where x = 2.

b.

Find the equation of the function which passes through the point (-1, 1) and whose gradient function is

= 8x3 + 6x2 - 4x - 1.

YEAR 12 MATHEMATICS

Calculus

c.

Find the area under the curve, y = x3 + 2 for the values of x between 0 and 2.

d.

Find the co ordinates of the point on the curve y =

YEAR 12 MATHEMATICS

where the gradient is

.

51

52

Calculus

QUESTION TWO Find the area between the x - axis and the curve y = (x + 1)(x - 4) = x2 - 3x - 4 for values of x between 0 and 5.

YEAR 12 MATHEMATICS

Calculus

53

QUESTION THREE 3.

An electronic powered model boat is being sailed on a small lagoon. Its velocity, in cm/s is given by: v = 18 + 15t - 3t2 for 0 # t # 6 where t is the time in seconds after the boat is started. a. After 2 seconds the boat is 65 cm from its owner who is controlling it from shore. How far was the boat from the owner at the start?

b. Use calculus to find the maximum velocity of the boat.

YEAR 12 MATHEMATICS

54

Calculus

QUESTION FOUR A manufacturer produces car polish in tin cans which have a volume of 335 cm3. Find the radius of the tin can which requires the least amount of metal. Note - for a cylinder V = πr2h and SA = 2πr2 + 2πrh.

YEAR 12 MATHEMATICS

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P

C1

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Coordinate Geometry

55

MATHEMATICS 2.4 ACHIEVEMENT STANDARD 90287 Use coordinate geometry methods


C

find the mid-point between 2 points

C

find the distance between 2 points

C

find the equation of a line

C

find the equation of a parallel line

C

find the equation of a perpendicular line

C

find the coordinates of the point of intersection

From straightforward contexts in two dimensions

to

of 2 lines

C

find equations of medians, perpendicular

situations with more complexity

bisectors and altitudes including C

formulate a proof (i.e. geometric cases relying on the above techniques

C

prove points are collinear

C

prove more challenging situations

C

solve more challenging contextual problems

YEAR 12 MATHEMATICS

three dimensional possibilities

and

extended chains of reasoning.

56

Coordinate Geometry

COORDINATE GEOMETRY - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. 1.

Find the midpoint between: a. (0, 5) and (4, 11) b. (-1.2, 6) and (4.4, -7.6)

2.

Find the distances between the points in Question 1.

3.

What is the gradient of the line 2y - 3x = 7 ?

YEAR 12 MATHEMATICS

Coordinate Geometry

4.

Find the equation of the line which passes through (-3, 2) and (1, 5).

5.

A line

has the equation 2x + 5y - 8 = 0.

Write the equation of a line (in the form ax + by + c = 0) which is: a. Parallel to

and passing through (1, -2).

b. Perpendicular to

YEAR 12 MATHEMATICS

and passing through (5, 6).

57

58

6.

Coordinate Geometry

Find the points of intersection of these pairs of lines: a. 3x + 4y = 10 5x + 3y = 13 b. 0.1x - 0.2y - 0.7 = 0 0.4x + 0.3y - 0.6 = 0

YEAR 12 MATHEMATICS

Coordinate Geometry

7.

P(1, 5), Q(3, 2) and R(-3, -1) are the vertices of a triangle. a. Find the equation of the median drawn from R to the midpoint of b. Find the equation of the altitude from P to

.

c. Find the equation of the perpendicular bisector of

YEAR 12 MATHEMATICS

.

.

59

60

8.

Coordinate Geometry

Prove that the triangle ªOPQ is isosceles.

YEAR 12 MATHEMATICS

Coordinate Geometry

61

COORDINATE GEOMETRY - PRACTICE TEST 1 A training track for horses has the shape shown. Use coordinate geometry techniques to solve all questions. Use the axes on the grid below to help answer these questions. Note the grid lines are 20 m apart.

YEAR 12 MATHEMATICS

62

Coordinate Geometry

QUESTION ONE a.

Calculate the distance along the length of the back straight between N(4, 11) and S(-10, -2).

b.

Find the equation of the line along this back straight.

c.

A fence line passes through the point (-2, 8) and follows a path parallel to the line y =

x + 5.

Find the equation of this fence line.

YEAR 12 MATHEMATICS

Coordinate Geometry

QUESTION TWO A drain runs along a straight line equidistant between the points (1, -2) and (3, -4). Find the equation of the line which the drain follows.

YEAR 12 MATHEMATICS

63

64

Coordinate Geometry

QUESTION THREE The ends of the back straight N(4, 11) and S(-10, -2) form a triangle with a trough at T(2, 0). Find the equation of the median of this triangle through N(4, 11).

YEAR 12 MATHEMATICS

Coordinate Geometry

65

QUESTION FOUR The equation of the road between the ends S(-10, -2) on the back straight, and P(-2, -10) on the front straight is x + y + 12 = 0. The altitude of the triangle SPT, through the horse trough (2, 0) is given by the equation: x - y - 2 = 0. Calculate the length of the altitude of the triangle SPT through vertex T(2, 0).

YEAR 12 MATHEMATICS

66

Coordinate Geometry

QUESTION FIVE The farmer who owns the property wishes to move the back straight of the track so that it now runs along the line y =

x + 8.

Calculate the closest distance this new piece of track comes to a new trough planned at the position (6, 0).

YEAR 12 MATHEMATICS

Coordinate Geometry

67

COORDINATE GEOMETRY - PRACTICE TEST 2 Part of a mini golf course is shown, with the first five tees (T, symbol !) and holes (H, symbol F). A plan is on the office wall, set to a grid system. The office has co-ordinates (0, 0) and some of the other tees and holes have been given coordinates. Two electrical cables run under the pond, from T3 to T2 and from L3 to L4 and are shown by dashed lines.

Every unit represents one metre. The diagram is not drawn to scale. YEAR 12 MATHEMATICS

68

Coordinate Geometry

QUESTION ONE a.

A solar light, L2, is located halfway between the second tee T2 (25, 15) and the second hole H2 (11, 28). Find the coordinates of L2.

b.

What is the equation of the line from T2 (25, 15) to T3 (1, 24)?

c.

The equation of the path from hole two at H2 to the third tee, T3 is y =

Write the equation of a line which is parallel to at

x-

.

and which passes through the bend

point, B(10, 26).

YEAR 12 MATHEMATICS

Coordinate Geometry

69

QUESTION TWO An old cable running from the bend at C(19, 12), under the pond, meets the service path at the midpoint between the two tees, T1(11, -2) and T3(1, 24). What is the length of this underground cable?

YEAR 12 MATHEMATICS

70

Coordinate Geometry

QUESTION THREE Consider a triangle formed by the three points T3, H2 and T2. Show how the equation of the altitude of the triangle T3, H2, T2 which passes through the vertex at T2 is 5x + 2y - 155 = 0.

YEAR 12 MATHEMATICS

Coordinate Geometry

71

QUESTION FOUR The path for the fifth hole has two ‘legs’. The first leg starts at T5(9, 0) and runs perpendicular to the line T1 H1 until it gets to the bend at D. From D, the second leg runs along a path which is perpendicular to the line BH3, ending at the hole H5 (3, 15). What are the co-ordinates of the bend at D?

YEAR 12 MATHEMATICS

72

Coordinate Geometry

QUESTION FIVE Another light is going to be located halfway between T3 and B so that the area of the north side of the pond can be lit up. A cable will run from this point and be connected to the existing cable which runs between T3 and T2. What will be the shortest distance between the new light and the line T3 T2?

YEAR 12 MATHEMATICS


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Sample Statistics

73

MATHEMATICS 2.5 ACHIEVEMENT STANDARD 90288 Select a sample and use this to make an inference about the population


C

select a sample from a population (possibly supplied)

C

provide evidence of the method

C

use appropriate sample statistics such as: mean median, quartiles, standard deviation, and proportions from the selected sample

C

describe the sampling method so that another

From more straightforward inferences taken from a sample about the population to

person could repeat the process C

comment on whether the sample is truly

fuller justification of the method(s) used

representative, or shows bias C

refer to sample statistics (and suitable graphs) to help justify the above position

C

when evaluating the sampling process, consider

leading to a more critical evaluation of the whole sampling process

limitations of, and possible improvements to reliability

and the results.

C

evaluate the accuracy of the results, considering an improved interval for the question

C

YEAR 12 MATHEMATICS

refer to how the data is distributed

74

Sample Statistics

SAMPLE STATISTICS - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. 1.

Find the mode and the mean of this data set: 109, 59, 130, 40, 42, 33, 32, 40, 71, 138, 203, 162, 84, 136, 154, 39, 55, 36, 30, 40, 33, 36, 38, 87, 119, 36.

2.

The number of nose-to-tail accidents during Labour weekend along a particular stretch of motorway is given below for each direction. The results have been recorded over the last 2 decades. North Bound Lanes: 14, 11, 25, 19, 10, 8, 23, 34, 21, 15, 20, 12, 25, 11, 13, 31, 10, 5, 33, 22 South Bound Lanes: 16, 18, 8, 14, 24, 2, 18, 13, 11, 21, 29, 17, 5, 16, 14, 20, 10, 9, 16, 30 a. Create a back to back stem and leaf plot for each data set. b. List the five point summary for each data set (high, median, quartiles, low). c. Form box and whisker plots for each on the same grid. d. Calculate the mean number for each direction.

YEAR 12 MATHEMATICS

Sample Statistics

3.

For the frequency table: x

1

2

3

4

5

y

13

9

4

2

2

a. Calculate the mean. b. Calculate the standard deviation of the data.

4.

The cost of a single person to stay one night at a motel in an east coast city varies considerably. This frequency table shows the tariffs which were charged in 2005. a. In which interval does the median price lie?

Price

Midpoint of Interval

Frequency

$50 -

$55

1

$60 -

$65

3

$70 -

$75

9

$80 -

$85

4

$90 -

$95

2

$100 -

$105

2

$110 -

$115

3

$120 - 130

$125

1

b. Estimate the mean price for a single person to stay in a motel. c. Draw a cumulative frequency curve of the data.

YEAR 12 MATHEMATICS

75

76

Sample Statistics

Each table below has columns to assist the manual calculation of the sample mean and sample standard deviation. a.

Complete each table, then use the formulae to find each sample statistic.

b.

Verify these answers, by using your calculator to find the values.

5.

6. x

Totals

(x - )2

x

f

25

11

2

14

8

7

15

13

7

16

16

9

21

9

22

7

25

5

x-

3

-5

6

-2

10

2

14

6

56

-

36

x.f

(x - )2

f.(x - )2

Totals

=

=

=

=

S =

S =

=

=

YEAR 12 MATHEMATICS

Sample Statistics

77

SAMPLE STATISTICS - PRACTICE TEST 1 Select a sample and use this to make an inference about the population. THE SITUATION A newspaper claims that recent significant earthquakes seemed to strike more often in the early hours of the morning rather than at any other time of the day. YOUR TASK Your local newspaper editor has asked you to check this claim and estimate which times of the day of an arbitrarily chosen year had the most quakes. YOU NEED TO: a. Choose a sampling method and use it to obtain a representative sample of at least 30 significant earthquakes. b.

Describe your sampling process clearly so that someone else can follow it.

c.

Justify your choice of sampling method by describing the decisions made and the reason for these decisions.

d.

List the data for your sample that you gathered from the data sheet.

e.

Explain whether your sample is actually representative of your population or not. (You do not have to select another sample if it is not.)

f.

Check appropriate statistics for your sample and use this to estimate when in the day there were significant earthquakes.

g.

Write a short paragraph outlining what you have found (from your estimates), and what you could conclude about when a significant earthquake occurred. Comment on the reliability of your estimate and therefore your conclusion.

h.

Evaluate the sampling and statistical processes you have used. Comment on things such as: C

reliability of your sampling process

C

limitations of your sampling process

C

the accuracy of your estimate - when an earthquake of this size was most likely

C

distribution of the data

You need at least three valid statements.

YEAR 12 MATHEMATICS

78

Sample Statistics

DATA SHEET - Significant Earthquakes of the World - 2005 (of magnitude 6.5 or greater and/or causing fatalities, injuries or substantial damage) Data from US Geological Survey, Earthquake Hazards Program: http://earthquake.usgs.gov/eqcenter/eqarchives/significant/sig_2005.php Downloaded 13 Jan 2006. Date and Time

Magnitude

Date and Time

Magnitude

Jan 01, 0625 Jan 10, 1847

6.7 5.4

off West coast of N. Sumatra N. Iran

Jun 04, 1450 Jun 06, 0741

6.1 5.7

E. New Guinea, PNG E. Turkey

Jan 10, 2348 Jan 12, 0840

5.5 6.8

W. Turkey Central-mid Atlantic Ridge

Jun 13, 2244 Jun 14, 1710

7.8 6.8

Tarapaca, Chile Aleutian Is, Alaska

Jan 16, 2017 Jan 19, 0611

6.6 6.6

Yap, Micronesia off E. Honshu, Japan

Jun 15, 0250 Jun 15, 1952

7.2 6.5

off N. Californian Coast off Aisen coast, Chile

Jan 23, 2010 Jan 25, 1630

6.3 4.8

Sulawesi, Indonesia Yunnan, China

Jun 16, 2053 Jun 17, 0621

4.9 6.7

greater L.A. area, California off N. Californian coast

Jan 25, 1644

5.9

Turkey - Iraq border

Jun 20, 0403

4.7

off W. Honshu, Japan

Feb 02, 0555 Feb 05,0334

4.8 6.6

Java, Indonesia Anatchan, N.Mariana Is

Jul 02, 0216 Jul 05, 0152

6.6 6.7

off Nicaragua Coast Nias Region Indonesia

Feb 05, 1223 Feb 08, 1448

7.1 6.8

Celebos Sea Vanuatu

Jul 05, 1653 Jul 23, 0734

2.7 6.0

S. Africa off S. Honshu, Japan

Feb 14, 2338 Feb 15, 1442

6.1 6.6

S. Xinjiang China Kepulauan, Indonesia

Jul 24, 1542 Jul 25, 1543

7.3 5.0

Nicobar Is, India Heilongjiang, China

Feb 15, 1946 Feb 16, 2027

5.5 6.6

off S. Honshu, Japan S. mid-Atlantic Ridge

Aug 05, 1414

5.2

Yunnan, China

Feb 19, 0004 Feb 22, 0225

6.5 6.4

Sulawesi, Indonesia Central Iran

Aug 13, 0458 Aug 16, 0246

4.8 7.2

Yunnan, China off E. Honshu, Japan

Feb 26, 1256

6.8

Simeulue, Indonesia

Aug 21, 0229

5.1

off W. Honshu, Japan

Mar 02, 1042 Mar 02, 1112

7.1 4.9

Banda Sea Pakistan

Sep 09, 0726 Sep 24, 1924

7.7 5.6

New Ireland region, PNG Ethiopa

Mar 05, 1906 Mar 09, 1015

5.8 5.0

Taiwan S.Africa

Sep 26, 0155 Sep 29, 1550

7.5 6.7

N. Peru New Britain region, PNG

Mar 12, 0736 Mar 14, 0155

5.7 5.8

E.Turkey E. Turkey

Oct 01, 2219

5.3

S. Peru

Mar 14, 0943 Mar 20, 0153

4.9 6.6

Maharashtra, India Kyushu, Japan

Oct 08, 0350 Oct 15, 0424

7.6 5.2

Pakistan SW Kashmir

Mar 21, 1223 Mar 28, 1609

6.9 8.7

Salta, Argentina N.Sumatra, Indonesia

Oct 15, 1551 Oct 16, 0705

6.5 5.1

NE of Taiwan E. Honshu, Japan

Apr 10, 1029

6.7

Kepulauan, Indonesia

Oct 20, 2140 Oct 27, 1118

5.9 4.2

off W. Turkey Guangxi, China

Apr 10, 1114 Apr 11, 1220

6.5 6.7

Kepulauan, Indonesia N. Coast of N.Guinea

Oct 29, 0405

6.5

S.E. Indian Ridge

Apr 11, 1708 Apr 19, 2111

6.8 5.5

S.E. of Loyalty Islands Kyushu, Japan

Nov 06, 0211 Nov 08, 0754

5.2 5.1

Pakistan S.China Sea

May 01, 1623

4.5

Kyushu, Japan

Nov 14, 2138 Nov 17, 1926

7.0 6.9

off E.Honshu, Japan Potosi, Bolivia

May 03, 0721 May 05, 1912

4.9 6.5

W. Iran S. of Panama

Nov 19, 1410 Nov 26, 0049

6.5 5.2

Simeulue, Indonesia Huber-Jiangxi, China

May 12, 1115 May 14, 0505

6.5 6.8

Pacific-Antartic Ridge Nias Region, Indonesia

Nov 27, 1022

6.0

S. Iran

May 16, 0354 May 19, 0154

6.6 6.9

S. of Kermadec Islands Nias Region, Indonesia

Dec 02, 1313 Dec 05, 1219

6.5 6.8

off E. Honshu, Japan L. Tanganyila Region, Tanzania

May 23, 0609

4.3

S. Africa

Dec 11, 1420 Dec 12, 2147

6.6 6.6

N. Britain Region, PNG Hindu Kush region, Afghanistan

Dec 13, 0316 Dec 24, 0201

6.7 4.5

Fiji region W. Honshu, Japan

Location

Location

YEAR 12 MATHEMATICS

Sample Statistics

YEAR 12 MATHEMATICS

79

80

Sample Statistics

YEAR 12 MATHEMATICS

Sample Statistics

81

SAMPLE STATISTICS - PRACTICE TEST 2 THE SITUATION A marine farm, about 800m offshore, has ropes anchored to the sea floor attached to floating longlines at the surface. Shellfish grow along each rope and these are shown as numbered spaces on the map (of the farm layout) below. 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

214

213

186

185

158

157

130

129

102

101

74

73

46

215

212

187

184

159

156

131

128

103

100

75

72

47

216

211

188

183

160

155

132

127

104

99

76

71

48

217

210

189

182

161

154

133

126

105

98

77

70

49

218

209

190

181

162

153

134

125

106

97

78

69

50

219

208

191

180

163

152

135

124

107

96

79

68

51

220

207

192

179

164

151

136

123

108

95

80

67

52

221

206

193

178

165

150

137

122

109

94

81

66

53

222

205

194

177

166

149

138

121

110

93

82

65

54

223

204

195

176

167

148

139

120

111

92

83

64

55

224

203

196

175

168

147

140

119

112

91

84

63

56

225

202

197

174

169

146

141

118

113

90

85

62

57

226

201

198

173

170

145

142

117

114

89

86

61

58

227

200

199

172

171

144

143

116

115

88

87

60

59

228

229

230

231

232

233

234

235

236

237

238

239

240

241

255

254

253

252

251

250

249

248

247

246

245

244

243

242

YEAR 12 MATHEMATICS

82

Sample Statistics

YOUR TASK The owners of the marine farm have asked you to help estimate the average mass of shell fish per rope in the farm. By selecting a sample of the ropes: 1.

Choose a sampling method and use it to design a sampling process to obtain a representative sample of at least 30 shellfish ropes. Explain your process clearly so that someone else could follow it.

2.

Justify your choice of sampling method taking into account the population and considerations of bias. You may like to describe any decisions you made and the reasons for these decisions.

3.

Use your sampling process to select a sample of shellfish ropes.

4.

List the data for your sample which you gathered from the data sheet.

5.

Explain whether your sample is representative of your population or not. You do not have to select another sample if it is not.

6.

Calculate appropriate statistics for your whole sample.

7.

Use your sample and the statistics you have calculated to: a. Estimate the average mass of shellfish per rope for the whole farm. b. Estimate a suitable measure of spread for the mass of shellfish per rope for the whole farm.

8.

Use your estimate to inform the marine farm owners of your conclusion as to the average mass of shellfish per rope in the farm. Justify your estimates and therefore your conclusion.

9.

Evaluate the sampling process you used. Comment on things like: C

the reliability of your sampling process

C

limitations of your sampling process and ways to improve it

C

accuracy or appropriateness of your estimate

C

the distribution of your data

You need at least 3 valid statements.

YEAR 12 MATHEMATICS

Sample Statistics

83

Table showing mass (in kg) of shellfish on each rope.

1

404

38

488

75

165

112

358

149

159

186

370

223

429

2

413

39

187

76

168

113

269

150

480

187

208

224

446

3

160

40

449

77

508

114

504

151

394

188

139

225

331

4

451

41

397

78

227

115

291

152

257

189

520

226

314

5

337

42

309

79

207

116

189

153

187

190

357

227

180

6

477

43

357

80

186

117

164

154

248

191

499

228

401

7

499

44

255

81

395

118

310

155

378

192

174

229

360

8

186

45

185

82

459

119

195

156

389

193

411

230

219

9

445

46

509

83

191

120

479

157

135

194

504

231

208

10

108

47

185

84

247

121

178

158

407

195

463

232

352

11

216

48

307

85

279

122

484

159

220

196

458

233

267

12

173

49

345

86

468

123

207

160

249

197

285

234

409

13

286

50

175

87

247

124

161

161

281

198

308

235

190

14

191

51

143

88

366

125

252

162

192

199

348

236

167

15

458

52

309

89

336

126

264

163

257

200

240

237

480

16

145

53

365

90

207

127

518

164

294

201

348

238

375

17

425

54

506

91

225

128

210

165

485

202

257

239

385

18

447

55

361

92

275

129

158

166

139

203

381

240

251

19

306

56

490

93

177

130

195

167

460

204

190

241

310

20

327

57

440

94

164

131

338

168

501

205

507

242

428

21

240

58

336

95

458

132

406

169

409

206

214

243

228

22

385

59

359

96

493

133

311

170

350

207

460

244

177

23

438

60

159

97

456

134

411

171

167

208

338

245

201

24

447

61

498

98

168

135

403

172

506

209

464

246

355

25

447

62

291

99

501

136

367

173

298

210

208

247

196

26

405

63

155

100

305

137

490

174

373

211

447

248

265

27

248

64

158

101

413

138

362

175

149

212

191

249

408

28

318

65

429

102

307

139

176

176

334

213

268

250

506

29

165

66

370

103

361

140

195

177

381

214

452

251

177

30

316

67

476

104

489

141

461

178

393

215

295

252

236

31

379

68

257

105

472

142

486

179

232

216

208

253

325

32

354

69

187

106

468

143

348

180

407

217

465

254

297

33

445

70

184

107

175

144

278

181

446

218

260

255

434

34

245

71

344

108

343

145

411

182

160

219

162

35

202

72

440

109

399

146

287

183

155

220

259

36

407

73

187

110

516

147

396

184

274

221

416

37

500

74

410

111

479

148

404

185

362

222

382

YEAR 12 MATHEMATICS

84

Sample Statistics

YEAR 12 MATHEMATICS

Sample Statistics

YEAR 12 MATHEMATICS

85

86

Sample Statistics

YEAR 12 MATHEMATICS

5


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Probability & Normal Distribution

87

MATHEMATICS 2.6 ACHIEVEMENT STANDARD 90289 Simulate probability situations and apply the normal distribution


C

design a simulation process using, for example, coins, dice or random numbers

C

use z = transformation)

C

find expected numbers

C

apply theoretical techniques using: probability trees, tables, informal conditional probability

C

make predictions from simulations

C

apply the normal distribution using more than one z-value

C

From

(the standard normal

find expected numbers from theoretical

a straightforward simulation process and normal distribution calculations to using theoretical probability

probability and normal distribution cases and C

inverse normal distribution problems

C

interpret results from normal distribution cases and make recommendations

C

relate the results of one simulation to a second

the normal distribution in contexts requiring several steps

simulation C

combine theoretical and experimental probabilities

C

discuss any limitations of the model (or process) used

YEAR 12 MATHEMATICS

and further interpretation of the model and the results.

88


PROBABILITY & NORMAL DISTRIBUTION - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. 1.

Find the probabilities of each event: a. Choosing a vowel out of all the letters in the name “WAIKAREMOANA”. b. Rolling a die and not selecting a prime number. c. Picking a Jack or a Queen or a King from a shuffled deck of cards (Jokers removed).

2.

A wheel is divided into 5 sections labelled A, B, C, D and E. The wheel is spun around its centre, then a dart is thrown at it. i. What is the probability that the dart lands in: a.

Region A?

b.

Regions C or E?

c.

Not in region B?

d.

In regions A, B or D?

ii. If the wheel was spun 150 times, how many hits would you expect in region B?

YEAR 12 MATHEMATICS


3.

89

Two dice are thrown and the sums of the top two faces are recorded. a. What is the chance that the sum is an even number? b. What is the probability that the sum is 7? c. What is the probability that the sum has double digits? d. What is the probability that the sum is a square number? e. Find the probability that the sum is a multiple of three. (First draw a table of possibilities.)

4.

Niko is a promising young school athlete who has been training on these throwing events - discus, shot put and hammer throw for the local championships. He knows that he has a 70% chance of winning the discus, an 80% chance of winning the shot put and a 75% chance of winning the hammer throw. The order of the events is as above. Assume the result of each event is independent of the others. a. Draw a probability tree of the possible outcomes. b. What is the probability that Niko wins all three events? c. Find the probability that Niko wins any two of the three events. d. Given that Niko wins the first event (discus) what is the chance that he loses the next two?

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90

5.


A paint manufacturer supplies tins of paint which are normally distributed with a mean volume of 4120 ml and a standard deviation of 60ml. a. What is the probability that a randomly chosen tin of paint has between 4000 ml and 4180 ml? b. If the manufacturer claims that each tin contains 4 l, what is the chance that a randomly selected tin has less than 4 l.

YEAR 12 MATHEMATICS


6.

Use the Normal Distribution table (page 171) to help answer these questions. a. Find:

b. Calculate:

c. A normal distribution has a mean of 55 and a standard deviation of 8. Use the conversion formula to find: i.

Pr( x > 61 ).

ii.

Pr( 50 < x < 66 ).

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91

92


PROBABILITY & NORMAL DISTRIBUTION PRACTICE TEST 1 Show ALL working. There are two parts to this activity: Section A

Requires you to design and carry out a simulation to answer questions.

Section B

Tests your knowledge of the Normal Distribution. For each question you should write correct probability statements and show working to support your answer.

These formulae may be useful: z=

or

z=

The Good Garden Bag Company provides large bags for people to put their garden clippings into. These large bags are then collected once a month and replaced by an empty bag. The clippings are tipped into a truck which compacts then transports the clippings to a composting site. The collectors on each truck monitor the clippings they pick up - they suspect that 10% of the bags will have bamboo, while 25% will contain flax. When either bamboo or flax is discovered the homeowner is given a warning letter which advises them to refrain from placing either of these (forbidden) items in their bag. The Good Garden Bag company asks you to investigate the situation. They believe that the presence of bamboo and flax are independent of each other. (Both bamboo and flax are not wanted since these two plants do not break down very readily and they may jam the auger at the composting plant). SECTION A Design a way to simulate the bag collection of a randomly selected truck, to find out how many of the next 80 homeowners will need to be given the written warning letter. You need to: 1.

a. Describe a method you use in sufficient detail so that another person could repeat it again with your help. b. Carry out at least 80 trials of the simulation. c. Record the result of each trial of the simulation, e.g. in a table. d. Use the results of your simulation to find the number of homeowners who will receive a letter of warning for placing these materials in their bags.

YEAR 12 MATHEMATICS


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94

2.


In a normal day a driving team could collect 200 bags. a. Use theoretical probability to calculate the expected number of households who would receive a warning letter. b. Use theoretical probability to calculate how many of the 200 households would be expected to have both bamboo and flax in their garden bags. c. Use the results of your simulation to find the expected number of homeowners who should get the letter if 200 households have their bags collected. d. Use the results of your simulation to find the expected number of households of the next 200 that would get a letter for having both flax and bamboo in their bags.

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3.

95

Compare the results of your simulation with the theoretical probability. Make at least one comment about your simulation. You could comment on any similarities or differences between the simulation results and the theoretical probability, or you could comment on ways in which your simulation could be improved so that it is a better model.

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SECTION B The operators also gather evidence on the masses of the bags they collect. The maximum limit for each bag should be 100 kg because manoeuvring these down driveways and paths to a truck can be very challenging. Analysis of the results showed that the masses of bags were normally distributed with a mean of 84 kg and a standard deviation of 8.5 kg. Note: A suitably shaded design or use of proper notation is the minimum working expected. 4.

Find the probability that a randomly chosen bag: a. weighs between 84 kg and 90 kg b. weighs under 94 kg c. weighs under the maximum.

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5.

97

a. Find the percentage of bags: i.

which weighed between 100 kg and 105 kg

ii.

which weighed over 105 kg, so that the homeowner receives a letter reminding them of the weight limit.

b. Bags which weigh over 104 kg are monitored for statistical purposes. Out of a 4 day collection of 850 bags, how many would be expected to be over 104 kg?

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6.


Very light bags are considered by some drivers as a waste of time, since they still require one person to collect, replace and load the bag. If 8 bags out of a sample of 175 were found to be “too light”, use the Normal Distribution and this information to describe a “very light bag” in terms of its mass.

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PROBABILITY & NORMAL DISTRIBUTION PRACTICE TEST 2 Show ALL working. Vai is collecting a set of 5 movie character wrist bands which can be found inside packets of Revita cordial drinks. The manufacturer of Revita experienced some packing machine malfunctions with 30% of the packs containing 2 wrist bands instead of 1 band. 1.

Design a simulation to predict the number of packets of Revita that are required to obtain a full set of movie character wrist bands. Describe this simulation in sufficient detail so that another person could repeat it without your help.

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100


2.

Carry out the simulation and record your results.

3.

Use your simulation to work out how many packets of Revita are needed on average to get a complete set.

YEAR 12 MATHEMATICS


4.

101

In Vai’s class there are 28 students altogether (including Vai) trying to collect the 5 wrist bands. Use the results of your simulation to estimate: a. How many students will have 4 different wristbands in the set after collecting 5 wristbands. b. How many students would be expected to have the full set of 5 wrist bands once they have collected less than or equal to 10 wristbands?

5.

After a while, 25 of the 28 students in Vai’s class have only one wrist band to collect to make the complete set. a. Use theoretical probability to predict how many of these 25 students will complete their set with the next packet of Revita. b. Use theoretical probability to predict how many of these 25 students will complete their set given that the next packet contains two wrist bands.

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The weights of Revita Packs of cordial are normally distributed with a mean of 315 grams and a standard deviation of 6 grams. 6.

What is the probability that a pack of Revita cordial weighs between 315 grams and 323 grams?

7.

What proportion of Revita packets weigh less than 323 grams?

A large carton delivered to a local supermarket contains 950 packets of Revita cordial. 8.

How many of these packets would you expect to weigh less than 310 grams?

YEAR 12 MATHEMATICS


9.

How many of these packets would you expect to weigh between 310 grams and 321 grams?

10.

What is the chance that a packet of Revita cordial will weigh between 321 and 324.5 grams?

11.

The manufacturer of Revita regularly checks the machines which weigh the packets of cordial. Packets under 300 grams or over 325 grams are rejected. What proportion of the packets will the manufacturer reject?

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104

12.


One day 599 packets are checked and 3 are found to be very light. If the machine is functioning normally, what is the maximum weight of a very light packet according to this sample?

13.

Calculate the mean weight that the machine needs to be set at so that 85% of Revita cordial packets exceed 308 grams? (Assume the same standard deviation.)

YEAR 12 MATHEMATICS


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Sequences

105

MATHEMATICS 2.7 ACHIEVEMENT STANDARD 90290 Solve straightforward problems involving arithmetic and geometric sequences


C

find general terms of an arithmetic progression (AP)

C

find general terms of a geometric progression (GP)

C

find partial sums of an AP

C

find partial sums of a GP

C

find the sum to infinity, S4 of a GP

C

use sigma (3) notation

C

manipulate formulae to find a, d or r

C

apply the techniques in contexts such as radio active decay, increasing/decreasing % and experiments which create sequences

C

use logarithmic equations to find n in GP’s

C

compare sequences

C

discuss long term effects from the results (process) used

YEAR 12 MATHEMATICS

Work with straightforward cases where a, d and r are evident

to situations where other techniques are used to solve problems which require further interpretation.

106

Sequences

SEQUENCES - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. 1.

Write the first four terms of the sequence whose nth term is defined by tn = n +

.

2.

Write down the next two terms of 0, 3, 8, 15, ......, .......

3.

Find the first term, a, and the common difference, d, for the arithmetic progression which has t4 = 4, t5 = 7 and t6 = 10.

4.

Find the eighth term and the sum of the first sixteen terms for the arithmetic progression 8, 14, 20, ......, .......

YEAR 12 MATHEMATICS

Sequences

5.

107

An Arithmetic Progression has t1 = 8 and t12 = 41. Find the general term, and the sum of the first 13 terms.

6.

How many terms of the series, 7+9+11+13+15+ ...... = 352 (i.e. find n when Sn = 352).

7.

For the Geometric Progression 3, 9, 27, ...... find the next two terms, and the eleventh term.

8.

A Geometric Progression has a fifth term of -80 and a sixth term of 160. Find an expression for the general term, tn.

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9.

Sequences

a. Find the sum to twelve terms of the series 3200 + 1600 + 800 + ...... (Leave your answer as a fraction.)

b. Find the sum to infinity for the same series.

10.

Evaluate: a.

b.

3k + 1

(2n - n2)

YEAR 12 MATHEMATICS

Sequences

SEQUENCES - PRACTICE TEST 1 Show ALL working. QUESTION ONE Carol spends 15 minutes texting her friends on the first day she bought her new mobile phone. She spends 19 minutes texting on Day 2. She spends 23 minutes texting on Day 3. She continues texting daily at the same rate. a.

How long will she spend texting on the sixteenth day?

b.

How much time will she spend texting on her new mobile phone over the first sixteen days?

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110

Sequences

QUESTION TWO Carol’s friend, Tony, spends 24 minutes texting his friends and family on the first day he got his mobile phone. Each day he increases the time spent texting by 5% from the day before (i.e. he spends 1.05 times as many minutes as he did the previous day). What is the total time Tony has spent on his phone in the first 20 days?

QUESTION THREE The Hong family lease a home security system and make payments every month. The payments reduce each month by the same percentage. They paid $68 in the second month. They paid $49.13 in the fourth month and $35.50 in the sixth month and so on. How much did they pay in the first month?

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Sequences

111

QUESTION FOUR Tony buys a DVD for $495. The DVD depreciates at the rate of 26% per year. How many years to the nearest year, will it take for the DVD to reduce to one-sixth of its original value?

QUESTION FIVE One of Tony’s uncles helps him to save money by employing him after school. In week 1 he pays Tony $96. Each week, the uncle reduces the hours and the pay by 17.5% so that Tony can return slowly to full time study. If this continued indefinitely, how much would Tony’s uncle pay him in total?

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Sequences

QUESTION SIX After several months of using her new mobile phone, Carol decides to reduce her texting by the same number of minutes each week. In Week 7, she texted for 595 minutes and by the end of Week 21, her total texting had amounted to 11,235 minutes. How many minutes did Carol text in Week 1 of her texting reduction plan?

YEAR 12 MATHEMATICS

Sequences

113

SEQUENCES - PRACTICE TEST 2 Show ALL working. QUESTION ONE The rungs of a triangular painting trestle decrease uniformly in length. The bottom rung is 88 cm and each successive rung is 2.75 cm shorter than the previous (lower) rung. If there are 13 rungs on each trestle, what is the length of the top rung?

QUESTION TWO A painter, Moe, has just bought some roller blades and is keen to practise as much as he can. On the first evening after work he roller blades 6 km. Each evening he increases this by 1.25 km more than the previous evening. If his first day of roller blading was on October 1, what was the total distance Moe would have roller bladed by the end of October (31 days)?

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Sequences

QUESTION THREE On one of the painting jobs he had, Moe noticed a Yucca plant on the sundeck of the house. The height of the Yucca was 95 cm and each week he noticed that it grew upwards by another 4% (i.e. 1.04 times taller). Calculate the height of the Yucca at the end of week 10 (the nearest cm).

QUESTION FOUR On another job, Moe saw the owner planting bamboo along a border. In week 1 it grew 48 cm after initially being at ground level. Each week the bamboo’s extra growth length is 12.5% less than the previous week. What is the maximum height the bamboo will ever grow?

YEAR 12 MATHEMATICS

Sequences

115

QUESTION FIVE In a park a fountain and statue are surrounded by circular concrete rings. Moe and his team must repaint the statue (S) with anti-graffiti paint. The concrete rings are centred around the fountain and have different concrete textures. The percentage increase in area from one ring to the next is constant. The first ring has an area of 201 m2 while the third ring has an area of 547.22 m2. What is the area of the outside ring (Ring 4)?

QUESTION SIX In the same park as the statue and fountain, a retaining wall has been constructed to hold in an earth bank. Twelve horizontal timber pieces have been used and each is 150 mm shorter than the one below. If 25.5 metres of timber was used altogether what must the length of the lowest piece be?

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Sequences

QUESTION SEVEN Moe plans to set up his own painting company so he has been looking for a tidy, recent model, second hand van to transport his equipment to jobs. He sees one that will be perfect and notes that there are two possible payment regimes. PLAN A pay a deposit of $4950 first month pay $300 each successive month pay $20 more than the previous month PLAN B pay a deposit of $7995 first month pay $615 each successive month pay $10 less than the previous month Calculate how many months it will take before the total paid into Plan A would be the same amount paid into Plan B.

YEAR 12 MATHEMATICS

5


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Trigonometry Problems

117

MATHEMATICS 2.8 ACHIEVEMENT STANDARD 90291 Solve trigonometry problems requiring modelling of practical situations


C

take measurements in a practical situation with suitable calculations to follow

C

From

calculations could include use of the Sine Rule, Cosine Rule and/or areas of triangles

measurements taken lead to

C

contexts to explore could be bearings, relative velocity, etc

C

find length and angles using: Sine Rule and/or Cosine Rule

C

straightforward calculations

find circular measures using: arc length and/or sector areas

to more complex trigonometry situations where

C

find triangular areas suitable models

C

use 2 dimensional representations of 3 dimensional situations

C

combine any of the above techniques to solve more integrated contextual problems

and rules are selected resulting in sensibly rounded solutions (in context) with appropriate units.

YEAR 12 MATHEMATICS

118


TRIGONOMETRY PROBLEMS - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. (Note, diagrams have not been drawn to scale.) 1.

Calculate the lengths or angles that have been marked. a. b.

c.

2.

Use the sine rule or cosine rule to find the missing lengths. a. b.

3.

Use the sine rule or cosine rule to find the missing angles. a. b.

4.

a. Convert these to radians. Give your answer to 2 decimal places. i. 30E

ii. 165E

iii. 238E

b. Change these to degrees. Give your answer to 1 decimal place. i. 1.06 rad

ii.

rad

iii. 5.89 rad

YEAR 12 MATHEMATICS


YEAR 12 MATHEMATICS

119

120

5.


Calculate the marked dimensions in these sectors. a. b.

c.

(Find θ in radians and degrees)

6.

Find the areas of these figures. a. b.

c.

d.

Find the shaded area.

7.

Three fisherman are angling along the banks of a river. Two of them, F and G, are on the same bank and are 50 metres apart. The third, H, is on the opposite bank of the river. It is known that pHFG is 62E and angle pFGH is 66E. How wide is the river estuary?

YEAR 12 MATHEMATICS


121

PRACTICAL TRIGONOMETRY PROBLEMS PRACTICE TEST 1 Show ALL working. This model assessment is in two parts. PART A 1. You and some other students go to a local park near your school. On a level piece of ground your teacher has marked out a large four sided area. You have been given tape measures, magnetic compasses, trundle wheel and alidade. 2.

From a reference point, X, inside this quadrilateral, you must begin your measurements. Note that the point X is not allowed on any sides nor any diagonals of the quadrilateral.

3.

You draw the quadrilateral and include point X.

4.

Using the equipment, from point X, complete a radial survey. Indicate the measurements you will need to take to calculate the area of the quadrilateral and mark these on your diagram.

NOTE Usually you need to draw your diagram from scratch. If you use the diagram given, the highest grade you will be awarded is ‘Achievement’. For the purpose of continuing PART A, assume that you have drawn the shape below (to scale) and it is 1: 1500.

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123

PART B Use the sketch and your measurements from PART A to complete this section of the task. You do not need any other measurements to complete this. 1.

Council drainage contractors are to check the old pipe which lies diagonally across the quadrilateral in Part A. Calculate the shortest distance between the pairs of opposite corners (i.e. the lengths of both diagonals of the quadrilateral).

2.

A memorial rose garden is going to be set up in the North-West corner of this quadrilateral. To assist with the planning, you are asked to find the size of the interior angle in that corner of the quadrilateral.

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3.


As the contractors explore the old drainage pipe, they find that there is a thick layer of sludge and sediment which covers the bottom of the pipe. The radius of the pipe (OP or OQ) is known to be 1.25 m. a. If the distance across the top of the sediment (i.e. chord

) is

2.24 m, then calculate the area of the cross-section of the sediment (shown by the shaded area on the diagram). b. Assume this measurement is the average width of the sediment in the pipe. Also assume that the pipe runs along the longer of the two diagonals you calculated in #1. What volume of sediment is the pipe holding? (Round this number to the nearest 10 units.)

YEAR 12 MATHEMATICS


125

PRACTICAL TRIGONOMETRY PROBLEMS PRACTICE TEST 2 Show ALL working. This model assessment is in two parts. PART A 1. As part of a measuring task, a class of Year 12 students met their teacher at the local ice skating rink. There they were paired up and given the measuring task, with a measuring tape and, of course, ice skates! An arbitrary point, Q, on the blue line was given to them by the teacher. They had to measure and record the lengths needed to find the angle pLQR. (Every student pair had a different location for Q.) A triangle is set with Q at one corner, on the blue line, and the other two corners, L and R, as the goal posts. NOTE:

If a pair of students couldn’t identify the measurements which they needed to take, they were supplied with a help sheet. This usually has the diagram of the triangle ªLQR, supplied with labels L, R and Q. If students use the help sheet, then the highest grade which they can be awarded for this task is ‘Achievement’.

At the end of the session the measurements are usually handed in to the teacher.

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PART B 1.

For the position given (to one pair of students) and the measurements provided, calculate the angle in the triangle on the blue line, pLQR.

2.

If a student was standing at point Q, looking towards the goal mouth, what would the “apparent width” of the posts be to her? i.e. find the length

on the diagram.

(Note: ªRXQ is not right angled but isosceles.)

YEAR 12 MATHEMATICS


3.

127

a. A student, Wayne (W) is 4.29 m from the right goal post, R. The distance between the goal posts (LR) is 1.83 m. The bearing of the left post, L, from Wayne is 093E and the bearing of the right post, R, from Wayne is 118E. Another student, Sheena (S) is standing further along the goal line, leaning on the rink wall.

What is the bearing of Sheena (S) from the left goal post? Note - round all angles to a nearest degree.

b. How far is Wayne from the left goal post?

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128

4.


Near each goal line there are two “face off” spots. Each “face off” spot is the centre of a circle with a 4.57 metre radius. A sponsor usually paints their logo or one of their products inside these circles before big games are played. One sponsor who imports sports gear has painted a sports helmet inside the circle, and centred it at the “face off” spot, O. The angle (below) pAOC = 2.17 radians, while AO = OC = 3.36 metres. Calculate the area inside the circle which has not been painted.

YEAR 12 MATHEMATICS


5.

129

Just as the students are completing their measurements, an ice hockey team arrives for a training session. The class decide to watch them practise. During one of the drills a player attempts to flick the puck into the back of the net. (The height of the goal is 1.22 m.) The player stands to the side of the goal mouth, so that the puck is: 3.5 metres

from the goal line.

4.9 metres

from the near post.

5.3 metres

from the far post.

The player aims the puck at a point, T, which is 30 cm below the top of the centre of the crossbar. Find the angle of elevation pTPC.

Note - in any triangle ªABC then AB2 + AC2 = 2AD2 + 2BD2

YEAR 12 MATHEMATICS


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130


MATHEMATICS 2.9 ACHIEVEMENT STANDARD 90292 Solve straightforward trigonometry equations


C

solve trigonometric equations using trigonometric graphs in degrees within the

From

range 0E # θ # 360E and in radians within the range π # θ # 2π

straightforward (simple step)

C

solve trig equations such as: 5 sin 3x = 4 on [0E, 360E] cos (x - 180E) = 0.5 on [-180E, 180E] 2 sin x - 1.5 = -1 on [0, 2π] using degrees or radians as directed

C

solve trigonometric equations in contexts such as tidal motion or wheel revolutions

equations to equations requiring several steps in contexts or situations involving

C

manipulate more complex trigonometric equations

C

more challenging manipulations.

solve problems in contexts, including mathematical situations

YEAR 12 MATHEMATICS

Trigonometric Equations

131

TRIGONOMETRIC EQUATIONS - Revision Summary The exercises in this section have been specifically chosen to underpin the Achievement Criteria that will be tested in this Achievement Standard. 1.

a. Change these degree measures to radians (to 2 dp). i.

30E

ii. 114E

iii.

268E

iii.

320E

b. Write each degree measure as a fraction of π. i.

30E

ii. 105E

c. Change these radian measures to degrees (to 1 dp where necessary). i.

0.92 radii.

YEAR 12 MATHEMATICS

2.63 rad

iii.

rad

132

2.


a. Draw i. and

ii.

y = 2 sin x. y = cos x - 1, on separate graphs.

On graph i. use 0 # x # 360E and on graph ii. use 0 # x # 2π.

b. On the first graph, draw the line y = 1.6 and find x where 2sin x = 1.6. On the second graph, draw the line y =

and find x where cos x - 1 =

.

YEAR 12 MATHEMATICS


3.

Solve these equations with a suitable method. a. 2 sin x = 1, on 0E # x # 360E b. 5 cos x - 2 = 1 on 0E # x # 2π c. cos 2x = 0.85 on 0 # x # 2π d. 2.5 sin 3x = -1.5 on 0E # x # 360E e. sin (x - 45E) = 0.4 on -270E # x # 270E

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133

134

4.


On the interval 0 # x # 4π, which values of x would provide the greatest and least values of y = 5 sin ?

YEAR 12 MATHEMATICS


TRIGONOMETRIC EQUATIONS PRACTICE TEST 1 Show ALL working. QUESTION ONE Solve the following trigonometric equations: a.

cos x = 0.4, 0E # x # 360E

b.

sin x + 4 = 3.09 on 0E # x # 360E

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135

136

c.


3 tan x = 4.8 on 0 # x # 2π

QUESTION TWO Solve cos 2x = 0.78, 0E # x # 360E.

YEAR 12 MATHEMATICS


QUESTION THREE Millie hears a “click - clack” sound while riding her bike. She dismounts and finds a small tack has embedded itself into the rubber tread. She knows that her bike wheel has a 66 cm diameter and once she is home, she turns it and observes how the tack on the tyre rotates around the central wheel hub. The height (H) of the tack relative to the centre hub may be given by: H = 33 sin (45t)E H = height in centimetres. t = time in seconds after the wheel begins to rotate. a.

After how many seconds will the tack first be 30 cm above the hub?

b.

How long will it take the tack to return to its starting position?

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137

138


QUESTION FOUR While watching her piano being tuned, Millie notices a tuning fork for the note Middle C. The pure tone of the tuning fork when struck, vibrates at 256 times per second. Millie believes these vibrations may be modelled by the trigonometric function y = 0.6 sin (512 π t), where the amplitude (loudness) is 0.6. The speed of the vibrations suggests the number of cycles per second cannot be detected by us. What is the least amount of time taken for the tuning fork to reach a loudness of 0.45?

YEAR 12 MATHEMATICS


139

QUESTION FIVE Over a long period of time, Millie has observed the depth (d) of water at Flint’s Stone, a large rock near the entrance to the local harbour. She believes that the tidal cycle is about 12½ hours. At low tide the water depth beside the rock is only 1.5 metres, but at high tide it has reached 6.1 m.

Millie believes this formula, d = 2.3 cos where:

+ 3.8, could model the depth around Flint’s Stone,

d = depth of the water in metres. t = time in hours after high tide.

Fishing boats may only enter the harbour when the water there is at least 3 metres deep. How long either side of high tide, can boats safely travel in and out of the harbour?

YEAR 12 MATHEMATICS

140


TRIGONOMETRIC EQUATIONS PRACTICE TEST 2 Show ALL working. Solve the following trigonometric equations: QUESTION ONE a.

cos θ = 0.4, 0E # x # 360E

b.

2.5 sin θ + 1 = 1.5, 0E # x # 360E

c.

tan θ + 2.7 = 1.1, 0 # θ # 2π

YEAR 12 MATHEMATICS


QUESTION TWO Solve the following trigonometric equations: a.

cos (θ - 30E) = -0.7, 0E # x # 360E

b.

sin 2θ = 0.61, 0 # θ # π

YEAR 12 MATHEMATICS

141

142


QUESTION THREE A patient with a fever is admitted to hospital for further observation. Her temperature varies from a low of 37E to a high of 40.2E Celcius.

A model of the temperature has been suggested as: T = 38.6 + 1.6 sin

where t is the time in days.

During which day, following her admission to hospital, does the patient’s temperature first reach 40E C?

YEAR 12 MATHEMATICS


143

QUESTION FOUR The eleven weeks from the middle of July until early October is a profitable time for a local winter sports retailer to sell snow boards. Sales of snow boards have been monitored over the same period of time for some years. They have developed this trig formula to model the numbers of snowboards that they sell over the 11 weeks.

S = 30 - 33 cos

, where week 1 # t # week 11.

According to the formula over how many weeks would they sell at least 45 snowboards?

YEAR 12 MATHEMATICS

Answers

144

THE ANSWERS Note: the answers have been checked and rechecked. If your answer differs from the one listed then check with your teacher, friends or write to Mahobe Resources. Due to space restrictions, we have not been able to set out all answers as fully as we would have liked. Remember - in the exam you should put each step of your answer on a separate line. Pages 6-10, ALGEBRA - Revision Summary 1. a. x =

b. r =

c. x =

or x = -11

2. a. c. 3. a. c.

12 - 9x 4x3 - 16x2 - 35x + 147 (x - 12)(x - 7) (a - v)(a + w)

4. a.

b. 3x3y5

=

d. x = 0,

, -8

b. 18x3 - 24x2 - 10x, b. (5x + 7)(2x - 3)

5. a. 35 = 243 b. i. log 5, ii. log 72, iii. log 30 c. 2 6. a. 3x - 1

d. x >-

8. a. x = 7, x = -6

c. x = -

, x = 0, x = 1

e. x = 0.45, x = -4.45 9. a. c. 10. a. c. 11. a.

(x - 12)(1350 + 24x)= (1800 + 20x)(x - 9) 24x2 + 1062x - 16200 = 20x2 + 1620x - 16200 4x2 - 558x = 0 2x(2x - 279) = 0 ˆ x = 0 or x = 139.5 Check: x = 139 tickets, PA < PB x = 139.5 tickets, PA = PB x = 140 tickets, PA > PB ˆ sell 140 tickets or more, price B is cheaper.

b. n =

c. x < 1

b. x =

,x=2

d. x = 1.15, x = -0.65 f.

x = -3.37, x = 2.37

7

x = 3 = 2187 b. x = =7 x = 2.67 (-4, 3) b. (-2, 3) (2, -2) and (3, 0) d. (-2, -1) and (1, 2) i. ª = 57 > 0 ˆ 2 real distinct roots ii. ª = 0 ˆ 1 real (repeated) root

b. (2n)2 - 4(2)(5) < 0 Y n2 < 10 ˆ -

86. • Ignore repeated numbers (®). • Stop when you have a list of approx 30 quakes. Number

Date

Time

R Magnitude

17

19/02

00 04

6.5

06

23/01

20 10

6.3

46

14/06

17 10

6.8

76

14/11

21 38

7.0

51

20/06

04 03

4.7

96

-

-

-

02

10/01

23 48

5.5

57

25/07

15 43

5.0

18

22/02

02 25

6.4

36

03/05

07 21

4.9

41

19/05

01 54

6.9

58

05/08

14 14

5.2

51 ®

-

-

-

75

8/11

07 54

5.1

46 ®

-

-

-

64

26/09

01 55

7.5

84

12/12

21 47

6.6

34

19/04

21 11

5.5

91

-

-

-

93

-

-

-

44

06/06

07 41

5.7

93

-

-

-

95

-

-

-

93

-

-

-

59

13/08

04 58

4.8

63

24/09

19 24

5.6

55

23/07

07 34

6.0

YEAR 12 MATHEMATICS

Answers

Number

Date

Time

R Magnitude

81

02/12

13 13

6.5

28

21/03

12 23

6.9

59 ®

-

-

-

57 ®

-

-

-

14

15/02

14 42

6.6

37

05/05

19 12

6.5

72

27/10

11 18

4.2

65

29/09

15 50

6.7

08

25/01

16 44

5.9

02 ®

-

-

-

52

02/07

02 16

6.6

99

-

-

-

94

-

-

-

76 ®

-

-

-

64 ®

-

-

-

83

11/12

14 20

6.6

13

14/02

23 38

6.1

03

12/01

08 04

6.8

Though it is physically possible to test all the items in this population of earthquakes, it is not essential, as a sample of about 30 is a useful representation of the population. Also, with the random function on the calculator (though it is not truly random) we are generally satisfied that each quake in the population had the same chance of selection in the sample. This simple random sampling method, though reliant on the generated random numbers, has provided 30 selections. If doubts rise over its authenticity, the exercise could readily be repeated with another simple random sample.

157

The evenness of this distribution would suggest that quakes can occur at any time including the early hours of the morning and the latest hours of the night. Though random sampling may be time consuming, we return to the theme, that ignoring the vagaries of calculator random programming, every earthquake in the population had the same chance of selection for the sample. Pearson developed a measure of skewness of a distribution. Skew

= = = -0.528

Since the co-efficient of skewness generally lies between -3 and +3, this suggests that the distribution of the sample times tend to negative skewness, so a few early hours are pulling the mean away from the median. Page 81 - 83 SAMPLE STATISTICS, PRACTICE TEST 2 Note, this is a possible solution only. The layout of the shellfish farm suggests that an ‘all in’ sample may not provide the most representative 30 selections from the population. One possibility is to look at five possible strata.

If this sample of 30 could be represented in a stem and leaf plot these sample statistics of earthquake times would result. LQ = 07 21, UQ = 19 12, median = ½ way between 13 13 hours and 14 14 hours (13 43 ½) The box and whisker plot of this sample would look like: Box Plot of Earthquake Times During Day (sample n = 30)

Based on these proportions: The top row should have The left group should have

From this sample it would appear that the times are skewed slightly to the later end of the day with a medium time of 13 : 43 : 30 The mean of the sample 0 . 12 26 and the standard deviation of the sample . 7hr 18 min. This would support the above result (box plot) with both the mean and median appearing as p.m. times for this sample. The distribution of the time in this sample appears to have a reasonably even spread, i.e. 00 h - 09 h 12 quakes 10 h - 19 h 12 quakes 20 h - 23 h 6 quakes

YEAR 12 MATHEMATICS

× 30 .5 in the sample × 30 .7 in the sample

The middle group should have

× 30 .7 in the sample

The right group should have

× 30 .8 in the sample

The bottom end should have

× 30 .3 in the sample 30 (Total)

Take a simple random sample of each strata / group in the population. Note each site is already numbered e.g. Top end: Assign the first two digits from the random number generator of your calculator starting at (e.g. 4+5) 9th number. Disregard any numbers > 45 or < 01 Ignore repeated numbers (®). Stop when we have 5 for this strata.

158

Answers

Page 81 - 83 (cont) e.g. 62 37 68 78 14 34 94 68 82 86 99 89 63 49 34 02 29 ˆ 37, 14, 34, 02, 29 for the top end Apply the same technique (using 3 digit numbers) to each of the other four strata, with sensibly applied numbering to suit each case. The goal is to finish with a total sample size of at least 30 shellfish sites

Visually this Box and Whisker Plot of the sample is:

Example of possible strata selections:

The distribution is skewed with the median placed in the upper end of the interquartile range. The middle 50% of the sample has a mass between 240 kg and 407 kg. The mean of the sample, 0 = 335.7 kg The sample standard deviation, s = 114.0 kg Using Pearson’s skewness rule:

Top End (001-045) #

Mass (kg)

37

500

14

191

34

245

02

413

29

165

Skew

=

Left Group (045 - 115) 77

508

59

359

49

345

107

175

47

185

109

399

88

366

Middle Group (116 - 171) 155

378

160

249

158

407

145

411

148

404

137

490

156

389

Right Group (172 - 227) 178

393

227

180

189 ®

520

180

407

174

373

196

458

188

139

200

240

Bottom End (228 - 255) 233

267

254

297

230

219

It is physically possible to have used the whole population, but it is too time consuming, and a sample such as this has attempted to represent the whole population. Also, with the areas of the shellfish farm split into strata, each sub-area is catered for and the randomness of the selection process, again is dependent on the generated random number function from the calculator.

=

= -0.889 The data is negatively skewed with the smaller masses pulling the mean away from the median. The masses of the shellfish lines are quite spread out. Both mean and median point towards the mid 300 kg mass, so if the owners of the farm were looking for a conservative estimate of the mass of their crop, they should tend towards the lower statistic of the two. Another sample could produce a similar result, or not, but the exercise could readily be repeated. It is important to consider the sites of the areas and which are closer to sea, to shore, to nutrients, affected more by weather etc. Therefore the strata sampling would still be a sensible option for this farm. Pages 88 - 91, PROBABILITY AND NORMAL DISTRIBUTION Revision Summary 1. a.

b.

=

c.

2. a.

=

b.

=

d.

=

e. 150 ×

=

c. = 25 times

3. Table of possibilities + 1 1 2 2 3 3 4 4 5 5 6 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

=

=

c.

=

a.

b.

5 6 7 8 9 10 11

6 7 8 9 10 11 12

d. Pr( 4 or 9) = e. Pr (3 or 6 or 9 or 12) =

=

The sample provides these statistics when ordered. LQ = 240 kg, median = 369.5 kg, UQ = 407 kg

YEAR 12 MATHEMATICS

Answers

Pages 88 - 91 cont 4. a.

= Pr (

)

= Pr( z > 0.75) = 0.5 - Pr ( 0 < z < 0.75) = 0.5 - 0.2734 = 0.2266 ii. Pr(50 < x < 66)

YEAR 12 MATHEMATICS

RAN#

RAN#

RAN#

RAN#

RAN#

745

-

754

-

854

-

097

-

259

-

636

-

155

B

348

-

643

-

247

-

970

-

632

-

580

-

012

F

791

-

685

-

331

-

550

-

355

-

292

-

211

F

761

-

630

-

671

-

175

B

728

-

867

-

515

F

511

F

445

-

033

-

744

-

954

-

850

-

773

-

257

-

763

-

127

B

944

-

557

-

801

F

926

-

556

-

923

F

719

F

814

F

423

F

303

F

795

-

237

-

541

-

744

-

739

-

986

-

301

F

464

-

460

-

121

B,F

007

F

688

-

938

-

683

-

592

-

790

-

167

B

583

-

730

-

623

F

189

B

863

-

421

F

793

-

329

-

361

-

865

-

684

-

254

-

569

-

315

F

897

-

d. Letter given to 21 households 2. One possible approach (apart from recognising independence) is a tree diagram.

160

Answers

Pages 92 - 98 (cont) a. P(get letter)

Pr(

= = 0.025 + 0.075 + 0.225 = 0.325, ˆ expect 200 x 0.325 = 65 letters b. P(both) = 0.025 ˆ expect 0.025 x 200 = 5 would have both c. From the simulation, 21 out of 80 would get a letter ˆ expect 200 x

) = Pr(1.882 < z < 2.471) = 0.4932 - 0.4700 = 0.0232 = 2.32%

ii. Pr(x > 105) = Pr(z >

)

= Pr(z > 2.471) = 0.5 - 0.4932 = 0.0068 = 0.68%

= 52

ˆ 52 - 53 households are to receive a letter. d. From the simulation, 1 out of 80 had both. ˆ expect 200 x

)

=2

ˆ 2 - 3 households with both 3. Note: Example comment only. The simulated results have lower values than the corresponding theoretical results. The random feature of a calculator is not truly random, but generated therefore a second sample of 80 numbers could have more than the theoretical values. Also there is no allowance in the theoretical case whether an operator who is monitoring these items, actually misses them in a particular bag (or bags)(i.e. human error).

= Pr(z > 2.353) = 0.5 - 0.4907 = 0.0093 Expect 850 x 0.0093 = 7.905 , i.e. 8 bags 6.

Section B 4. µ = 84 kg, σ = 8.5 kg Use inverse normal i.e. Pr(0 < x < W) = 0.4543 ˆ Pr(0 < z < ˆ a. Pr(84 < x < 90)

= Pr(0 < z