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
Below are the requirements of this Achievement Standard.
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
Below are the requirements of this Achievement Standard.
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
The DS-742ET Some advanced technology has gone into the Mahobe DS-742ET to make it one of the most powerful calculators available. If you use anything else then good luck!
planned orbit
eTOOL actual orbit
P
C1
C2
www.mahobe.co.nz.
MAHOBE
Coordinate Geometry
55
MATHEMATICS 2.4 ACHIEVEMENT STANDARD 90287 Use coordinate geometry methods
Below are the requirements of this Achievement Standard.
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
The DS-742ET Mahobe have added some amazing technology into their new eTool advanced scientific calculator. • Equation solving. • Enhanced statistics. • Improved powers and fraction display. This calculator is designed to handle even the toughest assignments. If you use any other calculator then good luck. With a Mahobe Resource you can have an added confidence that the answer will be correct.
eTOOL
MAHOBE
www.mahobe.co.nz.
Sample Statistics
73
MATHEMATICS 2.5 ACHIEVEMENT STANDARD 90288 Select a sample and use this to make an inference about the population
Below are the requirements of this Achievement Standard.
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
The DS-742ET Mahobe have added some amazing technology into their new eTool advanced scientific calculator. • Equation solving. • Enhanced statistics. • Improved powers and fraction display. This calculator is designed to handle even the toughest assignments. If you use any other calculator then good luck. With a Mahobe Resource you can have an added confidence that the answer will be correct.
eTOOL
MAHOBE
www.mahobe.co.nz.
Probability & Normal Distribution
87
MATHEMATICS 2.6 ACHIEVEMENT STANDARD 90289 Simulate probability situations and apply the normal distribution
Below are the requirements of this Achievement Standard.
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
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
Probability & Normal Distribution
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?
YEAR 12 MATHEMATICS
90
5.
Probability & Normal Distribution
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
Probability & Normal Distribution
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 ).
YEAR 12 MATHEMATICS
91
92
Probability & Normal Distribution
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
Probability & Normal Distribution
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93
94
2.
Probability & Normal Distribution
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.
YEAR 12 MATHEMATICS
Probability & Normal Distribution
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.
YEAR 12 MATHEMATICS
96
Probability & Normal Distribution
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.
YEAR 12 MATHEMATICS
Probability & Normal Distribution
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?
YEAR 12 MATHEMATICS
98
6.
Probability & Normal Distribution
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.
YEAR 12 MATHEMATICS
Probability & Normal Distribution
99
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.
YEAR 12 MATHEMATICS
100
Probability & Normal Distribution
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
Probability & Normal Distribution
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.
YEAR 12 MATHEMATICS
102
Probability & Normal Distribution
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
Probability & Normal Distribution
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?
YEAR 12 MATHEMATICS
103
104
12.
Probability & Normal Distribution
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
Below are the requirements of this Achievement Standard.
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.
YEAR 12 MATHEMATICS
108
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?
YEAR 12 MATHEMATICS
109
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?
YEAR 12 MATHEMATICS
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?
YEAR 12 MATHEMATICS
112
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)?
YEAR 12 MATHEMATICS
114
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?
YEAR 12 MATHEMATICS
116
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
Below are the requirements of this Achievement Standard.
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
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
Trigonometry Problems
YEAR 12 MATHEMATICS
119
120
5.
Trigonometry Problems
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
Trigonometry Problems
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.
YEAR 12 MATHEMATICS
122
Trigonometry Problems
YEAR 12 MATHEMATICS
Trigonometry Problems
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.
YEAR 12 MATHEMATICS
124
3.
Trigonometry Problems
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
Trigonometry Problems
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.
YEAR 12 MATHEMATICS
126
Trigonometry Problems
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
Trigonometry Problems
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?
YEAR 12 MATHEMATICS
128
4.
Trigonometry Problems
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
Trigonometry Problems
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
The DS-742ET Some advanced technology has gone into the Mahobe DS-742ET to make it one of the most powerful calculators available. If you use anything else then good luck!
planned orbit
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P
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MAHOBE
130
Trigonometry Problems
MATHEMATICS 2.9 ACHIEVEMENT STANDARD 90292 Solve straightforward trigonometry equations
Below are the requirements of this Achievement Standard.
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.
Trigonometry Problems
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
Trigonometric Equations
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
YEAR 12 MATHEMATICS
133
134
4.
Trigonometry Problems
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
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
YEAR 12 MATHEMATICS
135
136
c.
Trigonometry Problems
3 tan x = 4.8 on 0 # x # 2π
QUESTION TWO Solve cos 2x = 0.78, 0E # x # 360E.
YEAR 12 MATHEMATICS
Trigonometric Equations
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?
YEAR 12 MATHEMATICS
137
138
Trigonometry Problems
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
Trigonometric Equations
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
Trigonometry Problems
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
Trigonometric Equations
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
Trigonometry Problems
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
Trigonometric Equations
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
