MTN Learn - Department of Basic Education

MTN Learn .BUIFNBUJDT Grade 12

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GRADE 12 MATHEMATICS RADIO BROADCAST NOTES

CONTENTS

Introduction ...............................................................................................................

3

Broadcast Schedules................................................................................................

5

Preparing for Examinations.....................................................................................

6

Exam Techniques......................................................................................................

7

Exam Overview..........................................................................................................

9

Prelim Preparation ....................................................................................................

10

Spring School............................................................................................................ 37 Prelim Review............................................................................................................ 50 Exam Revision: Paper 1 ........................................................................................... 52 Exam Revision: Paper 2 ........................................................................................... 55 Listener’s Feedback Competition........................................................................... 61

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INTRODUCTION Have you heard about Mindset? Mindset Network, a South African non-profit organisation, was founded in 2002. Through our Mindset Learn programme, we develop and distribute high quality curriculum aligned educational resources for Grade 10 - 12. We make these materials available on TV (Dstv and Toptv channels 319), the Internet (www.mindset.co.za/learn) and as DVDs and books. At Mindset we are committed to helping South African learners succeed. This is why Mindset Learn is proud to offer Mindset Learn Xtra especially for Grade 10 – 12 learners. Learn Xtra offers you hundreds of hours of video and print support, live television shows between 4pm and 7pm every Monday to Thursday and a free Helpdesk where our expert teachers are on standby to help you. You can find out more about Mindset Learn Xtra at www.learnxtra.co.za. Learn Xtra also offers specific exam revision support. Every year we run Winter, Spring, Exam and Supplementary Schools to help you ace your exams. And now, Mindset is proud to announce a powerful partnership with MTN and the Department of Basic Education to bring you Mindset Learn Xtra Radio Revision powered by MTN – a full 3 months of radio programmes dedicated to supporting Grade 10 and 12 Mathematics, Physical Sciences and English First Additional Language.

Participating Community Radio Stations Listen to any of the following community radio stations to get your daily dose of expert tuition and exam preparation. Also listen out for on air competitions in which you can stand a chance of winning great prizes. 1. KZN: a. Hindvani Radio b. Maputaland Radio

91.5 fm (Durban) 102.3 fm ( rest of KZN) 170.6 fm

2. Limpopo: a. Sekgosese Radio b. Greater Tzaneen Radio c. Mohodi FM d. Moletsi e. Univen

100.3 fm 104.8 fm 98.8 fm 98.6 fm 99.8 fm

3. Eastern Cape: a. Vukani fm b. Fort Hare Community Radio c. Mdantsane fm d. Nkqubela fm e. Graaff Reinet

90.6 - 99.9 fm 88.2 fm 89.5 fm 97.0 fm 90.2 fm

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GETTING THE MOST FROM MINDSET LEARN XTRA RADIO REVISION In the Grade 12 Mathematics radio programme, we will focus on questions that come from recent previous exam papers. Please ask your teachers for copies of the exam papers or download the question papers and memos from our website or www.education.gov.za This booklet contains diagrams taken from the exam papers so that you will be able to follow what is said during the broadcast. Before you listen to the show, read through the questions for the show and try to answer them without looking up the solutions. If you have a problem and can’t answer any of the questions, don’t panic. Make a note of any questions you need answered. When listening to the show, compare your approach to the teacher’s. Don’t just copy the answers down but take note of the method used. Make sure you keep this booklet. You can re-do the questions you did not get totally correct and mark your own work. Remember that exam preparation also requires motivation and discipline, so try to stay positive, even when the work appears to be difficult. Every little bit of studying, revision and exam practice will pay off. You might benefit from working with a friend or a small study group as long as everyone is as committed as you are. Mindset believes that Mindset Learn Xtra Radio Revision will help you achieve the results you want. If you find Mindset Learn Xtra Radio Revision a useful way to revise and prepare for your exams, remember that we will be running the following other great Learn Xtra Exam Revision programmes on Dstv and Toptv channels 319 during rest of 2012:  

Grade 12 Mindset Learn Xtra Spring School: 1 October to 5October Grade 11 and 12 Mindset Learn Xtra Exam School: 22 October – 20 November

Both Spring School and Exam School will cover Mathematics, Physical Sciences, Life Sciences, Mathematical Literacy, English 1st Additional Language, Accounting and Geography.

Contact us We want to hear from you. So let us have your specific questions or just tell us what you think through any of the following: LearnXtra

[email protected]

@learnxtra

086 105 8262

www.learnxtra.co.za

Mindset Get the free app at pepclub.mobi

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BROADCAST SCHEDULE PRELIM PREPARATION Date

Time

Topic

27-Aug

17:00 -18:00

Maths 1: Sequences & Series

28-Aug

17:00 -18:00

Maths 1: Finance

29-Aug

17:00 -18:00

Maths 1: Functions

30-Aug

17:00 -18:00

Maths 1: Calculus

01-Sep

09:00 -10:00

Maths 1: Calculus Applications

03-Sep

17:00 -18:00

Maths 2: Data Handling

04-Sep

17:00 -18:00

Maths 2: Analytical Geometry

05-Sep

17:00 -18:00

Trigonometry ( Expressions, Equations & Graphs)

06-Sep

17:00 -18:00

Maths 2: Trigonometry (Identities)

08-Sep

09:00 -10:00

Maths 2: 2D & 3D Trigonometry

SPRING SCHOOL Date

Time

Topic

01-Oct

09:00 -10:00

Maths 1: Calculus

02-Oct

09:00 -10:00

Maths 2: Trigonometry

03-Oct

09:00 -10:00

Maths 1: Functions

04-Oct

09:00 -10:00


05-Oct

09:00 -10:00

Maths 1: Algebra

06-Oct

09:00 -10:00

Maths 2: Transformations and Trigonometry

PRELIM REVIEW Date

Time

Topic

08-Oct

17:00 -18:00

09-Oct

17:00 -18:00


10-Oct

17:00 -18:00

Maths 1: Functions

11-Oct

17:00 -18:00


13-Oct

09:00 -10:00

Maths 1: Calculus

Maths 1: Algebra

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EXAM SCHOOL Date

Time

Topic

31-Oct

17:00 -18:00

31-Oct

18:00 -19:00

Maths 1: Functions

01-Nov

17:00 -18:00

Maths 1: Calculus

01-Nov

18:00 -19:00

Maths 1: Calculus Applications

03-Nov

09:00 -10:00

Maths 2: Data Handling

03-Nov

10:00 - 11:00


04-Nov

17:00 -18:00


04-Nov

18:00 -19:00


Maths 1: Algebra

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PREPARING FOR EXAMINATIONS 1.

Prepare well in advance for all your papers and subjects. You need to start your planning for success in the final examination now. You cannot guarantee success if you only study the night before an exam.

2.

Write down the date of your prelim and final exam so that you can plan and structure a study time table for all your subjects.

3.

Set up a study time-table according to your prelim and final Grade 12 exam time-table and stick to your study schedule. If you study a small section every day, you will feel you have achieved something and you will not be as nervous by the time you have to go and write your first paper.

4.

Your study programme should be realistic. You need to spend no more than 2 hours per day on one topic. Do not try to fit too much into one session. When you cover small sections of work often, you will master them more quickly. The broadcast schedule may help you to make sure you have covered all the topics that are in the exam.

5.

When studying don’t just read through your notes or textbook. You need to be active by making summary checklists or mind maps. Highlight the important facts that you need to memorise. You may need to write out definitions and formulae a few times to make sure you can remember these. Check yourself as often as you can. You may find it useful to say the definitions out aloud.

6. Practise questions from previous examination papers. Follow these steps for using previous exam papers effectively:

● ● ●

● ● 7.

Take careful note of all instructions - these do not usually change. Try to answer the questions without looking at your notes or the solutions. Time yourself. You need to make sure that you complete a question in time. To work out the time you have, multiply the marks for a question by total time and then divide by the total number of marks. In most exams you need to work at a rate of about 1 mark per minute. Check your working against the memo. If you don’t understand the answer given, contact the Learn Xtra Help desk (email: [email protected]). If you did not get the question right, try it again after a few days.

Preparing for, and writing examinations is stressful. You need to try and stay healthy by making sure you maintain a healthy lifestyle. Here are some guidelines to follow: ● Eat regular small meals including breakfast. Include fruit, fresh vegetables, salad and protein in your diet. ● Drink lots of water while studying to prevent dehydration. ● Plan to exercise regularly. Do not sit for more than two hours without stretching or talking a short walk. ● Make sure you develop good sleeping habits. Do not try to work through the night before an exam. Plan to get at least 6 hours sleep every night.

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EXAM TECHNIQUES 1. Make sure you have the correct equipment required for each subject. You need to have at least one spare pen and pencil. It is also a good idea to put new batteries in your calculator before you start your prelims or have a spare battery in your stationery bag.

2.

Make sure you get to the exam venue early - don’t be late.

3.

While waiting to go into the exam venue, don’t try to cram or do last minute revision. Try not to discuss the exam with your friends. This may just make you more nervous or confused.

4.

Here are some tips as to what to do when you receive your question papers: Don’t panic, because you have prepared well. ● ● ● ● ● ● ● ● ● ● ●

You are always given reading time before you start writing. Use this time to take note of the instructions and to plan how you will answer the questions. You can answer questions in any order. Time management is crucial. You have to make sure that you answer all questions. Make notes on your question paper to plan the order for answering questions and the time you have allocated to each one. It is a good idea always to underline the key words of a question to make sure you answer it correctly. Make sure you look any diagrams and graph carefully when reading the question. Make sure you check the special answer sheet too. When you start answering your paper, it is important to read every question twice to make sure you understand what to do. Many marks are lost because learners misunderstand questions and then answer incorrectly. Look at the mark allocation to guide you in answering the question. When you start writing make sure you number your answers exactly as they are in the questions. Make sure you use the special answer sheet to answer selected questions. Think carefully before you start writing. It is better to write an answer once and do it correctly than to waste time rewriting answers. DO NOT use correction fluid (Tippex) because you may forget to write in the correct answer while you are waiting for the fluid to dry. Rather scratch out a wrong answer lightly with pencil or pen and rewrite the correct answer. Check your work. There is usually enough time to finish exam papers and it helps to look over your answers. You might just pick up a calculation error.

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EXAM OVERVIEW MATHEMATICS PAPER 1

3HRS

TOTAL MARKS: 150

Number & Number Relations Patterns & Sequences

±30 marks

Annuities & Finance

±15 marks

Functions & Algebra Functions & Graphs

±35 marks

Algebra, Equations & Inequalities

±20 marks

Calculus

±35 marks

Linear Programming

±15 marks

MATHEMATICS PAPER 2

3HRS

TOTAL MARKS: 150

Space, Shape and Measurement Co-ordinate Geometry

±40 marks

Transformations

±25 marks

Trigonometry

±60 marks

Data Handling and Probability ±25 marks

Data Handling

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RADIO BROADCAST prelim preparation sTUDY NOTES GRADE 12 Mathematics

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ON AIR COMPETITION QUESTIONS FOR PRELIM PREPARATION WEEK 1 (26 AUG – 2 SEPT) Question The function f(x) = -2x³ + 4x² + 8x – 6 describes the volume of boxes that can be made from a sheet of paper. What is the largest volume of a box described by the function f?

WEEK 2 (26 AUG – 2 SEPT) Question If sin 61° = √p , what is the value of cos 73° cos 15° + sin 73° sin 15° in terms of p

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27-AUG

17:00 -18:00


MATHS 1: SEQUENCES & SERIES

STUDY NOTES Arithmetic Sequences and Series An arithmetic sequence or series is the linear number pattern. We have a formula to help us determine any specific term of an arithmetic sequence. We also have formulae to determine the sum of a specific number of terms of an arithmetic series. The formulae are as follows:

Tn a  (n  1)d n S  2a  (n  1)d  n 2 n Sn  a  l  2

where a first term and d constant difference where  a first term and d constant difference where l is the last term

Geometric Sequences and Series A geometric sequence or series is an exponential number pattern. We have a formula to help us determine any specific term of a geometric sequence. We also have formulae to determine the sum of a specific number of terms of a geometric series. The formulae are as follows:

Tn  ar n1 Sn

a(r n  1) where r  1 r 1

Convergent Geometric Series Consider the following geometric series:

1 1 1 1     ................ 2 4 8 16

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We can work out the sum of progressive terms as follows:

1  0,5 2 1 1 3 S2     0,75 2 4 4

S1

1 1 1 7     0,875 2 4 8 8 1 1 1 1 15 S4       0,9375 2 4 8 16 16

S3 

(Start by adding in the first term) (Then add the first two terms) (Then add the first three terms) (Then add the first four terms)

If we continue adding progressive terms, it is clear that the decimal obtained is getting closer and closer to 1. The series is said to converge to 1. The number to which the series converges is called the sum to infinity of the series. There is a useful formula to help us calculate the sum to infinity of a convergent geometric series. The formula is S 

a 1 r

If we consider the previous series It is clear that a  a 1 r 1 2  1  S 1 1 2

1 1 1 1     ................ 2 4 8 16

1 1 and r  2 2

S 

A geometric series will converge only if the constant ratio is a number between negative one and positive one. In other words, the sum to infinity for a given geometric series will exist only if 1  r  1 If the constant ratio lies outside this interval, then the series will not converge. For example, the geometric series 1  2  4  8  16  ............ will not converge since the sum of the progressive terms of the series diverges.

QUESTION FOR DISCUSSION Nov. 2011 P1 Question 2, 3 & 4

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28-AUG

17:00 -18:00


MATHS 1: FINANCE

STUDY NOTES Future Value Annuity Formula This formula deals with saving money for the future. Remember that the value of n represents the number of payments and not necessarily the duration of the investment.

x (1  i )n  1 F  i where: x  equal and regular payment per period n  number of payments i

interest rate as a decimal 

r 100

Present Value Annuity Formula This formula deals with loans. There must always be a gap between the loan and the first payment for the formula to work.

P

x 1  (1  i )  n 

where:

i

x  equal and regular payment per period n  number of payments i interest rate as a decimal 

r 100

QUESTION FOR DISCUSSION Nov. 2011 P1 Question 7 & 12

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29-AUG

17:00 -18:00


MATHS 1: FUNCTIONS

STUDY NOTES How to tell if a Graph is a Function For every input of a function there is a unique output. To test if a graph is a function, we use a ruler to perform the “vertical line test” on a graph to see whether it is a function or not. Hold a clear plastic ruler parallel to the y -axis, i.e. vertical. Move it from left to right over the axes. If the ruler only ever cuts the curve in one place only throughout the movement from left to right, then the graph is a function. If the ruler ever passes through two or more points on the graph, the graph will not be a function. Rules for Sketching Parabolas of the Form

y  a ( x  p )2  q

 

The value of a tells us if the graph is concave ( a  0 ) or convex ( a  0 ). The equation of the axis of symmetry of the graph is obtained by putting the expression x p 0 and solving for x.



The axis of symmetry passes through the x-coordinate of turning point of the parabola.



The graph of y  a( x  p)  q is obtained by shifting the graph of units to the left or right and then q units up or down.

2



 

If



If



If



If

y  ax 2

by p

p  0 , the shift is left. p  0 , the shift is right. q  0 , the shift is upwards. q  0 , the shift is downwards.

The y-coordinate of the turning point is q. The y-intercept of the graph can be determined by putting x  0 . The x-intercept(s) of the graph can be determined by putting y  0 .

Rules for Sketching Hyperbolas of the Form 1. Determine the shape:

a0

y

a q x p

a0

(The dotted lines are the asymptotes)

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2. Write down the asymptotes and draw them on a set of axes. Vertical asymptote: Horizontal asymptote: x p 0 3. Plot four graph points on your set of axes. 4. Determine the y-intercept: let x  0 . 5. Determine the x-intercept: let y  0 . 6. Draw the newly formed graph. Rules for Sketching Hyperbolas of the Form

yq

y  ab x p  q

1. Write down the horizontal asymptote and draw it on a set of axes: Horizontal asymptote: y  q 2. Plot three graph points on your set of axes. 3. Draw the newly formed graph. Logarithmic Functions The inverse of the exponential function is called the logarithmic function. x y Consider the function y  a . The inverse of this graph is x  a y It is now possible to make y the subject of the formula in the equation x  a by means of the concept of a logarithm. y If x  a , then it is clear from the definition of a logarithm that log a x  y . In other words, x 1 we can write the inverse of the function f ( x)  a as f ( x)  log a x . The inverse function is formed by reflecting the function across the line y=x

Note on Inverses of Functions The inverse of some functions are not functions but relations. The inverse of a parabola is not a function. However, if the domain of the original function is restricted then the inverse may be a function too. For example f(x) = x2 ,then the inverse of f, f-1(x) = ±√x, is not a function. But if f(x) has a restricted domain x≥ 0 or x ≤ 0, then the inverse will be a function.

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QUESTION FOR DISCUSSION Nov. 2011 P1 Question 5 & 6 Graph for Question 6 (Nov. 2011 P1)

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30-AUG

17:00 -18:00


MATHS 1: CALCULUS

STUDY NOTES The most important fact in calculus is that the gradient of the tangent to a curve at a given point is the gradient of the curve at that point. Other words for gradient are: rate of change, derivative, slope Symbols for Gradient are:

f ( x)

dy dx

Dx

f (a ) is the gradient of f at x  a

f (a ) is the y-value corresponding to x  a

f (a)

m   f (a ) t m f

Average Gradient The average gradient (or average rate of change) of a function f between x  a and x  b is the gradient of the line joining the points on the graph of the function. We say that the average gradient of f over the interval is the gradient of the line ab. Gradient of a Curve at a Point using First Principles The formula to determine the gradient of a function from first principles is given by the following limit:

f ( x)  lim

h0

f ( x  h)  f ( x ) h

Gradient of a Function using the Rules of Differentiation You will be required to use the following rules of differentiation to determine the gradient of a function. n

Rule 1 If f ( x) ax , then f ( x) a.nx   Rule 2 If f ( x) ax  , then f ( x) a

n1

Rule 3  If f ( x) number,  then f ( x) 0 Determining the Equation of the Tangent to a Curve at a point The gradient of the tangent to a curve at a point is the derivative at that point. The equation is given by y  y1  m( x  x1 ) where ( x1 ; y1 ) is the point of tangency and m  f ( x1 )

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QUESTION FOR DISCUSSION Nov. 2011 P1 Question 8 & 9 Graph for Question 9 (Nov. 2011 P1)

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01-SEP

09:00 -10:00


MATHS 1: CALCULUS APPLICATIONS

STUDY NOTES Rules for Sketching the Graph of a Cubic Function The graph of the form f ( x)  ax3  bx 2  cx  d is called a cubic function. The main concepts involved with these functions are as follows: Intercepts with the Axes For the y-intercept, let x  0 and solve for y For the x-intercepts, let y  0 and solve for x (You might have to use the factor theorem here.) Stationary Points Determine f ( x) , equate it to zero and solve for x. Then substitute the x-values of the stationary points into the original equation to obtain the corresponding y-values. If the function has two stationary points, establish whether they are maximum or minimum turning points. Points of Inflection If the cubic function has only one stationary point, this point will be a point of inflection that is also a stationary point. For points of inflection that are not stationary points, find f ( x) , equate it to 0 and solve for x. Alternatively, simply add up the x-coordinates of the turning points and divide by 2 to get the x-coordinate of the point of inflection.

QUESTION FOR DISCUSSION Nov. 2011 P1 Question 10 Graph for Question 10 (Nov. 2011 P1)

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03-SEP

17:00 -18:00


MATHS 2: DATA HANDLING

STUDY NOTES Mean The mean of a set of data is the average. To get the mean, you add the scores and divide by the number of scores. Mode This is the most frequently occurring score. Quartiles Quartiles are measures of dispersion around the median which is a good measure of central tendency. The median divides the data into two halves. The lower and upper quartiles further subdivide the data into quarters. There are three quartiles: The Lower Quartile ( Q1 ):

This is the median of the lower half of the values.

The Median (M or Q 2 ):

This is the value that divides the data into halves.

The Upper Quartile ( Q3 ):

This is the median of the upper half of the values.

If there is an odd number of data values in the data set, then the specific quartile will be a value in the data set. If there is an even number of data values in the data set then the specific quartile will not be a value in the data set. A number which will serve as a quartile will need to be inserted into the data set (the average of the two middle numbers). Range The range is the difference between the largest and the smallest value in the data set. The bigger the range, the more spread out the data is. The Inter-quartile Range (IQR) The difference between the lower and upper quartile is called the inter-quartile range. Five Number Summaries The Five Number Summary uses the following measures of dispersion:  Minimum: The smallest value in the data  Lower Quartile: The median of the lower half of the values  Median: The value that divided the data into halves  Upper Quartile: The median of the upper half of the values  Maximum: The largest value in the data Box and Whisker Plots A Box and Whisker Plot is a graphical representation of the Five Number Summary. Box Whisker Whisker

Minimum

Lower Quartile

Median

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 Median: The value that divided the data into halves  Upper Quartile: The median of the upper half of the values GRADE 12 MATHEMATICS MINDSET Maximum: The LEARN XTRA largest value in the data

RADIO BROADCAST NOTES

Box and Whisker Plots A Box and Whisker Plot is a graphical representation of the Five Number Summary. Box Whisker Whisker

Minimum

Lower Quartile

Median

Upper Quartile

Maximum

Standard Deviation using a Table The standard deviation (SD) can be determined by using the following formula:

SD 

 ( x  x )2 n

Scatter Plots and Lines and Curves of Best Fit Plotting data on a scatter plot diagram will show trends in the data. Data could follow a linear, quadratic or exponential trend.

Linear

Quadratic

Exponential

The Normal Distribution Curve 1. 2. 3.

4.

5.

6.

The mean, median and mode have the same value. An equal number of scores lie on either side of the mean. The majority of scores (99,7%) lie within three standard deviations from the mean, i.e. in the interval ( x  3s ; x  3s) where x represents the mean and s represents the standard deviation. About 95% of scores lie within two standard deviations from the mean, i.e. in the interval ( x  2s ; x  2s) where x represents the mean and s represents the standard deviation. About two-thirds of scores (68%) lie within one standard deviation from the mean, i.e. in the interval ( x  s ; x  s) where x represents the mean and s represents the standard deviation. The smaller the standard deviation, the thinner and taller the bell shape is. The bigger the standard deviation, the wider and flatter the bell shape is.

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68% 95% 99,7%

x  3s x  2s

x s

x s

x

x  2s

x  3s

QUESTION FOR DISCUSSION Nov. 2011 P2 Question 1, 2, 3 and 4 Graph for Question 3 (Nov. 2011 P2)

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04-SEP

17:00 -18:00


MATHS 2: ANALYTICAL GEOMETRY

STUDY NOTES If AB is the line segment joining the points A( xA ; yA ) and B( xB ; yB ) , then the following formulas apply to line segment AB: The Distance Formula

AB2  ( xB  xA )2  ( yB  yA )2 or AB 

( xB  xA )2  ( yB  yA )2

The Midpoint Formula

 x  x y  yB  M A B ; A  where M is the midpoint of AB. 2   2 The Gradient of a Line Segment Joining Two Points

y  yA Gradient of AB  B xB  xA

Parallel Lines Parallel lines have equal gradients. If AB||CD then mAB  mCD Perpendicular Lines The product of the gradients of two perpendicular lines is 1 . If AB  CD, then

mAB  mCD  1

y  yA  m  x  xA 

The Equation of the Line Inclination of a Line

tan  mAB If mAB  0, then  is acute

If mAB  0, then  is obtuse Collinear Points (A, B and C)

mAB  mBC or mAC  mAB

or Circles and Tangents to Circles

mAC  mBC

The equation of a circle centre the origin is given by:

x2  y 2  r2

The equation of a circle centre (a ; b) is given by:

r2  x  a 2   y  b 2 

The radius is perpendicular to the tangent:

mradius  mtangent  1 24

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QUESTION FOR DISCUSSION Nov. 2011 P2 Question 5 Graph for Question 5 (Nov. 2011 P2)

Graph for Question 6 (Nov. 2011 P2)

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05-SEP

17:00 -18:00


TRIGONOMETRY (EXPRESSIONS, EQUATIONS & GRAPHS)

STUDY NOTES Trigonometric Theory

 sin 

sin   cos  

y x y  cos   tan  r r x

sin  

90   90  

cos   tan  

tan  

( x ; y) 180  

 360  

180  

sin  

sin  

cos  

cos  

tan  

tan  

Reduction Rules sin(180  )  sin 

sin(180  )   sin 

sin(360  )   sin 

cos(180  )   cos 

cos(180  )   cos 

cos(360  )  cos 

tan(180  )   tan 

tan(180  )  tan 

tan(360  )   tan 

sin(90   ) cos 

sin(90   ) cos 

cos(90   ) sin 

cos(90  )   sin 

sin()   sin 

cos()  cos 

tan()   tan 

Whenever the angle is greater than 360 , keep subtracting 360 from the angle until you get an angle in the interval 0 ;360 .

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Special Angles Triangle A

Triangle B 45

2

60

2

1

1

30

45

1

3

From Triangle A we have:

From Triangle B we have:

1 2

sin 45 

1  2

2 2

sin 30 

cos 45 

1  2

2 2

cos30 

tan 45 

1  1 1

tan 30 

3 2 1 3

and

sin 60 

3 2

and

cos60 

1 2

tan 60 

and

3  1

3

For the angles  0; 90;180;270;360 the diagram below can be used. 90

y

A(0 ;2)

B(1 ; 3) C( 2 ; 2)

2

G(  2 ; 0)

60

2

r2

45

D( 3 ; 1)

30

180

E(2 ; 0)

x

0 360

F(0 ;  2) 270

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TRIG GRAPHS Sine Function Parent function:

f(x) = sin x for the domain: [00; 3600]

Shape:

Wave-like shape, starting at the origin

Intercepts:

y-intercept = 0 x-intercept = 00, 1800, 3600 (every 1800 starting at 00)

Domain:

The domain is usually limited to the interval [00; 3600] Infinite angles are possible as a line centred at the origin on the Cartesian plane can be rotated many times. Rotating the line anti-clockwise gives positive angles, and rotating clockwise gives negative angles.

Period:

This corresponds to one rotation. The sine function repeats itself every 3600.

Range:

Minimum value: -1 when the angle x is 2700 or -2700 Maximum value: 1 when the angle x is 900 or -900 [-1; 1]

Amplitude:

For the parent function the amplitude is 1. It is half the range.

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MINDSET LEARN XTRA General form


f(x) = a sin x + q a – amplitude. For the parent function a = 1 The bigger the value of a, the bigger the maximum value will be. The graph is stretched away from the x-axis (rest position). Changing a does not change the x-intercepts when q =0.

q – rest position: For the parent function q = 0, this value shifts the whole graph vertically up when it is positive and down when it is negative. The q value changes the position of the rest position and will change the value of the intercepts

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Cosine Function Parent function:

f(x) = cos x for the domain: [00; 3600]

Shape:

Wave-like shape but when x = 00 the graph is a 1

Intercepts:

y-intercept = 1 x-intercept = 900, 2700, (every 1800 starting at 900)

Domain:

The domain is usually restricted to the interval [00; 3600] or [-3600; 3600]

Period:

This corresponds to one rotation. The cosine function repeats itself every 3600.

Range:

Minimum value: -1 when the angle x is 900 or -900 Maximum value: 1 when the angle x is 00 ; 3600 and -3600 [-1; 1]

Amplitude:

For the parent function the amplitude is 1. It is half the range.

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MINDSET LEARN XTRA General form


f(x) = a cos x + q a – amplitude. For the parent function a = 1 The bigger the value of a, the bigger the maximum value will be. The graph is stretched away from the x-axis (rest position). Changing a does not change the x-intercepts when q =0. q – rest position: For the parent function q = 0 This value shifts the whole graph vertically up when it is positive and down when it is negative. The q value changes the position of the rest position and will change the value of the intercepts

Tangent Function Parent function:

f(x) = tan x for the domain: [00; 3600]

Shape:

Not wave-like shape. Long thin curve that is repeated

Intercepts:

y-intercept = 0 x-intercept = 00, 1800, (every 1800 starting at 00)

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MINDSET LEARN XTRA Domain:


The domain is usually restricted to the interval [00; 3600] or [-3600; 3600] For x = ±900 and ±2700, the function is undefined.

Period:

The tangent function repeats itself every 1800, starting at -900 to 900

Range:

(-∞;∞) The minimum and maximum occur at the asymptotes at ±900 and ±2700. ( Every 1800 starting at 900)

Amplitude:

Since the tangent function is not a wave-like graph, it does not have an amplitude. However, for the parent function when x = 450, the value of the function is 1

General form

f(x) = a tan x + q For the parent function a = 1 a > 1 stretches the graph away from the x-axis a < 1 pulls the graph closer to the x-axis

Changing a does not change the x-intercepts or the asymptotes when q =0. q – rest position: For the parent function q = 0 This value shifts the whole graph vertically up when it is positive and down when it is negative. The q value changes the position of the rest position and will change the value of the intercepts but not the asymptotes.

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QUESTION FOR DISCUSSION Nov. 2011 P2 Question 9 and 10 Graph for Question 9 (Nov. 2011 P2)

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Graph for Question 10 (Nov. 2011 P2)

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06-SEP

17:00 -18:00


MATHS 2: TRIGONOMETRY (IDENTITIES)

STUDY NOTES Trig Identities

 cos2   sin 2  1  tan 

sin  cos 

Compound Angle Identities

Double Angle Identities

sin(A   B) sin A cos B  cos A sin B

2 cos 2   sin  cos 2

sin(A   B) sin A cos B  cos A sin B

sin 2 2sin  cos 

cos(A   B) cos A cos B  sin A sin B

cos 2   sin 2   cos 2 2 cos 2   1   2 1  2sin 

cos(A   B) cos A cos B  sin A sin B

QUESTION FOR DISCUSSION Nov. 2011 P2 Question 12

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08-SEP 09:00 -10:00 MATHS 2: 2D & 3D TRIGONOMETRY 08-SEP 09:00 -10:00 MATHS 2: 2D & 3D TRIGONOMETRY STUDY NOTES STUDY NOTES Solving Two-Dimensional Problems using the Sine, Cosine and Area Rules Solving Two-Dimensional Problems using the Sine, Cosine and Area Rules









The sine-rule can be used when the following is known in the triangle: The sine-rule can be used and when the following is known in the triangle: - more than 1 angle a side - more -than 1 angle a side(not included) 2 sides and and an angle - 2 sides and an angle (not included)

sin A sin B sin C sin A  sin B  sin C  a b c a The bcosine-rule c can be used when the following is known of the triangle:  The cosine-rule - 3 sides can be used when the following is known of the triangle: - 3 sides - 2 sides and an included angle - 2 sides 2and an angle 2 included 2

2a 2b  c  2bc cos A a2  bThe  carea  2bc cos A of any triangle can be found when at least two sides an included angle a



The area of any triangle can be found when at least two sides an included angle are known known

1 Area of ABC ab sin C 1  Area of ABC  ab sin2C 2

QUESTION FOR DISCUSSION QUESTION FOR DISCUSSION Nov. 2011 P2 Question 11 Nov. 2011 P2 Question 11 Diagram for Question 11 (Nov. 2011 P2) Diagram for Question 11 (Nov. 2011 P2)

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RADIO BROADCAST SPRING SCHOOL GRADE 12 Mathematics

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SPRING SCHOOL 01-OCT

09:00 -10:00

MATHS 1: CALCULUS

STUDY TIPS Sketching a Cubic Graph This is done in 3 steps. 1  find the x-intercepts by using factor theorem to find 1 factor x-a such that f(a) = 0. Then use long division or synthetic division to factorise f(x) and find other x-intercepts 2 get stationary points by setting derivative = 0 and solving for x. Then sub back into original to get y-values 3 get y-intercept by making x=0 Put all the points together on the axes and join them in a logical way. Cubic graphs examples:

Points of Inflection This is where the curve has a change in concavity or inflection. To get points of inflection we use f’’(x) = 0. Note that a stationary point is only a point of inflection if both f’(x) and f’’(x) =0 at that point. Where we have f’’(x)>0 the curve is concave up and f’’(x) 0, without a calculator, evaluate sinθ. We have an equation, a restriction and are asked to find the value of something else  a typical Pythagoras question! Method: Get the trig ratio on the LHS (sinθ = -5/3 y/r), then calculate. the quad using the restriction

sinθ < 0 and cosθ > 0

S

A√

T√

C

Now use a diagram and Pythagoras to√√ get the missing side and give it the correct sign.

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Trig Identities To prove a left hand side equals a right hand side, start with the more complicated side. Tools you can use are: Change to sin and cos Use squared Ids and double angle formulas Find LCD where needed Factorise If you get stuck with the LHS, try the RHS and see!

-

Sin Rule In a solution of Δ s question, use the sin rule to find a missing side or angle only if you have either 2 angles and 1 side, or 2 sides and an angle that is opposite one of the known sides. (Note: if the side opposite the given angle is the smaller of the 2 sides, there are 2 solutions) Area Rule To use the area rule you need to know 3 things: 2 sides and an included angle. √ area rule

√

√

Cos Rule In a solution of Δ s question use the cos rule -

To find the side opposite a given angle when we have 2 sides and an included angle To find an angle when we have 3 sides given √

cos rule

√

 missing side √

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QUESTION FOR DISCUSSION Feb. 2011 P2 Question 10 and 11 Graph for Question 11 (Feb. 2011 P2)

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03-OCT

09:00 -10:00


MATHS 1 FUNCTIONS

STUDY TIPS Functions and Inverses Remember the inverse is the reflection of a curve about the line y=x. To get the equation y-1 we swap x and y, then make y the subject of the formula. Note that the domain of y becomes the range of y-1 and range of y becomes domain of y-1. If y = x2, x = y2  y = ± x

QUESTION FOR DISCUSSION Feb. 2011 P1 Question 5 and 6

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Graph for Question 11 (Feb. 2011 P1)

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04-OCT

09:00 -10:00


MATHS 2 ANALYTICAL GEOMETRY

STUDY TIPS The Distance Formula If you are given the distance between two points AB, you can use the distance formula but remember to square both sides. E.g. if AB = 2 2 where A(a, 2) & B(-3, 5) Since AB = 2 2  AB2 = ( 2 2 )2  (a+3)2+(2-5)2 = 8, simply solve for a

Putting an Equation of a Circle into Standard Form. This is done by completing the square. E.g. x2+y2 -4x +8y -1 =0 Write as x2 -4x +y2+8y =1. Now halve the b term and square it to create 2 trinomials. x2 -4x +(4)-(4)+ y2+8y+(16)-(16) =1 . Now factorise the two trinomials taking the –ve terms to the RHS. (x-2)2 + (y+4)2 = 1+20 so r = 21 and the centre is (2, -4) Angle between 2 Lines Use the fact that gradient m=tanθ where θ is read anti-clockwise from +ve x-axis. So m1=tan-1(θ1) & m2=tan-1(θ2). Now get the angle by using bigger angle minus smaller angle.  α= θ1 – θ2 α θ2

θ1

Note: if gradient is negative, use (1800-ref angle) to get θ A Tangent to a Circle A tangent is just a straight line so you need m & c. Steps -

Find the gradient of the radius from centre to the point of contact (where tang meets circle) Get gradient of tangent using fact that grad. tang is  to grad. radius Now you know m in y = mx+c Next using equations of straight lines, sub in point of contact to get c and done!

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QUESTIONS FOR DISCUSSION Feb. 2011 P2 Question 4, 5 and 6 Graph for Question 4 (Feb. 2011 P2)

Graph for Question 6 (Feb. 2011 P2)

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05-OCT

09:00 -10:00


MATHS 1: ALGEBRA

STUDY TIPS Log Equations Solve for x in a log equation. If there is a constant term, multiply it by logaa (where a is the base in the rest of the question). Then use log laws to get single logs on both sides so that you can drop logs. Remember restrictions at the end! Logk(x-1) = Logk5  x-1 =5 and x-1>0 Solve for x – Inequalities Look out for inequalities! You should be thinking number line and critical values. Steps: - Bring everything to one side and factorise. - Get your CVs and put them on a number line with solid or open dots - Choose values in between the numbers on the number line to decide if the function values are +ve or –ve in that region and use this to get your final answer. Future Value Annuities Whether the deposits are made at the start or end of each period will affect the number of deposits which earn interest or not. For example, if payments of R100 are made at the end of each year for 5 years, the a in the geometric formula for Sn will be 100. If they are made at the start of each year, then a = 100(i+1) where i=%/100. Look out for this in the question! The Sum to Infinity Formula Note that this formula only works with geometric series where the r-value is -1