MTN Learn - Maths · PDF fileMTN . Learn . MINDSET ... At Mindset we are committed to helping...

MTN Learn

Grade 12

radio support notes

GRADE12MATHEMATICSRADIOBROADCASTNOTES

2 MTN Learn

MINDSETLEARNXTRA

powered by

CONTENTS

Introduction

Broadcast Schedules

Preparing for Examinations

Exam Techniques

Exam Overview

Prelim Preparation

Spring School

Prelim Review

Exam Revision: Paper 1

Exam Revision: Paper 2

Listener’s Feedback Competition

............................................................................................................... 3

................................................................................................ 5

..................................................................................... 6

...................................................................................................... 7

.......................................................................................................... 9

.................................................................................................... 10

............................................................................................................ 37

............................................................................................................ 50

........................................................................................... 52

........................................................................................... 55

........................................................................... 61


3 MTN Learn

MINDSETLEARNXTRA

powered by

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 91.5 fm (Durban) 102.3 fm ( rest of KZN) b. Maputaland Radio 170.6 fm

2. Limpopo:

a. Sekgosese Radio 100.3 fm b. Greater Tzaneen Radio 104.8 fm c. Mohodi FM 98.8 fm d. Moletsi 98.6 fm e. Univen 99.8 fm

3. Eastern Cape:

a. Vukani fm 90.6 - 99.9 fm b. Fort Hare Community Radio 88.2 fm c. Mdantsane fm 89.5 fm d. Nkqubela fm 97.0 fm e. Graaff Reinet 90.2 fm


4 MTN Learn

MINDSETLEARNXTRA

powered by

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


5 MTN Learn

MINDSETLEARNXTRA

powered by

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 Maths 2: Analytical Geometry

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







6 MTN Learn

MINDSETLEARNXTRA

powered by

EXAM SCHOOL Date Time Topic



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 Maths 2: Analytical Geometry

04-Nov 17:00 -18:00 Maths 2: Trigonometry

04-Nov 18:00 -19:00 Maths 2: Trigonometry


7 MTN Learn

MINDSETLEARNXTRA

powered by

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:

● 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.

7. 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.


8 MTN Learn

MINDSETLEARNXTRA

powered by

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.


9 MTN Learn

MINDSETLEARNXTRA

powered by

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

Data Handling ±25 marks


10 MTN Learn

MINDSETLEARNXTRA

powered by

prelim preparationsTUDY NOTESGRADE12Mathematics

RADIO BROADCAST


11 MTN Learn

MINDSETLEARNXTRA

powered by

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


12 MTN Learn

MINDSETLEARNXTRA

powered by

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:

T ( 1) where first term and constant difference

S 2 ( 1) where first term and constant difference2

S where is the last term2

n

n

n

a n d a dn a n d a d

n a l l

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:

1T

( 1)S where 11

nn

n

n

ar

a r rr

Convergent Geometric Series

Consider the following geometric series:

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


13 MTN Learn

MINDSETLEARNXTRA

powered by

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

11S 0,52

(Start by adding in the first term)

21 1 3S 0,752 4 4

(Then add the first two terms)

31 1 1 7S 0,8752 4 8 8

(Then add the first three terms)

41 1 1 1 15S 0,93752 4 8 16 16

(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 S1

ar

If we consider the previous series 1 1 1 1 ................2 4 8 16

It is clear that 12

a and 12

r

S1

12S 111

2

ar

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 1r

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


14 MTN Learn

MINDSETLEARNXTRA

powered by

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.

(1 ) 1F

where:equal and regular payment per periodnumber of payments

interest rate as a decimal100

nx i

i

xn

ri

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.

1 (1 )

P

where:equal and regular payment per periodnumber of payments

interest rate as a decimal100

nx i

i

xn

ri


Nov. 2011 P1 Question 7 & 12


15 MTN Learn

MINDSETLEARNXTRA

powered by

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 2( )y a x p q

The value of a tells us if the graph is concave ( 0a ) or convex ( 0a ). The equation of the axis of symmetry of the graph is obtained by putting the expression

0x p and solving for x. The axis of symmetry passes through the x-coordinate of turning point of the parabola.

The graph of 2( )y a x p q is obtained by shifting the graph of 2y ax by p units to the left or right and then q units up or down. If 0p , the shift is left. If 0p , the shift is right. If 0q , the shift is upwards. If 0q , the shift is downwards.

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

The x-intercept(s) of the graph can be determined by putting 0y .

Rules for Sketching Hyperbolas of the Form ay q

x p

1. Determine the shape: 0a 0a

(The dotted lines are the asymptotes)


16 MTN Learn

MINDSETLEARNXTRA

powered by

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

Consider the function xy a . The inverse of this graph is yx a

It is now possible to make y the subject of the formula in the equation yx a by means of the concept of a logarithm.

If yx a , then it is clear from the definition of a logarithm that loga x y . In other words,

we can write the inverse of the function ( ) xf x a as 1( ) logaf x 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.


17 MTN Learn

MINDSETLEARNXTRA

powered by


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


18 MTN Learn

MINDSETLEARNXTRA

powered by

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: ( ) D xdyf xdx

( ) is the gradient of at

( ) is the -value corresponding to

f a f x af a y x a

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:

0

( ) ( )( ) limh

f x h f xf xh

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.

Rule 1 1If ( ) , then ( ) .n nf x ax f x a nx Rule 2 If ( ) , then ( )f x ax f x a Rule 3 If ( ) number, then ( ) 0f x f x 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 1 1( )y y m x x where 1 1( ; )x y is the point of tangency

and 1( )m f x

( )f a ( )t fm m f a


19 MTN Learn

MINDSETLEARNXTRA

powered by


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


20 MTN Learn

MINDSETLEARNXTRA

powered by

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 3 2( )f x ax bx 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 0x and solve for y For the x-intercepts, let 0y 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)


21 MTN Learn

MINDSETLEARNXTRA

powered by

Whisker WhiskerBox

Minimum Lower Quartile Upper QuartileMedian Maximum

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 ( 1Q ): This is the median of the lower half of the values. The Median (M or 2Q ): This is the value that divides the data into halves. The Upper Quartile ( 3Q ): 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.


22 MTN Learn

MINDSETLEARNXTRA

powered by

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

2( )SD

x xn

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. The mean, median and mode have the same value. 2. An equal number of scores lie on either side of the mean. 3. The majority of scores (99,7%) lie within three standard deviations from the mean,

i.e. in the interval ( 3 ; 3 )x s x s where x represents the mean and s represents the standard deviation.

4. About 95% of scores lie within two standard deviations from the mean, i.e. in the interval ( 2 ; 2 )x s x s where x represents the mean and s represents the standard deviation.

5. 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. 6. 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.

Whisker WhiskerBox

Minimum Lower Quartile Upper QuartileMedian Maximum

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 ( 1Q ): This is the median of the lower half of the values. The Median (M or 2Q ): This is the value that divides the data into halves. The Upper Quartile ( 3Q ): 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.


23 MTN Learn

MINDSETLEARNXTRA

powered by


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

68%

95%

99,7%

3x s 3x s2x s 2x sx s x sx


24 MTN Learn

MINDSETLEARNXTRA

powered by

04-SEP 17:00 -18:00 MATHS 2: ANALYTICAL GEOMETRY

STUDY NOTES

If AB is the line segment joining the points A AA( ; )x y and B BB( ; )x y , then the following formulas apply to line segment AB: The Distance Formula

B A B A2 2 2AB ( ) ( )x x y y

or B A B A2 2AB ( ) ( )x x y y

The Midpoint Formula

A B A BM ;2 2

x x y y

where M is the midpoint of AB.

The Gradient of a Line Segment Joining Two Points

Gradient of B A

B AAB y y

x x

Parallel Lines

Parallel lines have equal gradients. If AB||CD then AB CDm m Perpendicular Lines The product of the gradients of two perpendicular lines is 1 . If ABCD, then

AB CD 1m m The Equation of the Line A Ay y m x x

Inclination of a Line ABtan m

AB

AB

If 0, then is acuteIf 0, then is obtuse

mm

Collinear Points (A, B and C) AB BCm m

AC AB

AC BC

or or

m mm m

Circles and Tangents to Circles

The equation of a circle centre the origin is given by: 2 2 2x y r

The equation of a circle centre ( ; )a b is given by: 2 2 2x a y b r The radius is perpendicular to the tangent: radius tangent 1m m


25 MTN Learn

MINDSETLEARNXTRA

powered by


Nov. 2011 P2 Question 5 Graph for Question 5 (Nov. 2011 P2)

Graph for Question 6 (Nov. 2011 P2)


26 MTN Learn

MINDSETLEARNXTRA

powered by

05-SEP 17:00 -18:00 TRIGONOMETRY (EXPRESSIONS, EQUATIONS & GRAPHS)

STUDY NOTES

Trigonometric Theory

Reduction Rules

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

sin(180 ) sin sin(180 ) sin sin(360 ) sincos(180 ) cos cos(180 ) cos cos(360 ) costan(180 ) tan tan(180 ) tan tan(360 ) tansin(90 ) cos sin(90 ) coscos(90 ) sin cos(9

0 ) sin

sin( ) sin cos( ) cos tan( ) tan

sincostan

sincostan

sincostan

sincostan

90 90

180

180 360

sin cos tany x yr r x

( ; )x y


27 MTN Learn

MINDSETLEARNXTRA

powered by

Special Angles Triangle A Triangle B

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

45

45

1

1

2 60

30

12

3

From Triangle A we have:

1 2sin 4522

1 2cos4522

1tan 45 11

From Triangle B we have:

1sin302

and 3sin 60

2

3cos302

and 1cos602

1tan303

and 3tan60 3

1

y

x30

45

60

90

180

270

3600

2

22r

A(0 ;2) B(1; 3)

C( 2 ; 2)

D( 3 ;1)

E(2 ; 0)

F(0 ; 2)

G( 2 ; 0)


28 MTN Learn

MINDSETLEARNXTRA

powered by

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.


29 MTN Learn

MINDSETLEARNXTRA

powered by

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


30 MTN Learn

MINDSETLEARNXTRA

powered by

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


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.


31 MTN Learn

MINDSETLEARNXTRA

powered by

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


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


32 MTN Learn

MINDSETLEARNXTRA

powered by

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.


33 MTN Learn

MINDSETLEARNXTRA

powered by


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


34 MTN Learn

MINDSETLEARNXTRA

powered by

Graph for Question 10 (Nov. 2011 P2)


35 MTN Learn

MINDSETLEARNXTRA

powered by

06-SEP 17:00 -18:00 MATHS 2: TRIGONOMETRY (IDENTITIES) STUDY NOTES Trig Identities

2 2 sincos sin 1 tancos

Compound Angle Identities Double Angle Identities


Nov. 2011 P2 Question 12

2 2

2 2

2

2

cos sin cos 2sin 2 2sin cos

cos sin

cos 2 2cos 1

1 2sin

sin(A B) sin Acos B cos Asin Bsin(A B) sin Acos B cos Asin Bcos(A B) cos Acos B sin Asin Bcos(A B) cos Acos B sin Asin B


36 MTN Learn

MINDSETLEARNXTRA

powered by

08-SEP 09:00 -10:00 MATHS 2: 2D & 3D TRIGONOMETRY

STUDY NOTES

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:

- more than 1 angle and a side - 2 sides and an angle (not included) sin A sin B sin C

a b c

The cosine-rule can be used when the following is known of the triangle: - 3 sides - 2 sides and an included angle

2 2 2 2 cosAa b c bc The area of any triangle can be found when at least two sides an included angle are

known 1Area of ABC sin C2

ab


Nov. 2011 P2 Question 11 Diagram for Question 11 (Nov. 2011 P2)

08-SEP 09:00 -10:00 MATHS 2: 2D & 3D TRIGONOMETRY

STUDY NOTES

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:

- more than 1 angle and a side - 2 sides and an angle (not included) sin A sin B sin C

a b c

The cosine-rule can be used when the following is known of the triangle: - 3 sides - 2 sides and an included angle

2 2 2 2 cosAa b c bc The area of any triangle can be found when at least two sides an included angle are

known 1Area of ABC sin C2

ab


Nov. 2011 P2 Question 11 Diagram for Question 11 (Nov. 2011 P2)


37 MTN Learn

MINDSETLEARNXTRA

powered by

SPRING SCHOOL GRADE12Mathematics

RADIO BROADCAST


38 MTN Learn

MINDSETLEARNXTRA

powered by

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 the curve is concave down

Tangent to a Curve

A tangent is just a straight line so you need m & c.

Steps:

- Find the gradient using f’(x value of point of contact) = gradient of tangent (i.e. substitute x into derivative to get m)

- Now you know m in y = mx+c - If you only have the x value, sub into original equationn to get y - Next, using equations of straight lines, substitute in point of contact to get c.

Rates of Change and Application of Calculus.

Know the formulas for perimeter, area and volume of the basic shapes, e.g. spheres, cubes, rectangles, triangular prisms, circles, trapeziums etc. For application questions you need to create equations using the given information. Then take what you are maximising/minimising and find the derivative of that eqn = 0. Solve for the unknown. This will be the maximising or minimising value.


39 MTN Learn

MINDSETLEARNXTRA

powered by


Feb. 2011 P1 Question 9, 10 and11

Graph for Question 11 (Feb. 2011 P1)


40 MTN Learn

MINDSETLEARNXTRA

powered by

02-OCT 09:00 -10:00 MATHS 2: TRIGONOMETRY STUDY TIPS Trig Graphs

It is useful to know the effects of parameters on a trig graph.

Consider: y = Asin (Bx-C)+D

A - Changes the amplitude (Shape change)

B – Changes the period (shape change) where period = 360/A

C - Horizontal shift (position change) e.g. sin(x-30) moves the graph of sinx by 30o to the right

D - Vertical shift (position change) e.g. sin(x)-1 moves the graph of sinx by 1 unit down

Trig Equations

This is when you are asked to solve for the angle. Think general solutions and reference angles.

Tips: - Bring all to one side - Change to sin and cos - Use squared Ids and double angle formulas - Find a way to factorise! - Get simple trig expressions where you can find ref angles and general solutions

from them. Pythagoras Questions If 3sinθ + 5 = 0 and cosθ > 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

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

S A√

T√ C

√√


41 MTN Learn

MINDSETLEARNXTRA

powered by

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 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 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

Δ

Δ

√

√

√

√

√

√


42 MTN Learn

MINDSETLEARNXTRA

powered by


Feb. 2011 P2 Question 10 and 11



43 MTN Learn

MINDSETLEARNXTRA

powered by

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


Feb. 2011 P1 Question 5 and 6


44 MTN Learn

MINDSETLEARNXTRA

powered by



45 MTN Learn

MINDSETLEARNXTRA

powered by

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 = 22 where A(a, 2) & B(-3, 5)

Since AB = 22 AB2 = ( 22 )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

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!

θ 1 θ 2

α


46 MTN Learn

MINDSETLEARNXTRA

powered by

QUESTIONS FOR DISCUSSION

Feb. 2011 P2 Question 4, 5 and 6




47 MTN Learn

MINDSETLEARNXTRA

powered by

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<r<1. This is called the criteria for a convergent series. The formula is Sn = a/1-r and is not given in the formula sheet. A common example of a convergent series question is that of a bouncing ball coming to rest. The distance the ball travels in total is given by the Sn formula.

Quadratic Series If there is a constant 2nd difference between successive terms of a sequence, then it is a quadratic sequence. To find Tn use Tn = an2+bn+c.

Solve a by using 2nd difference = 2a. Solve b by using T2-T1 = 3a+b where you now know a, Solve c by using T1 = a+b+c where you now know a and b

∑ Questions There are generally 2 types of sigma questions.

- Questions where you are looking for n. In this case the question will normally say

where the question is given as an equation and you have to use the Sn

formula to solve for n.

- The question is simply given as an evaluate

10

1)(

xxF where

n is given and you must use the Sn formula knowing n to simplify the expression. QUESTION FOR DISCUSSION Feb. 2011 P1 Question 1, 2, 3 and 4

n

xkxF

1)(


48 MTN Learn

MINDSETLEARNXTRA

powered by

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<r<1. This is called the criteria for a convergent series. The formula is Sn = a/1-r and is not given in the formula sheet. A common example of a convergent series question is that of a bouncing ball coming to rest. The distance the ball travels in total is given by the Sn formula.

Quadratic Series If there is a constant 2nd difference between successive terms of a sequence, then it is a quadratic sequence. To find Tn use Tn = an2+bn+c.

Solve a by using 2nd difference = 2a. Solve b by using T2-T1 = 3a+b where you now know a, Solve c by using T1 = a+b+c where you now know a and b

∑ Questions There are generally 2 types of sigma questions.

- Questions where you are looking for n. In this case the question will normally say

where the question is given as an equation and you have to use the Sn

formula to solve for n.

- The question is simply given as an evaluate

10

1)(

xxF where

n is given and you must use the Sn formula knowing n to simplify the expression. QUESTION FOR DISCUSSION Feb. 2011 P1 Question 1, 2, 3 and 4

n

xkxF

1)(


49 MTN Learn

MINDSETLEARNXTRA

powered by

06-OCT 09:00 -10:00 MATHS 2: TRANSFORMATIONS AND TRIGONOMETRY

STUDY TIPS

Translation – Rotation about the Origin If the rotation is clockwise by θ, use an angle of β = 360 - θ in the formula for the image of (x, y). Remember this formula is on the formula sheet: x =xcosβ – ysinβ and y’ = ycosβ+xsinβ. Remember also: A(x,y) A’(-y, x) after rotation of 90o about origin B(x,y) B’(-x,-y) after rotation of 180o about origin


Feb. 2011 P2 Question 7, 8 and 9



50 MTN Learn

MINDSETLEARNXTRA

powered by

pRELIM REVIEWGRADE12Mathematics

RADIO BROADCAST


51 MTN Learn

MINDSETLEARNXTRA

powered by

PRELIM REVIEW During this week we will be answering questions from various provincial preliminary examination papers. Please email questions that you need help with to [email protected] before the radio broadcast. You can also phone in on the helpdesk number, 086 1058 262, to speak to a teacher and to answer the daily on air competition question. We will post questions we receive from learners on the Facebook events page so that you can follow.







52 MTN Learn

MINDSETLEARNXTRA

powered by

exam revisionPAPER 1

GRADE12Mathematics

RADIO BROADCAST


53 MTN Learn

MINDSETLEARNXTRA

powered by

EXAM REVISION: PAPER 1 For the two days before you write the final exam, we will work through the solutions to questions taken from the Feb / Mar 2012 exam paper. Please call in or send an email to let us know with which questions you are struggling. 31-Oct 17:00 -18:00 Maths 1 Algebra

Questions for discussion Feb 2012 P1 Questions 1, 2, 3, 7 and 12

31-Oct 18:00 -19:00 Maths 1 Functions

Questions for discussion Feb 2012 P1 Questions 4, 5 and 6

Table for Question 4 (Feb 2012 P1)

Graph for Question 5 (Feb 2012 P1)


54 MTN Learn

MINDSETLEARNXTRA

powered by

01-Nov 17:00 -18:00 Maths 1 Calculus

Questions for discussion Feb 2012 P1 Questions 8 and 9


01-Nov 18:00 -19:00 Maths 1 Calculus Applications



55 MTN Learn

MINDSETLEARNXTRA

powered by

exam revisionPAPER 2

GRADE12Mathematics

RADIO BROADCAST


56 MTN Learn

MINDSETLEARNXTRA

powered by

EXAM REVISION: PAPER 2 For the two days before you write the final exam, we will work through the solutions to questions taken from the Feb / Mar 2012 exam paper. Please call in or send an email to let us know with which questions you are struggling.

03-Nov 09:00 -10:00 Maths 2 Data Handling

Questions for discussion Feb 2012 P2 Questions 1, 2, 3 and 4

Table for Question 2 (Feb 2012 P2)



57 MTN Learn

MINDSETLEARNXTRA

powered by

03-Nov 10:00 - 11:00 Maths 2 Analytical Geometry




03-Nov 10:00 - 11:00 Maths 2 Analytical Geometry





58 MTN Learn

MINDSETLEARNXTRA

powered by



59 MTN Learn

MINDSETLEARNXTRA

powered by

04-Nov 17:00 -18:00 Maths 2 Trigonometry





60 MTN Learn

MINDSETLEARNXTRA

powered by

04-Nov 18:00 -19:00 Maths 2 Trigonometry


Diagram for Question 12 (Feb 2012 P2)


61 MTN Learn

MINDSETLEARNXTRA

powered by

lISTENER fEEDBACKcOMPETITION

RADIO BROADCAST


62 MTN Learn

MINDSETLEARNXTRA

powered by

LISTENER’S FEEDBACK COMPETITION Complete and return this form and stand a chance to win an Apple iPad 3.

1. What grade are you in? Grade 10 Grade 12

2. How did you hear about Mindset Learn Xtra Radio Revision (tick all that apply)? A friend My school/teacher The radio A newspaper

The Mindset website The Mindset Facebook page

3. To which Mindset Learn Xtra Radio Revision programmes did you listen? Please tick all boxes that apply.

Grade 12 Grade 10

Mathematics Mathematics

Physical Sciences Physical Sciences

English First Additional Language English First Additional Language

4. Please rate how much you agree with each of the following statements by placing a tick in the relevant box where 1 is strongly disagree and 5 is strongly agree.

1 2 3 4 5

The radio broadcasts were generally clear and free from interference

The Learner Guide book was helpful.

The pace of the radio shows was just right – not too fast and not too slow.

The presenters spoke clearly and were easy to understand.

The content presented on TV and in the Learner Guide was at the right level – not too easy and not too difficult.

Topics were covered fully and all important areas were dealt with.

Working through exam questions helped me understand the content better.

The radio shows were enjoyable and informative.

I would recommend this revision programme to other learners.

The revision was very helpful in preparing me for the exams.

I feel more confident about writing exams now than I did.

5. Had you ever heard of Mindset, Mindset Learn or Mindset Learn Xtra before this broadcast?

Yes No


63 MTN Learn

MINDSETLEARNXTRA

powered by

5. Had you ever heard of Mindset, Mindset Learn or Mindset Learn Xtra before this broadcast?

Yes No

6. Had you ever used any Mindset content before broadcast? Yes No

7. Have you ever visited the Mindset Learn website (www.mindset.co.za/learn)? Yes No

8. Have you ever visited the Mindset Learn Facebook page www.facebook.com/learnxtra)?

Yes No

9. Are there any other general comments you would like to make?

_______________________________________________________________________

Your name: _________________________________________________________________________

Your contact number: __________________________________________________________________

Your address: _________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

Competition closes: 9 November 2012 at 12 noon.

Entries to be sent to [email protected], 086 101 6463 or Mindset Network – PO Box 4635, Randburg. 2125.

You can also provide your feedback and enter the competition at www.mindset.co.za/learn/win.

Terms and conditions: Full competition terms and conditions available at www.mindset.co.za/learn/win.

MTN Learn - Maths · PDF fileMTN . Learn . MINDSET ... At Mindset we are committed to helping...

Documents

Transcript of MTN Learn - Maths · PDF fileMTN . Learn . MINDSET ... At Mindset we are committed to helping...