Learn Xtra Winter School Gr12 Maths P2 Learner Guide

Mathematics Paper 2 Exam Revision Learner’s Guide Winter School July 2012

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Contents

Introduction .................................................................................................... 2

Programme Outline ........................................................................................ 4

Winter School Broadcast Schedule ................................................................ 5

Other Exam Revision Programmes ................................................................ 6

Mathematics Exam Preparation ..................................................................... 9

Topic Tips for Co-ordinate Geometry & Transformations ............................. 13

Topic 1: Co-ordinate Geometry & Transformations ...................................... 14

Topic Tips for Trigonometric Equations & Identities ..................................... 17

Topic 2: Trigonometric Equations & Identities .............................................. 18

Solutions to Topic 1: Co-ordinate Geometry & Transformations .................. 20

Solutions to Topic 2: Trigonometric Equations & Identities........................... 26


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Introduction

Have you heard about Mindset? Mindset Network, a South African non-profit organisation, was founded in 2002. We develop and distribute quality and contextually relevant educational resources for use in the schooling, health and vocational sectors. We distribute our materials through various technology platforms like TV, the Internet (www.mindset.co.za/learn) and on DVDs. The materials are made available in video, print and computer-based multimedia formats. At Mindset we are committed to innovation. Over the last three years, we successfully ran a series of broadcast events leading up to and in support of the NSC examinations. Now we are proud to announce our 2012 edition of Learn Xtra Winter School Exam Revision, in partnership with the National Department of Basic Education. Once again, Winter School supports seven key subjects - Mathematics, Physical Sciences, Life Sciences, Mathematical Literacy, English 1st Additional Language, Accounting and Geography. During Winter School, you will get exam overviews, study tips on each of the topics we cover, detailed solutions to selected questions from previous examination papers and live interactive programmes so you can work through more exam questions with our experienced teachers. Connect with us now

Learnxtra

[email protected]

@learnxtra

086 105 8262

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Mindset Get the free app at pepclub.mobi

Getting the most from Winter School Before you watch the broadcast of a topic, read through the questions for the topic and try to answer them without looking up the solutions. If you get stuck and can’t complete the answer don’t panic. Make a note of any questions you have. When watching the Topic session, compare the approach you took to what the teacher does. Don’t just copy the answers down but take note of the method used. Make sure you keep this booklet after Winter School. You can re-do the exam questions you did not get totally correct and mark your own work by looking up the solutions at the back of the booklet. 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 may benefit from working with a friend or a small study group, as long as everyone is as committed as you are. Mindset believes that the 2012 Winter School programme will help you achieve the results you want.

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If you find Winter School a useful way to revise and prepare for your exams, remember that we will be running the following other great Exam revision programmes during 2012:

Grade 12 Learn Xtra Term 2 repeat: 23rd June to 1st July

Grade 11 Learn Xtra Winter School: 9th July to 12th July

Grade 10 Learn Xtra Winter School: 13th July to 15th July

Grade 12 Learn Xtra Spring School – 1st October to 5th October

Grade 11 and 12 Learn Xtra Exam School – 22nd October – 20th November


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Programme Outline

The Mindset Learn Xtra Grade 12 Winter School is designed to focus on two subjects each day. For each subject you will find the following sessions:

Examination Overview This is a 15 minute session that gives details of what you can expect in each examination paper. Practical guidelines are also given on how to prepare for the day of the exam.

Topics Tips In this session you will be given a 15 minutes summary of the key ideas you need to know, common errors and study hints to help you prepare for your exams.

Topic Session An expert teacher will work through specially selected questions from previous exam papers.

Live Show This is your chance to ask your own questions. So submit your question to the Helpdesk so we can do it for you on air. All questions you submit will be answered within 48 hours as normal. Submit your questions through the following channels.

Learnxtra

[email protected]

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086 105 8262

Winter School Competition In each Winter School Live Show, we will be giving away exciting prizes including Mindset content, CDs and Mindsetter t-shirts. To win, all you need to do is watch and participate in the Live Shows through Facebook, Twitter, Peptx, or email. Winners will be announced at the end of each show.

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Winter School Broadcast Schedule

Monday 2nd July Tuesday 3rd July Wednesday 4th July Thursday 5th July Friday 6th July

09h00 Maths Paper 1: Functions

English FAL: Paper 1 Physical Sciences Paper 1: Mechanics

Maths Paper 2: Co-ordinate Geometry and Transformations

Physical Sciences Paper 2: Organic Chemistry

12h30 Lunch

13h00 Maths Literacy: Shape and Space

Accounting: Cash flow and interpretation of Ratios

Life Sciences Paper 1: DNA & RNA

Geography: Climatology

Life Sciences Paper 2: Life Processes

16h30 Maths Paper 1: Linear Programming

English FAL: Paper 3 Physical Sciences Paper 1: Waves, Sound and Light

Maths Paper 2: Trigonometry – Equations and Identities

Physical Sciences Paper 2: Rates and Chemical Equilibrium

18h30 Broadcast Ends


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Other Exam Revision Programmes

Grade 12 Learn Xtra Term 2 Repeat

A repeat of some of the crucial Term 2 Learn Xtra topics.

Date Maths (09h00) Physical Sciences (10h30)

Life Sciences (12h00)

Maths Literacy (13h30)

Accounting (15h00) Geography (16h30)

23rd

June Trigonometry

Organic Molecules Biodiversity Part 1 Reading from Graphs and Trends

Cash Flow Statements & ratios

Structural land forms

24th

June Trigonometry Identities / Transformations

Organic Molecules: Functional groups

Biodiversity Part 2 Using Graphs Debtors/creditors reconciliations

Geographic information systems

25th

June Functions & Graphs Organic Reactions Plant responses to the environment

Mapwork Part 1 Bank reconciliations Map interpretation climate / geomorphology

26th

June Inverse Functions & Graphs

Electromagnetic Spectrum

Nervous system Mapwork Part 2 Cash budgets/Projected income statement

People and Places: Processes & Spatial Patterns

27th

June Graphs: from linear to log

Photoelectric Effect Endocrine system and Thermo regulation

Space and Shape Part 1

Vat/Stock valuation Sustainability

28th

June Introduction to Differential Calculus

Wavefronts Eye and Ear Space and Shape Part 2

Manufacturing – Production Cost Statements/Break-even analysis calculation

Urban settlements

29th

June Calculus Applications Doppler Effect

General reproduction Space and Shape Part 3

Consolidations – Cash flows/Ratios/Corporate governance

People and their Needs: Structure of the Economy

30th

June Calculus Applications Pt 2

2D & 3D Optical Phenomena

Male And Female Reproductive Structures

Data Handling – Collecting and Organising Data

Consolidations – Financials /Asset disposal/ledgers/Audit report

Transport and trade

1st

July Live (Repeated on 16 Jun)

Live (Repeated on 16 Jun)



Consolidation- Bank reconciliation/internal controls/stock systems

Globalisation


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Grade 11 Winter School

A repeat of some of the crucial Term 2 Learn Xtra topics with 2 brand new LIVE shows every day.

Monday 9th July Tuesday 10th July Wednesday 11th July Thursday 12th July

09h00 Maths: Linear Graphs and Parabolas

Physical Sciences: Electric Circuits

Maths: Analytical Geometry

Physical Sciences: Chemical Bonding

10h30 Maths: Hyperbola, Exponential and Trig Graphs

Physical Sciences: Electromagnetism

Maths: Trigonometry Physical Sciences: Kinetic Theory of Gasses

12h00 Maths: Functions and Algebra LIVE

Physical Sciences: Electricity and Magnetism LIVE

Maths: Analytical Geometry and Trigonometry LIVE

Physical Sciences: Chemistry LIVE

13h00 Lunch

13h30 Maths Lit: Ration, Proportion and Rate

Life Sciences: The Transport System Part 1

Maths Lit: Graphs Live Sciences: Viruses and Bacteria

15h00 Maths Lit: Percentage in Context

Life Sciences: The Transport System Part 2

Maths Lit: Solving Real World Problems

Life Sciences: Fungi and Protists

16h30 Maths Lit: Percentage, Ratio, Proportion and Rate LIVE

Life Sciences: Life Processes in Plants and Animals LIVE

Maths Lit: Graphs, Tables and equations LIVE

Life Sciences: Life at the Molecular, Cellular and Tissue Level LIVE

17h30 Maths: Number Patterns Life Sciences: Excretion Maths Lit: Financial Maths Physical Sciences: Energy and Chemical Change



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Grade 10 Winter School

A repeat of some of the crucial Term 2 Learn Xtra topics.

Friday 13th July Saturday 14th July Sunday 15th July

09h00 Maths: Trigonometry in Triangles Maths: Trig Equations Maths: Trig Functions

10h30 Physical Sciences: Transverse Pulses & Waves

Physical Sciences: Longitudinal Waves & Sound

Physical Sciences: Electromagnetic Radiation

12h00 Life Sciences: Molecules of Life Life Sciences: Cells - The Basic Units of Life

Life Sciences: Cell Division – Mitosis

13h30 Maths Lit: Numbers Maths Lit: Graphs that tell a story Maths Lit: Personal & Household Finance

15h00 Maths: Algebraic Expressions Maths: Exponents Maths: Numbers & Number Patterns

16h30 Physical Sciences: Tools of Science Physical Sciences: Matter Physical Sciences: Microscopic view of Matter



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Mathematics Exam Preparation

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. Here are some guidelines

to help you in your preparations.

Mark the date of your Exam today Mathematics Paper 1 09:00 – 12:00 28 October 2011 Mathematics Paper 2 09:00 – 12:00 31 October 2011

Mark the dates of the Learn Xtra Matric Revision programmes

Spring School – 1st October to 5th October

Exam School – 22nd October – 20th November

Practice questions from past examination papers

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. For a 8 mark question you have only 6,6 minutes! That’s less than a mark a minute. To work out the time you have multiple the marks by 150 (total marks)/180 (total time).

Check your working with the memo. If you don’t understand the answer given contact the Learn Xtra Help desk.

If you did not get the question right make sure you try it again after a few days!

Know what’s in the each Exam and the mark breakdown

Mathematics Paper 1 3hrs Total marks: 150 o Number & Number Relations

Patterns & Sequences ±30 marks Annuities & Finance ±15 marks

o 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 o Space, Shape and Measurement

Co-ordinate Geometry ±40 marks Transformations ±25 marks Trigonometry ±60 marks

o Data Handling and Probability Data Handling ±25 marks

Additional Notes

All aspects relating to trigonometric graphs will be examined in Paper 2 only.

Make sure you can derive the derivate from first principles for:

and


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Make sure you know the properties of geometric figures such as isosceles triangles, equilateral triangles, right angled triangles, the kite, parallelogram, rectangle, rhombus and square. You can be asked to confirm the properties of these figures in Co-ordinate Geometry questions.

Tips for Paper 2

These are additional exam tips you can share with your learners:

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: o bring all to one side o change to sin and cos o use squared Ids and double angle formulas o find a way to factorise! o 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 pythag question!

Method: Get the trig ratio on the LHS (sinθ = -5/3 y/r) then calc the quad using the restriction

sinθ < 0 and cosθ > 0

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

Trig identities Proving 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!

S A

√

T

√ C

√√


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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 given opposite on of the known sides. (Note if the side opp 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 opp 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

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

Find the gradient of the radius from centre to the pt of contact (where tang meets circle)

Get grad of tangent using fact that grad tang is to grad radius

Now you know m in y = mx+c

Next using eqns of straight lines, sub in pt of contact to get c and done!

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

o A(x,y) A’(-y, x) after rotation of 90o about origin

o B(x,y) B’(-x,-y) after rotation of 180o about origin

The Distance formula If you are given the distance between to 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 now simply solve for a.

Putting an equation of a circle into standard form 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)

Δ

Δ

√

√

√

√

√

√


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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.

θ 1 θ 2

α


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Topic Tips for Co-ordinate Geometry & Transformations

Overview

Straight lines on Cartesian plane

Circles on Cartesian Plane

Transformations Straight lines on Cartesian plane Know how to apply formulae

Gradient

Perpendicular Lines

Mid-point

Distance between two points Circles on Cartesian plane

Circle with centre at origin

Circle with centre not at the origin

Circle & tangent

Transformations Know how to apply the following transformations to a point or a shape

Translation

Reflection

Rotation

Enlargement Study Hints

Be familiar with the data sheet

Write up a summary of all the formulae one a single A4 page including the formulae not on the data sheet

Revise the geometric properties of triangles and quadrilaterals


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1

2

BPQ

Topic 1: Co-ordinate Geometry & Transformations

Question 1

The diagram below shows the points P(0 ; 2) and Q(4 ; 0). Point A is the midpoint of PQ. The line AB is perpendicular to PQ and intersects the x-axis at G and the y-axis at B. 1.1 Show that the gradient of PQ is (1) 1.2 Determine the coordinates of A. (2) 1.3 Determine the equation of the line AB (5) 1.4 Calculate the length of BQ. (3) 1.5 Show that is isosceles. (2) 1.6 If PBQR is a rhombus, determine the coordinates of R. (3) [16]


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Question 2

The straight line AB has the equation 5 3 5 0y x . Another straight line CD is drawn to

intersect AB at P(5 ; 4) such that the acute angle between AB and CD is 45 .

2.1 Determine the gradient of the line CD. (5) 2.2 Hence, or otherwise, determine the equation of the line CD. (2) [7]

Question 3

3.1 Determine the centre and radius of the circle with the equation defined by 2 2 8 4 38 0x y x y (4)

3.2 A second circle has the equation 2 2( 4) ( 6) 26x y . Calculate the distance

between the centres of the two circles. (2) 3.3 Hence, show that the circles intersect each other. (3) 3.4 Show that the circles intersect along the line 4y x . (4)

[13]


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Question 4

The point ( ; 2)x is rotated about the origin through an angle of 150 in an anticlockwise

direction to give the point ( 3 ; )y . Calculate the values of x and y. [5]

Question 5

In the diagram below MNP is given with vertices M(– 5 ; 2), N (6 ; 4) and P(2 ; – 4).

MNP is enlarged by a factor of 1,5 to .

5.1 Draw on the grid provided. (3) 5.2 Write down the vales of:

5.2.1 (2)

5.2.2 (2)

5.3 If the above transformation is applied to n more times to get the image

, write down the value of . (2)

[9]

/// PNM

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

x

y

O

/// PNM

// NM

MN

/// PNΔMarea

ΔMNParea

MNP

////// PNΔM////// PNΔMarea

ΔMNParea

M(– 5 ; 2)

P(2 ; – 4)

N(6 ; 4)


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Topic Tips for Trigonometric Equations & Identities

Overview


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Topic 2: Trigonometric Equations & Identities

Question 1

Adapted from February, 2011,DOE Paper 2, 10.4

Determine the general solution of: (7)

Question 2

Adapted from November, 2009, DOE Paper 2, 9.3

Determine the general solution of: (7)

Question 3

Adapted from October, 2009, DOE Paper 2, 5.1 and 5.2 The following identity is given:

3.1 Give the values of A for which this identity will be undefined. (2) 3.2 Prove the following identity (6)

Question 4

Adapted from February/March, 2011, DOE Paper 2, 10.5.2 Prove that:

for all other values of x. (5)

Question 5

Adapted from November, 2008, Western Cape DOE Paper 2, 4.1 Simplify the following to its simplest form:

(5)


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Question 6

Adapted from June, 2008, DOE Paper 2, 4.2

Given that sin27 express each of the following in term of t:

6.1 (2)

6.2 (3)

6.3 (2)

6.4 (3)

Question 7

Adapted from November, 2008, DOE Western Cape Paper 2, 5.1 Answer the following without the use of the calculator:

Given that:

and

, where , calculate:

7.1 (6)

7.2 (3)


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Solutions to Topic 1: Co-ordinate Geometry & Transformations

Question 1

1.1 PQ

2 0 1

0 4 2m

substitution (1)

1.2 0 4 2 0A ; A(2 ;1)

2 2

x-coordinate y-coordinate

(2)

1.3 AB PQ

AB

. 1

2

1 2( 2)

1 2 4

2 3

m m

m

y x

y x

y x

AB PQ. 1m m

AB 2m

equation of line substitution 2 3y x

(5)

1.4

2 2 2

2

B(0 ; 3)

Q(4 ; 0)

BQ (0 4) ( 3 0)

BQ 25

BQ 5

B(0 ; 3)

substitution answer

(3)

1.5 BP 5

But BQ 5

BP BQ

BPQ is isosceles

BP 5 BP BQ

(2)

1.6 If PBQR is a rhombus, then A is the midpoint of BR. Let the coordinates of R be ( ; )x y

0 32 and 1

2 2

4 and 5

R(4 ; 5)

x y

x y

A is the midpoint x-coordinate y-coordinate

(3)

[16]


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Question 2

2.1

AB

5 3 5 0

5 3 5

31

5

3

5

Let be the inclination of AB

3tan

5

30,96

Let be the inclination of CD

45 30,96 75,96

gradient of CD tan 75,96 4

y x

y x

y x

m

AB

3

5m

3

tan5

30,96

75,96

CD 4m

(5)

2.2 4 4( 5)

4 4 20

4 16

y x

y x

y x

substitution equation

(2)

[7]

Question 3

3.1 2 2

2 2

2 2

2 2

8 4 38 0

8 4 38

8 16 4 4 38 16 4

( 4) ( 2) 58

Centre ( 4 ; 2)

radius 58

x y x y

x x y y

x x y y

x y

completing the square factor form centre radius

(4)

3.2 Centre of second circle: (4 ; 6)

2 2Distance ( 4 4) ( 2 6) 128

(4 ; 6)

128

(2)

3.3 sum of radii 58 26 12,71

Distance between centres 128 11,31

sum of radii distance between centres

circles intersect

sum of radii conclusion

(3)


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3.4 Check that the line 4y x cuts the two circles at

the same point. 2 2

2 2

2 2

2

2

8 4 38 0

( 4) 8 4( 4) 38 0

8 16 8 4 16 38 0

2 4 6 0

2 3 0

( 3)( 1) 0

3 or 1

x y x y

x x x x

x x x x x

x x

x x

x x

x x

2 2

2 2

2 2

2 2

2

2

( 4) ( 6) 26

( 4) ( 4 6) 26

8 16 ( 2) 26

8 16 4 4 26

2 4 6 0

2 3 0

( 3)( 1) 0

3 or 1

x y

x x

x x x

x x x x

x x

x x

x x

x x

substitution answer substitution answer

(4)

[13]


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Question 4

3 cos150 2sin150

3 ( cos30 ) 2sin 30

3 cos30 2sin 30

3 13 2

2 2

6 3 2

3 4

4

3

sin150 2cos150

4sin 30 2cos30

3

4 1 32

2 23

23

3

2 3 3

3

2 3

3

x

x

x

x

x

x

x

y x

y

y

y

y

y

1

3y

3 cos150 2sin150x

3 1

3 22 2

x

4

3x

sin150 2cos150y x

1

3y

[5]


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Question 5

5.1

5.1 see diagram above coordinates of new points

(3)

5.2.1 2 2 2

2

/ /

/ /

MN (6 ( 5)) (4 2)

MN 125

MN 125

3M N 125

2

MN 125 2

3 3M N 1252

MN 125

/ /

MN 2

3M N

(2)

5.2.2

/ / / 2

area ΔMNP 1 1 4

9 9area ΔM N P 342

4

9

(2)


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5.3 1

// // //

area ΔMNP 4

9area ΔM N P

n

answer (2) [9]


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Solutions to Topic 2: Trigonometric Equations & Identities

Question 1

Reference angle: Reference angle: 0

Question 2

A

C T

S A

C T

S


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Question 3

3.1

3.2


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Question 4


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Question 5

Question 6

6.1

6.2

6.3


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6.4

Question 7

and

and

7.1

C T

A S

T

S

C

A

-3 5

4

-8

17

15


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7.2


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Notes


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