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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
helpdesk@learnxtra.co.za
@learnxtra
086 105 8262
www.learnxtra.co.za
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
helpdesk@learnxtra.co.za
www.learnxtra.co.za
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)
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
19h00 Broadcast Ends
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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
18h00 Broadcast Ends
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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
Mathematics Paper 2 Exam Revision Learner’s Guide Winter School July 2012
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Notes
Mathematics Paper 2 Exam Revision Learner’s Guide Winter School July 2012
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