GRADE/GRADE 12 SEPTEMBER 2016 MATHEMATICS P2 / WISKUNDE V2 MEMORANDUM
|
|
- Jade Simon
- 7 years ago
- Views:
Transcription
1 NATIONAL SENIOR CERTIFICATE GRADE/GRADE SEPTEMBER 06 MATHEMATICS P / WISKUNDE V MEMORANDUM MARKS / PUNTE: 50 This memorandum consists of 6 pages. Hierdie memorandum bestaan uit 6 bladsye.
2 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION / VRAAG Day/Dag Weight / Gewig correct points 5-7 correct points plotting all points 4 punte korrek 5 7 punte korrek alle punte afgesteek. a 4,84 b 5,83 y 4,84 5,83x.3 x ; y 8;5;75,5.4 y-int 4,84 4,84 5,83x 80 A B equation / vergelyking 8 ;5;75,5 and/en y-int / y-afsnit 4,84 regression line / regressielyn () substitution / vervanging 5,83x 44,84 x 7,69 answer / antwoord On the morning of the 8 th day the bar of soap will be less than 80 grams. Op die oggend van die 8 ste dag sal steen seep minder as 80 gram wees. ().5 r 0, 998 answer / antwoord ().6 Very strong negative correlation. Baie sterk negatiewe answer / antwoord korrelasie. () [] Kopiereg voorbehou Blaai om asseblief
3 (EC/SEPTEMBER 06) WISKUNDE V 3 QUESTION / VRAAG learners / leerders answer / antwoord (). 3 pages / bladsye answer / antwoord ().3 x 8, 9 answer / antwoord.4 3, answer / antwoord.5 8,9 3,; 8,9 3, 5,07;4,3 8 learners are outside one standard deviation 8 leerders is buite een standaardafwyking interval / interval 8 learners / leerders 8,57% () () ,0% [8] Copyright reserved Please turn over
4 4 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 3 / VRAAG mpr m SP PT SQ y 4 x y x 3.3 Equation of PR / Vergelyking van PR y x 4 y x 3 x x 3 4x 4 x 6 5x 0 x y subst. P and R into correct formula verv. van P en R in korrekte formule m PR m SQ subst. m and Q into correct formula verv. van m en Q in korrekte formule y x substituting m and P into equation of a str. line / vervanging van m en P in die vergelyking van reguitlyn equation of PR / vergelyking van PR equating PR and SQ / gelykstel van PR en SQ x-value / x-waarde y-value / y-waarde () OR/OF substituting m and R into equation of a Kopiereg voorbehou Blaai om asseblief
5 (EC/SEPTEMBER 06) WISKUNDE V 5 y 5 x 4 y0 x 4 y x6 y x3 x x 3 4x 4 x 6 5x 0 x y 3.4 x y4 x 5 and/ en y 8 S ( 5 ; 8) 3.5 SQ PT Area PQS unit / eenhede str. line / vervanging van m en R in die vergelyking van reguitlyn equation of PR / vergelyking van PR equating PR and SQ / gelykstel van PR en SQ x-value / x-waarde y-value / y-waarde substituting into correct formula vervanging in korrekte formule x-value / x-waarde y-value / y-waarde subt. into correct form / verv. in korrekte formule SQ = 6 5 PT = 5 Subt into correct form. / verv. in korrekte formule 5 units / eenhede (5) OR/OF area of PQS = ST PT unit / eenhede SQ = 6 5 ST = 3 5 PT = 5 subst into form / verv. in formule 5 units / eenhede (5) [8] Copyright reserved Please turn over
6 6 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 4 / VRAAG 4 4. x 8x 6 y x 4 y 3 M 4 ; y 3 y y 3 tan/ rkl 6y y Q (0; ) 3 mradius 4 0 m y x 0 y x [tangent radius] [raaklyn completing square kwadraatsvoltooiing x 4 y 3 0 x-coordinate / x-koördinaat y-coordinate / y-koördinaat (4) subst. x = 0 into circle equation verv. x = 0 in sirkel vergelyking y 3 4 y m rad m / m raaklyn = tan subst. mtan/ rkl and Q into correct form. / verv. mtan/ rkl en Q in korrekte formule. equation / vergelyking 4.4 y 6 answer / antwoord () (4) Kopiereg voorbehou Blaai om asseblief
7 (EC/SEPTEMBER 06) WISKUNDE V x 5 x 5 U ; 6 6 mau mad 0 0 mau mda AU DA Â 90º DQˆ U 90º [tangent radius] / [raaklyn radius] 6 = x + 5 x s QUAD is a cyclic quad.[opp. add up to 80º] QUAD is ʼn koordevierhoek [ teenoorst. e se som is 80º] m AU m AD m m AU DA Â 90 D Q ˆU 90 R () (6) [0] Copyright reserved Please turn over
8 8 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 5 / VRAAG sin cos.9 5. sin 5 sin 5 answer t cos 9 cos 38 cos9 cos9 sin 74 sin 74 4 OR/OF cos 38 cos 38 t cos(80 x).sin(80 x).sin 74 sin( x 360).sin 37.sin 53.sin( x 90) ( cos x).sin x.sin 74 sin x.sin 37.cos 37.( cos x) cos.9 cos 38 cos.9 cos 9 simplification / vereenvoudiging answer / antwoord (4) cos x sin x sin x cos37 cos x sin 74 answer / antwoord cos(80 x).sin(80 x).sin sin( x360 ).sin 37.sin 53.sin( x90 ) ( cos x).sin x.sin sin x.sin 37.cos 37.( cos x) 0 sin sin 37.cos37 4sin 37.cos sin 37.cos cos x sin x sin x cos37 cos x sin 37 0 answer / antwoord (7) Kopiereg voorbehou Blaai om asseblief
9 (EC/SEPTEMBER 06) WISKUNDE V cos x 0 cos x 0 cos x x 0 k.360 x k x 0 k.80; k x k; 5.3. L.H.S/LK sinx cosx sin x cos x sin x sin x sin x sin x sin x sin x sin x k (4) sin x removing brackets verwyder hakies sin x [] Copyright reserved Please turn over
10 0 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 6 / VRAAG sin x 60 sin 90 x x x 360. k or/ of x x 360. k or/ x k of x k 0 0 x 0 0. k ; k Z x 30 ; 0 ; OR/OF cos x cos 30 x x 30 x 360. k or 3x k or x x 30; 0; g: f: x 30 x k 360. k ; k co-ratio / kofunksie both gen. solns beide algemene oplossings co-ratio / kofunksie both gen. solns beide algemene oplossings (5) x-intercept. x-afsnit y-intercept. y-afsnit shape / vorm x-intercept. x-afsnit y-intercept. y-afsnit shape / vorm (6) answer / antwoord () 6.4 h x cos x 90 sin x substitution / vervanging sin x () [4] Kopiereg voorbehou Blaai om asseblief
11 (EC/SEPTEMBER 06) WISKUNDE V QUESTION 7 / VRAAG NP ON NP 77,86 PQ OP cos(0) using Pyth theorem correctly korrekte gebruik van stelling van Pythagoras answer / antwoord subst. into cosine rule verv. in cosinus formule () answer / antwoord PQ 78, ,86 77, cos Nˆ Nˆ 60.0 substitution / vervanging (77,86)(77,86) Nˆ () () [6] Copyright reserved Please turn over
12 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 8 / VRAAG 8 8. Bisects the chord / Halveer die koord answer / antwoord () Qˆ 90 [ in semi circle.] / [ in halwe sirkel] s Lˆ Qˆ 90 [corresp., QN LO] e [ooreenk., QN LO] QL = LP [line from centre perp. to chord] [lynstuk vanaf middelpt is loodreg op koord] 8.. 8ML 8..3 OP OL LP 6ML 9ML 49 7ML 49 ML 7 S R S/R R (4) MF () using Pyth correctly 4ML 3ML 7 korrekte gebruik van Pythagoras simplification / vereenvoudiging ML [9] Kopiereg voorbehou Blaai om asseblief
13 (EC/SEPTEMBER 06) WISKUNDE V 3 QUESTION 9 / VRAAG 9 9. Dˆ 3 30 [ s opp equal sides] /[ e teenoor gelyke sye] S R Ô 0 [sum of s of a.] / [som v/d e S/R van ʼn ] 9. 0 Â 60 [ at centre = at circumf.] / S R [middelpunts = omtrekshoek] () 9.3 Ĉ 0 [opp. s of cyclic quad.] / [teenoorst. e van k.v] S R () 9.4 A Dˆ B 70 [tan chord theorem.] / [raaklyn koord stelling] S R () [9] Copyright reserved Please turn over
14 4 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 0 / VRAAG 0 0. constr. konstr. Û 90 [tan radius.] / [raaklyn radius] Û Ẑ 90 [ in semi circle] / [ in halwe sirkel] Ŝ 80 (90 Û ) 90 [sum of s of a ] [som van die e van ʼn ] Ŝ Û Ŝ Ŷ [ s in same segment] / [ e in dieselfde segment] Û Ŷ S/R S/R S/R S/R (5) Kopiereg voorbehou Blaai om asseblief
15 (EC/SEPTEMBER 06) WISKUNDE V Â x [ s opp. = sides] / [ e teenoor gelyke sye] S/R S/R S/R Ĉ Â x [tan chord theo.] / [raaklyn koord stelling] Â Ĉ [tan chord theo.] / [raaklyn koord stelling] Ĉ Ŝ [alt. s, CB TS] / [verw. e, CB TS] S/R Ĉ Tˆ [corresp s, CB TS] / [ooreenk. e, CB TS] S/R (5) 0.. Ŝ Tˆ x [proven in 0..] / [bewys in 0..] S CS = CT [sides opp. = s ] / [sye teenoor gelyke hoeke] R () 0..3 AR AT S/R [line to one side of a ] BR CT [lynstuk aan een sy van ʼn ] CS = CT CS = CT [proved in 0..] / [bewys in 0..] 3 AT 4 AT 6cm substitution vervanging AT (4) [6] Copyright reserved Please turn over
16 6 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION / VRAAG F R B A C Z D P. Â Ĉ [tan chord theo.] / [raaklyn koord stelling] S R Ẑ [tan chord theo] / [raaklyn koord stelling] S/R Ĉ Ẑ [both = x] / [beide = x] BC RZ [corresp. s =] / [ooreenk. e =] R (4). Ẑ Pˆ [ s in same segment] / [ e in dieselfde segment] S/R Ĉ [corresp. s ; BC RZ] / [ooreenk. e S ; BC RZ] BC is a tangent to circle ACP [conv. of tan chord theorem] BC is ʼn raaklyn aan sirkel ACP [ omgekeerde van raaklyn koord stelling] R.3 Bˆ Dˆ [ext. of a cyclic quad.] S R [buite van koordevierhoek] Rˆ Bˆ [corresp. s, BC RZ] [ooreenk. e, BC RZ] S/R S/R.4 Rˆ Dˆ Pˆ Ẑ s [ in same segment] [ e in dieselfde segment] Â Ĉ 4 [3 rd ] / [3 de ] RZA DPC [equiangular or ] [gelykhoekig of ] ZA RA PC DC [similar s ] / [gelykvormige e ] ZADC RA....() PC AR AZ AB AC [line to one side of a ] [lynstuk aan een sy van ʼn ] AZ AB AR...() AC ZADC AZ AB PC AC DC AC PC AB N R S/R ZADC RA PC S/R AZ AB RA AC simplification (5) (5) [7] TOTAL/TOTAAL: 50 Kopiereg voorbehou Blaai om asseblief
GRADE/GRAAD 12 SEPTEMBER 2014 MATHEMATICS P2/WISKUNDE V2 MEMORANDUM
NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 12 SEPTEMBER 2014 MATHEMATICS P2/WISKUNDE V2 MEMORANDUM MARKS/PUNTE: 150 This memorandum consists of 12 pages./ Hierdie memorandum
More informationDepartment of Mathematics and Applied Mathematics Departement Wiskunde en Toegepaste Wiskunde
Department of Mathematics and Applied Mathematics Departement Wiskunde en Toegepaste Wiskunde MATHEMATICS COMPETITION WISKUNDE KOMPETISIE GRADES 0 AND GRADE 0 EN SEPTEMBER 04 SEPTEMBER 04 TIME: HOURS TYD:
More informationNATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 12 SEPTEMBER 2014 MATHEMATICS P1/WISKUNDE V1 MEMORANDUM
NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 12 SEPTEMBER 2014 MATHEMATICS P1/WISKUNDE V1 MEMORANDUM MARKS/PUNTE: 150 Hierdie memorandum bestaan uit 16 bladsye./ This memorandum
More informationNATIONAL SENIOR CERTIFICATE NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 12
NATIONAL SENI CERTIFICATE NASIONALE SENI SERTIFIKAAT GRADE/GRAAD MATHEMATICS P/WISKUNDE V NOVEMBER 0 MEMANDUM MARKS/PUNTE: 50 This memorandum consists of 3 pages. Hierdie memorandum bestaan uit 3 bladsye.
More informationNATIONAL SENIOR CERTIFICATE GRADE/GRAAD 12
NATIONAL SENI CERTIFICATE GRADE/GRAAD MATHEMATICS P/WISKUNDE V EXEMPLAR 0/MODEL 0 MEMANDUM MARKS: 0 PUNTE: 0 This memandum consists of pages. Hierdie memandum bestaan uit bladse. Mathematics P/Wiskunde
More informationUNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
VAN/SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS VOORNAME/FIRST NAMES: WTW 162 DYNAMICAL PROCESSES EKSAMEN
More informationName Date Class. Lines and Segments That Intersect Circles. AB and CD are chords. Tangent Circles. Theorem Hypothesis Conclusion
Section. Lines That Intersect Circles Lines and Segments That Intersect Circles A chord is a segment whose endpoints lie on a circle. A secant is a line that intersects a circle at two points. A tangent
More informationFind all of the real numbers x that satisfy the algebraic equation:
Appendix C: Factoring Algebraic Expressions Factoring algebraic equations is the reverse of expanding algebraic expressions discussed in Appendix B. Factoring algebraic equations can be a great help when
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationClass-10 th (X) Mathematics Chapter: Tangents to Circles
Class-10 th (X) Mathematics Chapter: Tangents to Circles 1. Q. AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches
More informationGeometry Unit 5: Circles Part 1 Chords, Secants, and Tangents
Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents Name Chords and Circles: A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.
More informationEquation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1
Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical
More informationShape, Space and Measure
Name: Shape, Space and Measure Prep for Paper 2 Including Pythagoras Trigonometry: SOHCAHTOA Sine Rule Cosine Rule Area using 1-2 ab sin C Transforming Trig Graphs 3D Pythag-Trig Plans and Elevations Area
More informationMATHEMATICS Grade 12 EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014
EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014 Checklist Make sure you learn proofs of the following theorems: The line drawn from the centre of a circle perpendicular to a chord bisects the chord The angle
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel International GCSE Mathematics A Paper 3HR Centre Number Tuesday 6 January 015 Afternoon Time: hours Candidate Number Higher Tier Paper Reference
More informationDepartment of Mathematics and Applied Mathematics Departement Wiskunde en Toegepaste Wiskunde
Department of Mathematics and Applied Mathematics Departement Wiskunde en Toegepaste Wiskunde MATHEMATICS COMPETITION WISKUNDE KOMPETISIE GRADES 8 AND 9 GRADE 8 EN 9 SEPTEMBER 204 SEPTEMBER 204 TIME: 2
More informationTRIGONOMETRY Compound & Double angle formulae
TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae
More informationPractice Test Answer and Alignment Document Mathematics: Geometry Performance Based Assessment - Paper
The following pages include the answer key for all machine-scored items, followed by the rubrics for the hand-scored items. - The rubrics show sample student responses. Other valid methods for solving
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 4H Centre Number Monday 1 January 015 Afternoon Time: hours Candidate Number
More informationName: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More informationNational Quali cations SPECIMEN ONLY. Forename(s) Surname Number of seat. Date of birth Day Month Year Scottish candidate number
N5 SQ9/N5/0 Date Not applicable Duration hour FOR OFFICIAL USE National Quali cations SPECIMEN ONLY Mark Mathematics Paper (Non-Calculator) *SQ9N50* Fill in these boxes and read what is printed below.
More informationGeometry Chapter 10 Study Guide Name
eometry hapter 10 Study uide Name Terms and Vocabulary: ill in the blank and illustrate. 1. circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More informationNational 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Edexcel IGCSE Centre Number Mathematics A Paper 3H Monday 6 June 2011 Afternoon Time: 2 hours Candidate Number Higher Tier Paper Reference 4MA0/3H You must have:
More informationWEDNESDAY, 2 MAY 1.30 PM 2.25 PM. 3 Full credit will be given only where the solution contains appropriate working.
C 500/1/01 NATIONAL QUALIFICATIONS 01 WEDNESDAY, MAY 1.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper 1 (Non-calculator) 1 You may NOT use a calculator. Answer as many questions as you can. Full
More informationSection 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18
Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel International GCSE Mathematics A Paper 1FR Centre Number Wednesday 14 May 2014 Morning Time: 2 hours Candidate Number Foundation Tier Paper Reference
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationTUESDAY, 6 MAY 9.00 AM 9.45 AM. 2 Full credit will be given only where the solution contains appropriate working.
X00//0 NATIONAL QUALIFICATIONS 04 TUESDAY, 6 MAY 9.00 AM 9.45 AM MATHEMATICS INTERMEDIATE Units, and Paper (Non-calculator) Read carefully You may NOT use a calculator. Full credit will be given only where
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationAlgebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids
Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?
More informationwww.sakshieducation.com
LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c
More informationSample Problems. Practice Problems
Lecture Notes Circles - Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More information1. Introduction circular definition Remark 1 inverse trigonometric functions
1. Introduction In Lesson 2 the six trigonometric functions were defined using angles determined by points on the unit circle. This is frequently referred to as the circular definition of the trigonometric
More informationHigh School Geometry Test Sampler Math Common Core Sampler Test
High School Geometry Test Sampler Math Common Core Sampler Test Our High School Geometry sampler covers the twenty most common questions that we see targeted for this level. For complete tests and break
More informationStraight Line. Paper 1 Section A. O xy
PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of
More informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationPaper Reference. Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Centre No. Candidate No. Paper Reference 1 3 8 0 4 H Paper Reference(s) 1380/4H Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Higher Tier Friday 11 June 2010 Morning Time: 1 hour 45 minutes
More informationExercise Set 3. Similar triangles. Parallel lines
Exercise Set 3. Similar triangles Parallel lines Note: The exercises marked with are more difficult and go beyond the course/examination requirements. (1) Let ABC be a triangle with AB = AC. Let D be an
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationAP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:
AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationTesting for Congruent Triangles Examples
Testing for Congruent Triangles Examples 1. Why is congruency important? In 1913, Henry Ford began producing automobiles using an assembly line. When products are mass-produced, each piece must be interchangeable,
More informationEUCLIDEAN GEOMETRY: (±50 marks)
ULIN GMTRY: (±50 marks) Grade theorems:. The line drawn from the centre of a circle perpendicular to a chord bisects the chord. 2. The perpendicular bisector of a chord passes through the centre of the
More informationVersion 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC4. (Specification 6360) Pure Core 4. Final.
Version.0 General Certificate of Education (A-level) January 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together
More informationINTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE
INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE ABSTRACT:- Vignesh Palani University of Minnesota - Twin cities e-mail address - palan019@umn.edu In this brief work, the existing formulae
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationUnit 3: Circles and Volume
Unit 3: Circles and Volume This unit investigates the properties of circles and addresses finding the volume of solids. Properties of circles are used to solve problems involving arcs, angles, sectors,
More informationWednesday 5 November 2014 Morning
H Wednesday 5 November 2014 Morning GCSE MATHEMATICS B J567/03 Paper 3 (Higher Tier) * 1 1 8 3 2 9 5 6 3 5 * Candidates answer on the Question Paper. OCR supplied materials: None Other materials required:
More informationDear Accelerated Pre-Calculus Student:
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
More informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationMonday 11 June 2012 Afternoon
THIS IS A NEW SPECIFICATION H Monday 11 June 2012 Afternoon GCSE MATHEMATICS A A502/02 Unit B (Higher Tier) *A517000612* Candidates answer on the Question Paper. OCR supplied materials: None Other materials
More informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationGEOMETRY B: CIRCLE TEST PRACTICE
Class: Date: GEOMETRY B: CIRCLE TEST PRACTICE Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the measures of the indicated angles. Which statement
More informationWednesday 6 November 2013 Morning
H Wednesday 6 November 2013 Morning GCSE MATHEMATICS B J567/03 Paper 3 (Higher Tier) *J540550313* Candidates answer on the Question Paper. OCR supplied materials: None Other materials required: Geometrical
More informationAnalytical Geometry (4)
Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line
More informationWednesday 15 January 2014 Morning Time: 2 hours
Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 4H Centre Number Wednesday 15 January 2014 Morning Time: 2 hours Candidate Number
More informationTOPPER Sample Paper - I. Class : XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100
TOPPER Sample Paper - I Class : XI MATHEMATICS Questions Time Allowed : 3 Hrs Maximum Marks: 100 1. All questions are compulsory.. The question paper consist of 9 questions divided into three sections
More informationCircle Theorems. This circle shown is described an OT. As always, when we introduce a new topic we have to define the things we wish to talk about.
Circle s circle is a set of points in a plane that are a given distance from a given point, called the center. The center is often used to name the circle. T This circle shown is described an OT. s always,
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationSan Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS
San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors
More informationNATIONAL SENIOR CERTIFICATE GRADE/GRAAD 11
NATIONAL SENIOR CERTIFICATE GRADE/GRAAD MATHEMATICS P/WISKUNDE V NOVEMBER 04 MEMORANDUM MARKS/PUNTE: 50 This memorandum consists of 6 pages. Hierdie memorandum bestaan uit 6 bladsye. Mathematics/P/Wiskunde/V
More informationInvention of the plane geometrical formulae - Part I
International Journal of Scientific and Research Publications, Volume 3, Issue 4, April 013 1 ISSN 50-3153 Invention of the plane geometrical formulae - Part I Mr. Satish M. Kaple Asst. Teacher Mahatma
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationTuesday 6 November 2012 Morning
H Tuesday 6 November 2012 Morning GCSE MATHEMATICS A A502/02 Unit B (Higher Tier) *A516821112* Candidates answer on the Question Paper. OCR supplied materials: None Other materials required: Geometrical
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationMathematics (Project Maths)
2010. M128 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination Mathematics (Project Maths) Paper 2 Ordinary Level Monday 14 June Morning 9:30 12:00 300 marks Examination
More informationGeneral Certificate of Secondary Education January 2014. Mathematics Unit T3 (With calculator) Higher Tier [GMT31] FRIDAY 10 JANUARY, 9.15am 11.
Centre Number 71 Candidate Number General Certificate of Secondary Education January 2014 Mathematics Unit T3 (With calculator) Higher Tier [GMT31] MV18 FRIDAY 10 JANUARY, 9.15am 11.15 am TIME 2 hours,
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationGeneral Certificate of Secondary Education November 2012. Mathematics (Linear) B 4365 Paper 2 Higher Tier. Final. Mark Scheme
General Certificate of Secondary Education November 2012 Mathematics (Linear) B 4365 Paper 2 Higher Tier Final Mark Scheme Mark schemes are prepared by the Principal Examiner and considered, together with
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationPaper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 3 (Non-Calculator) Monday 6 June 2011 Afternoon Time: 1 hour 45 minutes
Centre No. Candidate No. Paper Reference 1 3 8 0 3 H Paper Reference(s) 1380/3H Edexcel GCSE Mathematics (Linear) 1380 Paper 3 (Non-Calculator) Higher Tier Monday 6 June 2011 Afternoon Time: 1 hour 45
More informationMark Scheme (Results) June 2011. GCSE Mathematics (1380) Paper 3H (Non-Calculator)
Mark Scheme (Results) June 011 GCSE Mathematics (1380) Paper 3H (Non-Calculator) Edexcel is one of the leading examining and awarding bodies in the UK and throughout the world. We provide a wide range
More informationMATHEMATICS Unit Pure Core 2
General Certificate of Education January 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Wednesday 9 January 2008 1.30 pm to 3.00 pm For this paper you must have: an 8-page answer
More informationGeometry Regents Review
Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest
More informationUnit 10 Geometry Circles. NAME Period
Unit 10 Geometry Circles NAME Period 1 Geometry Chapter 10 Circles ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1. (10-1) Circles and Circumference
More informationPaper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Friday 10 June 2011 Morning Time: 1 hour 45 minutes
Centre No. Candidate No. Paper Reference 1 3 8 0 4 H Paper Reference(s) 1380/4H Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Higher Tier Friday 10 June 2011 Morning Time: 1 hour 45 minutes
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationBlue Pelican Geometry Theorem Proofs
Blue Pelican Geometry Theorem Proofs Copyright 2013 by Charles E. Cook; Refugio, Tx (All rights reserved) Table of contents Geometry Theorem Proofs The theorems listed here are but a few of the total in
More informationNational Quali cations 2015
N5 X747/75/01 TUESDAY, 19 MAY 9:00 AM 10:00 AM FOR OFFICIAL USE National Quali cations 015 Mark Mathematics Paper 1 (Non-Calculator) *X7477501* Fill in these boxes and read what is printed below. Full
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationUnit 2 - Triangles. Equilateral Triangles
Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics
More information