# Mathematics (Project Maths Phase 3)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 2014. M329 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2014 Mathematics (Project Maths Phase 3) Paper 1 Higher Level Friday 6 June Afternoon 2:00 4: marks Running total Examination number Centre stamp For examiner Question Mark Total Grade

2 Instructions There are two sections in this examination paper. Section A Concepts and Skills 150 marks 6 questions Section B Contexts and Applications 150 marks 3 questions Answer all nine questions. Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. There is space for extra work at the back of the booklet. You may also ask the superintendent for more paper. Label any extra work clearly with the question number and part. The superintendent will give you a copy of the Formulae and Tables booklet. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination. You will lose marks if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. Answers should be given in simplest form, where relevant. Write the make and model of your calculator(s) here: Leaving Certificate 2014 Page 2 of 19 Project Maths, Phase 3

3 Section A Concepts and Skills 150 marks Answer all six questions from this section. Question 1 (a) The graph of a cubic function f (x) cuts the x-axis at x 3, x 1 and x 2, and the y-axis at (0, 6), as shown. Verify that f (x) can be written as f ( x) x 3 2x 2 5x 6. (25 marks) y f (x) 5 x (b) (i) The graph of the function g ( x) 2x 6 intersects the graph of the function f ( x ) above. Let f ( x) g( x) and solve the resulting equation to find the co-ordinates of the points where the graphs of f ( x ) and gx ( ) intersect. (ii) Draw the graph of the function g ( x) 2x 6 on the diagram above. page running Leaving Certificate 2014 Page 3 of 19 Project Maths, Phase 3

4 Question 2 Let z1 1 2 i, where i (a) The complex number z 1 is a root of the equation 2z 7z 16z Find the other two roots of the equation. (25 marks) (b) (i) Let, 1 1 z1 is the conjugate of. 1 1 z 1 and w on the Argand diagram and label each point. Im(z) Re(z) (ii) Find the measure of the acute angle, zwz, formed by joining 1 1 z1 to w to z 1 on the diagram above. Give your answer correct to the nearest degree. Leaving Certificate 2014 Page 4 of 19 Project Maths, Phase 3

5 Question 3 (a) Prove, by induction, that the sum of the first n natural numbers, n n n, is. 2 (25 marks) (b) Hence, or otherwise, prove that the sum of the first n even natural numbers, n, is n 2 n. (c) Using the results from (a) and (b) above, find an expression for the sum of the first n odd natural numbers in its simplest form. page running Leaving Certificate 2014 Page 5 of 19 Project Maths, Phase 3

6 Question 4 (a) Differentiate the function 2x 2 3x 6 with respect to x from first principles. (25 marks) (b) 2x Let f( x), x 2, x. Find the co-ordinates of the points at which the slope of the x 2 tangent to the curve y f (x) is 1. 4 Leaving Certificate 2014 Page 6 of 19 Project Maths, Phase 3

7 Question 5 (a) Find 5cos3 x dx. (25 marks) (b) The slope of the tangent to a curve y f (x) at each point ( x, y ) is 2x 2. The curve cuts the x-axis at ( 2, 0). (i) Find the equation of f ( x ). (ii) Find the average value of f over the interval 0 x 3, x. page running Leaving Certificate 2014 Page 7 of 19 Project Maths, Phase 3

8 Question 6 The th n n term of a sequence is Tn ln a, where a > 0 and a is a constant. (a) (i) Show that T, T, and 1 2 T 3 are in arithmetic sequence. (25 marks) (ii) Prove that the sequence is arithmetic and find the common difference. (b) Find the value of a for which T1 T2 T3 T98 T99 T Leaving Certificate 2014 Page 8 of 19 Project Maths, Phase 3

9 (c) Verify that, for all values of a, T T T T 100d T T T T, where d is the common difference of the sequence. page running Leaving Certificate 2014 Page 9 of 19 Project Maths, Phase 3

10 Section B Contexts and Applications 150 marks Answer all three questions from this section. Question 7 (a) Three natural numbers a, b and c, such that (40 marks) 2 2 a b c 2, are called a Pythagorean triple. (i) 2 2 Let a 2n 1, b 2n 2n and c 2n 2n 1. Pick one natural number n and verify that the corresponding values of a, b and c form a Pythagorean triple. (ii) 2 2 Prove that a 2n 1, b 2n 2 n and c 2n 2n 1, where n, will always form a Pythagorean triple. Leaving Certificate 2014 Page 10 of 19 Project Maths, Phase 3

11 (b) ADEC is a rectangle with AC = 7 m and AD = 2 m, as shown. B is a point on [AC] such that AB = 5 m. P is a point on [DE] such that DP = x m. (i) D Let f ( x) PA PB PC. 2 Show that f( x) 3x 24x 86, for 0 x 7, x. A B C 5 m 2 m 2 m x m P E (ii) The function f (x) has a minimum value at x = k. Find the value of k and the minimum value of f (x). page running Leaving Certificate 2014 Page 11 of 19 Project Maths, Phase 3

12 Question 8 In 2011, a new footbridge was opened at Mizen Head, the most south-westerly point of Ireland. The arch of the bridge is in the shape of a parabola, as shown. The length of the span of the arch, [ AB ], is 48 metres. (50 marks) (0, 5) D C E A(0, 0) B(48, 0) (a) Using the co-ordinate plane, with A (0, 0) and B (48, 0), the equation of the parabola is 2 y 0 013x 0 624x. Find the co-ordinates of C, the highest point of the arch. (b) The perpendicular distance between the walking deck, [DE], and [AB] is 5 metres. Find the co-ordinates of D and of E. Give your answers correct to the nearest whole number. Leaving Certificate 2014 Page 12 of 19 Project Maths, Phase 3

13 (c) Using integration, find the area of the shaded region, ABED, shown in the diagram below. Give your answer correct to the nearest whole number. D C E A B (d) Write the equation of the parabola in part (a) in the form are constants. y k p x h 2 ( ), where k, p, and h (e) Using what you learned in part (d) above, or otherwise, write down the equation of a parabola 2 for which the coefficient of x is 2 and the co-ordinates of the maximum point are ( 3, 4). page running Leaving Certificate 2014 Page 13 of 19 Project Maths, Phase 3

14 Question 9 (60 marks) kt Ciarán is preparing food for his baby and must use cooled boiled water. The equation y Ae describes how the boiled water cools. In this equation: t is the time, in minutes, from when the water boiled, y is the difference between the water temperature and room temperature at time t, measured in degrees Celsius, A and k are constants. The temperature of the water when it boils is (a) 100 C and the room temperature is a constant 23 C. Write down the value of the temperature difference, y, when the water boils, and find the value of A. y = A = (b) After five minutes, the temperature of the water is 88 C. Find the value of k, correct to three significant figures. (c) Ciarán prepares the food for his baby when the water has cooled to 50 C. How long does it take, correct to the nearest minute, for the water to cool to this temperature? Leaving Certificate 2014 Page 14 of 19 Project Maths, Phase 3

15 (d) Using your values for A and k, sketch the curve f kt ( t) Ae for 0 t 100, t. 80 y t (e) (i) On the same diagram, sketch a curve g( t) Ae, showing the water cooling at a faster rate, where A is the value from part (a), and m is a constant. Label each graph clearly. mt (ii) Suggest one possible value for m for the sketch you have drawn and give a reason for your choice. page running Leaving Certificate 2014 Page 15 of 19 Project Maths, Phase 3

16 (f) (i) Find the rates of change of the function f (t) after 1 minute and after 10 minutes. Give your answers correct to two decimal places. (ii) Show that the rate of change of f (t) will always increase over time. Leaving Certificate 2014 Page 16 of 19 Project Maths, Phase 3

17 You may use this page for extra work. page running Leaving Certificate 2014 Page 17 of 19 Project Maths, Phase 3

18 You may use this page for extra work. Leaving Certificate 2014 Page 18 of 19 Project Maths, Phase 3

19 You may use this page for extra work. page running Leaving Certificate 2014 Page 19 of 19 Project Maths, Phase 3

20 Leaving Certificate 2014 Higher Level Mathematics (Project Maths Phase 3) Paper 1 Friday 6 June Afternoon 2:00 4:30

### Mathematics (Project Maths Phase 3)

013. M37 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 013 Mathematics (Project Maths Phase 3) Paper 1 Ordinary Level Friday 7 June Afternoon :00 4:30 300

### Coimisiún na Scrúduithe Stáit State Examinations Commission. Leaving Certificate Examination 2015. Mathematics

015. M7 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 015 Mathematics Paper 1 Ordinary Level Friday 5 June Afternoon :00 4:30 300 marks Running total Examination

### Mathematics (Project Maths Phase 3)

01. M37 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination, 01 Mathematics (Project Maths Phase 3) Paper 1 Ordinary Level Friday 8 June Afternoon :00 4:30 300 marks

### Mathematics (Project Maths Phase 2)

2012. S234 Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination, 2012 Mathematics (Project Maths Phase 2) Paper 1 Higher Level Friday 8 June Afternoon 2:00 to 4:30

### Mathematics (Project Maths)

2010. M130 S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination Sample Paper Mathematics (Project Maths) Paper 2 Higher Level Time: 2 hours, 30 minutes 300 marks

### Mathematics (Project Maths Phase 3)

2014. M328 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2014 Mathematics (Project Maths Phase 3) Paper 2 Ordinary Level Monday 9 June Morning 9:30 12:00 300

### Mathematics (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

### Cork Institute of Technology. CIT Mathematics Examination, Paper 1 Sample Paper A

Cork Institute of Technology CIT Mathematics Examination, 2015 Paper 1 Sample Paper A Answer ALL FIVE questions. Each question is worth 20 marks. Total marks available: 100 marks. The standard Formulae

### Mathematics (Project Maths Phase 1)

2012. M128 S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination, 2012 Sample Paper Mathematics (Project Maths Phase 1) Paper 2 Ordinary Level Time: 2 hours, 30

### Cork Institute of Technology. CIT Mathematics Examination, Paper 2 Sample Paper A

Cork Institute of Technology CIT Mathematics Examination, 2015 Paper 2 Sample Paper A Answer ALL FIVE questions. Each question is worth 20 marks. Total marks available: 100 marks. The standard Formulae

### Mathematics (Project Maths Phase 3)

2013. M328 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2013 Mathematics (Project Maths Phase 3) Paper 2 Ordinary Level Monday 10 June Morning 9:30 12:00

### Coimisiún na Scrúduithe Stáit State Examinations Commission. Leaving Certificate Examination Mathematics

2016. M30 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2016 Mathematics Paper 2 Higher Level Monday 13 June Morning 9:30 12:00 300 marks Examination number

### MATHEMATICS: PAPER I. 5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.

NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 015 MATHEMATICS: PAPER I Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 1 pages and an Information

### Mathematics (Project Maths Phase 1)

2011. S133S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination Sample Paper Mathematics (Project Maths Phase 1) Paper 2 Ordinary Level Time: 2 hours 300 marks Running

### Catholic Schools Trial Examination 2004 Mathematics

0 Catholic Trial HSC Examination Mathematics Page Catholic Schools Trial Examination 0 Mathematics a If x 5 = 5000, find x correct to significant figures. b Express 0. + 0.. in the form b a, where a and

### cos Newington College HSC Mathematics Ext 1 Trial Examination 2011 QUESTION ONE (12 Marks) (b) Find the exact value of if. 2 . 3

1 QUESTION ONE (12 Marks) Marks (a) Find tan x e 1 2 cos dx x (b) Find the exact value of if. 2 (c) Solve 5 3 2x 1. 3 (d) If are the roots of the equation 2 find the value of. (e) Use the substitution

### Mathematics (Project Maths)

Pre-Leaving Certificate Examination Mathematics (Project Maths) Paper 2 Higher Level February 2010 2½ hours 300 marks Running total Examination number Centre stamp For examiner Question Mark 1 2 3 4 5

### Mathematics (Project Maths)

2010. M128 S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination Sample Paper Mathematics (Project Maths) Paper 2 Ordinary Level Time: 2 hours, 30 minutes 300 marks

### Coimisiún na Scrúduithe Stáit State Examinations Commission. Junior Certificate Examination 2015 Sample Paper. Mathematics

2015. S34 S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination 2015 Mathematics Paper 1 Higher Level Time: 2 hours, 30 minutes 300 marks Examination number Running

### Mathematics (Project Maths Phase 2)

2013. M230S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certiﬁcate Examination 2013 Sample Paper Mathematics (Project Maths Phase 2) Paper 2 Higher Level Time: 2 hours, 30 minutes

### x = y + 2, and the line

WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first

### IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed. Name: Date: Class:

IB Math Standard Level Year 2: May 11, Paper 2, TZ 1 IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed Name: Date: Class: 1. The following diagram shows triangle ABC. AB = 7 cm, BC = 9 cm

### MATHEMATICS Unit Pure Core 2

General Certificate of Education June 2006 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Monday 22 May 2006 9.00 am to 10.30 am For this paper you must have: * an 8-page answer book

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and

### National Quali cations 2015

H National Quali cations 05 X77/76/ WEDNESDAY, 0 MAY 9:00 AM 0:0 AM Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only

### Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.

Lecture 17 MATH10070 - Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx

### Mathematics (Project Maths Phase 1)

2013. S135 S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination, 2013 Sample Paper Mathematics (Project Maths Phase 1) Paper 2 Higher Level Time: 2 hours, 30 minutes

### AQA Level 2 Certificate FURTHER MATHEMATICS

AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

### MATHEMATICS: PAPER II

NATIONAL SENIOR CERTIFICATE EXAMINATION SUPPLEMENTARY 2014 MATHEMATICS: PAPER II Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 10 pages, a

### Mathematics (Project Maths Phase 1)

2011. S135S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination 2011 Sample Paper Mathematics (Project Maths Phase 1) Paper 2 Higher Level Time: 2 hours, 30 minutes

### Calculus II MAT 146 Integration Applications: Area Between Curves

Calculus II MAT 46 Integration Applications: Area Between Curves A fundamental application of integration involves determining the area under a curve for some interval on the x- or y-axis. In a previous

### Applications of Integration Day 1

Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

### Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}

Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following

### MATHEMATICS 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

### *X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) MONDAY, 21 MAY 1.00 PM 2.30 PM NATIONAL QUALIFICATIONS 2012

X00//0 NTIONL QULIFITIONS 0 MONY, MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (40 marks) Instructions for completion

### 2012 HSC Mathematics Sample Answers

2012 HSC Mathematics Sample Answers When examination committees develop questions for the examination, they may write sample answers or, in the case of some questions, answers could include. The committees

### GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

### Mathematics Paper 1 (Non-Calculator)

H National Qualifications CFE Higher Mathematics - Specimen Paper A Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full

### Dear 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

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Time: 1 hour 10 minutes

[C00/SQP48] Higher Time: hour 0 minutes Mathematics Units, and 3 Paper (Non-calculator) Specimen Question Paper (Revised) for use in and after 004 NATIONAL QUALIFICATIONS Read Carefully Calculators may

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### x 2 would be a solution to d y

CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y

### Practice Problems for Exam 1 Math 140A, Summer 2014, July 2

Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other

### AP CALCULUS AB 2008 SCORING GUIDELINES

AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### Mathematics Extension 1

Girraween High School 05 Year Trial Higher School Certificate Mathematics Extension General Instructions Reading tjmc - 5 mjnutcs Working time- hours Write using black or blue pen Black pen is preferred

### Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

### Differential Equations

Differential Equations A differential equation is an equation that contains an unknown function and one or more of its derivatives. Here are some examples: y = 1, y = x, y = xy y + 2y + y = 0 d 3 y dx

### AP Calculus AB 2010 Free-Response Questions Form B

AP Calculus AB 2010 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

### Paper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 3 (Non-Calculator) Thursday 5 November 2009 Morning Time: 1 hour 45 minutes

Centre No. Candidate No. Paper Reference 1 3 8 0 3 H Surname Signature Paper Reference(s) 1380/3H Edexcel GCSE Mathematics (Linear) 1380 Paper 3 (Non-Calculator) Higher Tier Thursday 5 November 2009 Morning

### South 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

### MATHEMATICS Unit Pure Core 1

General Certificate of Education June 2009 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 1 MPC1 Wednesday 20 May 2009 1.30 pm to 3.00 pm For this paper you must have: an 8-page answer book

### Trigonometry. Week 1 Right Triangle Trigonometry

Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### Mathematics. Total marks 100. Section I Pages marks Attempt Questions 1 10 Allow about 15 minutes for this section

04 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators ma be

### C100/SQP321. Course Assessment Specification 2. Specimen Question Paper 1 5. Specimen Question Paper Specimen Marking Instructions Paper 1 23

00/SQP Maths Higher NTIONL QULIFITIONS ontents Page ourse ssessment Specification Specimen Question Paper 5 Specimen Question Paper 7 Specimen Marking Instructions Paper Specimen Marking Instructions Paper

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Senior Math Circles February 4, 2009 Conics I

1 University of Waterloo Faculty of Mathematics Conics Senior Math Circles February 4, 2009 Conics I Centre for Education in Mathematics and Computing The most mathematically interesting objects usually

### Higher Mathematics Homework A

Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(-3,1) and B(5,-3) 2. Find the equation of the tangent to the circle

### 1.6. Determine a Quadratic Equation Given Its Roots. Investigate

1.6 Determine a Quadratic Equation Given Its Roots Bridges like the one shown often have supports in the shape of parabolas. If the anchors at either side of the bridge are 4 m apart and the maximum height

### Calculus Card Matching

Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information

### AP Calculus AB 2004 Free-Response Questions

AP Calculus AB 2004 Free-Response Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be

### Student Activity: To investigate the Average Value of a Function

Student Activity: To investigate the Average Value of a Function Use in connection with the interactive file, Average Value 3, on the Student s CD. 1. Click all the boxes in the interactive file. Move

### Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned

### Contents. Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!)

Page Contents Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!) Systematic Search for a Change of Sign (Decimal Search) Method Explanation

### leaving certificate Active Maths 3 Old Syllabus Strand 5 Ordinary Level - Book 1 - y 1 - x 1 y 2 x 2 m = 2πr Oliver Murphy

leaving certificate Active Maths Ordinary Level - Book Old Syllabus Strand 5 m = y - y x - x πr πr Oliver Murphy Editors: Priscilla O Connor, Sarah Reece Designer: Liz White Layout: Compuscript Illustrations:

### MATHEMATICS Compulsory Part PAPER 2

015-DSE MATH CP PAPER LONGMAN MATHEMATICS SERIES HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION 015 MOCK PAPER MATHEMATICS Compulsory Part PAPER Time allowed: 1 hour 15 minutes INSTRUCTIONS 1 Read

### http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

### AP Calculus AB 2006 Scoring Guidelines

AP Calculus AB 006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

### ALGEBRA 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

### AP Calculus AB. Practice Exam. Advanced Placement Program

Advanced Placement Program AP Calculus AB Practice Exam The questions contained in this AP Calculus AB Practice Exam are written to the content specifications of AP Exams for this subject. Taking this

### 2015 Mathematics. New Higher Paper 1. Finalised Marking Instructions

National Qualifications 0 0 Mathematics New Higher Paper Finalised Marking Instructions Scottish Qualifications Authority 0 The information in this publication may be reproduced to support SQA qualifications

### HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 1999 MATHEMATICS PAPER 2

99-CE MATHS Paper HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 999 MATHEMATICS PAPER 5 am 45 pm (½ hours) Subject Code 80 Read carefully the instructions on the Answer

### Conics. Find the equation of the parabola which has its vertex at the origin and its focus at point F in the following cases.

Conics 1 Find the equation of the parabola which has its vertex at the origin and its focus at point F in the following cases. a) F(, 0) b) F(0,-4) c) F(-3,0) d) F(0, 5) In the Cartesian plane, represent

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### NORTH SYDNEY GIRLS HIGH SCHOOL

NORTH SYDNEY GIRLS HIGH SCHOOL 0 TRIAL HSC EXAMINATION Mathematics General Instructions Reading Time 5 minutes Working Time hours Write using black or blue pen. Board-approved calculators may be used A

### Paper Reference. Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may be used.

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 18 May 2009 Afternoon Time: 1 hour 45

### GRADE 12 SEPTEMBER 2014 MATHEMATICS P2

NATIONAL SENIOR CERTIFICATE GRADE 1 SEPTEMBER 014 MATHEMATICS P MARKS: 150 TIME: 3 hours *MATHE* This question paper consists of 15 pages, including diagram sheets and 1 information sheet. MATHEMATICS

### 1.7. Solve Linear-Quadratic Systems. Investigate A

1.7 Solve Linear-Quadratic Systems Marina is a set designer. She plans movie sets using freehand sketches and her computer. In one scene, a banner will hang across a parabolic archway. To make it look

### Test Two Review Cal 2

Name: Class: Date: ID: A Test Two Review Cal 2 Short Answer 1. Set up the definite integral that gives the area of the region bounded by the graph of y 1 x 2 2x 1 and y 2 2x.. Find the area of the region

### Advanced Higher Mathematics Course Assessment Specification (C747 77)

Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational

### GEOMETRY (Common Core)

GEOMETRY (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY (Common Core) Wednesday, August 12, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### TOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM

Content Area: Mathematics Course Title: Precalculus Grade Level: High School Right Triangle Trig and Laws 3-4 weeks Trigonometry 3 weeks Graphs of Trig Functions 3-4 weeks Analytic Trigonometry 5-6 weeks

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### Math 115 HW #8 Solutions

Math 115 HW #8 Solutions 1 The function with the given graph is a solution of one of the following differential equations Decide which is the correct equation and justify your answer a) y = 1 + xy b) y