Edexcel past paper questions. Core Mathematics 4. Parametric Equations
|
|
- Kristian Booth
- 6 years ago
- Views:
Transcription
1 Edexcel past paper questions Core Mathematics 4 Parametric Equations Edited by: K V Kumaran kvkumaran@gmail.com C4 Maths Parametric equations Page 1
2 Co-ordinate Geometry A parametric equation of a curve is one which does not give the relationship between x and y directly but rather uses a third variable, typically t, to do so. The third variable is known as the parameter. A simple example of a pair of parametric equations: x = 5t + 3 y = t 2 + 2t Converting to Cartesian You need to be able to find the Cartesian equation of the curve from parametric equations, that is the equation that relates x and y directly. To do this you need to eliminate the parameter. The easiest way to do this is to rearrange on parametric equation to get the parameter as the subject and then substitute this into the other equation. A circle with an origin (a, b) has the parametric equation: x = a + rcosθ y = b + rsinθ You can use the result sin 2 θ + cos 2 θ = 1 to derive these. As before, θ is the parameter instead of t in the equations. You need to be able to recognise these as parametric equations of circles in the exam. C4 Maths Parametric equations Page 2
3 C4 Maths Parametric equations Page 3
4 Example The curve C is described by the parametric equations x = 5cost, y = cos2t, 0 t π a) Find a Cartesian equation for the curve C. b) Draw a sketch of the curve C. a) From C3 again you should remember that: Therefore: cos2t cos 2 t sin 2 t 2cos 2 t 1 y = 2cos 2 t 1 If x = 5cost then cost = x 5 So finally y = 2 2x 25-1 b) This is simply a quadratic that is symmetrical about the y axis and intercepts with the y axis at C4 Maths Parametric equations Page 4
5 Example C4 Maths Parametric equations Page 5
6 The parametric equations of a curve are x = 14 sin t, y = 14t cos t π dy where 0 < t <. Find in terms of t, and hence show that the gradient of the 2 dx curve is zero where tan t 1 t dy Since the curve is given parametrically we can use the chain rule to find dx dy dy dt dx dt dx So by differentiating the parametric: x = 14 sin t dx dy = 14 cos t dt y = 14t cos t (a Product) = 14 cos 14t sin t dt dy dy dt dx dt dx dy dx 14 cost - 14t sin t 1 ttant 14cost When the gradient is zero 1 ttan t 0 tan t 1 t This equation could be solved by iterative methods (C3). C4 Maths Parametric equations Page 6
7 Example 4 A curve is given by the parametric equations x = 7 sin 3 t, y = 6 cos 2t, 0 < t < 4 Show that dx dy By chain rule 7 sin t 8 dx dx dt dy dt dy dx 21sin 2 tcost dt dy 12sin2t don't forget the 2. dt By C3 trig identities sin 2t = 2 sin t cos t 2 dx dx dt 21sin t cos t dy dt dy 24sin tcost dx 7 sin t dy 8 The final example in this section deals with tangents and normals to curves. Example 5 The curve C is described by the parametric equations x = tan t y = sin 2t t 2 2 π a) Find the gradient of the curve at the point P where t= 3 b) Find the equation of the normal to the curve at P. a) Find the gradient of the curve at the point P where t= 3 C4 Maths Parametric equations Page 7
8 Using chain rule: dx dy sec 2 t 2cos2t dt dt dx dx dt dy dt dy Let t= 3 π dy 2cos 2t 2cos 2 tcos2t 2 dx sec t 2 Grad = 2 cos cos b) Find the equation of the normal to the curve at P. We are asked for the equation of the normal therefore the gradient will be 4 (why?). Using t= 3 the x and y coordinates are 3 and 23 respectively. Using y = mx + c 3 = c 2 c = Therefore the equation of the normal is y 4x Differentiating a x C4 Maths Parametric equations Page 8
9 This function describes growth and decay, and its derivative gives a measure of the rate of change of this growth/decay. Since y = a x, taking logs of both sides gives ln y = ln a x = x ln a. Using implicit differentiation to differentiate ln y: 1 dy = ln a y dx dy dx = y ln a = ax ln a This result needs to be learn, and is not given in the formula sheet. C4 Parametric differentiation past paper questions C4 Maths Parametric equations Page 9
10 1. A curve has parametric equations x = 2 cot t, y = 2 sin 2 t, 0 < t 2 (a) Find an expression for y dx d in terms of the parameter t. (4) (b) Find an equation of the tangent to the curve at the point where t = 4 (4) (c) Find a cartesian equation of the curve in the form y = f(x). State the domain on which the curve is defined. (4) (C4 June 2005, Q6.) 2. Figure 2 y O x The curve shown in Figure 2 has parametric equations x = sin t, y = sin t, 6 < t <. 2 2 (a) Find an equation of the tangent to the curve at the point where t = 6 (6) (b) Show that a cartesian equation of the curve is C4 Maths Parametric equations Page 10
11 y = 3. A curve has parametric equations 3 1 x + (1 x 2 ), 1 < x < (3) (C4 June 2006, Q4.) x = 7 cos t cos 7t, y = 7 sin t sin 7t, < t <. 8 3 (a) Find an expression for dy dx in terms of t. You need not simplify your answer. (3) (b) Find an equation of the normal to the curve at the point where t = 6 Give your answer in its simplest exact form. 4. A curve has parametric equations (6) (C4 Jan 2007, Q3.) x = tan 2 t, y = sin t, 0 < t < 2 (a) Find an expression for dy dx in terms of t. You need not simplify your answer. (3) (b) Find an equation of the tangent to the curve at the point where t = 4 Give your answer in the form y = ax + b, where a and b are constants to be determined. (5) (c) Find a cartesian equation of the curve in the form y 2 = f(x). (4) 5. (C4 June 2007, Q6.) C4 Maths Parametric equations Page 11
12 Figure 3 The curve C shown in Figure 3 has parametric equations x = t 3 8t, y = t 2 where t is a parameter. Given that the point A has parameter t = 1, (a) find the coordinates of A. (1) The line l is the tangent to C at A. (b) Show that an equation for l is 2x 5y 9 = 0. (5) The line l also intersects the curve at the point B. (c) Find the coordinates of B. (6) (C4 Jan 2009, Q7.) C4 Maths Parametric equations Page 12
13 6. Figure 2 Figure 2 shows a sketch of the curve with parametric equations x = 2 cos 2t, y = 6 sin t, 0 t 2 (a) Find the gradient of the curve at the point where t = 3 (4) (b) Find a Cartesian equation of the curve in the form y = f(x), k x k, Stating the value of the constant k. (c) Write down the range of f(x). (4) (2) (C4 June 2009, Q5.) C4 Maths Parametric equations Page 13
14 7. A curve C has parametric equations x = sin 2 t, y = 2 tan t, 0 t < 2 (a) Find y dx d in terms of t. (4) The tangent to C at the point where t = 3 cuts the x-axis at the point P. (b) Find the x-coordinate of P. (6) (C4 June 2010, Q4.) 8. The curve C has parametric equations Find x = ln t, y = t 2 2, t > 0. (a) An equation of the normal to C at the point where t = 3, (b) A Cartesian equation of C. (6) (3) (C4 Jan 2011, Q6.) C4 Maths Parametric equations Page 14
15 9. Figure 3 Figure 3 shows part of the curve C with parametric equations x = tan, y = sin, 0 < 2 The point P lies on C and has coordinates 1 3, 3 2 (a) Find the value of at the point P. (2) The line l is a normal to C at P. The normal cuts the x-axis at the point Q. (b) Show that Q has coordinates (k 3, 0), giving the value of the constant k. (6) (C4 June 2011, Q7.) C4 Maths Parametric equations Page 15
16 10. Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x = 4 sin (a) Find an expression for 6 y dx t, y = 3 cos 2t, 0 t < 2. d in terms of t. (3) (b) Find the coordinates of all the points on C where y dx d = 0. (5) (C4 Jan 2012, Q5.) C4 Maths Parametric equations Page 16
17 11. Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x = 3 sin 2t, y = 4 cos 2 t, 0 t. (a) Show that d y dx = k 3 tan 2t, where k is a constant to be determined. (5) (b) Find an equation of the tangent to C at the point where t = 3 Give your answer in the form y = ax + b, where a and b are constants. (c) Find a Cartesian equation of C. (4) (3) (C4 June 2012, Q6.) C4 Maths Parametric equations Page 17
18 12. Figure 2 Figure 2 shows a sketch of part of the curve C with parametric equations x = t, y = 2 t 1. The curve crosses the y-axis at the point A and crosses the x-axis at the point B. (a) Show that A has coordinates (0, 3). (b) Find the x-coordinate of the point B. (c) Find an equation of the normal to C at the point A. (2) (2) (5) (C4 Jan 2013, part of Q5.) C4 Maths Parametric equations Page 18
19 13. A curve C has parametric equations x = 2sin t, y = 1 cos 2t, t 2 2 (a) Find d y dx at the point where t = 6 (b) Find a cartesian equation for C in the form (4) stating the value of the constant k. y = f(x), k x k, (3) (c) Write down the range of f(x). (2) (C4 June 2013, Q4) 14. Figure 2 Figure 2 shows a sketch of the curve C with parametric equations 3 x 27sec t, y 3tan t, 0 t 3 C4 Maths Parametric equations Page 19
20 (a) Find the gradient of the curve C at the point where t = 6 (4) (b) Show that the cartesian equation of C may be written in the form y stating values of a and b ( x 9), a x b (3) (C4 June 2013_R, part of Q7) 15. Figure 3 Figure 3 shows a sketch of the curve C with parametric equations x 4cos t, y = 2sin t, 0 t 2π 6 (a) Show that (b) Show that a cartesian equation of C is x + y = 2 3 cos t (3) (x + y) 2 + ay 2 = b where a and b are integers to be determined. (2) (C4 June 2014, Q5) C4 Maths Parametric equations Page 20
21 16. Figure 3 The curve shown in Figure 3 has parametric equations x = t 4 sin t, y = 1 2 cos t, 2 2 t 3 3 The point A, with coordinates (k, 1), lies on the curve. Given that k > 0 (a) find the exact value of k, (b) find the gradient of the curve at the point A. There is one point on the curve where the gradient is equal to 1 2 (2) (4) (c) Find the value of t at this point, showing each step in your working and giving your answer to 4 decimal places. [Solutions based entirely on graphical or numerical methods are not acceptable.] (6) (C4 June 2014_R, Q8) C4 Maths Parametric equations Page 21
22 17. A curve C has parametric equations x = 4t + 3, y = 4t , t 0. 2t (a) Find the value of simplest form. dy dx at the point on C where t = 2, giving your answer as a fraction in its (3) (b) Show that the Cartesian equation of the curve C can be written in the form y = x 2 ax b, x 3, x 3 where a and b are integers to be determined. (3) (C4 June 2015, Q5) C4 Maths Parametric equations Page 22
Edexcel Core Mathematics 4 Parametric equations.
Edexcel Core Mathematics 4 Parametric equations. Edited by: K V Kumaran kumarmaths.weebly.com 1 Co-ordinate Geometry A parametric equation of a curve is one which does not give the relationship between
More informationIntegration - Past Edexcel Exam Questions
Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More informationC3 Revision Questions. (using questions from January 2006, January 2007, January 2008 and January 2009)
C3 Revision Questions (using questions from January 2006, January 2007, January 2008 and January 2009) 1 2 1. f(x) = 1 3 x 2 + 3, x 2. 2 ( x 2) (a) 2 x x 1 Show that f(x) =, x 2. 2 ( x 2) (4) (b) Show
More informationQuestions Q1. The function f is defined by. (a) Show that (5) The function g is defined by. (b) Differentiate g(x) to show that g '(x) = (3)
Questions Q1. The function f is defined by (a) Show that The function g is defined by (b) Differentiate g(x) to show that g '(x) = (c) Find the exact values of x for which g '(x) = 1 (Total 12 marks) Q2.
More informationAlgebra and functions; coordinate geometry in the (x, y) plane; sequences and series; differentiation; integration; vectors.
Revision Checklist Unit C4: Core Mathematics 4 Unit description Assessment information Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; differentiation; integration;
More informationBook 4. June 2013 June 2014 June Name :
Book 4 June 2013 June 2014 June 2015 Name : June 2013 1. Given that 4 3 2 2 ax bx c 2 2 3x 2x 5x 4 dxe x 4 x 4, x 2 find the values of the constants a, b, c, d and e. 2. Given that f(x) = ln x, x > 0 sketch
More informationCoordinate goemetry in the (x, y) plane
Coordinate goemetr in the (x, ) plane In this chapter ou will learn how to solve problems involving parametric equations.. You can define the coordinates of a point on a curve using parametric equations.
More informationCore Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationNOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system.
NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system. Duplicating, selling, or otherwise distributing this product
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS A2 level Mathematics Core 3 course workbook 2015-2016 Name: Welcome to Core 3 (C3) Mathematics. We hope that you will use this workbook to give you an organised set of notes for
More informationC3 PAPER JUNE 2014 *P43164A0232* 1. The curve C has equation y = f (x) where + 1. (a) Show that 9 f (x) = (3)
PMT C3 papers from 2014 and 2013 C3 PAPER JUNE 2014 1. The curve C has equation y = f (x) where 4x + 1 f( x) =, x 2 x > 2 (a) Show that 9 f (x) = ( x ) 2 2 Given that P is a point on C such that f (x)
More information*P46958A0244* IAL PAPER JANUARY 2016 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA. 1. f(x) = (3 2x) 4, x 3 2
Edexcel "International A level" "C3/4" papers from 016 and 015 IAL PAPER JANUARY 016 Please use extra loose-leaf sheets of paper where you run out of space in this booklet. 1. f(x) = (3 x) 4, x 3 Find
More informationA2 MATHEMATICS HOMEWORK C3
Name Teacher A2 MATHEMATICS HOMEWORK C3 Mathematics Department September 2016 Version 1 Contents Contents... 2 Introduction... 3 Week 1 Trigonometric Equations 1... 4 Week 2 Trigonometric Equations 2...
More informationC3 papers June 2007 to 2008
physicsandmathstutor.com June 007 C3 papers June 007 to 008 1. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04* physicsandmathstutor.com June 007 x + 3 9+
More informationParametric Equations and Polar Coordinates
Parametric Equations and Polar Coordinates Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. Another
More informationIntegration by parts Integration by parts is a direct reversal of the product rule. By integrating both sides, we get:
Integration by parts Integration by parts is a direct reversal of the proct rule. By integrating both sides, we get: u dv dx x n sin mx dx (make u = x n ) dx = uv v dx dx When to use integration by parts
More informationSolutionbank Edexcel AS and A Level Modular Mathematics
Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x
More informationPLC Papers. Created For:
PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use
More informationCore Mathematics 2 Trigonometry
Core Mathematics 2 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Trigonometry 2 1 Trigonometry Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.
More informationOCR A2 Level Mathematics Core Mathematics Scheme of Work
OCR A Level Mathematics Core Mathematics Scheme of Work Examination in June of Year 13 The Solomen press worksheets are an excellent resource and incorporated into the SOW NUMERICAL METHODS (6 ) (Solomen
More informationThe above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ.
Math 7A Practice Midterm III Solutions Ch. 6-8 (Ebersole,.7-.4 (Stewart DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam. You
More informationCore A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document
Core A-level mathematics reproduced from the QCA s Subject criteria for Mathematics document Background knowledge: (a) The arithmetic of integers (including HCFs and LCMs), of fractions, and of real numbers.
More informationC4 "International A-level" (150 minute) papers: June 2014 and Specimen 1. C4 INTERNATIONAL A LEVEL PAPER JUNE 2014
C4 "International A-level" (150 minute) papers: June 2014 and Specimen 1. C4 INTERNATIONAL A LEVEL PAPER JUNE 2014 1. f(x) = 2x 3 + x 10 (a) Show that the equation f(x) = 0 has a root in the interval [1.5,
More informationBHASVIC MαTHS. Skills 1
Skills 1 Normally we work with equations in the form y = f(x) or x + y + z = 10 etc. These types of equations are called Cartesian Equations all the variables are grouped together into one equation, and
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationCreated by T. Madas. Candidates may use any calculator allowed by the regulations of this examination.
IYGB GCE Mathematics MP Advanced Level Practice Paper M Difficulty Rating:.8750/1.176 Time: hours Candidates may use any calculator allowed by the regulations of this examination. Information for Candidates
More informationVCE. VCE Maths Methods 1 and 2 Pocket Study Guide
VCE VCE Maths Methods 1 and 2 Pocket Study Guide Contents Introduction iv 1 Linear functions 1 2 Quadratic functions 10 3 Cubic functions 16 4 Advanced functions and relations 24 5 Probability and simulation
More informationCentre No. Candidate No. Paper Reference(s) 6665 Edexcel GCE Core Mathematics C3 Advanced Level Mock Paper
Paper Reference (complete below) Centre No. Surname Initial(s) 6 6 6 5 / 0 1 Candidate No. Signature Paper Reference(s) 6665 Edexcel GCE Core Mathematics C3 Advanced Level Mock Paper Time: 1 hour 30 minutes
More informationCore Mathematics C4 Advanced Level
Paper Reference(s) 6666/0 Edexcel GCE Core Mathematics C4 Advanced Level Thursday June 0 Afternoon Time: hour 0 minutes Materials required for examination Mathematical Formulae (Pink) Items included with
More informationA2 MATHEMATICS HOMEWORK C3
Name Teacher A2 MATHEMATICS HOMEWORK C3 Mathematics Department September 2014 Version 1.1 Contents Contents... 2 Introduction... 3 HW1 Algebraic Fractions... 4 HW2 Mappings and Functions... 6 HW3 The Modulus
More informationREVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ
REVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ INVERSE FUNCTIONS Two functions are inverses if they undo each other. In other words, composing one function in the other will result in simply x (the
More informationCore Mathematics 3 Trigonometry
Edexcel past paper questions Core Mathematics 3 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Maths 3 Trigonometry Page 1 C3 Trigonometry In C you were introduced to radian measure
More informationHEINEMANN HIGHER CHECKLIST
St Ninian s High School HEINEMANN HIGHER CHECKLIST I understand this part of the course = I am unsure of this part of the course = Name Class Teacher I do not understand this part of the course = Topic
More information11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes
Parametric Differentiation 11.6 Introduction Sometimes the equation of a curve is not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this Section we see how to calculate the
More informationH2 MATHS SET D PAPER 1
H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationx n+1 = ( x n + ) converges, then it converges to α. [2]
1 A Level - Mathematics P 3 ITERATION ( With references and answers) [ Numerical Solution of Equation] Q1. The equation x 3 - x 2 6 = 0 has one real root, denoted by α. i) Find by calculation the pair
More information1.1 GRAPHS AND LINEAR FUNCTIONS
MATHEMATICS EXTENSION 4 UNIT MATHEMATICS TOPIC 1: GRAPHS 1.1 GRAPHS AND LINEAR FUNCTIONS FUNCTIONS The concept of a function is already familiar to you. Since this concept is fundamental to mathematics,
More informationPaper Reference. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Level. Monday 12 June 2006 Afternoon Time: 1 hour 30 minutes
Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Level Monday 12 June 2006 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical
More informationPaper Reference. Core Mathematics C3 Advanced. Wednesday 20 January 2010 Afternoon Time: 1 hour 30 minutes. Mathematical Formulae (Pink or Green)
Centre No. Candidate No. Surname Signature Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Wednesday 20 January 2010 Afternoon Time: 1 hour 30 minutes Materials required for examination
More informationFall 2013 Hour Exam 2 11/08/13 Time Limit: 50 Minutes
Math 8 Fall Hour Exam /8/ Time Limit: 5 Minutes Name (Print): This exam contains 9 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information
More informationCore Mathematics C34
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C34 Advanced Tuesday 20 June 2017 Afternoon Time: 2 hours 30 minutes
More informationJUST THE MATHS UNIT NUMBER ORDINARY DIFFERENTIAL EQUATIONS 1 (First order equations (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 5. ORDINARY DIFFERENTIAL EQUATIONS (First order equations (A)) by A.J.Hobson 5.. Introduction and definitions 5..2 Exact equations 5..3 The method of separation of the variables
More information1 Exam 1 Spring 2007.
Exam Spring 2007.. An object is moving along a line. At each time t, its velocity v(t is given by v(t = t 2 2 t 3. Find the total distance traveled by the object from time t = to time t = 5. 2. Use the
More informationCore 3 (A2) Practice Examination Questions
Core 3 (A) Practice Examination Questions Trigonometry Mr A Slack Trigonometric Identities and Equations I know what secant; cosecant and cotangent graphs look like and can identify appropriate restricted
More informationMATH 152, Fall 2017 COMMON EXAM II - VERSION A
MATH 15, Fall 17 COMMON EXAM II - VERSION A LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited.. TURN OFF cell phones
More informationG H. Extended Unit Tests A L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests
M A T H E M A T I C S H I G H E R Higher Still Advanced Higher Mathematics S T I L L Extended Unit Tests A (more demanding tests covering all levels) Contents Extended Unit Tests Detailed marking schemes
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More informationCore Mathematics C4. You must have: Mathematical Formulae and Statistical Tables (Pink)
Write your name here Surname Other names Pearson Edexcel GCE Centre Number Core Mathematics C4 Advanced Candidate Number Friday 23 June 2017 Morning Time: 1 hour 30 minutes Paper Reference 6666/01 You
More informationEdexcel New GCE A Level Maths workbook
Edexcel New GCE A Level Maths workbook Straight line graphs Parallel and Perpendicular lines. Edited by: K V Kumaran kumarmaths.weebly.com Straight line graphs A LEVEL LINKS Scheme of work: a. Straight-line
More informationExam 1 Review SOLUTIONS
1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make
More informationSolutions to O Level Add Math paper
Solutions to O Level Add Math paper 04. Find the value of k for which the coefficient of x in the expansion of 6 kx x is 860. [] The question is looking for the x term in the expansion of kx and x 6 r
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationsec x dx = ln sec x + tan x csc x dx = ln csc x cot x
Name: Instructions: The exam will have eight problems. Make sure that your reasoning and your final answers are clear. Include labels and units when appropriate. No notes, books, or calculators are permitted
More informationPossible C4 questions from past papers P1 P3
Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationSANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
More informationC3 A Booster Course. Workbook. 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. (3) b) Hence, or otherwise, solve the equation
C3 A Booster Course Workbook 1. a) Sketch, on the same set of axis the graphs of y = x and y = 2x 3. b) Hence, or otherwise, solve the equation x = 2x 3 (3) (4) BlueStar Mathematics Workshops (2011) 1
More informationPhysicsAndMathsTutor.com
. A curve C has parametric equations x = sin t, y = tan t, 0 t < (a) Find in terms of t. (4) The tangent to C at the point where t = cuts the x-axis at the point P. (b) Find the x-coordinate of P. () (Total
More informationPaper Reference. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Level. Thursday 18 January 2007 Afternoon Time: 1 hour 30 minutes
Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Level Thursday 18 January 2007 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical
More informationPaper Reference. Core Mathematics C3 Advanced Level. Thursday 18 January 2007 Afternoon Time: 1 hour 30 minutes. Mathematical Formulae (Green)
Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Level Thursday 18 January 007 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical
More informationRegent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1
Regent College Maths Department Core Mathematics Trapezium Rule C Integration Page Integration It might appear to be a bit obvious but you must remember all of your C work on differentiation if you are
More informationHSC Marking Feedback 2017
HSC Marking Feedback 017 Mathematics Extension 1 Written Examination Question 11 Part (a) The large majority of responses showed correct substitution into the formula x = kx +lx 1 k+l given on the Reference
More information2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2
29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with
More informationTrue or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ
Math 90 Practice Midterm III Solutions Ch. 8-0 (Ebersole), 3.3-3.8 (Stewart) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam.
More informationMAS153/MAS159. MAS153/MAS159 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS hours. Mathematics (Materials) Mathematics For Chemists
Data provided: Formula sheet MAS53/MAS59 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics (Materials Mathematics For Chemists Spring Semester 203 204 3 hours All questions are compulsory. The marks awarded
More informationEdexcel New GCE A Level Maths workbook Circle.
Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint
More informationNewbattle Community High School Higher Mathematics. Key Facts Q&A
Key Facts Q&A Ways of using this booklet: 1) Write the questions on cards with the answers on the back and test yourself. ) Work with a friend who is also doing to take turns reading a random question
More information1. Compute the derivatives of the following functions, by any means necessary. f (x) = (1 x3 )(1/2)(x 2 1) 1/2 (2x) x 2 1( 3x 2 ) (1 x 3 ) 2
Math 51 Exam Nov. 4, 009 SOLUTIONS Directions 1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work shown.. Any numerical answers should be left in exact form,
More informationChapter 9 Overview: Parametric and Polar Coordinates
Chapter 9 Overview: Parametric and Polar Coordinates As we saw briefly last year, there are axis systems other than the Cartesian System for graphing (vector coordinates, polar coordinates, rectangular
More informationPhysicsAndMathsTutor.com
. A curve C has parametric equations x sin t, y tan t, 0 t < (a) Find in terms of t. (4) The tangent to C at the point where t cuts the x-axis at the point P. (b) Find the x-coordinate of P. () (Total
More informationSec 4 Maths. SET A PAPER 2 Question
S4 Maths Set A Paper Question Sec 4 Maths Exam papers with worked solutions SET A PAPER Question Compiled by THE MATHS CAFE 1 P a g e Answer all the questions S4 Maths Set A Paper Question Write in dark
More information10.1 Review of Parametric Equations
10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations
More informationa k 0, then k + 1 = 2 lim 1 + 1
Math 7 - Midterm - Form A - Page From the desk of C. Davis Buenger. https://people.math.osu.edu/buenger.8/ Problem a) [3 pts] If lim a k = then a k converges. False: The divergence test states that if
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More informationab = 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 informationMath 113 Final Exam Practice
Math Final Exam Practice The Final Exam is comprehensive. You should refer to prior reviews when studying material in chapters 6, 7, 8, and.-9. This review will cover.0- and chapter 0. This sheet has three
More informationabc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS
More informationSolutions to Test #1 MATH 2421
Solutions to Test # MATH Pulhalskii/Kawai (#) Decide whether the following properties are TRUE or FALSE for arbitrary vectors a; b; and c: Circle your answer. [Remember, TRUE means that the statement is
More informationLecture Wise Questions from 23 to 45 By Virtualians.pk. Q105. What is the impact of double integration in finding out the area and volume of Regions?
Lecture Wise Questions from 23 to 45 By Virtualians.pk Q105. What is the impact of double integration in finding out the area and volume of Regions? Ans: It has very important contribution in finding the
More informationSUBJECT: ADDITIONAL MATHEMATICS CURRICULUM OUTLINE LEVEL: 3 TOPIC OBJECTIVES ASSIGNMENTS / ASSESSMENT WEB-BASED RESOURCES. Online worksheet.
TERM 1 Simultaneous Online worksheet. Week 1 Equations in two Solve two simultaneous equations where unknowns at least one is a linear equation, by http://www.tutorvista.com/mat substitution. Understand
More informationLevel 3, Calculus
Level, 4 Calculus Differentiate and use derivatives to solve problems (965) Integrate functions and solve problems by integration, differential equations or numerical methods (966) Manipulate real and
More information11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes
Parametric Differentiation 11.6 Introduction Often, the equation of a curve may not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this section we see how to calculate the
More informationCalculus I Sample Exam #01
Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6
More informationCore Mathematics C4 Advanced
Centre No. Candidate No. Paper Reference(s) 6666/01 Edexcel GCE Core Mathematics C4 Advanced Tuesday 16 June 2015 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae
More informationCore Mathematics C4 Advanced
Centre No. Candidate No. Paper Reference(s) 6666/01 Edexcel GCE Core Mathematics C4 Advanced Tuesday 16 June 015 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae
More informationQuestion. [The volume of a cone of radius r and height h is 1 3 πr2 h and the curved surface area is πrl where l is the slant height of the cone.
Q1 An experiment is conducted using the conical filter which is held with its axis vertical as shown. The filter has a radius of 10cm and semi-vertical angle 30. Chemical solution flows from the filter
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More informationDifferentiating Functions & Expressions - Edexcel Past Exam Questions
- Edecel Past Eam Questions. (a) Differentiate with respect to (i) sin + sec, (ii) { + ln ()}. 5-0 + 9 Given that y =, ¹, ( -) 8 (b) show that = ( -). (6) June 05 Q. f() = e ln, > 0. (a) Differentiate
More informationPaper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours
1. Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Mark scheme Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question
More informationC-1. Snezana Lawrence
C-1 Snezana Lawrence These materials have been written by Dr. Snezana Lawrence made possible by funding from Gatsby Technical Education projects (GTEP) as part of a Gatsby Teacher Fellowship ad-hoc bursary
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
More informationHigher Mathematics Course Notes
Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that
More informationFINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed.
Math 150 Name: FINAL - PART 1 MATH 150 SPRING 2017 KUNIYUKI PART 1: 135 POINTS, PART 2: 115 POINTS, TOTAL: 250 POINTS No notes, books, or calculators allowed. 135 points: 45 problems, 3 pts. each. You
More informationCore Mathematics C3 Advanced
Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Friday 12 June 2015 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More information