Newbattle Community High School Higher Mathematics. Key Facts Q&A
|
|
- Peter Booth
- 5 years ago
- Views:
Transcription
1 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 and answering. 3) Ask a friend or family member** to test you by reading questions (on the left-hand side) to you. The questions are on the left-hand side of each page and the answers are on the right. **If the person who is testing you has not done Higher Maths recently (or ever!), they may need some help reading the questions, so some mathematical symbols have been written out phonetically (in a smaller bold underlined font) to help them. Some formulae in questions are printed large. These can be held up or shown.
2 Key Revision from National 5 Essential This knowledge should already be known from National 5 Mathematics and is still required for Higher. Learn these facts as soon as possible. You will need them to pass the unit assessment. Important A/B Learn these facts. You will need them to pass the final exam. You will need these facts to pass the course with an A or B but if you are just aiming to pass you may not need them. All Units: Equations and Graphs 1) How do you find the y intercepts of a graph? Substitute x = 0 into its equation ) How do you find the roots of a graph? Substitute y = 0 into its equation and solve 3) What three points do you need to indicate on the sketch of any graph? (if they exist) a) Stationary points b) Roots c) y intercept D Watkins 016 Page 1
3 All Units: Trigonometric Graphs and Equations 4) For which values of x between 0 and 360 is sin x (sine x) equal to zero? 5) For which values of x between 0 and 360 is cos x equal to zero? 6) For which values of x between 0 and 360 is tan x equal to zero? 0, 180 and and 70 0, 180 and 360 7) What is the formula for sine in a right-angled triangle? Opposite over Hypotenuse 8) What is the formula for cos in a right-angled triangle? Adjacent over Hypotenuse 9) What is the formula for tan in a right-angled triangle? Opposite over Adjacent 10) What is sin 30? (sine thirty degrees) 11) What is cos 30? (cos thirty degrees) 1) What is tan 30? (tan thirty degrees) (root three over two) (one over root three) 13) What is sin 45? (sine forty five degrees) 1 (one over root two) 14) What is cos 45? (cos forty five degrees) 1 (one over root two) 15) What is tan 45? (tan forty five degrees) 1 16) What is cos 60? (cos sixty degrees) 17) What is sin 60? (sine sixty degrees) 1 3 (root three over two) 18) What is tan 60? (tan sixty degrees) 3 (root three) 19) What is cos 0? (cos zero degrees) 1 0) What is sin 90? (sine ninety degrees) 1 1) What is cos180? (cos one eighty degrees) 1 ) How do you change from radians to degrees? Divide by Pi and multiply by 180 D Watkins 016 Page
4 3) How do you change from degrees to radians? Divide by 180 and multiply by Pi 4) What is 180 degrees in radians? (Pi) 5) What is 360 degrees in radians? (Two Pi) 6) What is 90 degrees in radians? (Pi over two) 7) What is (Pi) radians in degrees? 180 8) What is (Two Pi) radians in degrees? 360 9) What is (Pi over Two) radians in degrees? 90 30) What is the formula for tan? sin x cos x (sine over cos) 31) How is the range of sin x and cos x restricted? It is between 1 and 1 3) How do you find the amplitude of a trigonometric function from its equation? 33) How do you find the frequency of a trigonometric function from its equation? It s the number in front of sin, cos or tan It s the number in front of x D Watkins 016 Page 3
5 Applications 1.1: The Straight Line y y1 34) What is the formula for the gradient m x x1 between two points? (y two minus y one over x two minus x one) 35) What is the general formula for the equation of a straight line? 36) How do you find the gradient of a straight line if you only know its equation? 37) What is the formula linking the gradient to the angle that a line makes with the x axis? 38) What is the rule for the gradient of parallel lines? 39) What do perpendicular gradients multiply to give? 40) What two things do you to do find a perpendicular gradient? 41) What is the distance formula? 4) What are the two properties of the perpendicular bisector of a line? 43) What are the two properties of the median of a triangle through point A? 44) What are the two properties of the altitude of a triangle through point A? 45) How do you find the gradient of an altitude of a triangle through point A? 46) How do you find the gradient of the median of a triangle through point A? y b m( x a) y minus b equals m brackets x minus a a) Rearrange to make y the subject b) The gradient is the coefficient of x m tan m equals tan theta They are the same 1 a) Flip it upside down b) Change the sign ( x x ) ( y y ) 1 1 Square root of x two minus x one squared add y two minus y one squared a) It goes through the midpoint of the line b) It is perpendicular to the line a) It goes through point A b) It goes through the midpoint of the opposite side a) It goes through point A b) It is perpendicular to the opposite side a) Find the gradient of the side opposite A b) Flip and change sign a) Find the midpoint of the side opposite A b) Find the gradient from the midpoint to A D Watkins 016 Page 4
6 Applications 1.: Circles 47) What does it mean if two circles are congruent? 48) What does it mean if two circles are concentric? 49) If you are asked to write down the equation of a circle, which one should you use? 50) How can you tell if an equation is a valid equation of a circle? 51) What are the three steps to find the coordinates where a line meets a circle? 5) How do you show that a line is a tangent to a circle? 53) How do you show that a line and a circle do not intersect 54) How do you find the gradient of a tangent to a circle? They have the same radius They have the same centre The one with brackets in The radius must be positive a) make y (or x) the subject of the straight line b) substitute this into the circle equation c) solve the resulting quadratic equation Show there is only one point of intersection Alternative answer: show the discriminant of the equation of intersection is zero Show there are no points of intersection Alternative answer: show the discriminant of the equation of intersection is negative It is perpendicular to the radius 55) What is a common tangent? A line which is a tangent to two circles 56) How do I show that two circles have two points of intersection? 57) How do I show that two circles do not intersect? 58) How do I show that two circles touch in one point? Show that the distance between the two centres is less than the sum of the two radii Show that the distance between the two centres is greater than the sum of the two radii Show that the distance between the two centres is equal to the sum of the two radii D Watkins 016 Page 5
7 Applicatons 1.3: Sequences 59) If a limit exists, what do we know about a? It must be between 1 and 1 60) What is the formula for the limit of a recurrence relation? 61) What sentence do you HAVE to write for a communication mark when you are finding a limit? 6) How can you find the values that you multiply and add in a recurrence relation when you know the first three terms? b L (L equals b over 1 minus a) 1 a A limit exists because a is, which is between 1 and 1 Make two simultaneous equations and solve D Watkins 016 Page 6
8 Expressions and Formulae 1.3: Functions and Related Graphs 63) What is the domain of a function? What goes INTO the function 64) What is the range of a function? What comes OUT of the function 65) What is the turning point of the quadratic function ax ( p) q? a x plus p squared plus q 66) What two possible reasons are there for a value of x not being in the domain of a function? ( p, q) a) Square rooting a negative number b) Dividing by zero 67) If a function contains a square root, its domain is restricted. The expression under the square root is 0 (greater than or equal to zero) 68) If a function includes dividing, its domain is restricted. The expressions in the denominator are 69) How has the graph of y f( x) k been transformed? (y equals f of x, plus k) 70) How has the graph of y f( x k) been transformed? (y equals f of brackets x plus k) 71) How has the graph of y kf( x) been transformed? (y equals k f of x) 7) How has the graph of y f( kx) been transformed? (y equals f of k x) 73) How has the graph of y f ( x) been transformed? (y equals minus f of x) 74) How has the graph of y f( x) been transformed? (y equals f of minus x) 75) How can you show that two functions f and g are inverses? 76) What are the two steps for finding the inverse of a function? 0 (not equal to zero) It has been moved up by k units (or down if k is negative) It has been moved to the left by k units (or to the right if k is negative) It has been stretched vertically by scale factor k It has been compressed horizontally by scale factor k It has been reflected in the x axis Alternative answer: it has been reflected upside down It has been reflected in the y axis Alternative answer: it has been reflected left to right Show that f ( gx ( )) x (f of g of x equals x) a) Switch x and y in the formula b) Change the subject of the formula to y D Watkins 016 Page 7
9 Relationships and Calculus 1.1a: Quadratic Functions 77) Write down the quadratic formula 78) How can we find the minimum/maximum of a quadratic graph from its equation? x b b 4ac a Complete the square 79) What is formula for the discriminant? b 4ac (b squared minus 4 a c) 80) If an equation has real and equal roots, what do we know about the discriminant? 81) If an equation has real and distinct roots, what do we know about the discriminant? 8) If an equation has real roots, what do we know about the discriminant? 83) If an equation has no real roots, what do we know about the discriminant? 84) How do we solve a quadratic inequality? 85) How do we show that a line is tangent to a parabola? It is equal to zero (= 0) It is greater than zero (> 0) It is greater than or equal to zero ( 0) It is less than zero (< 0) Sketch the parabola and use the sketch to write down the values of x for which the graph is above or below the axes Show they only have one point of intersection Alternative answer: show the discriminant of the equation of intersection is zero Relationships and Calculus 1.1b: Quadratic Functions 86) How do you find the remainder when a polynomial is divided by x a?(x minus a) 87) When doing synthetic division, what do we call the number in the box at the end? 88) When factorising a polynomial, what do you HAVE to write for a communication mark after synthetic division? 89) If a function has a repeated root, what does the graph look like at that point? Use synthetic division (with a) The remainder The remainder is zero, so is a factor It is tangent to the x axis at that root D Watkins 016 Page 8
10 Relationships and Calculus 1.3 and Applications 1.4: Differentiation 90) What are the two basic steps for differentiating powers? a) Multiply by the old power b) Take one away from the power 91) How do you find the derived function? Differentiate 9) How do you find the derivative? Differentiate 93) How do you find a rate of change? Differentiate 94) How do we find the gradient of a tangent at a particular value of x? Differentiate and substitute in x 95) What do we know about dy dx function is increasing? 96) What do we know about dy dx function is decreasing? 97) What do we know about dy dx function is stationary? (dee y by dee x) if a (dee y by dee x) if a (dee y by dee x) if a It is greater than zero ( > 0 ) It is less than zero ( < 0 ) It is equal to zero ( = 0 ) 98) What are the four types of stationary point? 99) If you are trying to find a maximum or minimum something in a particular interval, what three points do you check? 100) What sentence do you HAVE to write for communication marks in an exam in a stationary points question? a) Maximum c) Rising point of inflection b) Minimum d) Falling point of inflection a) Stationary points b) The start value c) The end value dy A stationary point exists when 0 dx (dee y by dee x equals zero) (or f ( x) 0 (f dash x equals zero) ) 101) When is a function not differentiable? If it is not defined at a point 10) What do you get if you differentiate sine? cos 103) What do you get if you differentiate cos? sine 104) What do you have to remember about angles when differentiating sine or cos? 105) What are the three steps for differentiating a n bracket ( ax b)? (a x plus b to the power n) Angles must be in radians and not degrees a) Multiply by the old power b) Take one off the power c) Differentiate the bracket and multiply at the front by your answer D Watkins 016 Page 9
11 (ignore these next two questions if you were taught to use nature tables instead of the second derivative) 106) If 107) If d y (dee two y by dee x squared) is positive, what dx does this tell us about a stationary point? d y (dee two y by dee x squared) is negative, what dx does this tell us about a stationary point? It is a minimum It is a maximum Relationships and Calculus 1.4 and Applications 1.4: Integration 108) What are the three steps for basic integration? a) Add one to the power, b) Divide by the new power c) Add C (or add a constant ) 109) If we know dy (dee y by dee x), how do we dx find the original equation? 110) How do we find the area between two curves? 111) What must we remember for indefinite integrals? 11) How do you find the area underneath a curve? 113) What do you always need when the question asks you to find an area? 114) What do we have to remember when finding an area below the x-axis? a) Integrate b) Substitute in a point on the curve to find C Integral of top take away bottom + C a) Integrate the equation of the curve b) Substitute in the limits A sketch of the graphs to check if the area is above the axis, below the axis or split The answer will be negative, and we have to deal with this appropriately 115) What are the three steps to remember when the area is partly above and partly below the x-axis? 116) What do you get if you integrate sine? cos plus C 117) What do you get if you integrate cos? sine plus C 118) How do you integrate a bracket n ( ax b)? (a x plus b to the power n) 119) What do you have to remember about angles when integrating sine or cos? a) Calculate out the areas above and below the x-axis separately b) For the bit where the integral is negative say so the area is (positive answer) c) Add 1. Add one to the power. Divide by the new power 3. Differentiate the bracket and divide by your answer Angles must be in radians and not degrees D Watkins 016 Page 10
12 Relationships and Calculus 1. and Expressions and Functions 1.: Trigonometric Expressions and Equations 10) What should you always check after finishing a trig equation? 11) If you know solutions to a trig equation between 0 and 360, how can you find other solutions? 1) What are the three steps to solve an equation containing cos x or sin x? (cos squared x or sine squared x) 13) In a wave function question, what is the formula for k? Should the final answer be in degrees or radians? Add or take away multiples of 360 a) Rearrange to get zero on the righthand side b) Factorise it c) Use each bracket to make a new equation k a b (k equals square root of a squared plus b squared) 14) In a wave function question, what is the formula for a? tan a ksin a kcos a (tan a equals k sine a over k cos a) Expressions and Functions 1.4: Vectors FORMULA SHEET a b a b cos or a b ab 1 1 ab ab ) What are the three steps for finding the magnitude of a vector? 16) What is the formula for the components of a vector from A to B? a) Square all the components b) Add them c) Square root b a 17) How do you show that vectors u and v are parallel? Show that u kv for some k 18) What does it mean if three points are collinear? They all lie in a straight line 19) How do you show three points are collinear? a) Show two vectors joining them are parallel b) State that they have a common point 130) What is the condition for two vectors to be perpendicular? Their dot product is zero D Watkins 016 Page 11
13 131) What two things do you HAVE to state for a communication mark when showing points are collinear 13) What formula do you need to find the angle between two vectors? 133) When talking about the angle between two vectors, how should the vectors be positioned? 134) How do you show that two vectors a and b are perpendicular? 135) What is a a equal to? (a dot a) 136) How do you expand the brackets in a ( b c )? (a dot brackets b plus c) a) That the vectors are parallel b) And that they contain a common point a b a b cos (a dot b equals magnitude of a times magnitude of b times cos theta) Either both pointing outwards Or both pointing inwards Show that a b 0 (a dot b equals zero) a (the magnitude of a squared) a b a c (a dot b plus a dot c) Expressions and Formulae 1.1: Exponential and Logarithmic Functions 137) What is e correct to two decimal places? 7 138) In an exponential equation represent? 139) In an exponential equation represent? kt Ae, what does A kt Ae, what does k The starting value The percentage increase/decrease (as a decimal) 140) What two letter abbreviation do we use for the logarithm to the base e? 141) What is the inverse function of y x a? y equals a to the power of x 14) What is the inverse function of y log a x? 143) What is the inverse function of y equals log to the base a of x y x e? y equals e to the power of x 144) What is the inverse function of y ln x? y equals L N x LN y log a x y equals log to the base a of x y x a y equals a to the power of x y ln x y x e y equals e to the power of x D Watkins 016 Page 1
14 145) What is the value of log 1? (for any base) 0 146) Which two coordinates will an exponential graph (base a) always pass through? 147) Which two coordinates will a logarithmic graph (base a) always pass through? 148) How do you solve an equation where x is in the x power? (eg 4 10) 149) How do you solve an equation with a log on the lefthand side? 150) How do you solve an equation containing the number e? 151) Laws of logs: When you add two logs with the same base, what happens to the numbers? 15) Laws of logs: When you take away two logs with the same base, what happens to the numbers? 153) Laws of logs: How do you deal with a power inside a log? 154) A straight line graph has log x or log y on the axes. What are the three steps to find its equation? ( 0,1) and ( 1, a ) ( 1,0) and (a,1) Take logs of both sides then bring x down to the front Take powers of the right hand side By taking ln (natural log) of both sides Multiply them Divide them Bring the power down to the front a) Find the equation of the line using y = mx + c b) Replace x and y with whatever is on the axes c) Use laws of logs to rearrange D Watkins 016 Page 13
Higher 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 informationNewbattle Community High School National 5 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 National 5 Maths to take turns reading a
More information2 2xdx. Craigmount High School Mathematics Department
Π 5 3 xdx 5 cosx 4 6 3 8 Help Your Child With Higher Maths Introduction We ve designed this booklet so that you can use it with your child throughout the session, as he/she moves through the Higher course,
More informationΠ xdx cos 2 x
Π 5 3 xdx 5 4 6 3 8 cos x Help Your Child with Higher Maths Introduction We ve designed this booklet so that you can use it with your child throughout the session, as he/she moves through the Higher course,
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 informationKey Facts and Methods
Intermediate Maths Key Facts and Methods Use this (as well as trying questions) to revise by: 1. Testing yourself. Asking a friend or family member to test you by reading the questions (on the lefthand
More informationHigher Mathematics Skills Checklist
Higher Mathematics Skills Checklist 1.1 The Straight Line (APP) I know how to find the distance between 2 points using the Distance Formula or Pythagoras I know how to find gradient from 2 points, angle
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 informationZETA MATHS. Higher Mathematics Revision Checklist
ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions
More informationBrief Revision Notes and Strategies
Brief Revision Notes and Strategies Straight Line Distance Formula d = ( ) + ( y y ) d is distance between A(, y ) and B(, y ) Mid-point formula +, y + M y M is midpoint of A(, y ) and B(, y ) y y Equation
More informationKing s Year 12 Medium Term Plan for LC1- A-Level Mathematics
King s Year 12 Medium Term Plan for LC1- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives
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 information1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.
Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationThe Not-Formula Book for C1
Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) May 2010
Link to past paper on OCR website: http://www.mei.org.uk/files/papers/c110ju_ergh.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or
More informationOutline schemes of work A-level Mathematics 6360
Outline schemes of work A-level Mathematics 6360 Version.0, Autumn 013 Introduction These outline schemes of work are intended to help teachers plan and implement the teaching of the AQA A-level Mathematics
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 informationCandidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.
Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;
More informationMathematics (MEI) Advanced Subsidiary GCE Core 1 (4751) June 2010
Link to past paper on OCR website: www.ocr.org.uk The above link takes you to OCR s website. From there you click QUALIFICATIONS, QUALIFICATIONS BY TYPE, AS/A LEVEL GCE, MATHEMATICS (MEI), VIEW ALL DOCUMENTS,
More informationA-Level Notes CORE 1
A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. (a) 5
More informationMEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions
MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms
More information4 The Trigonometric Functions
Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater
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 informationPre-Calculus MATH 119 Fall Section 1.1. Section objectives. Section 1.3. Section objectives. Section A.10. Section objectives
Pre-Calculus MATH 119 Fall 2013 Learning Objectives Section 1.1 1. Use the Distance Formula 2. Use the Midpoint Formula 4. Graph Equations Using a Graphing Utility 5. Use a Graphing Utility to Create Tables
More informationPure Mathematics P1
1 Pure Mathematics P1 Rules of Indices x m * x n = x m+n eg. 2 3 * 2 2 = 2*2*2*2*2 = 2 5 x m / x n = x m-n eg. 2 3 / 2 2 = 2*2*2 = 2 1 = 2 2*2 (x m ) n =x mn eg. (2 3 ) 2 = (2*2*2)*(2*2*2) = 2 6 x 0 =
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 informationModel Paper WITH ANSWERS. Higher Maths
Model Paper WITH ANSWERS Higher Maths This model paper is free to download and use for revision purposes. The paper, which may include a limited number of previously published SQA questions, has been specially
More informationFree download from not for resale. Apps 1.1 : Applying algebraic skills to rectilinear shapes.
Free download from, not for resale. Apps 1.1 : Applying algebraic skills to rectilinear shapes. Gradients m = tanθ Distance Formula Midpoint Formula Parallel lines Perpendicular lines y = mx + c y - b
More informationFinal Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14
Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)
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 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 informationMathematics AQA Advanced Subsidiary GCE Core 1 (MPC1) January 2010
Link to past paper on AQA website: http://store.aqa.org.uk/qual/gce/pdf/aqa-mpc1-w-qp-jan10.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are
More information1. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y 3x
MATH 94 Final Exam Review. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y x b) y x 4 c) y x 4. Determine whether or not each of the following
More informationNATIONAL QUALIFICATIONS
Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -
More informationLearning Objectives These show clearly the purpose and extent of coverage for each topic.
Preface This book is prepared for students embarking on the study of Additional Mathematics. Topical Approach Examinable topics for Upper Secondary Mathematics are discussed in detail so students can focus
More informationMock Final Exam Name. Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) A) {- 30} B) {- 6} C) {30} D) {- 28}
Mock Final Exam Name Solve and check the linear equation. 1) (-8x + 8) + 1 = -7(x + 3) 1) A) {- 30} B) {- 6} C) {30} D) {- 28} First, write the value(s) that make the denominator(s) zero. Then solve the
More informationCore 1 Module Revision Sheet J MS. 1. Basic Algebra
Core 1 Module Revision Sheet The C1 exam is 1 hour 0 minutes long and is in two sections Section A (6 marks) 8 10 short questions worth no more than 5 marks each Section B (6 marks) questions worth 12
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More informationSummer Packet A Math Refresher For Students Entering IB Mathematics SL
Summer Packet A Math Refresher For Students Entering IB Mathematics SL Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school
More informationFinal Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i
Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add
More information( and 1 degree (1 ) , there are. radians in a full circle. As the circumference of a circle is. radians. Therefore, 1 radian.
Angles are usually measured in radians ( c ). The radian is defined as the angle that results when the length of the arc of a circle is equal to the radius of that circle. As the circumference of a circle
More informationTrigonometry Self-study: Reading: Red Bostock and Chandler p , p , p
Trigonometry Self-study: Reading: Red Bostock Chler p137-151, p157-234, p244-254 Trigonometric functions be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant, secant,
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationYEAR 12 - Mathematics Pure (C1) Term 1 plan
Week YEAR 12 - Mathematics Pure (C1) Term 1 plan 2016-2017 1-2 Algebra Laws of indices for all rational exponents. Use and manipulation of surds. Quadratic functions and their graphs. The discriminant
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 1 C1 2015-2016 Name: Page C1 workbook contents Indices and Surds Simultaneous equations Quadratics Inequalities Graphs Arithmetic series
More informationPre-calculus 12 Curriculum Outcomes Framework (110 hours)
Curriculum Outcomes Framework (110 hours) Trigonometry (T) (35 40 hours) General Curriculum Outcome: Students will be expected to develop trigonometric reasoning. T01 Students will be expected to T01.01
More informationARE YOU READY FOR CALCULUS?? Name: Date: Period:
ARE YOU READY FOR CALCULUS?? Name: Date: Period: Directions: Complete the following problems. **You MUST show all work to receive credit.**(use separate sheets of paper.) Problems with an asterisk (*)
More informationQuadratics. SPTA Mathematics Higher Notes
H Quadratics SPTA Mathematics Higher Notes Quadratics are expressions with degree 2 and are of the form ax 2 + bx + c, where a 0. The Graph of a Quadratic is called a Parabola, and there are 2 types as
More informationDIFFERENTIATION AND INTEGRATION PART 1. Mr C s IB Standard Notes
DIFFERENTIATION AND INTEGRATION PART 1 Mr C s IB Standard Notes In this PDF you can find the following: 1. Notation 2. Keywords Make sure you read through everything and the try examples for yourself before
More informationSt Peter the Apostle High. Mathematics Dept.
St Peter the postle High Mathematics Dept. Higher Prelim Revision 6 Paper I - Non~calculator Time allowed - hour 0 minutes Section - Questions - 0 (40 marks) Instructions for the completion of Section
More informationNational Quali cations
H 2018 X747/76/11 National Quali cations Mathematics Paper 1 (Non-Calculator) THURSDAY, 3 MAY 9:00 AM 10:10 AM Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given
More information1 Triangle ABC has vertices A( 1,12), B( 2, 5)
Higher Mathematics Paper : Marking Scheme Version Triangle ABC has vertices A(,), B(, ) A(, ) y and C(, ). (a) (b) (c) Find the equation of the median BD. Find the equation of the altitude AE. Find the
More informationMEI Core 2. Sequences and series. Section 1: Definitions and Notation
Notes and Eamples MEI Core Sequences and series Section : Definitions and Notation In this section you will learn definitions and notation involving sequences and series, and some different ways in which
More informationS56 (5.1) Polynomials.notebook August 25, 2016
Q1. Simplify Daily Practice 28.6.2016 Q2. Evaluate Today we will be learning about Polynomials. Q3. Write in completed square form x 2 + 4x + 7 Q4. State the equation of the line joining (0, 3) and (4,
More informationMay 2015 Timezone 2 IB Maths Standard Exam Worked Solutions
May 015 Timezone IB Maths Standard Exam Worked Solutions 015, Steve Muench steve.muench@gmail.com @stevemuench Please feel free to share the link to these solutions http://bit.ly/ib-sl-maths-may-015-tz
More informationStraight Line. SPTA Mathematics Higher Notes
H Straight Line SPTA Mathematics Higher Notes Gradient From National 5: Gradient is a measure of a lines slope the greater the gradient the more steep its slope and vice versa. We use the letter m to represent
More informationIYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
IYGB Special Paper U Time: 3 hours 30 minutes Candidates may NOT use any calculator Information for Candidates This practice paper follows the Advanced Level Mathematics Core Syllabus Booklets of Mathematical
More informationMathematics 1 Lecture Notes Chapter 1 Algebra Review
Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to
More informationPearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0)
Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0) Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) First teaching from September 2017 First certification from June 2018 2
More informationWe look forward to working with you this school year!
Name: Summer Review Packet for Students Entering IB MATH SL Year 2 Directions: Complete all pages in this review. Show all work for credit. You may get video help and instruction via YouTube for the topics
More information2001 Higher Maths Non-Calculator PAPER 1 ( Non-Calc. )
001 PAPER 1 ( Non-Calc. ) 1 1) Find the equation of the straight line which is parallel to the line with equation x + 3y = 5 and which passes through the point (, 1). Parallel lines have the same gradient.
More informationSince x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)
SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2
More informationNational 5 Learning Checklist - Relationships
National 5 Learning Checklist - Relationships Topic Skills Extra Stud / Notes Straight Line Gradient Represented b m Measure of steepness of slope Positive gradient the line is increasing Negative gradient
More informationFunctions and their Graphs
Chapter One Due Monday, December 12 Functions and their Graphs Functions Domain and Range Composition and Inverses Calculator Input and Output Transformations Quadratics Functions A function yields a specific
More informationAnswers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16)
Answers for NSSH eam paper type of questions, based on the syllabus part (includes 6) Section Integration dy 6 6. (a) Integrate with respect to : d y c ( )d or d The curve passes through P(,) so = 6/ +
More informationCore Mathematics C1 (AS) Unit C1
Core Mathematics C1 (AS) Unit C1 Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation. Graphs of functions; sketching curves defined by simple equations.
More informationA Library of Functions
LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us
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 informationC100/SQP321. Course Assessment Specification 2. Specimen Question Paper 1 5. Specimen Question Paper Specimen Marking Instructions Paper 1 23
C00/SQP Maths Higher NTIONL QULIFICTIONS Contents Page Course ssessment Specification Specimen Question Paper 5 Specimen Question Paper 7 Specimen Marking Instructions Paper Specimen Marking Instructions
More informationAlgebra One Dictionary
Algebra One Dictionary Page 1 of 17 A Absolute Value - the distance between the number and 0 on a number line Algebraic Expression - An expression that contains numbers, operations and at least one variable.
More informationAssessment Report. Level 2, Mathematics
Assessment Report Level 2, 2006 Mathematics Manipulate algebraic expressions and solve equations (90284) Draw straightforward non-linear graphs (90285) Find and use straightforward derivatives and integrals
More information2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW
FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.
More informationMaths Pack. Distance Learning Mathematics Support Pack. For the University Certificates in Astronomy and Cosmology
Maths Pack Distance Learning Mathematics Support Pack For the University Certificates in Astronomy and Cosmology These certificate courses are for your enjoyment. However, a proper study of astronomy or
More informationREQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS
REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study
More informationADDITIONAL MATHEMATICS
ADDITIONAL MATHEMATICS GCE NORMAL ACADEMIC LEVEL (016) (Syllabus 4044) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE
More informationThe Not-Formula Book for C2 Everything you need to know for Core 2 that won t be in the formula book Examination Board: AQA
Not The Not-Formula Book for C Everything you need to know for Core that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationNAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018
NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 A] Refer to your pre-calculus notebook, the internet, or the sheets/links provided for assistance. B] Do not wait until the last minute to complete this
More informationIntegers, Fractions, Decimals and Percentages. Equations and Inequations
Integers, Fractions, Decimals and Percentages Round a whole number to a specified number of significant figures Round a decimal number to a specified number of decimal places or significant figures Perform
More informationPure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions
Pure Mathematics Year (AS) Unit Test : Algebra and Functions Simplify 6 4, giving your answer in the form p 8 q, where p and q are positive rational numbers. f( x) x ( k 8) x (8k ) a Find the discriminant
More informationWhen using interval notation use instead of open circles, and use instead of solid dots.
P.1 Real Numbers PreCalculus P.1 REAL NUMBERS Learning Targets for P1 1. Describe an interval on the number line using inequalities. Describe an interval on the number line using interval notation (closed
More informationNational Quali cations
H 2017 X747/76/11 FRIDAY, 5 MAY 9:00 AM 10:10 AM National Quali cations Mathematics Paper 1 (Non-Calculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
hsn.uk.net Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still
More information2016 Notes from the Marking Centre - Mathematics
2016 Notes from the Marking Centre - Mathematics Question 11 (a) This part was generally done well. Most candidates indicated either the radius or the centre. Common sketching a circle with the correct
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 informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For
More informationEdexcel past paper questions. Core Mathematics 4. Parametric Equations
Edexcel past paper questions Core Mathematics 4 Parametric Equations Edited by: K V Kumaran Email: kvkumaran@gmail.com C4 Maths Parametric equations Page 1 Co-ordinate Geometry A parametric equation of
More informationAdd Math (4047/02) Year t years $P
Add Math (4047/0) Requirement : Answer all questions Total marks : 100 Duration : hour 30 minutes 1. The price, $P, of a company share on 1 st January has been increasing each year from 1995 to 015. The
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationTABLE OF CONTENTS 2 CHAPTER 1
TABLE OF CONTENTS CHAPTER 1 Quadratics CHAPTER Functions 3 CHAPTER 3 Coordinate Geometry 3 CHAPTER 4 Circular Measure 4 CHAPTER 5 Trigonometry 4 CHAPTER 6 Vectors 5 CHAPTER 7 Series 6 CHAPTER 8 Differentiation
More information7.1 Indefinite Integrals Calculus
7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential
More informationALGEBRA II (COMMON CORE) FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM
ALGEBRA II (COMMON CORE) FACTS YOU MUST KNOW COLD FOR THE REGENTS EXAM NYS Mathematics Regents Preparation Created by Trevor Clark Algebra II [Common Core] Regents Exam Study Guide ALGEBRA & FUNCTIONS
More informationS4 (4.3) Quadratic Functions.notebook February 06, 2018
Daily Practice 2.11.2017 Q1. Multiply out and simplify 3g - 5(2g + 4) Q2. Simplify Q3. Write with a rational denominator Today we will be learning about quadratic functions and their graphs. Q4. State
More informationReview Notes for IB Standard Level Math
Review Notes for IB Standard Level Math 1 Contents 1 Algebra 8 1.1 Rules of Basic Operations............................... 8 1.2 Rules of Roots..................................... 8 1.3 Rules of Exponents...................................
More informationALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
More informationA-Level Maths Revision notes 2014
A-Level Maths Revision notes 2014 Contents Coordinate Geometry... 2 Trigonometry... 4 Basic Algebra... 7 Advanced Algebra... 9 Sequences and Series... 11 Functions... 12 Differentiation... 14 Integration...
More informationSummer Packet Honors PreCalculus
Summer Packet Honors PreCalculus Honors Pre-Calculus is a demanding course that relies heavily upon a student s algebra, geometry, and trigonometry skills. You are epected to know these topics before entering
More informationAP Calculus I and Calculus I Summer 2013 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS?
Name: AP Calculus I and Calculus I Summer 0 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS? Calculus is a VERY RIGOROUS course and completing this packet with your best effort will help you
More information