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.
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
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 360 90 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) 1 3 1 3 (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
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
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
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
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
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
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
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
(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
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 3 3 15) 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
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
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