New Higher Past Papers by Topic 2015 Trigonometry Straight Line Functions and graphs -Including quadratics The Circle Differentiation Vectors Recurrence Relations Logs & Exponentials Polynomials Wave Function Integration Duncanrig Secondary - Maths Department

Higher Mindmaps by Topic Straight Line Trigonometry Composite Functions The Circle Differentiation Vectors Recurrence Relations Logs & Exponential Polynomials Wave Function Integration Duncanrig Secondary - Maths Department

Straight Line Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Functions and Graphs and Quadratic Theory Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Differentiation Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Recurrence Relations Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Polynomials Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Integration Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Trigonometry Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Circle Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Vectors Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Logs and Exponentials Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Wave function Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department

Mind Maps Duncanrig Secondary - Maths Department

Parallel lines have same gradient Distance between 2 points m < 0 m = undefined Terminology Median – midpoint Bisector – midpoint Perpendicular – Right Angled Altitude – right angled m1.m2 = -1 m > 0 m = 0 Possible values for gradient Form for finding line equation y – b = m(x - a) (a,b) = point on line Straight Line y = mx + c Parallel lines have same gradient m = gradient c = y intercept (0,c) For Perpendicular lines the following is true. m1.m2 = -1 θ m = tan θ Duncanrig Secondary - Maths Department Mindmaps

+ - - + f(x) f(x) f(x) Remember we can combine these together !! flip in y-axis Move vertically up or downs depending on k f(x) f(x) - Stretch or compress vertically depending on k y = f(x) ± k y = f(-x) Remember we can combine these together !! y = kf(x) Graphs & Functions y = -f(x) y = f(kx) y = f(x ± k) f(x) f(x) flip in x-axis Stretch or compress horizontally depending on k - f(x) + Move horizontally left or right depending on k Duncanrig Secondary - Maths Department Mindmaps

= + x x g(x) = g(f(x)) f(x) = x2 - 4 g(f(x)) = y = f(x) g(x) = f(g(x)) 1 x g(f(x)) f(x) = x2 - 4 But y = f(x) is x2 - 4 1 x2 - 4 g(f(x)) = 1 y x y = f(x) Restriction Domain Range x2 - 4 ≠ 0 A complex function made up of 2 or more simpler functions Similar to composite Area (x – 2)(x + 2) ≠ 0 x ≠ 2 x ≠ -2 Composite Functions 1 x = + But y = g(x) is g(x) = 1 x f(g(x)) 1 x 2 - 4 f(x) = x2 - 4 f(g(x)) = x 1 x2 - 4 y = g(x) y2 - 4 Rearranging Restriction x2 ≠ 0 Domain Range Duncanrig Secondary - Maths Department Mindmaps

Basics before Differentiation/ Integration Format for Differentiation / Integration Surds Indices Basics before Differentiation/ Integration Division Working with fractions Adding Multiplication Subtracting Duncanrig Secondary - Maths Department Mindmaps

+ - f(x) = axn then f’x) = anxn-1 -1 2 5 x Max Differentiation Nature Table Equation of tangent line Leibniz Notation -1 2 5 + - x f’(x) Max Straight Line Theory Gradient at a point f’(x)=0 Stationary Pts Max. / Mini Pts Inflection Pt Graphs f’(x)=0 Derivative = gradient = rate of change Differentiation of Polynomials f(x) = axn then f’x) = anxn-1 Duncanrig Secondary - Maths Department Mindmaps

Graphs Trig Harder functions Use Chain Rule Differentiations Rules of Indices Polynomials Differentiations Factorisation Graphs Real life Meaning Stationary Pts Mini / Max Pts Inflection Pts Rate of change of a function. Tangent equation Gradient at a point. Straight line Theory Duncanrig Secondary - Maths Department Mindmaps

f(x) =2x2 + 4x + 3 f(x) =2(x + 1)2 - 2 + 3 f(x) =2(x + 1)2 + 1 Completing the square f(x) = a(x + b)2 + c Easy to graph functions & graphs 1 4 5 2 -2 -2 -4 -2 Factor Theorem x = a is a factor of f(x) if f(a) = 0 1 2 1 f(x) =2x2 + 4x + 3 f(x) =2(x + 1)2 - 2 + 3 f(x) =2(x + 1)2 + 1 (x+2) is a factor since no remainder If finding coefficients Sim. Equations Discriminant of a quadratic is b2 -4ac Polynomials Functions of the type f(x) = 3x4 + 2x3 + 2x +x + 5 Degree of a polynomial = highest power Tangency b2 -4ac > 0 Real and distinct roots b2 -4ac < 0 No real roots b2 -4ac = 0 Equal roots Duncanrig Secondary - Maths Department Mindmaps

b (1 - a) Un+1 = aUn + b a > 1 then growth a < 1 then decay Limit L is equal to Given three value in a sequence e.g. U10 , U11 , U12 we can work out recurrence relation U11 = aU10 + b (1 - a) L = b U12 = aU11 + b Use Sim. Equations a = sets limit b = moves limit Un = no effect on limit Recurrence Relations next number depends on the previous number Un+1 = aUn + b |a| > 1 |a| < 1 a > 1 then growth a < 1 then decay Limit exists when |a| < 1 + b = increase - b = decrease Duncanrig Secondary - Maths Department Mindmaps

IF f’(x) = axn Then I = f(x) = f(x) g(x) b A= ∫ f(x) - g(x) dx a Remember to change sign to + if area is below axis. f(x) Remember to work out separately the area above and below the x-axis . g(x) A= ∫ f(x) - g(x) dx b a Integration is the process of finding the AREA under a curve and the x-axis Area between 2 curves Finding where curve and line intersect f(x)=g(x) gives the limits a and b Integration of Polynomials IF f’(x) = axn Then I = f(x) = Duncanrig Secondary - Maths Department

Duncanrig Secondary - Maths Department Special case Duncanrig Secondary - Maths Department Mindmaps

Duncanrig Secondary - Maths Department Double Angle Formulae sin2A = 2sinAcosA cos2A = 2cos2A - 1 = 1 - 2sin2A = cos2A – sin2A Addition Formulae sin(A ± B) = sinAcosB cosAsinB cos(A ± B) = cosAcosB sinAsinB C A S T 0o 180o 270o 90o xo 4 2 The exact value of sinx Trig Formulae and Trig equations 3cos2x – 5cosx – 2 = 0 Let p = cosx 3p2 – 5p - 2 = 0 (3p + 1)(p -2) = 0 sinx = 2sin(x/2)cos(x/2) p = cosx = 1/3 cosx = 2 sinx = 2 (¼ + √(42 - 12) ) x = cos-1( 1/3) x = no soln sinx = ½ + 2√15) x = 109.5o and 250.5o Duncanrig Secondary - Maths Department Mindmaps

sin x cos x tan x ÷ π 360o 90o 180o 270o 360o 360o 180o 90o Rearrange into sin = Find solution in Basic Quads Remember Multiple solutions Exact Value Table C A S T 0o 180o 270o 90o ÷180 then X π degrees radians Basic Strategy for Solving Trig Equations ÷ π then x 180 Trigonometry sin, cos , tan 360o 1 -1 Amplitude Complex Graph 90o 2 -1 1 180o 270o 360o 3 Period sin x Basic Graphs Amplitude 1 -1 360o Amplitude 1 -1 180o 90o Period Period y = 2sin(4x + 45o) + 1 Period cos x Max. Value =2+1= 3 Period = 360 ÷4 = 90o tan x Duncanrig Secondary - Maths Department Mini. Value = -2+1 -1 Amplitude = 2 Mindmaps

Addition Scalar product Component form Magnitude Vector Theory same for subtraction Addition 2 vectors perpendicular if Scalar product Component form Magnitude Basic properties scalar product Vector Theory Magnitude & Direction Q Notation B P a Component form A Vectors are equal if they have the same magnitude & direction Unit vector form Duncanrig Secondary - Maths Department Mindmaps

b a c b a θ Tail to tail Vector Theory Magnitude & Direction C B C n B Angle between two vectors properties Vector Theory Magnitude & Direction Section formula C A B O A B C n c Points A, B and C are said to be Collinear if B is a point in common. m b a Duncanrig Secondary - Maths Department Mindmaps

y = logax y = ax a0 = 1 a1 = a y = axb y = abx To undo log take exponential y = ax y x To undo exponential take log y x (a,1) loga1 = 0 (1,a) a0 = 1 (0,1) logaa = 1 (1,0) a1 = a Basic log graph Basic exponential graph log A + log B = log AB log A - log B = log B A log (A)n = n log A Logs & Exponentials Basic log rules y = axb Can be transformed into a graph of the form y = abx Can be transformed into a graph of the form (0,C) log y log x (0,C) log y x log y = x log b + log a log y = b log x + log a Y = mX + C Y = mX + C Y = (log b) X + C Y = bX + C C = log a m = log b C = log a m = b Duncanrig Secondary - Maths Department Mindmaps

trigonometric identity a = k cos β Square and add then f(x) = a sinx + b cosx Compare coefficients compare to required trigonometric identity a = k cos β Square and add then square root gives b = k sin β f(x) = k sin(x + β) = k sinx cos β + k cosx sin β Process example Divide and inverse tan gives Wave Function a and b values decide which quadrant transforms f(x)= a sinx + b cosx into the form Write out required form OR Related topic Solving trig equations Duncanrig Secondary - Maths Department Mindmaps