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Higher Maths

Higher Maths. Question Types. You need to learn basic movements. Steps : Outside function stays the same EXCEPT replace x terms with a ( ) 2. Put inner function in bracket. Exam questions normally involve two movements. Composite Functions. Remember order BODMAS. Sketching

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Higher Maths

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  1. Higher Maths Question Types

  2. You need to learn basic movements • Steps : • Outside function stays • the same EXCEPT replace • x terms with a ( ) • 2. Put inner function in bracket Exam questions normally involve two movements Composite Functions Remember order BODMAS Sketching Graphs • Restrictions : • Denominator NOT ALLOWED • to be zero • CANNOT take the square root • of a negative number Functions & Graphs TYPE questions (Trig , Quadratics)

  3. f’(x) > 0 f’(x) < 0 Function Increasing or Decreasing Basic and Format Max / Mini in closed intervals Differentiation TYPE questions (Fractions / Surds /Indices) Gradient f’(x) • Steps : • Find Max / Mini points • Find end values • Decide Max / Mini Points Stationary Points Optimization Tangent Line • Steps : • Differentiate • f’(x) = 0 (statement) • Factorise • Nature Table • Sub x = to original • equation to find • y coordinate. • Steps : • Differentiate • Sub x = into f’(x) • to find gradient • Use a point on the line • and y – b = m(x – a)

  4. Steps : • Setup recurrence relation • State if limit exists • Find limit • Steps : • Using information given • setup two equations • Use simultaneous • equation method to find • constants Finding Constants Wordy question Recurrence Relations TYPE questions (Fractions / Sim Equations)

  5. f(x) = a(x + b)2 + c e.g. use coefficients to factorise further if possible !! -2 1 4 5 2 -2 -4 -2 Factor Theorem 2x2 - 8x + 9 1 2 1 0 (x+2) is a factor since no remainder Remember to answer question f(x) = ( ) ( ) ( ) 2x2 - 8x + 9 Completing the square 2(x2 - 4x) + 9 Factorising cubic's polynomials 2(x - 2)2 + 9 - 8 Quadratic Theory questions (Circle, Function Graphs) f(x) = 2(x - 2)2 + 1 simultaneous equations Sketch See Function & Graphs Harder Finding coefficients Discriminant b2 – 4ac 3 scenarios • Steps • Identify a , b and c. • Discriminant .... = 0 • and factorise. • Sketch and identify • solution based on • question asked. > 0 Harder discriminant = 0 tangent !!! < 0 (1 - 2k)x2 - 5kx - 2k > 0

  6. -1 2 Simple Area under the curve Area above & below x-axis Basic Integration TYPE questions (Fractions / Surds /Indices) Original Equation 0 1 4 Do separately and remember statement for below x-axis AT = A1 + A2 Area between two curves • Steps : • For limits make • equal to each other. • 2. Integrate • Top – (bottom) 3 -2

  7. 5 4 Substitution and solving β α Sub for cos2x 3 12 Factorise • Steps • Pythagoras Theorem • Expansion • SOHCAHTOA • Solve. Solve Expansion With Triangles See A3 sheet given out in Unit 1 For more solving techniques Trigonometry TYPE questions (Quadratic, Function Graphs) Exact values and radians !!! Sketching Basic f(x) = sinx f(2x) = sin2x Equation from Graph and solving 3f(2x) = 3sin2x f(2x) + 1 = 3sin2x + 1 • Steps • Write down equation using graph • Using balance method to solve See A3 sheet given out in Unit 1 For more solving techniques

  8. Does circles touch externally or internally ? radius = (a,b) centre = (-g,-f) (x - a)2 + (y - b)2 = r2 x2 + y2 + 2gx + 2fy + c =0 Finding centre and radius from circle equation Is equation a circle ? Equation from graph r > 0 Circle TYPE questions (Straight Line , Quadratics) Intersection points between line and circle Equation of tangent (a,b) • Steps • Sub line equation y = ... • into circle. • Discriminant to establish • how many points. • Factorise for x coordinates • and sub into line equation • for y coordinates 3 possible scenarios • Steps • Find gradient of • centre to point • Use m1 x m2 = -1 • to find gradient of line • Use y – b = m(x - a)

  9. b Tail to tail θ a Angle between two vectors properties Vector Theory Magnitude & Direction Section formula C B C n A B c m Points A, B and C are said to beCollinearif Parallel AND B is a point in common. b A a O

  10. Harder functions Use Chain Rule Differentiations Question Type see Basic Differentiation. Differentiation Further Calculus Integration Trig Integration Question Type see Basic Integration.

  11. Remember ln e-kt = -kt log A + log B = log AB Solving Exp Equations (half – life) Solving Log Equations A log A - log B = log log (A)n = n log A B loga1 = 0 Logs & Exp Question Types Functions & Graphs Straight Line logaa = 1 Graph 2 Graph 1 y = axb y = abx log y log y log y = b log x + log a log y = x log b + log a (0,C) (0,C) Y = mX + C Y = mX + C log x x Y = bX + C Y = (log b) X + C C = log a m = b C = log a m = log b

  12. f(x) = a sinx + b cosx compare to required trigonometric identities f(x) = k sin(x + β) = k sinx cos β + k cosx sin β Changing format Part (a) of question Wave Function Type Questions (Functions & Graphs) Find Max / Mini Value Normally Part (b) of question Solving Equation Normally Part (b) of question UNIT 2 Sketching Wave Function Normally Part (b) of question UNIT 1 3sin(x + 45o) = 1 Arrange into x = ....... S A T C

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