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Higher Maths Question Types You need to learn basic movements Exam questions normally involve two movements Remember order BODMAS Sketching Graphs Steps : 1. Outside function stays the same EXCEPT replace x terms with a ( ) Composite Functions 2. Put inner function in bracket Restrictions : Functions & Graphs TYPE questions (Trig , Quadratics) 1. Denominator NOT ALLOWED to be zero 2. CANNOT take the square root of a negative number f ( x) x3 x 3x OR y x3 x 3x Function Increasing or Decreasing Basic and Format f’(x) Gradient f’(x) > 0 Differentiation TYPE questions (Fractions / Surds /Indices) Stationary Points Optimization Steps : 1. Differentiate 2. f’(x) = 0 (statement) 3. Factorise 4. Nature Table 5. Sub x = to original equation to find y coordinate. f’(x) < 0 Max / Mini in closed intervals Steps : 1. Find Max / Mini points 2. Find end values 3. Decide Max / Mini Points Tangent Line Steps : 1. Differentiate 2. Sub x = into f’(x) to find gradient 3. Use a point on the line and y – b = m(x – a) Steps : 1. Setup recurrence relation 2. State if limit exists 3. Find limit Wordy question Steps : Finding Constants Recurrence Relations TYPE questions (Fractions / Sim Equations) 1. Using information given setup two equations 2. Use simultaneous equation method to find constants f(x) = a(x + b)2 + c 2x2 - 8x + 9 2x2 - 8x 2(x2 - 4x) +9 +9 2(x - 2)2 - 8 + 9 e.g. -2 1 4 5 2 use coefficients to -2 -4 -2 factorise further Factor Theorem if possible !! 1 2 1 0 (x+2) is a factor Remember to since no remainder answer question Completing the square f(x) = ( ) ( ) ( ) Factorising cubic's polynomials f(x) = 2(x - 2)2 + 1 Sketch See Function & Graphs Discriminant b2 – 4ac 3 scenarios > 0 = 0 tangent !!! <0 Quadratic Theory questions (Circle, Function Graphs) Harder discriminant (1 - 2k)x2 - 5kx - 2k > 0 Harder Finding coefficients simultaneous equations Steps 1. Identify a , b and c. 2. Discriminant .... = 0 and factorise. 3. Sketch and identify solution based on question asked. 3 x3 x 3x -1 dx 2 0 Simple Area under the curve 1 Basic Find original equation given 4 Area above & below x-axis Do separately and remember statement for below x-axis Integration TYPE questions Original Equation 1 (Fractions / Surds /Indices) AT = A1 + A2 dy 2 x 1 and passes through (0,1) dx y 2 x 1 dx y x2 x c To find c sub x 0 y 1 1 02 0 c 2 c 1 y x x 1 Steps : Area between two curves 1. For limits make equal to each other. 2. Integrate Top – (bottom) -2 3 4 α 3 cos2x 3cos x 1 0 5 β Substitution and solving 12 Steps 1. Pythagoras Theorem Expansion 2. Expansion With Triangles 3. SOHCAHTOA 4. Solve. Trigonometry TYPE questions Sketching y 3sin 2 x 1 f(x) = sinx (Quadratic, Function Graphs) Sub for cos2x Factorise Solve See A3 sheet given out in Unit 1 For more solving techniques Basic Exact values and radians !!! f(2x) = sin2x 3f(2x) = 3sin2x f(2x) + 1 = 3sin2x + 1 Equation from Graph and solving Steps 1. Write down equation using graph 2. Using balance method to solve See A3 sheet given out in Unit 1 For more solving techniques radius = (a,b) 2 2 g f c centre = (-g,-f) (x - a)2 + (y - b)2 = r2 Equation from graph Intersection points between line and circle 3 possible scenarios x2 + y2 + 2gx + 2fy + c =0 Finding centre and radius from circle equation Circle TYPE questions (Straight Line , Quadratics) Steps 1. Sub line equation y = ... into circle. 2. Discriminant to establish how many points. 3. Factorise for x coordinates and sub into line equation for y coordinates Does circles touch externally or internally ? externally Dc1c2 r1 r2 internally Dc1c2 r2 r1 Is equation a circle ? r>0 Equation of tangent (a,b) Steps 1. Find gradient of centre to point 2. Use m1 x m2 = -1 to find gradient of line 3. Use y – b = m(x - a) b Tail to tail a (b c ) a b a c a b b a a b cos a b Vector Theory Magnitude & Direction Section formula B A B m Points A, B and C are said to be Collinear if A AB kBC Parallel AND B is a point in common. a Angle between two vectors properties C θ b C n c b a n m a c m n m n O d d d (inside)n (outside the bracket) (inside the bracket) dx dx dx d 3 Harder functions ( x 2 x5 )2 2( x3 2 x5 )(3x 2 10 x 4 ) Use Chain Rule dx d sin ax a cos ax dx Differentiations Question Type see Basic Differentiation. d cos ax a sin ax dx Differentiation Further Calculus Integration 1 sin ax dx a cos ax c 1 cos ax dx sin ax c a Trig Integration Question Type see Basic Integration. n 1 ( ax b ) n ( ax b ) dx c a(n 1) 7 (2 x 3) 6 (2 x 3) dx c 14 Remember ln e-kt = -kt log A + log B = log AB log A - log B = log A B n log (A) = n log A loga1 = 0 logaa = 1 Solving Exp Equations (half – life) Solving Log Equations Logs & Exp Question Types Functions & Graphs Straight Line Graph 2 Graph 1 y = axb y = abx log y log y = x log b + log a (0,C) x Y = mX + C Y = (log b) X + C C = log a m = log b log y = b log x + log a Y = mX + C Y = bX + C C = log a m=b log y (0,C) log x 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) Solving Equation Normally Part (b) of question Normally Part (b) of question UNIT 2 3sin(x + 45o) =1 Arrange into x = ....... Find Max / Mini Value Sketching Wave Function S A T C Normally Part (b) of question UNIT 1