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Maths GCSE Linear HIGHER – Things to Remember Remember that the information below is always provided on the formula sheet at the start of your exam paper In addition to these formulae, you also need to learn other facts and formulae When you first open your exam paper, you should write down all the key formulae and information that you have revised for the exam before you start answering questions The following pages list the key information you need to remember…… Converting between fractions, decimals and percentages: F → D → P ÷ x 100 e.g. 2 = 2 ÷ 5 = 0.4, and 0.4 x 100 = 40% 5 5 of 24 = 24 ÷ 8 x 5 = 3 x 5 = 15 8 30% of 64 = 3 lots of 10% of 60 = 3 x 6.4 = 19.2 8 as a percentage of 20 = 8_ = _40 = 40% 20 100 Reverse Percentages After an increase of 20%, the new amount is 120% of the original amount After a decrease of 10%, the new amount is 90% of the original amount Ratio To split £160 in the ratio 3 : 5, you need to split it into 3 + 5 = 8 parts…….. £160 ÷ 8 = £20 And then find what 3 parts and 5 parts are……… 3 x £20 = £60 5 x £20 = £100 Standard Form Numbers 3 x 107 = 30 000 000 460 000 = 4.6 x 105 2.4 x 10-5 = 0.000024 0.000000081 = 8.1 x 10-8 nth term of a sequence the sequence 3, 11, 19, 27……. has a common difference of 8, so the nth term must include 8n but the sequence for 8n is 8, 16, 24, 32….. So the sequence 3, 11, 19, 27……. must be 8n – 5 Expressions, Identities Equations Formulae and An Expression has letters and numbers, but no equal sign An Equation has an equal sign and usually just one letter A Formula has an equal sign and will have more than one letter An Identity will have the identity sign (≡) instead of an equal sign Solving Equations and Re-arranging Formulae Remember to always do the same to all terms on both sides of the equal sign Try to eliminate negatives, brackets and fractions first The equation of a straight line graph y = mx + c m is the gradient of the graph c is the y-intercept (where the line crosses the y axis) Expanding and Factorising To expand a bracket, multiply all the terms inside the bracket e.g. 3(x + 7) = 3x + 21 x(x – 4) = x2 – 4x Factorising is the opposite of expanding e.g. 6x – 15 = 3(2x – 5) x2 + 8x = x(x + 8) To expand a double bracket, use the FOIL method e.g. (x +3)(x + 7) = x2 + 7x + 3x + 21 = x2 + 10x + 21 To factorise a quadratic expression such as x2 + 11x + 24, you need two numbers that multiply to give 24 and add to give 11, i.e. 3 and 8, so: x2 + 11x + 24 = (x +3)(x + 8) Solving Quadratic Equations If a quadratic equation factorises, it can be solved like this: x2 + 3x - 28 = 0 (x + 7) (x - 4) = 0 x = -7 or x = 4 If a quadratic equation of the form ax2 + bx + c = 0 cannot be factorised, it can be solved by using the quadratic formula: A quadratic equation can also be solved by using the method of completing the square, when the quadratic equation is rewritten in the form (x + p)2 + q = 0 [from this form of the equation we know that the minimum point on the quadratic graph is (-p, q)] Laws of Indices ax x a y = a x + y e.g. 35 x 37 = 312 ax ÷ a y = a x – y e.g. x8 ÷ x3 = x5 a-x = 1 ax e.g. 3-2 = 1 32 a½ = √a e.g. 6½ = √6 a⅔ = (3√a)2 e.g. 8⅔ = (3√8)2 = 22 = 4 Surds √a x √b = √(ab) e.g. √3 x √5 = √15 √a ÷ √b = √(a/b) e.g. √18 ÷ √3 = √6 When finding MEAN FROM A TABLE, remember to MULTIPLY the data by the frequency (then sum of data ÷ sum of frequencies) *for GROUPED DATA, find the MID-POINTS first Remember to use the Handling Data cycle when describing how to test a hypothesis: 1. state what data to collect and how to collect it 2. state what TYPE of graph or average to use 3. state that you will compare your results to the hypothesis PROBABILITY OR means ADD the probabilities (OR → +) AND means MULTIPLY the probabilities (AND → x) CUMULATIVE FREQUENCY (Curves and Box Plots) Median = ½ (read off from halfway up the frequency axis) Lower Quartile (LQ) = ¼ (read off from ¼ of the way up the frequency axis) Upper Quartile (UQ) = ¾ (read off from ¾ of the way up the frequency axis) Interquartile Range (IQR) = UQ – LQ When asked to COMPARE data or graphs, try to compare an AVERAGE (MEAN OR MEDIAN) and the INTERQUARTILE RANGE HISTOGRAMS – remember that the AREA of the bar represents the frequency, so: HEIGHT OF BAR = FREQUENCY ÷ CLASS WIDTH STRATIFIED SAMPLING Take a sample in proportion to the different group sizes Group size Total population size x sample size Perimeter and Area Area of a Triangle = (base x height) ÷ 2 Area of a Circle = π x radius x radius or A = π r² Circumference of a Circle = π x diameter or C = πD Parts of a Circle Arc Length (a fraction of the circumference) Length of an Arc = angle of arc x πD 3600 Sector Area (a fraction of the area of a circle) Area of a Sector = angle of sector x πr2 3600 Angles Don’t forget: Angles on a straight line add to 1800 Angles at a point add to 3600 Angles in a triangle add to 1800 Angles in a quadrilateral add to 3600 Angles on parallel lines: Angles in Polygons Bearings – remember: North is 00 Measure angle clockwise You need to learn these imperial to metric conversions: 5 miles = 8 km 1 inch = 2.5 cm (so, 4 inches = 10 cm and 12 inches = 30 cm – a large ruler and 40 inches = 100 cm, or 1 m) 2.2 pounds = 1 kg (or 1 mile = 1.6 km) (or 2 lb ≈ 1 kg) 1 gallon = 4.5 litres (1 gallon = 8 pints) 1.75 pints = 1 litre ("a litre of water is a pint and three quarters") Speed, Distance and Time Density, Mass and Volume Transforming Shapes (TERRy) Translate – describe how far across and how far up or down(use a vector) Enlarge – describe scale factor and centre of enlargement Rotate – describe angle and direction of rotation and rotation Reflect – describe the equation of the mirror line Y centre of Circle Theorems In a Right Angled Triangle: Pythagoras’ Theorem (to find a missing side) c2 = a 2 + b 2 (c is the hypotenuse, the longest side) a2 = c2 – b2 b2 = c2 – a2 (a and b are the shorter sides) SOH CAH TOA (to find a side or an angle) Sin θ = opposite hypotenuse Cos θ = adjacent hypotenuse Tan θ = opposite adjacent In NON Right Angled Triangles: C Use the Sine Rule or Cosine Rule to find missing sides or angles in NON Right Angled Triangles A The Sine rule a b c sin A sin B sin C is on the exam paper formula sheet (this can be used to find missing sides or angles) The Cosine rule to find a side, a2 = b2 + c2 – 2bc cos A is on the formula sheet, but you need to learn the find an angle: Cosine Rule to cos A = b2 + c2 – a2 2bc b a c B