Divit’s Cumulative Revision Sheet Laws of Indices y y y a b (y ) y a b Surds a b a b y y y y 1 a 1 y n n y b b ya y a ab b 0 a a a √𝑎 × √𝑏 = √𝑎𝑏 Rationalise the denominator 4 ×√2 √2 ×√2 √𝑎 × 𝑏 = 𝑏√𝑎 = 4√2 2 = 2√2 Converting recurring decimals to fractions 𝐿𝑒𝑡 𝑥 = 𝑡ℎ𝑒 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑜𝑢𝑡 … × 10, 100 𝑜𝑟 1000 (check how many digits recur) Subtract by aligning the decimal points e.g. Upper Bounds & Lower Bounds The lower and upper bounds are the ± margins for rounding to the value (𝑥). 𝑀𝑎𝑟𝑔𝑖𝑛𝑠 = ± ℎ𝑎𝑙𝑓 𝑤ℎ𝑎𝑡 𝑖𝑡 ℎ𝑎𝑠 𝑏𝑒𝑒𝑛 𝑟𝑜𝑢𝑛𝑑𝑒𝑑 𝑡𝑜 𝑙𝑜𝑤𝑒𝑟𝑏𝑜𝑢𝑛𝑑 ≤ 𝑥 < 𝑢𝑝𝑝𝑒𝑟𝑏𝑜𝑢𝑛𝑑 e.g. 𝑥 = 0.2666 … 10𝑥 = 2.6666 10𝑥 − 𝑥 = 2.6666 − 0.2666 9𝑥 = 2.4 24 𝑥= 90 Direct/Inverse Proportion 𝑦 = 𝑘𝑥 2 ↔ y is (directly) proportional to 𝑥 2 𝑘 𝑦 = 2 ↔ y is inversely proportional to 𝑥 2 𝑥 lower and upper bounds of 5 are 4.5 and 5.5 respectively. (4.5 ≤ 5 < 5.5) Histograms – the area of the bars represents the 𝐹𝑟𝑒𝑞 frequency. Frequency Density is 𝐹𝐷 |𝐶𝑊 Cumulative Frequency - plot the upper bound of the class interval and the frequency. Box Plots Estimating Mean from a table Intervals Frequency Midpoint x F Sum of this Sum of this Mean = 𝑠𝑢𝑚 𝑜𝑓 (𝑚𝑖𝑑×𝑓𝑟𝑒𝑞) 𝑠𝑢𝑚 𝑜𝑓 𝑓𝑟𝑒𝑞 Frequency Polygons – plot the midpoint and the frequency/ Pythagoras’ Theorem – for right-angled triangles Square, add and square-root for the longest side Square, subtract and square-root for a shorter side Basic Trigonometry – for right-angled triangles only 𝑂𝑝𝑝 SOH sin 𝜃 | 𝐻𝑦𝑝 *press shift to find angles* CAH 𝐴𝑑𝑗 cos 𝜃 | 𝐻𝑦𝑝 TOA 𝑂𝑝𝑝 tan 𝜃 | 𝐴𝑑𝑗 𝐷 Speed is distance/time 𝑆| 𝑇 𝑀 Density is mass/volume 𝐷| 𝑉 𝜃 Comparing datasets – comment on an average (median or mean) and the spread (IQR or range). Parts of a circle Area of a circle is 𝜋𝑟 2 𝜃 *for sectors × 360 Circumference is 𝜋𝑑 𝜃 *for arcs × 360 Area of a circle is 𝜋𝑟 2 *for sectors × 360 Circumference is 𝜋𝑑 𝜃 *for arcs × 360 Transformations of shapes Rotation about a point, 90° (anti)clockwise or 180° Reflection through a line *look out for 𝑦 = 𝑥 or 𝑦 = −𝑥 𝑥 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 Translation through a vector ( ) 𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 Enlargement from a point, by a scale factor * if fraction: shape gets smaller * if negative: shape inverted through the centre Constructions Perpendicular Bisector Straight Line Geometry Angle Bisector 𝑦 = 𝑚𝑥 + 𝑐 (m is gradient, c is y-intercept) Find c by substituting 𝑥, 𝑦 𝑎𝑛𝑑 𝑚 Gradient = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑦 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑥 = 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 Midpoint = add the x-coordinates and divide by 2 add the y-coordinates and divide by 2 * to construct from a point, start with compass on that point and mark onto the line first. Percentages The Nth Term 𝑑𝑛 + 𝑜 (coefficient of n is the common difference and add the zero’th term) The multiplier always goes with the change Increase = higher multiplier and vice versa HCF and LCM HCF = common prime factors LCM = HCF × leftovers Simple Interest New price = original × multiplier To find an original price, divide by the multiplier Division and Multiplication 0.8 × 0.12 = 0.096 (3 decimal places in total) 0.8 × 100 80 ÷ 4 10 1 = = =3 0.12 × 100 12 ÷ 4 3 3 Compound interest New amount = original × 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟 𝑦𝑒𝑎𝑟𝑠 Quadratic Equations To factorise, check the sum-product 𝑥 2 − 5𝑥 + 6 sum = -5 and product = 6 (𝑥 − 3)(𝑥 − 2) Inequalities 0≤𝑥<4 For quadratics with a co-efficient of 𝑥 2 3𝑥 2 + 8𝑥 − 3 sum = 8 and product = -9 (3×-3) 3𝑥 2 + 9𝑥 − 1𝑥 − 3 split the middle term 3𝑥(𝑥 + 3) − 1(𝑥 + 3) factorise the first 2 and last 2 (3𝑥 − 1)(𝑥 + 3) factorise again The difference of two squares 𝑥 2 − 64 = (𝑥 + 8)(𝑥 − 8) 4𝑥 2 − 9𝑦 2 = (2𝑥 + 3𝑦)(2𝑥 − 3𝑦) 𝑥>2 𝑥≥2 To draw a region, use a table of values to draw the straight lines. When you multiply or divide both sides of an inequality by a negative, flip the sign. 𝑣 = 𝜋𝑟 2 ℎ 𝑆𝐴 = 2𝜋𝑟ℎ 4 𝑣 = 3 𝜋𝑟 3 𝑆𝐴 = 4𝜋𝑟 2 Imperial to Metric Conversions Imperial Metric Measurements 1 inch (1”) 1 foot (1 ft) 5 miles (5 mi.) Measurements 2.5 centimeter (2.5 cm) 30 centimeters (30 cm) 8 kilometers (8 km) Weights 1 pound (1 lb) 2.2 pounds (2.2 lbs) 1 stone (1 st.) Weights 450 grams (450 g) 1 kilogram (1 kg) 6.3 kilograms (6.3 kg) Capacity 1 pint (1 pt) 1 gallon (1 gal) 1.75 pints (1 pt) Capacity 570 millilitres (570 ml) 4.5 litres (4.5 l) 1 litre (1 l) Prime Factor Decomposition Prime Factor Decomposition is the breakdown of positive integers into prime numbers. This is done through many steps of factorisation. e.g. 24 = 4 × 6 = 2 × 2 × 2 × 3 = 23 × 3 Stem & Leaf Diagrams Stem and Leaf Diagrams are used to organise data sets by separating the place values in the number. All Stem and Leaf Diagrams MUST have a key. e.g. 2 | 1 = 21 Mean – The mean is the total of all the values, divided by the number of values Median – The median is the middle number in a list of numbers ordered from lowest to highest Mode – The mode is the value that appears most often in a set of data Range – The range is the difference between the lowest value and the highest value Correlation refers to the dependance of 2 different variables when compared together. e.g. if x increases y increases; this is positive correlation Ratios Simultaneous Equations Simplification – Divide both parts by the same number until it cant be further divided Bring 1 variable to a side in 1 equation and then plug into second equation to solve. E.g. 4𝑥 − 2𝑦 = 4 5𝑥 + 3𝑦 = 16 Using Ratios – To find proportional values divide the number by the corresponding part of the ratio. E.g. 9 : ? using the ratio 3 : 4 9 = 3, 4 × 3 = 12 3 Ratios <-> Fractions – To convert from a Ratio to a Fraction add all the parts together and then use the part in the question as the numerator. 3 E.g. 3 ∶ 5, 3 + 5 = 8 ∴ 3 ∶ 5 → 8 2𝑦 = 4𝑥 − 4 𝑦 = 2𝑥 − 2 5𝑥 + 3(2𝑥 − 2) = 16 11𝑥 − 6 = 16 11𝑥 = 22 𝑥=2 There is the subtraction method aswell, however this is superior. Angle Rules Sum of interior angles in polygon = (𝑛 − 2) × 180 360 One exterior angle of a regular polygon = 𝑛 Properties of Parallelograms and Rhombuses In a parallelogram all the opposite sides and angles are the same value. In a rhombus all sides are the same value and opposite angles are the same value. Kite Rules - The diagonals of the kite intersect at 90° - The lines BC and CD are equal and BA and AD are also equal - The angle B = D Proof of Exterior Angle of a Triangle Angle x is an exterior angle of the triangle: The exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices. In other words, x = a + b in the diagram. Proof: The angles in the triangle add up to 180 degrees. So a + b + y = 180. The angles on a straight line add up to 180 degrees. So x + y = 180. Therefore y = 180 − x. Putting this into the first equation gives us: a + b + 180 − x = 180. Therefore a + b = x after rearranging. This is what we wanted to prove. Proof of Angles in a Triangle Construct a line parallel to BC and label the new angles x and y. Since angle x and c are alternate they are equal and sicne angle y and b are alternate they are equal. Since angles on a straight line equal 180°; 𝑥 + 𝑦 + 𝑧 = 180°. Therefore angles in a triangle equal 180° Bearings Bearings are calculated from north. If a bearing is less than 100° then it should have a 0 at the front. Transformations There are 4 types of translations. Enlargements and Dilations use scale factors to transform the shape whereas translations use vectors. e.g. means translate the shape 4 squares to the right and 3 squares down. Scale Factors In this image the blue triangle is enlarged by a scale factor of -2 around the point P to make the pink triangle.
