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• • Natural numbers: • Used for counting purposes • Made up off all possible rational and irrational numbers Integer: • • • Prime numbers: • Divisible only by itself and one • 1 is not a prime number Rational numbers: • • A whole number Can be written as a fraction Irrational numbers: • Cannot be written as a fraction e.g. 𝜋 Page 2 • • Highest Common Factor and Lowest Common Multiple: • HCF = product of common factors of x and y • LCM = product of all items in Venn diagram Prime Factorization: finding which prime numbers multiply together to make the original number Page 3 • • Limits of accuracy: • The degree of rounding of a number • E.g. 2.1 to 1 d.p. 2.05 ≤ 𝑥 < 2.15 Standard form: • 104 = 10000 10−1 = 0.1 • 103 = 1000 10−2 = 0.01 • 102 = 100 10−3 = 0.001 • 101 = 10 10−4 = 0.0001 • 100 = 1 10−5 = 0.00001 Page 4 • • Ratio • Used to describe a fraction • E.g. 3 : 1 Foreign exchange • Money changed from one currency to another using proportion • E.g. Convert $22.50 to Dinars $1 : 0.30KD $22.50 : 6.75KD • Map scales • Using proportion to work out map scales 1km = 1000m 1m = 100cm 1cm = 10mm Page 5 • • Direct variation: • 𝑦 is proportional to 𝑥 • 𝑦∝𝑥 𝑦 = 𝑘𝑥 Inverse variation: • 𝑦 is inversely proportional to 𝑥 • 𝑦∝ 1 𝑥 𝑦= 𝑘 𝑥 Page 6 • • Percentage: • Convenient way of expressing fractions • Percent means per 100 Percentage increase or decrease: • • • 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 = 𝐴𝑐𝑡𝑢𝑎𝑙 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐴𝑚𝑜𝑢𝑛𝑡 Simple interest: 𝑃𝑅𝑇 100 • 𝐼= • 𝑃 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑅 = 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑇 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑇𝑖𝑚𝑒 𝑅 = 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑛 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑇𝑖𝑚𝑒 Compound interest: 𝑅 𝑛 100 • 𝐴 = 𝑃 1+ • 𝑃 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 Page 7 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑇𝑖𝑚𝑒 • 𝑆𝑝𝑒𝑒𝑑 = • 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑝𝑒𝑒𝑑 = • Units of speed: km/hr m/s • Units of distance: km m • Units of time: hr sec • 𝑘𝑚/ℎ𝑟 × 5 18 = 𝑚/𝑠𝑒𝑐 • 𝑚/𝑠𝑒𝑐 × 18 5 = 𝑘𝑚/ℎ𝑟 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒 Page 8 • • Function notation: • 𝑓: 𝑥 → 2𝑥 − 1 • Function 𝑓 such that 𝑥 maps onto 2𝑥 − 1 Composite function • Given two functions 𝑓 𝑥 and 𝑔 𝑥 , the composite function of 𝑓 and 𝑔 is the function which maps 𝑥 onto 𝑓 𝑔 𝑥 • 𝑓 2 • Substitute 𝑥 = 2 and solve for 𝑓 𝑥 • 𝑓𝑔(𝑥) • 𝑓 −1 (𝑥) • • Substitute 𝑥 = 𝑔 𝑥 Let 𝑦 = 𝑓(𝑥) and make 𝑥 the subject Page 9 Page 10 • General equation • • 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 Solve quadratics by • Trinomial factorization • Quadratic formula • 𝑥= • −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 IMPORTANT! • When question says “give your answer to two decimal places”, USE FORMULA! Page 11 • Gradient of a Straight Line: 𝑦 −𝑦 • 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 2 1 • Equation of Line: 𝑥2 −𝑥1 • 𝑦 = 𝑚𝑥 + 𝑐 • Find the gradient, 𝑚 • Find the 𝑦-intercept, 𝑐 • Midpoint of Graph: • • 𝑥1 +𝑥2 𝑦1 +𝑦2 , 2 2 Length between two points: • 𝑥1 − 𝑥2 2 + 𝑦1 − 𝑦2 2 Page 12 𝑓 𝑥 =1 𝑓 𝑥 =𝑥 𝑓 𝑥 = 𝑥3 𝑓 𝑥 = 1 𝑥 𝑓 𝑥 = 𝑥2 𝑓 𝑥 = 1 𝑥2 Page 13 • From O to A : Uniform speed • From B to C : Uniform speed (return journey) • From A to B : Stationery (speed = 0) • Gradient = speed Page 14 • From O to A : Uniform speed • From A to B : Constant speed (acceleration = 0) • From B to C : Uniform deceleration / retardation • Area under a graph = distance travelled. • Gradient = acceleration. • If the acceleration is negative, it is called deceleration or retardation. (The moving body is slowing down.) Page 15 • Sum of angles at a point =360 • Angles on a straight line = 180 • Sum of angles in a triangle =180 • For regular polygon • • 360 𝑛 • External angles = • Internal angles = 180 − Corresponding angles • Alternate angles • Co-interior angles • Exterior angle=sum of interior opposite ∠ 360 𝑛 For irregular polygon: • Sum of exterior angles =360 • Sum of interior angles =180(n-2) Vertically opposite angles • Page 16 • A line of symmetry divides a two-dimensional shape into two congruent (identical) shapes. • A plane of symmetry divides a three-dimensional shape into two congruent solid shapes. • The number of times shape fits its outline during a complete revolution is called the order of rotational symmetry. Shape Number of Lines of Symmetry Order of Rotational Symmetry Square 4 4 Rectangle 2 2 Parallelogram 0 2 Rhombus 2 2 Trapezium 0 1 Kite 1 1 Equilateral triangle 3 3 Regular hexagon 6 6 Page 17 • Rectangle: Opposite sides parallel and equal, all angles 90°, diagonals bisect each other. • Parallelogram : Opposite sides parallel and equal, opposite angles equal, diagonals bisect each other • Rhombus: A parallelogram with all sides equal, opposite angles equal, diagonals bisect each other • Trapezium: One pair of sides parallel • Kite: Two pairs of adjacent sides equal, diagonals meet at right angles bisecting one of them Page 18 = universal set ∉ = does not belongs to ∪ (union) = all the elements ⊆ = Subset ∩ (intersection) = common elements 𝐴′ = compliment of A Ø or { } = empty set n(A) = the number of elements in A. ∈ = belongs to 𝐴 ∩ 𝐵 is shaded 𝐴 ∪ 𝐵 is shaded ⊂ ‘is a subset of’ 𝑏∈𝑋 = {a, b, c, d, e} A’ is shaded Page 19 Angle at centre = twice Angle subtended by same arc Angles in semicircle angle on circumference at circumference are equal are 90° Opposite angles in a cyclic quadrilateral = 180° Tangents from one point are equal Alternate segment ∠ between tangent and radius is 90° theorem Page 20 • Right angled triangles: • sin 𝑥 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 • cos 𝑥 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 • tan 𝑥 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 • For any other triangle: • Sine rule: • 𝑎 sin 𝑎 • = 𝑏 sin 𝑏 = 𝑐 sin 𝑐 Cosine rule • To find the angle given 3 sides • cos 𝑎 = One pair of information needed • 𝑏 2 +𝑐 2 −𝑎2 2𝑏𝑐 To find side given angle and two sides • 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝑎 Page 21 • Pythagoras theorem • To find hypotenuse • 𝑎2 + 𝑏 2 = 𝑐 2 • To find one of the shorter sides • 𝑎2 = 𝑐 2 − 𝑏 2 • 𝑏 2 = 𝑐 2 − 𝑎2 • Angle of elevation: • • Angle of depression: • • Angle above the horizontal line. Angle below the horizontal line. 1 2 Area of a triangle: 𝑎𝑏 sin 𝑐 Page 22 • • The bearing of a point B from another point A is: • An angle measured from the north at A. • In a clockwise direction. • Written as three-figure number (i.e. from 000 ° to 360°) Eg: The bearing of B from A is 050° Page 23 • Area: • Parallelogram = 𝑏 × ℎ • • 1 Triangle= 𝑏 × ℎ 2 1 Trapezium= 𝑎 + 2 2 Circle= 𝜋𝑟 • Sector= 𝜋𝑟 2 × • • OR 𝑎𝑏 sin 𝜃 𝑏 ℎ 𝜃 360 Volume and surface area: • • • Cylinder Sphere • 𝐶𝑢𝑟𝑣𝑒𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 2𝜋𝑟ℎ • 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 4𝜋𝑟 2 • 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ • 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3 • Cone • 𝐶𝑢𝑟𝑣𝑒𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 𝜋𝑟𝑙 • 𝑉𝑜𝑙𝑢𝑚𝑒 = 4 3 1 (𝜋𝑟 2 ℎ) 3 Hemisphere • 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 2𝜋𝑟 2 2 3 • 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3 Page 24 • Volume: • Capacity and Mass: • Connecting volume and capacity: • • 1𝑚𝑙 = 1𝑐𝑚3 • 1𝑘𝑙 = 1𝑚3 Density = 𝑀𝑎𝑠𝑠 𝑉𝑜𝑙𝑢𝑚𝑒 Page 25 • 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛 • 𝑎 𝑚 𝑏 • 𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛 • 𝑛 • 𝑎𝑚 𝑛 = 𝑎𝑚𝑛 • 𝑎0 = 1 • 𝑎−𝑛 = 1 𝑎𝑛 • 𝑎×𝑏 𝑚 • 𝑚 𝑎 𝑎𝑚 𝑏𝑚 = 𝑚 = 𝑎𝑛 • 𝑎× 𝑏= 𝑎×𝑏 • 𝑎 𝑏 • = 𝑎 2 𝑎 𝑏 =𝑎 = 𝑎𝑚 × 𝑏 𝑚 Exponential equations: • Equations involving unknown indices Page 26 • Condition 1: Given distance from a point • Condition 2: Given distance from a straight line • Condition 3: Equi-distant from two given points • Condition 4: Equi-distant from two intersecting lines Page 27 • A vector quantity has both magnitude and direction. • E.g. Vectors a and b represented by the line segments can be added using the parallelogram rule or the nose-to-tail method. • • Multiplication by a scalar: • A scalar quantity has a magnitude but no direction • The negative sign reverses the direction of the vector Column vector: • • • Top number is the horizontal component and bottom number is the vertical component Parallel vectors: • Vectors are parallel if they have the same direction • In general the vector 𝑘 𝑎 𝑏 is parallel to 𝑎 𝑏 Modulus of a vector: • In general, if 𝑥 = 𝑚 𝑛 , 𝑥 = (𝑚2 + 𝑛2 Page 28 • • • • Addition: 𝑝 𝑞 𝑎+𝑝 𝑏+𝑞 𝑎 𝑏 • + = 𝑟 𝑠 𝑐 𝑑 𝑐+𝑟 𝑑+𝑠 Multiplication by scalar 𝑎 𝑏 𝑘𝑎 𝑘𝑏 • 𝑘 = 𝑐 𝑑 𝑘𝑐 𝑘𝑑 Multiplication by vector: 𝑎𝑝 + 𝑏𝑟 𝑎𝑞 + 𝑏𝑠 𝑝 𝑞 𝑎 𝑏 • × = 𝑟 𝑠 𝑐𝑝 + 𝑑𝑟 𝑐𝑞 + 𝑑𝑠 𝑐 𝑑 • You can only multiply if no. of columns in left equals to no. of rows in right Determinant: • • Determinant = leading diagonal – secondary diagonal 𝑎 𝑏 • A = 𝐴 = 𝑎𝑑 − (𝑏𝑐) 𝑐 𝑑 Inverse: • • 1 To work out inverse, switch leading diagonal, negate secondary diagonal, multiply by 𝑎 1 𝑎 𝑏 𝑑 −𝑏 A = 𝐴−1 = (𝑎𝑑−𝑏𝑐) −𝑐 𝑐 𝑑 𝑎 Page 29 • Reflection: • • When describing a reflection, the position of the mirror line is essential. Rotation: • To describe a rotation, the centre of rotation, the angle of rotation and the direction of rotation are required. • • Translation: • • A clockwise rotation is negative and an anticlockwise rotation is positive. When describing a translation it is necessary to give the translation vector Enlargement: • To describe an enlargement, state the scale factor, K and the centre of enlargement • 𝑆𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 = • If K > 0, both object and image lie on same side of the centre of enlargement. • If K < 0, object and image lie on opposite side of the centre of enlargement. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 = 𝐾 2 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡 Page 30 • Shear: • To describe a shear, state; the shear factor, the invariant line and the direction of the shear • 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑚𝑜𝑣𝑒𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑠ℎ𝑒𝑎𝑟 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑓𝑖𝑥𝑒𝑑 𝑙𝑖𝑛𝑒 • 𝑆ℎ𝑒𝑎𝑟 𝑓𝑎𝑐𝑡𝑜𝑟 = • 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡 = 𝑎 𝑏 Stretch: • To describe a stretch, state; the stretch factor, the invariant line and the direction of the stretch 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶 ′ 𝑓𝑟𝑜𝑚 𝐴𝐵 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝐶 𝑓𝑟𝑜𝑚 𝐴𝐵 • 𝑆𝑡𝑟𝑒𝑡𝑐ℎ 𝑓𝑎𝑐𝑡𝑜𝑟 = • 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒 = 𝑆𝑡𝑟𝑒𝑡𝑐ℎ 𝑓𝑎𝑐𝑡𝑜𝑟 × 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡 Page 31 • • Reflection: 1 0 • 0 −1 • −1 0 • 0 1 1 0 • 0 −1 0 1 Reflection in the 𝑥 − 𝑎𝑥𝑖𝑠 Reflection in the 𝑦 − 𝑎𝑥𝑖𝑠 Reflection in the line 𝑦 = 𝑥 −1 0 Enlargement: 𝑘 0 • 0 𝑘 Reflection in the line 𝑦 = −𝑥 where k=scale factor and centre of enlargement = (0,0) Page 32 • • Rotation: 0 −1 • 1 0 • 0 −1 1 0 • −1 0 0 −1 Stretch: 1 0 • 0 𝑘 • 𝑘 0 0 1 Rotation 90° anticlockwise, centre (0,0) Rotation 90° clockwise, centre (0,0) Rotation 180° clockwise/anticlockwise, centre (0,0) Stretch factor k, invariant line x-axis & parallel to y-axis Stretch factor k, invariant line y-axis & parallel to y-axis Page 33 • Shear: 1 • 0 𝑘 1 Shear factor k, invariant line x-axis & parallel to x-axis 1 𝑘 0 1 Shear factor k, invariant line y-axis & parallel to y-axis • Page 34 • Histograms: • A histogram displays the frequency of either continuous or grouped discrete data in the form of bars. • The bars are joined together. • The bars can be of varying width. • The frequency of the data is represented by the area of the bar and not the height. • When class intervals are different it is the area of the bar which represents the frequency not the • height • Instead of frequency being plotted on the vertical axis, frequency density is plotted. • Class width = Interval • Frequency density = Height • 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝐶𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ × 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 Page 35 • Mean: • • • Median: • The middle value when the data has been written in ascending or descending order • Odd no. of values • Even no. of (add two values divide by 2) Most frequently occurring value Range: • • 5+1 = 3𝑟𝑑 𝑣𝑎𝑙𝑢𝑒 2 6+1 values = 3.5𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 2 Mode: • • 𝑆𝑢𝑚 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 Difference between highest and lowest values Estimated mean of grouped data: • Work out midpoints of each group and multiply by frequency • Divide by number of values Page 36 • Cumulative frequency is the total frequency up to a given point. • Inter-quartile range = 𝑢𝑝𝑝𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 − 𝑙𝑜𝑤𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 Page 37 • Probability is the study of chance, or the likelihood of an event happening. • Probability of an event = • If probability = 0, the event is impossible and if probability =1, the event is certain to happen • All probabilities lie between 0 and 1. • Exclusive events: 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 • Two events are exclusive if they cannot occur at the same time. • The OR Rule: • For exclusive events A and B • p(A or B) = p(A) + p(B) • Independent events: • Two events are independent if occurrence of one is unaffected by occurrence of other. • The AND Rule: • p(A and B) = p(A) × p(B) Page 38 • Linear sequences: • • Find common difference e.g. 3 then multiply by 𝑛 and work out what needs to be added Quadratic sequences: • Format: 𝑎𝑛2 + 𝑏𝑛 + 𝑐 𝑎+𝑏+𝑐 = 3𝑎 + 𝑏 = 2𝑎 = • • Work out the values and then place into formula to work out nth term formula Geometric progression: • Sequence where term has been multiplied by a constant to form next term • 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑜𝑓 𝐺. 𝑃. = 𝑎𝑟 (𝑛−1) • a = 1st term r = common difference Page 39