Emaths-Formula summary

Algebra:  𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐  (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2  (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏2  𝑎2 + 𝑏2 = (𝑎 + 𝑏)2 − 2𝑎𝑏  𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Quadratic Equations: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 Solve by formula: 𝑥 = −𝑏±√𝑏2−4𝑎𝑐 2𝑎 Complete the square: Make sure a = 1 𝑎𝑥2 + 𝑏𝑥 + ( 𝑏 2)2 − ( 𝑏 2)2 + 𝑐 = 0 Indices:  𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛  𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛  (𝑎𝑚)𝑛 = 𝑎𝑚𝑛  𝑎0 = 1  𝑎−𝑛 = 1 𝑎𝑛  (𝑎 × 𝑏)𝑚 = 𝑎𝑚 × 𝑏𝑚  ( 𝑎 𝑏)𝑚 = 𝑎𝑚 𝑏𝑚  ( √𝑎 𝑛 )𝑚 = 𝑎 𝑚 𝑛  √𝑎 × √𝑏 = √𝑎 × 𝑏  √ 𝑎 𝑏 = √𝑎 √𝑏  (√𝑎)2 = 𝑎 Variation: 𝑦 is proportional to 𝑥: 𝑦 = 𝑘𝑥 𝑦 is inversely proportional to 𝑥: 𝑦 = 𝑘 𝑥 Simple Interest - To find interest: 𝑖 = 𝑃𝑅𝑇 100 Compound Interest: 𝐴 = 𝑃(1 + 𝑟 100)𝑛 Conversion of units: km/hr x 5 18 = m/sec ; m/sec x 18 5 = km/hr  Pythagoras Theorem: 𝑐𝟐 = 𝑎𝟐 + 𝑏𝟐 Trigonometry: TOA CAH SOH Cosine Rule: Length of a side: 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴 Find an angle when all 3 sides are given: cos 𝐴 = 𝑏2 + 𝑐2 − 𝑎2 2𝑏𝑐 Co-ordinate Geometry: Eqn. of a straight line: 𝑦 = 𝑚𝑥 + 𝑐 Gradient of a straight line: 𝑚 = 𝑦2 − 𝑦1 𝑥2− 𝑥1 Midpoint: 𝑀 = ( 𝑥1+ 𝑥2 2 , 𝑦1 + 𝑦2 2 ) Distance between two points: 𝐴𝐵 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Matrices: Addition (𝑎 𝑏 𝑐 𝑑) + (𝑝 𝑞 𝑟 𝑠) = (𝑎 + 𝑝 𝑏 + 𝑞 𝑐 + 𝑟 𝑑 + 𝑠 ) Subtraction (𝑎 𝑏 𝑐 𝑑) − (𝑝 𝑞 𝑟 𝑠) = (𝑎 − 𝑝 𝑏 − 𝑞 𝑐 − 𝑟 𝑑 − 𝑠 ) Multiplication (𝑎 𝑏 𝑐 𝑑) × (𝑝 𝑞 𝑟 𝑠) = (𝑎𝑝 + 𝑏𝑟 𝑎𝑞 + 𝑏𝑠 𝑐𝑝 + 𝑑𝑟 𝑐𝑞 + 𝑑𝑠 ) 𝑘 × (𝑎 𝑏 𝑐 𝑑) = (𝑘𝑎 𝑘𝑏 𝑘𝑐 𝑘𝑑 ) Vectors: Parallelogram law of addition: 𝑂𝐵⃗⃗⃗⃗⃗ + 𝑂𝐴⃗⃗⃗⃗⃗ = 𝑂𝐶⃗⃗⃗⃗⃗ Polygons: Sum of Exterior angles = 360° One Exterior angle = 360° 𝑛 Sum of interior angles = (𝑛 − 2) × 180° Types of polygons: No. of sides 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 decagon Angle Properties of Triangle: Sum of all angles = 180° Exterior angle (x) = Sum of opposite interior angles (𝑎 + 𝑏) Properties of Circles: -------------------------------------------- -------------------------------------------- -------------------------------------------- -------------------------------------------- -------------------------------------------- Chord of a Circle: -------------------------------------------- Tangent to a Circle: -------------------------------------------- -------------------------------------------- Alternate Segment Theorem: ∠QAB = ∠ACB (p = q) 𝑇𝑎𝑛 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 = 𝑂 𝐴 𝐶𝑜𝑠 𝜃 = 𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒 = 𝐴 𝐻 𝑆𝑖𝑛 𝜃 = 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐻𝑦𝑝𝑜𝑡ℎ𝑒𝑛𝑢𝑠𝑒 = 𝑂 𝐻 Sine Rule: 𝑎 sin 𝐴 = 𝑏 sin 𝐵 = 𝑐 sin 𝐶 Triangular law of addition: 𝑂𝐴⃗⃗⃗⃗⃗ + 𝐴𝐶⃗⃗⃗⃗⃗ = 𝑂𝐶⃗⃗⃗⃗⃗ Angle at centre (2p) is twice angle at circumference (p) Angle in the same segment of a circle are equal Angle in a semicircle is a right angle. Opposite angles of a quadrilateral add up to 180o ∠A + ∠C = 180o ∠B + ∠D = 180o Exterior angle of a quadrilateral equals to interior opposite angle (∠b = ∠p) A line joining two points on a circle is called a chord (Line AB). Angle between tangent and radius drawn to contact ∠ABO or ∠OBC = 90° Any point outside of a circle, two tangents drawn to the circle = equal length (TA = TB) ∠QAB = ∠ACB (p = q) TC2 = AC x BC

Similar Plane Figures - Figures are similar only if their corresponding sides are proportional - their corresponding angles are equal Similar Solid Figures Solids are similar if their corresponding linear dimensions are proportional. Congruent Figures Congruent figures are exactly the same size and shape. 2 triangles are congruent if they satisfy any of the following: a. SSS property: All 3 sides of one triangle are equal to the corresponding sides of the other triangle. b. SAS property: 2 given sides and a given angle of one triangle are equal to the corresponding sides and angle of the other triangle. c. AAS property: 2 given angles and a given side of one triangle are equal to the corresponding angles and side of the other triangle. d. RHS property: The hypothenuse and a given side of a right-angled triangle are equal to the hypothenuse and the corresponding side of the other right-angled triangle. Graphs of functions Positive Negative y = mx + c y = -mx + c 𝒚 = 𝑎𝑥𝟐 + 𝒃𝒙 + 𝒄 𝒚 = −𝑎𝑥𝟐 + 𝒃𝒙 + 𝒄 𝒚 = 𝑎𝑥𝟑 𝒚 = −𝑎𝑥𝟑 𝒚 = 𝑎𝑥−𝟏 𝒚 = −𝑎𝑥−𝟏 𝒚 = 𝑎𝑥−𝟐 𝒚 = −𝑎𝑥−𝟐 𝒚 = 𝒌𝑎𝒙 𝒚 = −𝑘𝑎𝒙 a>1 a>1 Graphs from complete the square 𝒚 = (𝑥 + 𝑎)𝟐 + 𝒉 𝒚 = −(𝑥 + 𝑎)𝟐 + 𝒉 min 𝑝𝑜𝑖𝑛𝑡 (−𝑎, ℎ) max 𝑝𝑜𝑖𝑛𝑡 (−𝑎, ℎ)

Area & Perimeter: Figure Area Perimeter/ Circumference Rectangle 𝑙 × 𝑏 2 (𝑙 + 𝑏) Square 𝑎 × 𝑎 4 × 𝑎 Parallelogram 𝑏 × ℎ 2(𝑎 + 𝑏) Triangle 1 2 × 𝑏 × ℎ Or 1 2 𝑎𝑏 sin 𝐶 𝑎 + 𝑏 + 𝑐 Trapezium 1 2 (𝑎 + 𝑏)ℎ Sum of all sides Circle 𝜋𝑟2 2𝜋𝑟 Semicircle 1 2 𝜋𝑟2 1 2 𝜋𝑑 + 𝑑 Sector 𝜋𝑟2 × 𝜃 360 Length of an arc = 2𝜋𝑟 × 𝜃 360 Surface Area & Volume: Figure Surface area Volume Cylinder Curved surface area = 2𝜋𝑟ℎ Total surface area = 2𝜋𝑟(ℎ + 𝑟) 𝜋𝑟2ℎ Cone Curved surface area = 𝜋𝑟𝑙 Where 𝑙 = √(𝑟2 + ℎ2) Total surface area = 𝜋𝑟(𝑙 + 𝑟) 1 3 𝜋𝑟2 Sphere 4𝜋𝑟2 4 3 𝜋𝑟3 Pyramid Base area + Area of shapes in the sides 1 3 × base area × perpendicular height Cubiod 2(𝑙𝑏 + 𝑏ℎ + 𝑙ℎ) 𝑙 × 𝑏 × ℎ Cube 6𝑙2 𝑙3 Hemisphere 2𝜋𝑟2 2 3 𝜋𝑟3 Sets: Statistics: Mean = ∑ 𝑓𝑥 ∑ 𝑓 = average Mode of a series of number = number which occurs most frequently Median = arrange series of numbers in ascending order and then choosing the number in the middle. Probability: Prob. Of an event = 𝑛𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑜.𝑜𝑓 𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 Exclusive event (events cannot occur at the same time) For exclusive event A & B: p(A or B) = p(A) + p(B) Independent event (events can occur at the same time) For independent event A & B: p(A and B) = p(A) × p(B) Ace Scorers Enrichment Centre Contact us @ 6289 4551 or 6440 0553 www.acescorers.com.sg Ace Scorers Enrichment Centre 2A Maju Avenue S556680 Contact us @ 6289 4551 or 6440 0553 Youtube Ace Scorers www.acescorers.com.sg Instagram @chernteng