Csec Mathematics Summary, Cheat Sheet of Mathematics

Download Csec Mathematics Summary and more Cheat Sheet Mathematics in PDF only on Docsity! 1 MATHEMATICS CSEC SUMMARY 2022 Section 1 – Number Theory and Computation Section 2 – Consumer Arithmetic Section 3 – Sets Section 4 – Measurements Section 5 – Statistics Section 6 – Algebra Section 7 – Relations, Functions and Graphs Section 8 – Geometry and Trigonometry Section 9 – Vectors and Matrices 2 Section 1 – Number Theory and Computation Sets of numbers: Natural numbers, N = {1, 2, 3,….} Whole numbers, W = {0, 1, 2, 3, …..} Integers, Z = { …, -2, -1, 0, 1, 2, …} Rational numbers, Q = { ! " , p and q are integers, q ≠ 0} Irrational numbers, 𝑄# = { , √2 , 𝜋, ….} Real numbers, R = Q ∪ 𝑄# Significant figures rules: 1. All non-zero numbers ARE significant 2. Zeros between two non-zero digits ARE significant 3. Leading zeros are NOT significant. .e.g. 0.0045 has 2 sig. fig. 4. Trailing zeros to the right of the decimal ARE significant. e.g. 45.00 has 4 sig. fig. 5. Trailing zeros in a whole number with decimal shown ARE significant. 6. Trailing zeros in a whole number with no decimal shown are NOT significant. Properties of numbers: a) Closure: If a,b ∈ 𝑅 then a*b ∈ 𝑅. (𝑏) Associative: (x + y) + z = x + (y + z) c) Commutative: x + y = y + x and x . y = y . x. (d) Distributive: x . (y + z) = x . y. + x . z e) Additive Identity: x + 0 = 0 + x = x. (f) Multiplicative Identity: x . 1 = 1 . x = x g) Additive Inverse: x + ( -x ) = ( - x ) + x = 0 (h) Multiplicative Inverse: x./# $ 0=/# $ 0. x =1 Ratios: A ratio of a : b : c implies that the fractions being shared are 𝒂 𝒂"𝒃"𝒄 ∶ 𝒃 𝒂"𝒃"𝒄 ∶ 𝒄 𝒂"𝒃"𝒄 5 Section 4 – Measurements Length Mass 10 mm = 1 cm 1g = 1000mg 100 cm = 1 m 1kg = 1000g 1000 mm = 1 m 1kg = 2.2lbs 1000 m = 1 km 1lb = 16 ounces Speed = 9-*%.'1+ %-7+ Units: ms-1 or kmh-1 Distance = speed x time Time = 9-*%'.1+ *!++9 6 Section 5 – Statistics Basic definitions o Population: The entire group being investigated o Sample: A subset of the population o Discrete data: Specific values only o Continuous data: Range of values o Raw data: Unordered info o Mean: Average value = ∑$ . (ungrouped) or ∑;$∑ ; (grouped data) o Median: Middle value from a set of ordered values o Mode: Most frequent value o Probability: Chances of an event occurring o Standard deviation: Gives a spread of the data. (how far away from mean) Types of statistical charts: To plot histograms, we need the class boundaries as shown below: 7 To construct a cumulative frequency graph and read off the Quartiles we do the following: Quartiles: Lower Quartile, Q1 = # < (𝑛 + 1) th term Median, Q2 = # = (𝑛 + 1) th term Upper Quartile, Q3 = > < (𝑛 + 1) th term Inter Quartile Range = Q3 – Q1 Semi-Inter Quartile Range = ?!3 ?" = 10 Composite Functions Inverse of a function Co-ordinate Geometry: Distance between two points: L(𝑥= − 𝑥#)= + (𝑦= − 𝑦#)= Mid-point: ( $$B$" = , C$BC" = ) Gradient: m = C$3C" $$3$" !'0'((+( (-.+* D'E+ +"F'( /0'9-+.%* [7"H 7$] !+0!+.9-1F('0 (-.+*,!0&9F1% &; /0'9-+.%* +"F'(3#. [7"7$H 3#] functions of functions, substitute one function into the next eg 𝑓(𝑥) = 2𝑥 − 1, 𝑔(𝑥) = 𝑥 4 𝑓P𝑠(𝑥)R = 𝑓 / 𝑥 40 = 2/ 𝑥 40 − 1 = 𝑥 2 − 1 𝑔P𝑓(𝑥)R = 𝑔(2𝑥 − 1) = 2𝑥 − 1 4 𝑓=(𝑥) = 𝑓P𝑓(𝑥)R 𝑓(2𝑥 − 1) = 2(2𝑥 − 1) − 1 = 4𝑥 − 2 − 1 = 4𝑥 − 3 𝑓(𝑥) 𝑓3#(𝑥) Steps 1)let y = f(x) 2) interchange x and y 3) Solve for y 4) 𝑦 = 𝑓3#(𝑥) 𝑒. 𝑔 𝑓(𝑥) = 2𝑥 − 1 𝑙𝑒𝑡𝑦 = 2𝑥 − 1 𝑖𝑛𝑡𝑒𝑟𝑐ℎ𝑎𝑛𝑔𝑒 𝑥 𝑎𝑛𝑑 𝑦 𝑥 = 2𝑦 − 1 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑦 2𝑦 − 1 = 𝑥 2𝑦 = 𝑥 + 1 𝑦 = 𝑥 + 1 2 𝑓3#(𝑥) = 𝑥 + 1 2 11 Equation of a line: y = mx + c m – gradient c – y-intercept (cuts the y-axis) To find the equation of a line: - find gradient of line - obtain a point on the line - substitute in 𝑐 = 𝑦 −𝑚𝑥 NB. - Solving equations simultaneously gives the points of intersection of the equations. Quadratic: General form: y = ax2 + bx + c [highest power of x is 2] To complete the square: y = a(x + h)2 + k. where h = @ =' and k = c – ah2 To sketch a quadratic: • Shape: minimum , a > 0 Maximum, a < 0 • Turning point : ( - h , k) • Maximum or minimum value is always k . • X-value which gives max or minimum value is – h. • X-intercepts: solve ax2 + bx +c = 0 • Y-intercept: (0,c) Inequalities: • Solve inequalities like equations, but • Change the inequality sign when ÷ by a negative • For < or ≤ : shade below the line • For > or ≥: shade above the line < less than / fewer than > greater than / more than ≤ at most / no more than ≥ at least / no less than 12 Section 8 – Geometry and Trigonometry Construction 900 and 600 15 Trigonometry 𝐴𝑟𝑒𝑎 = # = 𝑏 × ℎ Bearings 1) Start N 2) Move in a clockwise direction 3) Show all angles eg. B is on a bearing of 70° from A C is due south of B 70° 70° A B N N hyp adj opp 𝜃 Right – angled ℎ𝑦𝑝= = 𝑜𝑝𝑝= + 𝑎𝑑𝑗= sin 𝜃 = 8 < cos 𝜃 = L < tan 𝜃 = 8 L A B C c b a Cosine rule: more lengths than angles 𝑎= = 𝑏= + 𝑐= − 2𝑏𝑐 cos 𝐴 Sine rule: more angles than lengths ' MNOL = @ MNOP 𝐴𝑟𝑒𝑎 = # = 𝑎𝑏 sin 𝐶 16 Section 9 – Vectors and Matrices Vectors: Position vector, 𝑂𝑃kkkkk⃗ = /250 𝑃𝑂kkkkk⃗ = −/350 = /−3−50 Addition: / 2−30 + / 4 10 = / 2 + 4−3 + 10 = / 6−20 Subtraction: / 2−30 − / 4 10 = / 2 − 4−3 − 10 = /−2−40 Multiplying Vectors a) By a scalar If 𝑂𝑃kkkkk⃗ = /310 2 𝑂𝑃kkkkk⃗ = 2 /310 = /620 b) Two vectors If we have two vectors 𝑃k⃗ = /𝑎𝑏0 and 𝑄k⃗ = /𝑐𝑑0 then P.Q = ad + bc is called dot or scalar product P (3,5) 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑂𝑃 |𝑂𝑃| = L3= + 5= 17 Displacement Vectors If 𝐴𝐵kkkkk⃗ = p and 𝐴𝐶kkkkk⃗ = q 𝐵𝐶kkkkk⃗ = 𝐵𝐴kkkkk⃗ + 𝐴𝐶kkkkk⃗ = −𝑝 + 𝑞 alternate route from B to C parallel vector are multiples of each other a=kb Collinear B C A q p 𝐴𝐶kkkkk⃗ = 𝐴𝐵kkkkk⃗ + 𝐵𝐶kkkkk⃗ 𝐴𝐵kkkkk⃗ = 𝐴𝐶kkkkk⃗ + 𝐶𝐵kkkkk⃗ C B A To show collinear 1) show 𝐴𝐵kkkkk⃗ ∥ 𝐵𝐶kkkkk⃗ 2) Show 𝐴𝐵kkkkk⃗ + 𝐵𝐶kkkkk⃗ = 𝐴𝐶kkkkk⃗ 20 (3) Always rows in matrix one × (by) columns in matrix 2. (1) If 𝐴 = )2 −1 3 2 - and 𝐵 = )0 1 3 −1- 2 × × 2 ≡ 2 × 2 𝐴𝐵 = )2 −1 3 2 - )0 1 3 −1 - = D (2 × 10) + (−1)(3) (2)(1) + (−1)(−1) (3 × 0) + (2 × 3) (3)(1) + (2)(−1) E = )−3 3 6 1- 𝐵𝐴 = )0 1 3 −1- ) 2 −1 3 2 - = D (0 × 2) + (1 × 3) (0 × −1) + (1 × 2) (3 × 2) + (−1 × 3) (3 × −1) + (2 × −1)E = )3 2 3 −5- Determinant If 𝐴 = )𝑎 𝑏 𝑐 𝑑 - then the determinant, det 𝐴 𝑜𝑟 |𝐴| |𝐴| = 𝑎𝑑 − 𝑏𝑐 Example 𝐴 = )2 −1 3 4 - |𝐴| = (2)(4) − (−1)(3) = 8 + 3 = 11 𝐴 = ) 2 4 −1 −3- 𝐵 = )1 2 2 4- |𝐴| = (2)(−3) − (4)(−1) |𝐵| = (1)(4) − (2)(2) = −6 + 4 = 4 − 4 = −2 = 0 2 2 2 × 2 21 If det in ≠ 0 then matrix is said to be non-singular. If det = 0, then the matrix is Singular. If 𝐴 = )2 𝑃 3 1- in a Singular matrix, Find p Since matrix is singular |𝐴| = 0 = (2)(1) − 𝑝(3) 0 = 2 − 3𝑝 𝑝 = !" !# 𝑝 = " # If 𝐴 = D2 3 𝑝 4E is a singular matrix, Find p. Det 𝐴 = 0 = (2)(4) − (3)(𝑝) 0 = 8 − 3𝑝 3𝑝 = 8 𝑝 = $ # Inverse of a matrix If 𝐴 = )𝑎 𝑏 𝑐 𝑑 - then its inverse, 𝐴!% is 𝐴!% = % |'| ) 𝑑 −𝑏 −𝑐 𝑎 - NB. A must be non-singular Example. If 𝐴 = )2 −3 1 3 -, then det 𝐴 = (2)(3) − (−3)(1) = 6 + 3 = 9 ∴ 𝐴!% = 1 9 ) 3 3 −1 2- = L 3 9 3 9 −1 9 2 9 M 𝐴 = )2 −4 1 −3- 𝐵 = )2 6 1 3- Det 𝐴 = −6 − (−4) |𝐵| = (2)(3) − (6)(1) = −6 + 4 = 6 − 6 = −2 = 0 𝐴!% = % !" )−3 4 −1 2- ∴ 𝐵 is singular ∴ no inverse 1 0 = ∞ 22 Solving Simultaneous Matrix Method 2𝑥 + 𝑦 = 3 3𝑥 − 2𝑦 = 1 Write in matrix form So 𝑥 = 𝐴!%. 𝑏 )2 1 3 −2- ) 𝑥 𝑦- = ) 3 1- 𝐴 = ) 2 1 3 −2- 𝐴!% = % !( )−2 −1 −3 2 - 𝐴𝑥 = 𝑏 Since Where 𝑥 = 𝐴!% . 𝑏 𝐴, 𝑥 and 𝑏 are matrices. = % !( )−2 −1 −3 2 - . ) 3 1- = 1 −7 )−2 −1 −3 2 - ) 3 1- = 1 −7 )−6 + (−1) −9 + 2 - = 1 −7 )−7−7- = Q !( !( !( !( R ) 𝑥 𝑦- = )11- 𝐴𝑥 = 𝑏 (1) Matrix form (2) 𝐴3# (3) / 𝑥 𝑦0 = 𝐴3# . 𝑏