83 Mathematics Intermediate Tier

Higher Tier - Algebra revision Contents: Indices Expanding single brackets Expanding double brackets Substitution Solving equations Solving equations from angle probs Finding nth term of a sequence Simultaneous equations – linear Simultaneous equations – of each Inequalities Factorising – common factors Factorising – quadratics Factorising – grouping & DOTS Solving quadratic equations Using the formula Completing the square Rearranging formulae Algebraic fractions Curved graphs Graphs of y = mx + c Graphing inequalities Graphing simultaneous equations Graphical solutions to equations Expressing laws in symbolic form Graphs of related functions Kinematics Indices a x a c (F ) a t ÷ t 2e x 3ef 4xy ÷ 2xy b x ÷ x 5p qr x 6p q r Expanding single brackets x e.g Remember to multiply all the terms inside the bracket by the term immediately in front of the bracket 4(2a + 3) = 8a + 12 x If there is no term in front of the bracket, multiply by or -1 Expand these brackets and simplify wherever possible: 3(a - 4) = 3a - 12 6(2c + 5) = 12c + 30 -2(d + g) = -2d - 2g c(d + 4) = cd + 4c -5(2a - 3) = -10a + 15 a(a - 6) = a2 - 6a 4r(2r + 3) = 8r2 + 12r - (4a + 2) = -4a - - 2(t + 5) = -2t - 10 2(2a + 4) + 4(3a + 6) = 16a + 32 11 2p(3p + 2) - 5(2p - 1) = 6p2 - 6p + Expanding double brackets Split the double brackets into single brackets and then expand each bracket and simplify (3a + 4)(2a – 5) “3a lots of 2a – and lots of 2a – 5” = 3a(2a – 5) + 4(2a – 5) = 6a2 – 15a + 8a – 20 = 6a2 – 7a – 20 If a single bracket is squared (a + 5)2 change it into double brackets (a + 5)(a + 5) Expand these brackets and simplify : (c + 2)(c + 6) = c2 + 8c + 12 (2a + 1)(3a – 4) = 6a2 – 5a – (3a – 4)(5a + 7) = 15a2 + a – 28 (p + 2)(7p – 3) = 7p2 + 11p – (c + 7)2 = c2 + 14c + 49 (4g – 1)2 = 16g2 – 8g + Substitution 3a If a = , b = and c = find the value of : c 4b2 15 ab – 2c 26 a2 –3b 144 c(b – a) ac (3a)2 10 225 4bc (5b – ac) a 9.6 144 900 Now find the value of each of these expressions if a = - , b = 3.7 and c = 2/3 Solving equations Solve the following equation to find the value of x :  Take 4x from both sides 4x + 17 = 7x – 17 = 7x – 4x –  Add to both sides 17 = 3x – 17 + = 3x 18 = 3x  Divide both sides by Now solve these: 18 = x 2x + = 17 – x = 3 3x + = x + 15 6=x 4(x + 3) = 20 x=6 Some equations cannot be solved in this way and “Trial and Improvement” methods Find x to d.p if: Try x2 + 3x = 200 Calculation x = 10 (10 x 10)+(3 x 10) = 130 x = 13 (13 x 13)+(3 x 13) = 208 Comment Too low Too high etc Solving equations from angle problems Find the size of each angle 4y 2y 150 Rule involved: Angles in a quad = 3600 4y + 2y + y + 150 = 360 7y + 150 = 360 7y = 360 – 150 7y = 210 y = 210/7 Angles are: y = 300 300,600,1200,1500 y Find the value of v + 4v 2v + 39 Rule involved: “Z” angles are equal 4v + = 2v + 39 4v - 2v + = 39 2v + = 39 2v = 39 - 2v = 34 v = 34/2 v = 170 Check: (4 x 17) + = 73 , (2 x 17) + 39 = 73 Finding nth term of a simple sequence Position number (n) 6 10 12 This sequence is the times table shifted a little , , , 11 , 13 , 15 ,…… …… Each term is found by the position number times then add another So the rule for the sequence is nth term = 2n + 100th term = x 100 + = 203 Find the rules of these sequences And these sequences  1, 3, 5, 7, 9,… 2n  6, 8, 10, 12, 2n …… 5n  3, 8, 13, 18,…… 6n  20,26,32,38, 7n –  1, 4, 9, 16, 25, … +4 –  3, 6,11,18,27…… + 14 n2 n2 + -2n + 22 -3n + 43  20, 18, 16, 14, 20n - 14 Finding nth term of a more complex sequence n = , 13 , 26 , 43 , 64 ,…….…… +9 +13 +21 +17 2nd difference is means that the first term is 2n2 What’s left +4 +4 +4 2n2 = , , 18 , 32 , 50 ,…….…… = 2, 5, This sequence has a rule = 3n - , 11 , 14 ,…….…… So the nth term = 2n2 + 3n - Find the rule for these sequences (a) 10, 23, 44, 73, 110, … (b) 0, 17, 44, 81, 128, … (c) 3, 7, 17, 33, 55, …  (a) nth term = 4n2 + n +  (b) nth term = 5n2 + 2n –  (c) nth term = 3n2 – 5n + y Graphs of y = mx + c In the equation: Y = 3x + c y = mx + c m = the gradient (how far up for every one along) c = the intercept (where the line crosses the y axis) m x Graphs of y = mx + c Write down the equations of these lines: y x Answers: y=x y=x+2 y=-x+1 y = - 2x + y = 3x + x=4 y=-3 Graphing inequalities x = -2 y y=x y>3 y=3 Find the region that is not covered by these regions x≤ -2 y≤ x y>3 x≤ x y