TMJC H2 Chapter 4 Equations & Inequalities Discussion Solutions 2024

Chapter 4 Equations and Inequalities TMJC 2024 Page 1 of 22 H2 Mathematics (9758) Chapter 4 Equations and Inequalities Discussion Solutions Level 1 Questions 1 Solve the equation 221 11 2 0x x   algebraically. Q1    221 11 2 0 7 1 3 2 0 x x x x       1 2 or7 3x x  3 methods to solve quadratic equations algebraically: 1) Factorisation (shown in solution) 2) Complete the square (try on your own) 3) Quadratic Formula (try on your won)

Chapter 4 Equations and Inequalities TMJC 2024 Page 2 of 22 2 Using a graphing calculator, solve the following equations (a) 3 22 23 41 115 0x x x    , (b) 1 2 2ln 0 xx   . Q2 (a) 3 22 23 41 115 0x x x    Using GC,      1.46 3 s.f. or 4.87 3 s.f. or 8.08 3 s.f.x x x   (b) 1 2 2ln 0 xx   Using GC, x = 0.187 or x = 2.16 y x O 2.16 0.187 x = 0 Solving equations using a GC: - For polynomial equations (up to degree 10) we can use the PolyRootFinder in the APP. - We can use the graphical method (sketching one or two graphs) to determine the roots (x-value of the intersection points).

Chapter 4 Equations and Inequalities TMJC 2024 Page 3 of 22 3 Solve the system of linear equations: 31 2 4 1 2 3 4 1 2 3 4 1 2 3 4 10 4 2 0 xx x x x x x x x x x x x x x x                 Q3 31 2 4 1 2 3 4 1 2 3 4 1 2 3 4 10 4 2 0 xx x x x x x x x x x x x x x x                 Using GC, .x1 = 4, x2 = 3, x3 = 2, x4 = 1.

Chapter 4 Equations and Inequalities TMJC 2024 Page 4 of 22 4 By drawing number lines, simplify the following ranges of values of x: (a) 3 8 x   and 4x , (b) 3 8 x  or 2 5 x   . Q4 (a) 3 8 x   and 4x 4 8 x   (b) 3 8 x  or 2 5 x   2 8 x   5 Without the use of a calculator, solve the following inequalities (a) 2 5 4x x  , (b) 29 6 8 0x x   , (c) 2 7x  , (d) 2 1 3x  . Q5 (a) 2 5 4 2 5 5 2 x x x x       (b)    29 6 8 0 3 2 3 4 0 x x x x       2 4 or3 3x x  (c) 2 7x  7 2 7 9 5 x x        (d) 2 1 3x  2 1 3 or 2 1 3 2 2 or 2 4 1 or 2 x x x x x x         3 4 x 8 2 3 x 5 8 “and” means intersection of the two regions. “or” means union of the two regions.

Chapter 4 Equations and Inequalities TMJC 2024 Page 5 of 22 6 By sketching appropriate graphs, solve the following inequalities (a) 1 ln xx  , (b) 2 4 1 1 0x x    . Q6 (a)  1For ln , 1.76 3 s.f.x xx   (b)   2For 4 1 1 0, 0 0.586 or 3.41 4 3 s.f.x x x x        x y ݔ= 0 ݕ= 0 ݕ= 1 ݔ ݕ= lnݔ 1.76, 0.567) y x Method 1: Sketching of two graphs (both LHS and RHS) to solve the inequality. Method 2: Sketching of one graph to solve the inequality.

Chapter 4 Equations and Inequalities TMJC 2024 Page 6 of 22 Level 2 Questions 7 A VCR manufacturing company produces three models: model A, model B and model C. The time taken for the production, assembly and testing of each model is given in the table below. To minimize costs, the company decides that 385 hours should be allocated for production, 557 hours for assembly, and 128 hours for testing. How many of each model can the company produce if it wants to use up all the allocated time for each phase of the process? Model Production time in hrs Assembly time in hrs Testing time in hrs A 1.8 3.0 0.5 B 2.2 3.2 0.8 C 3.0 3.5 1.0 Q7 Let a, b, and c represents the number of VCRs models A, B, and C respectively. 1.8a + 2.2 b + 3.0 c = 385 no. of hrs for production 3.0a + 3.2 b + 3.5 c = 557 no. of hrs for assembly 0.5a + 0.8 b + 1.0 c = 128 no. of hrs for testing Using GC, a = 60, b = 85, c = 30. The company has to produce 60 of model A, 85 of model B and 30 of model C. 1) Identify unknowns 2) Form equations with the unknowns. (3 unknowns, 3 distinct equations)

Chapter 4 Equations and Inequalities TMJC 2024 Page 7 of 22 8 2008/HCI prelim/I/1 A student has been saving 10 cent, 20 cent and 50 cent coins in a moneybox. When she opened the box after one month, she found the amount saved is $15 and the number of 10 cent coins equals the total number of 20 cent and 50 cent coins. She also found that only half as many coins is needed to save the same amount using just 50 cent coins. Find the number of 10 cent, 20 cent and 50 cent coins in the moneybox. [4] Q8 2008 HCI Prelim/P1/1 Let x, y and z be the number of 10 cent, 20 cent and 50 cent coins respectively. Equation 1 10 20 50 1500 (1)x y z    Equation 2 0 (2) x y z x y z        Equation 3 50 15002 25 25 25 1500 (3) x y z x y z            Using GC, 30, 10, 20x y z   There are 30 10-cent coins, 10 20-cent coins and 20 50-cent coins. 1) Identify unknowns 2) Form equations with the unknowns. (3 unknowns, 3 distinct equations) Remember to arrange your equations in terms of ax by cz d   , where a, b, c, d are constants, before keying into GC. Conclude your answer.

Chapter 4 Equations and Inequalities TMJC 2024 Page 8 of 22 9 2015(9740)/I/1 A curve C has equation 2 ay bx cx   where a, b and c are constants. It is given that C passes through the points with coordinates  1.6, 2.4 and  0.7, 3.6 , and that the gradient of C is 2 at the point where 1x . (i) Find the values of a, b and c, giving your answers correct to 3 decimal places. [4] (ii) Find the x-coordinate of the point where C crosses the x-axis, giving your answer correct to 3 decimal places. [2] (iii) One asymptote of C is the line with equation 0x . Write down the equation of the other asymptote of C. [1] Q9 2015(9740)/I/1 (i) 3d 2d y ax bx   --- (*) Substitute 1x  into (*), 3d 2 (1) 2d 2 2 --- (1) y a bx a b       Substitute (1.6, 2.4) into 2 ay bx cx   , 2 1 1.6 2.4 --- (2)1.6 a b c   Substitute ( 0.7,3.6) into 2 ay bx cx   ,   2 1 0.7 3.6 --- (3) 0.7 a b c    Solving (1), (2) and (3) using GC, 3.59345 3.593 5.18691 5.187 correct to 3 d.p. 7.30274 7.303 a b c          When x = 1, gradient of C is 2

Chapter 4 Equations and Inequalities TMJC 2024 Page 9 of 22 (ii) Using GC, 0.589x   (3 d.p.) (iii) 5.187 7.303 5.19 7.30 (correct to 3 s.f.) y x y x     y x Use 5 s.f. values of a, b and c. (intermediate step use 5 s.f. ) The equations of asymptote of 2 ay bx cx   are y bx c  and 0x 

Chapter 4 Equations and Inequalities TMJC 2024 Page 10 of 22 10 Solve the following inequalities (a)    1 3 3 1 0x x x    , (b)      2 4 1 2 0x x x    , (c)    2 4 5 1 0x x x    . Q10 (a)    1 3 3 1 0x x x    1 or 1 33x x    (b)      2 4 1 2 0x x x    4 or 1 or 2x x x    (c)    2 4 5 1 0x x x    Since   22 4 5 2 1 0x x x      for all real values of x,  1 0x  1x  1 + 3 + − − – 4 1 + 2 − + − Put “closed circles” for the roots for inequalities of the form: 0 and 0 Put “closed circles” for the roots for inequalities of the form: 0 and 0