TMJC H2 Chapter 4 Equations & Inequalities Assignment Solutions 2024

Chapter 4 Equations and Inequalities TMJC 2024 Page 1 of 3 H2 Mathematics (9758) Chapter 4 Equations and Inequalities Assignment Suggested Solutions 1 2020Promo/NYJC/Q4 (i) Using an algebraic method, solve the inequality 2 6 124 xx −− , leaving your answers in exact form. [4] (ii) Hence, solve ( ) 2 6 1 2 4ln lnxx  −− . [3] Q1 Solution (i) 2 2 6 (2 4 ) 024 xx xx − − − −− 2 2 44 024 xx xx ++ −− 2 2 ( 2) 042 x xx + +− 2 2 ( 2) 0( 2) 6 x x + +− 2( 2) 0 ( 2 6)( 2 6) x xx +  + + + − ( ) ( ) 2( 2) 0 2 6 2 6 x xx +   − − − − − +   2 6 or 2 or 2 6x x x− − =− − + (ii) From (i), replace x with ln x . ln l 2 6 or nn o2 l2 r 6x x x− − =− − + 2 6 2 2 60 e or e or ex x x − − + −−   =  Alternatively, 0 0.0117 or 0.135 or 1.57x x x   =  −2 + ξ6 −2 − ξ6 −2 Note: the inequality to solve is ≥ 0, so x = −2, satisfies the inequality Factorise Solve using number line Bring everything to one side

Chapter 4 Equations and Inequalities TMJC 2024 Page 2 of 3 2 2020/EJC/I/Q4 (modified) (i) On the same axes, sketch the graphs of 1 ay x= − and y x a=− , where a is a constant such that 01 a . It is given that the two graphs intersect exactly twice. [3] (ii) Hence, solve the inequality 1 axa x− − . [3] Q2 Solution (i) Finding intersection points: 1 a xax =−− Since 1xa , ( )1 a xax =−− ( )( )1x x a a− − = ( ) 2 10x a x− + = ( )10x x a − + = 0    or   1x x a= = + When x = a + 1, y = 1 (ii) Hence from the graph, 0 1, 11 ax a x a xx−     + − Alternatively, 0 1     or   1   1x x a     + y x x = 1 (a, 0) (0, a) (a + 1, 1) y = 0 𝑦 = 𝑎 ȁ𝑥 − 1ȁ 𝑦 = ȁ𝑥 − 𝑎ȁ O Notice from the graph drawn, the 2nd intersection point occurs after x = 1

Chapter 4 Equations and Inequalities TMJC 2024 Page 3 of 3 3 2019/HCI/I/Q1 The number of units, D(x) of a particular product that people are willing to purchase per week in city A at a price $ x is given by the function 40320() g( )Dx x= , where g( x) is a quadratic polynomial in x. The following table shows t he number of units people are willing to purchase at different prices. x 5 8 10 D(x) 384 224 168 Find the number of units of the product that people are willing to purchase at a price of $18. [4] Q3 Solution 40320() g( ) 40320g( ) () Dx x x Dx = = 2 40320 , , ,()ax bx c a b c Dx+ + =  Given 2 403205 5 105 384a b c+ + = = -- (1) 2 403208 8 180 224a b c+ + = = -- (2) 2 4032010 10 240 168a b c+ + = = -- (3) Using GC, 1, 12, 20a b c= = = When x = 18, 2 40320(18) 7218 12(18) 20D == ++ units