ACJC 2023 JC1 H1 Maths Revision Set A

0 ANGLO-CHINESE JUNIOR COLLEGE MATHEMATICS DEPARTMENT MATHEMATICS 8865 Higher 1 JC 1 REVISION EXERCISE SET A Mar 2023 Topics Start Page Number Complete solutions will be uploaded into Math Google Site one week before CA1. Exponential & Logarithmic Functions 1 Graphing Techniques 2 Equations & Inequalities 5 ACJC CA1 past year papers 8 Name: _________________________________ H1 Maths Group: 1MAX ___ Form Class: ___________

1 EXPONENTIAL & LOGARITHMIC FUNCTIONS 1 (a) Solve 9lg( 8) lg 1 lg 24 xx    − + = +       . [4] (b) Solve the equation ( ) ( ) 153log12log 55 =−−+ xx [3] 2 SRJC 2016 Promo Q3 (a) Given that ln p = k, find ln (pe25) in terms of k. [2] (b) Given that 3e x – 4e –x = 11, find the exact value of x. [4] 3 CJC 2012 Promo Q3 (a) Find the exact value(s) of the equation xxx −=− e3e5e2 3 . [4] (b) Given that 3log =xa , find )(log 2axa . [2] 4 RI 2012 Promo Q2 Solve the simultaneous equations y = (1 + e x ) (1 − e x ), x = ln (y + 1). [5] 5 YJC 2013 Prelim Q2 Solve the simultaneous equations 34 464 x y= and lg( ) 1 lg( 1)y x x− = − − . [4] 6 TJC 2016 Promo Q3 Given that 2 44log log 0q q p− − = and log 2p q = , show that 44p p= . [3] (i) On the same set of axes, sketch the graphs of 4xy= and 4yx= . [1] (ii) Hence find the solutions of p and corresponding values of q . [2] 7 JJC 2013 Promo Q4 In a research project, 5 000 ants are put in a man -made habitat for the population to grow. After t days, the population of the ants, P (in thousands) is given by 45eAtPB=− + , for some constants A and B. Given that 10 days later, the population has increased to 35 000. (i) Show that B = 50 and ln 3Ak= , where k is a constant to be found. [4] (ii) Find the population of the ants after 20 days. [2] (iii) Find the population of the ants after a very long time. [1] (iv) Sketch the graph of P against t. [1]

2 8 RVHS 2016 Promo Q11 The temperature, C, of the water in a flask, t minutes after the start of heating is modelled by the equation ( )ln 2 1b k t = + + , where b and k are constants. Initially the temperature of the water is 20 C . After 2 minutes of heating, the temperature is 65 C . (i) Find the values of b and k. [4] (ii) Find the time it takes for the water to reach a temperature of 50 C . [3] (iii) Sketch the graph of the temperature of the water against time for 08 t , labelling clearly the end points of the graph. [3] (iv) State, with a reason, whether this model is suitable in the long term. [2] Answers 1 (a) x = 18 (b) −2 2 (a) k + 25 (b) ln 4 3 (a) ln 3 2x= (b) 13 4 x = 0, y = 0 5 x = 3.5, y = 7.5 6 (ii) 2, 16pq== ; 4, 216pq== 7 (i) 1 10k =− (ii) 45 000 (iii) 50 000 8 (i) 45 ln 5k = (ii) 0.962 min (iv) No Suggested reason: Boiling point of water is 100 C but the graph is