Raffles Institution 2024 Y3 SUPP WS Exponential and Logarithmic Eqns and Fn

Page 1 of 4 RAFFLES INSTITUTION MATHEMATICS DEPARTMENT 2024 YEAR 3 RP MATHEMATICS TOPIC 9:EXPONENTIAL AND LOGARITHMIC EQUATIONS & FUNCTIONS (MATHS 1 & MATHS 2) SUPPLEMENTARY WORKSHEET Name: Class: Sec 3 ( ) Date: 1 2022/Y3RP/M2/T2/Q1 Solve the equation ( ) ( )6 6 0.2 1log 2 3 4 log 4 log 25xx+ − − = . [3] [Ans: 76 15 ] 2 2022/Y3RP/M2/T2/Q2 Solve the equation 1 11 2 3 2 8 0 xxe e e ++ − − = , giving your answer(s) correct to 3 significant figures. [4] [Ans:1.39] 3 2022/Y3RP/M2/T2/Q3 Given that 2loga xm= and log a yn= , express 4logx y in terms of m and n . [4] [Ans: n m ] 4 2022/Y3RP/M2/T2/Q4 (i) Sketch the graph of ( )ln 2 3yx=+ , indicating clearly the asymptote and any intersection(s) with the axes. [2] (ii) Insert on your sketch the additional straight line graph required to obtain a graphical solution of the equation 3 23 xe x= + . [2] [Ans: ln 3yx=− + ] 5 2022/Y3RP/M2/T2/Q5 Carbon dating is a method used to estimate the age of archaeological artefacts. The age, W years old, of an artefact with a carbon ratio of N is given by ln 57000.693 NW =−  . (i) Find the age of an artefact, correct to the nearest 1000 years, with a carbon ratio of 0.48. [1] (ii) Express N in the form kWNe= , where k is a constant correct to 3 significant figures. [2] (iii) Hence, sketch the graph of N against W . [2] [Ans: (i) 6000 years old (ii) 0.000122WNe −= ] 6 2021/Y3RP/M2/T2/Q1 Solve 2 1lg(ln ) 2x = . [2] [Ans: 4.86 ] 7 2021/Y3RP/M2/T2/Q2 Solve ( )( ) 2 2 13 2 5 15x x x +−= . [4]

Page 2 of 4 [Ans: 0.683− ] 8 2021/Y3RP/M2/T2/Q3 Sketch the graph of ( )ln 5 2yx=− . Insert in your sketch the additional straight line required to obtain a graphical solution to the equation ( ) 33 52xex+ =− . [4] [Ans: 1 13yx=+ ] 9 2021/Y3RP/M2/T2/Q4 Solve the simultaneous equations 27 39 x y= , 3 3 3log 1 log log ( 4)x y x− + = + . [5] [Ans: x = 3, y = 7] 10 2021/Y3RP/M2/T2/Q5 Given that 8log xp= and 8log yq= , find (i) m in terms of p if 64 2 mx= , [2] (ii) log 4x y in terms of p and q. [3] [Ans: (i) 63mp=+ (ii) 46 3 q p + ] 11 2020/Y3RP/M2/T2/Q1 (i) State the equation of the asymptote of the graph of ( )ln 2 3yx=− . [1] (ii) To solve the equation 1 32xex+ += using the graph of ( )ln 2 3yx=− , a straight line graph needs to be inserted. Determine the equation of this straight line. [2] [Ans: (i) 2 3x= (ii) 11 22yx=+ ] 12 2020/Y3RP/M2/T2/Q4 Without using a calculator, solve the equation ( ) ( ) 2 16 27 22log 2 1 6log 9 log 5 3xx− + = − . [5] [Ans: 3 5x= ] 13 2020/Y3RP/M2/T2/Q5 (i) Solve the equation 3 61(2 ) 15 2x x++= . [3] (ii) Hence, without using a calculator, express 8log 72 in terms of x where x is the solution of the equation in (i). [2] [Ans: (i) 0.528 (ii) 2x + 1] 14 2019