CIE IGCSE ADDITIONAL MATHEMATICS (0606) TOPICAL PRACTICE QUESTIONS TOPIC 7: LOGARITHMIC AND EXPONENTIAL FUNCTIONS Compiled from: Paper 1 Variants 1, 2 and 3 2016- 2020 "Success doesn't come from what you do occasionally, but what you do consistently." Source: 0606/11/M/J/18 - Question No. 5 1 Page 1 The population, P, of a certain bacterium t days after the start of an experiment is modelled by P = 800ekt, where k is a constant. (i) State what the figure 800 represents in this experiment. [1] (ii) Given that the population is 20 000 two days after the start of the experiment, calculate the value of k. [3] (iii) Calculate the population three days after the start of the experiment. © UCLES 2018 [2] Compilation: www.mystudycompass.com Source: 0606/11/M/J/18 - Question No. 6 2 Page 2 (a) Write ^log 2 ph^log 3 2h + log 3 q as a single logarithm to base 3. [3] (b) Given that ^log a 5h2 - 4 log a 5 + 3 = 0 , find the possible values of a. [3] © UCLES 2018 Compilation: www.mystudycompass.com CIE IGCSE ADDITIONAL MATHEMATICS (0606) TOPICAL PRACTICE QUESTIONS TOPIC 7: LOGARITHMIC AND EXPONENTIAL FUNCTIONS Compiled from: Paper 2 Variants 1, 2 and 3 2016- 2020 "Success doesn't come from what you do occasionally, but what you do consistently." Source: 0606/21/M/J/16 - Question No. 3 1 Page 1 of 21 Do not use a calculator in this question. (i) Find the value of - log p p 2 . [1] J1 N (ii) Find lg K nO . L10 P [1] (iii) Show that lg 20 - lg 4 = ^lg yh2 , where y is a constant to be found. log5 10 (iv) Solve log r 2x + log r 3x = log r 600 . © UCLES 2016 [2] [2] Compilation: www.mystudycompass.com Source: 0606/21/M/J/20 - Question No. 8 2 (a) Expand (2 - x) 5, simplifying each coefficient. Page 2 of 21 [3] 5 5 e (2 - x) # e 80x = e -x . (b) Hence solve 4 e 10x + 32 © UCLES 2020 [4] Compilation: www.mystudycompass.com Source: 0606/21/O/N/16 - Question No. 3 3 Solve the equation © UCLES 2016 2 lg x - lg c Page 3 of 21 x + 10 m = 1. 2 [5] Compilation: www.mystudycompass.com Source: 0606/21/O/N/17 - Question No. 3 4 Solve the equation © UCLES 2017 log5 (10x + 5) = 2 + log5 (x - 7) . Page 4 of 21 [4] Compilation: www.mystudycompass.com Source: 0606/21/O/N/18 - Question No. 2 5 b x - 1l (a) Solve 3 2 = 10 . (b) Solve 2e 1 - 2y = 3e 3y + 2 . © UCLES 2018 Page 5 of 21 [3] [4] Compilation: www.mystudycompass.com Source: 0606/21/O/N/19 - Question No. 3 6 Page 6 of 21 (a) Solve e 2x + 1 = 3e 4 - 3x . [3] (b) Solve lg (y - 6) + lg (y + 15) = 2 . [5] © UCLES 2019 Compilation: www.mystudycompass.com Source: 0606/21/O/N/20 - Question No. 3 7 Write © UCLES 2020 3 lg x + 2 - lg y as a single logarithm. Page 7 of 21 [3] Compilation: www.mystudycompass.com Source: 0606/22/M/J/16 - Question No. 10 8 (c) Solve log2 (x + 1) - log2 x = 3 . © UCLES 2016 Page 8 of 21 [3] Compilation: www.mystudycompass.com Source: 0606/22/M/J/17 - Question No. 7 9 Page 9 of 21 (a) Given that a 7 = b , where a and b are positive constants, find, (i) log a b , [1] (ii) log b a . [1] 1 (b) Solve the equation log 81 y =- . 4 [2] 2 (c) Solve the equation © UCLES 2017 32 x - 1 = 16 . 2 4x [3] Compilation: www.mystudycompass.com Source: 0606/22/M/J/20 - Question No. 9 10 (a) Solve the equation (b) loga b - Page 10 of 21 9 5x = 243. 27 x - 2 [3] 1 = logb a, where a 2 0 and b 2 0. 2 Solve this equation for b, giving your answers in terms of a. © UCLES 2020 [5] Compilation: www.mystudycompass.com Source: 0606/22/O/N/16 - Question No. 3 11 Solve the equation © UCLES 2016 2 lg x - lg c Page 11 of 21 x + 10 m = 1. 2 [5] Compilation: www.mystudycompass.com Source: 0606/22/O/N/17 - Question No. 2 12 Solve the equation © UCLES 2017 2x 1.5 + 6x -0.5 = x. x 0.5 + 5x -0.5 Page 12 of 21 [5] Compilation: www.mystudycompass.com Source: 0606/22/O/N/18 - Question No. 4 13 Page 13 of 21 Solve (i) 2 3x - 1 = 6 , (ii) log 3 (y + 14) = 1 + © UCLES 2018 [3] 2 . log y 3 [5] Compilation: www.mystudycompass.com Source: 0606/22/O/N/20 - Question No. 2 14 Find the value of x such that © UCLES 2020 Page 14 of 21 1 x 4x+1 3 # 83 . = 32 2x-1 [4] Compilation: www.mystudycompass.com Source: 0606/22/O/N/20 - Question No. 10 Page 15 of 21 15 The number, b, of bacteria in a sample is given by b = P + Qe 2t , where P and Q are constants and t is time in weeks. Initially there are 500 bacteria which increase to 600 after 1 week. (a) Find the value of P and of Q. © UCLES 2020 [4] Compilation: www.mystudycompass.com Source: 0606/22/O/N/20 - Question No. 10 Page 16 of 21 (b) Find the number of bacteria present after 2 weeks. [1] (c) Find the first week in which the number of bacteria is greater than 1 000 000. [3] © UCLES 2020 Compilation: www.mystudycompass.com Source: 0606/23/M/J/16 - Question No. 3 16 Page 17 of 21 Do not use a calculator in this question. (i) Find the value of - log p p 2 . [1] J1 N (ii) Find lg K nO . L10 P [1] (iii) Show that lg 20 - lg 4 = ^lg yh2 , where y is a constant to be found. log5 10 (iv) Solve log r 2x + log r 3x = logr 600 . © UCLES 2016 [2] [2] Compilation: www.mystudycompass.com Source: 0606/23/M/J/17 - Question No. 1 17 Page 18 of 21 (a) Solve the equation 7 2x + 5 = 2.5, giving your answer correct to 2 decimal places. © UCLES 2017 [3] Compilation: www.mystudycompass.com Source: 0606/23/O/N/16 - Question No. 2 18 Solve the equation e 3x = 6e x . © UCLES 2016 Page 19 of 21 [3] Compilation: www.mystudycompass.com Source: 0606/23/O/N/20 - Question No. 8 19 Page 20 of 21 DO NOT USE A CALCULATOR IN THIS QUESTION. log 2 (y + 1) = 3 - 2 log2 x log2 (x + 2) = 2 + log2 y (a) Show that x 3 + 6x 2 - 32 = 0 . © UCLES 2020 [4] Compilation: www.mystudycompass.com Source: 0606/23/O/N/20 - Question No. 8 (b) Find the roots of x 3 + 6x 2 - 32 = 0 . Page 21 of 21 [4] (c) Give a reason why only one root is a valid solution of the logarithmic equations. Find the value of y corresponding to this root. [2] © UCLES 2020 Compilation: www.mystudycompass.com Source: 0606/11/M/J/19 - Question No. 7 3 Page 3 (a) Solve log 3 x + log 9 x = 12 . [3] 1 (b) Solve log 4 (3y 2 - 10) = 2 log 4 ( y - 1) + . 2 [5] © UCLES 2019 Compilation: www.mystudycompass.com Source: 0606/11/O/N/16 - Question No. 3 4 2 1 1 3 p 3 q-2 r 2 By using the substitution y = log x , or otherwise, find the values of x for which Given that = p a q b r c3, find the value of each of the integers a, b and c. 5 - 23 p (qr) 3 (log 3 x) 2 + log 3 x 5 - log 3 9 = 0 . © UCLES 2016 Page 4 [3] [6] Compilation: www.mystudycompass.com Source: 0606/11/O/N/18 - Question No. 7 5 (a) Express 2 + 3 lg x - lg y as a single logarithm to base 10. (b) (i) (ii) © UCLES 2018 Solve 6x + 7 - 3 = 0. x Page 5 [3] [2] Hence, given that 6 log a 3 + 7 - 3 log 3 a = 0 , find the possible values of a. [4] Compilation: www.mystudycompass.com Source: 0606/12/M/J/18 - Question No. 7 6 Page 6 t A population, B, of a particular bacterium, t hours after measurements began, is given by B = 1000e 4 . (i) Find the value of B when t = 0. [1] (ii) Find the time taken for B to double in size. [3] (iii) Find the value of B when t = 8. [1] © UCLES 2018 Compilation: www.mystudycompass.com Source: 0606/12/M/J/18 - Question No. 9 7 Page 7 (a) (i) Solve lg x = 3. [1] (ii) Write lg a - 2 lg b + 3 as a single logarithm. (b) (i) Solve x - 5 + [3] 6 = 0. x [2] (ii) Hence, showing all your working, find the values of a such that log4 a - 5 + 6 log a 4 = 0 . © UCLES 2018 [3] Compilation: www.mystudycompass.com Source: 0606/12/M/J/19 - Question No. 3 8 Page 8 The number, B, of a certain type of bacteria at time t days can be described by B = 200e 2t + 800e -2t . (i) Find the value of B when t = 0 . © UCLES 2019 [1] Compilation: www.mystudycompass.com Source: 0606/13/M/J/18 - Question No. 5 9 Page 9 The population, P, of a certain bacterium t days after the start of an experiment is modelled by P = 800ekt, where k is a constant. (i) State what the figure 800 represents in this experiment. [1] (ii) Given that the population is 20 000 two days after the start of the experiment, calculate the value of k. [3] (iii) Calculate the population three days after the start of the experiment. © UCLES 2018 [2] Compilation: www.mystudycompass.com Source: 0606/13/M/J/18 - Question No. 6 10 (a) Write ^log 2 ph^log 3 2h + log 3 q as a single logarithm to base 3. (b) Given that ^log a 5h2 - 4 log a 5 + 3 = 0 , find the possible values of a. © UCLES 2018 Page 10 [3] [3] Compilation: www.mystudycompass.com Source: 0606/13/M/J/20 - Question No. 2 11 (a) Given that log 2 x + 2 log 4 y = 8 , find the value of xy. (b) Using the substitution y = 2 x , or otherwise, solve 2 2x + 1 - 2 x + 1 - 2 x + 1 = 0 . © UCLES 2020 Page 11 [3] [4] Compilation: www.mystudycompass.com Source: 0606/13/O/N/16 - Question No. 5 12 5 (i) Given that log 9 xy = , show that log 3 x + log 3 y = 5. 2 (ii) Hence solve the equations [3] 5 log 9 xy = , 2 log 3 x # log 3 y =- 6 . © UCLES 2016 Page 12 [5] Compilation: www.mystudycompass.com Source: 0606/13/O/N/18 - Question No. 5 13 Page 13 (i) Show that log 9 4 = log 3 2 . [2] (ii) Hence solve log 9 4 + log 3 x = 3. [3] © UCLES 2018 Compilation: www.mystudycompass.com Source: 0606/13/O/N/19 - Question No. 8 14 Page 14 (a) Given that log a x = p and log a y = q, find, in terms of p and q, (i) log a axy 2, [2] (ii) log a e x3 o , ay [2] (iii) log x a + log y a. [1] (b) Using the substitution m = 3 x, or otherwise, solve 3 x - 3 1 + 2x + 4 = 0. © UCLES 2019 [3] Compilation: www.mystudycompass.com
