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Markscheme Review Log Exp 1a. [6 marks] Markscheme (i) (A1) A1 N2 (ii) evidence of doubling (A1) e.g. 560 setting up equation A1 e.g. , (A1) in the year 2007 A1 N3 [6 marks] 1b. [6 marks] Markscheme (i) (A1) (A1) A1 N3 (ii) A1 not doubled A1 N0 valid reason for their answer R1 e.g. 1 [6 marks] 1c. [5 marks] Markscheme (i) correct value A2 N2 e.g. , 91.4, (ii) setting up an inequality (accept an equation, or reversed inequality) M1 e.g. , finding the value (A1) after 10 years A1 N2 [5 marks] 2a. [2 marks] Markscheme interchanging x and y (seen anywhere) (M1) e.g. (accept any base) evidence of correct manipulation A1 e.g. , , , AG N0 [2 marks] 2b. [1 mark] Markscheme , A1 N1 [1 mark] 2c. [4 marks] 2 Markscheme METHOD 1 finding (seen anywhere) A1 attempt to substitute (M1) e.g. evidence of using log or index rule (A1) e.g. , A1 N1 METHOD 2 attempt to form composite (in any order) (M1) e.g. evidence of using log or index rule (A1) e.g. , A1 A1 N1 [4 marks] 3. [7 marks] Markscheme recognizing e.g. (seen anywhere) (A1) , recognizing (A1) e.g. correct simplification A1 3 e.g. , evidence of correct approach to solve (M1) e.g. factorizing, quadratic formula correct working A1 e.g. , A2 N3 [7 marks] 4a. [3 marks] Markscheme METHOD 1 recognizing that (M1) e.g. recognizing that (A1) e.g. A1 N2 METHOD 2 attempt to find the inverse of e.g. (M1) , substituting 1 and 8 (M1) e.g. , A1 N2 [3 marks] 4b. [4 marks] 4 Markscheme METHOD 1 recognizing that (M1) e.g. (A1) (accept ) A2 N3 METHOD 2 attempt to find inverse of (M1) e.g. interchanging x and y , substituting into correct inverse (A1) e.g. , A2 N3 [4 marks] 5a. [3 marks] Markscheme (i) interchanging x and y (seen anywhere) M1 e.g. correct manipulation A1 e.g. , AG N0 (ii) A1 N1 [3 marks] 5 5b. [4 marks] Markscheme collecting like terms; using laws of logs (A1)(A1) e.g. , , , simplify (A1) e.g. , A1 N2 [4 marks] 6a. [1 mark] Markscheme 5 A1 N1 [1 mark] 6b. [4 marks] Markscheme METHOD 1 (A1) (A1) (A1) , (accept ) A1 N3 METHOD 2 (A1) 6 (A1) (A1) , (accept ) A1 N3 [4 marks] 7a. [2 marks] Markscheme substituting (0, 13) into function M1 e.g. A1 AG N0 [2 marks] 7b. [3 marks] Markscheme substituting into e.g. A1 , evidence of solving equation (M1) e.g. sketch, using (accept ) A1 N2 [3 marks] 7c. [5 marks] Markscheme (i) 7 A1A1A1 N3 Note: Award A1 for , A1 for , A1 for the derivative of 3 is zero. (ii) valid reason with reference to derivative R1 N1 e.g. , derivative always negative (iii) A1 N1 [5 marks] 7d. [6 marks] Markscheme finding limits , (seen anywhere) A1A1 evidence of integrating and subtracting functions (M1) correct expression A1 e.g. , area A2 N4 [6 marks] 8a. [2 marks] Markscheme (A1) A1 N2 [2 marks] 8b. [2 marks] Markscheme evidence of using the derivative (M1) A1 N2 8 [2 marks] 8c. [4 marks] Markscheme METHOD 1 setting up inequality (accept equation or reverse inequality) A1 e.g. evidence of appropriate approach M1 e.g. sketch, finding derivative (A1) least value of k is 36 A1 N2 METHOD 2 , and A2 least value of k is 36 A2 N2 [4 marks] 9a. [1 mark] Markscheme A1 N1 [1 mark] 9b. [2 marks] Markscheme substitution into formula (A1) e.g. , A1 N2 [2 marks] 9 9c. [5 marks] Markscheme set up equation (M1) e.g. attempting to solve (M1) e.g. graph, use of logs correct working (A1) e.g. sketch of intersection, A1 correct time 18:33 or 18:34 (accept 6:33 or 6:34 but nothing else) A1 N3 [5 marks] 10a. [2 marks] Markscheme combining 2 terms (A1) e.g. , expression which clearly leads to answer given A1 e.g. , AG N0 [2 marks] 10b. [3 marks] Markscheme attempt to substitute either value into f (M1) e.g. , 10 , A1A1 N3 [3 marks] 10c. [6 marks] Markscheme (i) , A1A1 N1N1 (ii) A1A1A1 N3 Note: Award A1 for sketch approximately through , A1 for approximately correct shape, A1 for sketch asymptotic to the y-axis. 11 (iii) (must be an equation) A1 N1 [6 marks] 10d. [1 mark] Markscheme A1 N1 [1 mark] 10e. [4 marks] Markscheme 12 A1A1A1A1 N4 Note: Award A1 for sketch approximately through shape of the graph reflected over , A1 for approximately correct , A1 for sketch asymptotic to x-axis, A1 for point clearly marked and on curve. [4 marks] 11a. [4 marks] Markscheme attempt to apply rules of logarithms (M1) 13 e.g. , correct application of (seen anywhere) A1 e.g. correct application of (seen anywhere) A1 e.g. so (accept ) A1 N1 [4 marks] 11b. [3 marks] Markscheme transformation with correct name, direction, and value A3 e.g. translation by , shift up by , vertical translation of [3 marks] 12a. [4 marks] Markscheme evidence of substituting the point A (M1) e.g. manipulating logs A1 e.g. A2 N2 [4 marks] 12b. [5 marks] Markscheme 14 (i) (accept A1 N1 (ii) A1A1A1A1 N4 Note: Award A1 for asymptote at , A1 for an increasing function that is concave up, A1 for a positive x-intercept and a negative y-intercept, A1 for passing through the point [5 marks] 12c. [4 marks] Markscheme METHOD 1 recognizing that (R1) 15 . evidence of valid approach (M1) e.g. switching x and y (seen anywhere), solving for x correct manipulation (A1) e.g. A1 N3 METHOD 2 recognizing that (R1) identifying vertical translation (A1) e.g. graph shifted down 3 units, evidence of valid approach (M1) e.g. substituting point to identify the base A1 N3 [4 marks] Printed for American Community School at Beirut © International Baccalaureate Organization 2016 International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional® 16