5 The functions f and g are defined as follows. ) (f x x x 4 2 = + for x R ! ( ) g e x 1 x 2 = + for x R ! (a) Find the range of f. [2] (b) Write down the range of g. [1] (c) Find the exact solution of the equation ) ( fg x 21 = , giving your answer as a single logarithm. [4]

5 2 5 10 1 1 ... 3 3 9   − = − +     x x x 2 M1 allow one sign error or one arithmetic slip 3 27, 3 = = a a B1 Term in x: 2 3 5 3 3 − = a a b M1 For multiplying their terms, must have sum of 2 relevant products = b 18 = − b A1 Term in 2 : x ( ) 2 3 5 10 3 3 3 9 − + = a a a c M1 For multiplying their terms, must have sum of 3 relevant products = c This question appears to have a mismatch between the question text and the provided mark scheme. The mark scheme seems to correspond to a different question involving polynomial expansion and coefficient matching, not the functions f(x) and g(x) as stated in the question text. Therefore, I cannot provide a relevant note based solely on the given mark scheme for the stated question.

1 - 6 - 5 - 4 - 3 - 2 - 1 0 y x 1 2 3 4 5 6 5 10 15 20 (a) On the axes, draw the graphs of y x 5 3 2 = + - and y x 11 = - . [4] (b) Using the graphs, or otherwise, solve the inequality x x 11 5 3 2 1 - + - . [2]

1(a) 4 M1 for ∨ shape of 5 3 2 y x = + − with vertex at 2 , 5 3       A1 for correct graph with y-intercept ( ) 0, 7 M1 for correct straight line for 11 y x = − A1 for correct straight line with y-intercept ( ) 0,1 1 1(b) 2 x > or 2 x < − B2 Mark final answer for B2 B1 FT for exactly two correct critical values or correct FT critical values soi, FT dependent on at least M1 in (a)

1 x 0 - 2 - 1 1 2 3 4 5 6 7 8 - 2 2 4 6 8 10 y (a) On the axes draw the graphs of y x 5 = - and y x 6 2 7 = - - . [4] (b) Use your graphs to solve the inequality x x 5 6 2 7 2 - - - . [2]

1(a) 4 M1 for 5 = − y x : ∨ shape with vertex at (5, 0) A1 Correct graph with y-intercept at (0, 5) M1 for 6 2 7 = − − y x : ∧ shape with vertex at (3.5, 6) A1 Correct graph with y-intercept at (0, −1) 1(b) x 6 final answer B2 B1 for exactly two correct critical values or B1 FT for exactly two correct FT critical values soi, FT dependent on at least M1 in (a) If the CVs are decimal allow BOD for reasonable values

13 The functions f and g are defined, for x 0 2 , by ( ) f x x x 3 2 1 2 = - , ( ) g x x 1 = . (a) Find and simplify an expression for ( ) fg x . [2] (b) (i) Given that f 1 - exists, write down the range of f 1 - . [1] (ii) Show that ( ) f x px qx r 4 1 2 = + + - , where p, q and r are integers. [4]

13(a) [ ] fg( ) = x 2 1 2 1 1 3  −           x x oe M1 [ ] 2 2 fg( ) 3 − = x x x or 2 3 3 −x x A1 mark final answer 13(b)(i) f−1 > 0 B1 13(b)(ii) 2 2 3 1 0 − −= x xy or 2 2 3 1 0 − −= y xy B1 Correctly applies quadratic formula: [ ] 2 ( 3 ) ( 3 ) 4(2)( 1) 2(2) −− ± − − − = y y x oe or [ ] 2 ( 3 ) ( 3 ) 4(2)( 1) 2(2) −− ± − − − = x x y oe M1 FT their 2 2 3 1 0 − −= x xy or 2 2 3 1 0 − −= y xy with at most one sign error in the equation Justifies the positive square root at some point B1 2 1 3 9 8 f ( ) 4 − + + = x x x cao A1 must be a function of x

1 On the axes below, sketch the graph of ( )( )( ) y x x x 2 1 2 = - + + showing the coordinates of the points where the curve meets the axes. [3] O y x

1 B1 Shape B1 Correct x-coordinates B1 Correct y-coordinate and max in first quadrant

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Functions past paper questions

Revise O-Level Additional Mathematics (4037) — Functions with real Cambridge past-paper questions grouped by topic, each shown with its marking scheme. Working one topic at a time targets exactly the skill each question tests.

32 free questions from 2016–2021, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.