4 The first 3 terms in the expansion of a x x 1 3 3 5 + - ` b j l , in ascending powers of x, can be written in the form bx cx 27 2 + + , where a, b and c are integers. Find the values of a, b and c. [8]

4 ( ) 3 3 2 2 3 3 3   + = + + +   a x a a x ax x B1

4 (a) Find the first three terms, in ascending powers of x2, in the expansion of x 2 1 3 2 2 8 - e o . Write your coefficients as rational numbers. [3] (b) Find the coefficient of x2 in the expansion of x x x 2 1 3 2 2 1 2 8 2 - + e b o l . [3]

4(a) 2 4 1 7 256 24 36 − + x x 3 B1 for 1 256 B1 for 2 24 −x B1 for 4 7 36 x 4(b) 2 2 1 4 4 + + x x soi B1 Coefficient of 2 x 1 4 256 1 4 24 7 36 their their their their their   ×       + × −       +    M1 Allow one sign error, but must have 3 terms in 2 x only, with an attempt at addition. 25 576 A1

5 A geometric progression is such that its sum to 4 terms is 17 times its sum to 2 terms. It is given that the common ratio of this geometric progression is positive and not equal to 1. (a) Find the common ratio of this geometric progression. [3] (b) Given that the 6th term of the geometric progression is 64, find the first term. [2] (c) Explain why this geometric progression does not have a sum to infinity. [1]

5(a) ( ) ( ) 4 2 1 1 17 1 1 − − = − − a r a r r r M1 Allow equivalents Allow if ‘a’ terms missing (assume to have been cancelled) ( )( ) ( ) 2 2 2 1 1 17 1 − + = − r r r or better 4 2 17 16 0 − + = r r oe 3 2 16 16 0 + − − = r r r oe M1 Dep M1 for a correct simplified equation in r only r = 4 only, from correct working A1 5(b) 5 64 = ar M1 For use of 5 a r with their positive r 0.0625 = a or 1 16 A1 Must be exact A0 if 4 = r not from correct working in (a) 5(c) Because 1 > r oe B1 FT on their 1 > r Must have a value for r

9 (a) The first three terms of an arithmetic progression are , , 4 8 20 - . Find the smallest number of terms for which the sum of this arithmetic progression is greater than 2000. [4] (b) The 7th and 9th terms of a geometric progression are 27 and 243 respectively. Given that the geometric progression has a positive common ratio, find (i) this common ratio, [2] (ii) the 30th term, giving your answer as a power of 3. [2] (c) Explain why the geometric progression 1, sini, sin2i, … for r r 2 2 1 1 i - , where i is in radians, has a sum to infinity. [2]

9(a) 12 = d B1 ( ) ( ) 8 1 12 2000 2 −+ − > n n 2 3 5 1000 0 − − > n n M1 For use of sum formula to obtain a three term quadratic inequality or equation 5 25 12000 6 ± + = n 19.1 = n M1 Dep for attempt at critical value(s) using their quadratic, may be using a calculator, so may be implied by a correct answer of 20. 20 = n A1 9(b)(i) 3 = r 2 M1 For 6 27 = ar and 8 243 = ar with an attempt to eliminate a to obtain 2r . Allow other valid methods. 9(b)(ii) 26 3 2 B1 for 1 27 = a or 3 3− nfww 9(c) Common ratio or sinθ = r B1 May be implied by e.g. 1 1 sinθ − or 1 sin 1 sin θ θ − − n 1 sin 1 θ −< < or sin 1 θ < or 1 1 −< < r or 1 < r with no incorrect statements seen. B1 Dep on previous B1

1 Using the binomial theorem, expand ( ) 1 e x 2 4 + , simplifying each term. [2]

1 2 4 6 8 1 4e 6e 4e e + + + + x x x x B2 mark final answer for B2 B1 for any 3 correct simplified terms in a sum or all 5 simplified terms listed but not summed or for a correct, simplified expansion that is not their final answer or M1 for a correct unsimplified expansion e.g. 2 2 2 2 3 2 4 4 3 4 3 2 1 4e (e ) (e ) (e ) 2 6 × × × + + + + x x x x If 0 scored, SC1 for a complete, correct, simplified expansion as final answer found by multiplying out the brackets

Cambridge O Level Additional Mathematics · 4037

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

Revise O-Level Additional Mathematics (4037) — Series 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.

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