O-LevelAdditional MathematicsSeries4037/12 · May/June 2024 · Q1012 Marks

10 (a) The first 3 terms of an arithmetic progression are 3 sin 2x, 5 sin 2x, 7 sin2x.

(a)(i)Show that the sum to n terms of this arithmetic progression can be written in the form n(n+a) sin2x, where a is a constant.[3]
(a)(ii)Given that x = 2π/3, find the exact sum of the first 20 terms.[2]
(b)(i)The first 3 terms of a geometric progression are ln2y, ln4y², ln 16y⁴. Find the nth term of this geometric progression.[2]
(b)(ii)Find the sum to n terms of this geometric progression, giving your answer in its simplest form.[2]
(c)The first 3 terms of a different geometric progression are (2w-1/4), (2w-1/4)², (2w-1/4)³. Find the values of w for which this geometric progression has a sum to infinity.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] d = 2sin2x soi
[M1] S_n = n/2 (6sin2x+2(n-1)sin2x) or S_n = n/2 (3sin2x+(2n+1)sin2x) For correct use of the sum formula with their d
[A1] n(n+2)sin 2x
[M1] for use of their answer to (a), but must be a correct method, sin(4π/3) needs to be evaluated to give an exact answer.
[A1] -220√3 isw
[B1] r = 2
[B1] 2^(n-1) ln 2y or (ln2y)2^(n-1) or ln(2y)^(2^(n-1))
[M1] ln 2y ( (1-their r^n) / (1-their r) ) For correct use of sum formula with their non-logarithmic r
[A1] (2^n-1)ln2y or -1(1-2^n)ln2y or ln(2y)^(2^n-1) or -ln2y + 2^n ln 2y oe isw when appropriate A0 if denominator is still present.
[B1] r = 2w-1/4. May be seen in the sum to infinity formula
[B2] -1 < 2w-1/4 < 1 or -0.375 < w < 0.625 and 5/8 seen
[A1] 3/8 < w < 5/8

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.