O-LevelAdditional MathematicsSeries4037/11 · May/June 2023 · Q911 Marks
9
- (a)The first three terms of an arithmetic progression are ln q, ln q⁴ and ln q⁷, where q is a positive constant. The sum to n terms of this progression is 4845 ln q. Find the value of n.[3]
- (b)The first three terms of a geometric progression are p³ˣ, pˣ and p⁻ˣ, where p is a positive integer. Find the nth term of this progression giving your answer in the form p^(a+bn)x.[3]
- (c)The first three terms of a different geometric progression are (4/3)cos²3θ, (16/9)cos⁴3θ and (64/27)cos⁶3θ, for 0 < θ < π/3. Find the set of values of θ for which this progression has a sum to infinity.[5]
Mark scheme
Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.
- [3] B1 for n/2 (2 ln q + 3(n - 1)ln q) oe soi
- Allow if in indices form i.e.: n/2 (ln q² + (n - 1)ln q³)
- B1 for 3n² - n - 9690 = 0 oe soi
- B1 n = 57 cso
- [3] B1 Allow unsimplified
- B1 Allow unsimplified
- B1 p^(5-2n)x
- [5] B1 Allow unsimplified. Must be convinced it is the common ratio not just writing the first term e.g. r = (16/9)cos⁴3θ / (4/3)cos²3θ or seeing (4/3)cos²3θ
- B1 for (4/3)cos²3θ(*) -1 or (4/3)cos²3θ(*) > 1 or (4/3)cos²3θ(*) < -1 or (4/3)cos²3θ(*) = 0
- B1 for cos3θ(*) > √3/2 or cos3θ(*) < -√3/2 or cos3θ(*) = 0
- B1 Seeing 5π/18 or π/18 implied the first 3 marks
- B1 π/18 < θ < 5π/18
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