The first and second terms of an arithmetic progression are tanθ and sinθ respectively, where 0 < θ < ½π. For parts (b)(i) and (b)(ii), the first and second terms of a geometric progression are tanθ and sinθ respectively, where 0 < θ < ½π.

Mark scheme

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

1. [M1] [d =]sinθ – tanθ For subtraction of their first term from their second term. Condone incorrect evaluation before subtraction.
2. [A1] [d =]-√2/2 OE Sight of -0.29 AWRT can be awarded M1A1.
3. [M1] [S40 =]40/2 (2×tanθ+39(sinθ – tanθ)) Use of a correct formula for S40. Condone use of their, clearly identified, incorrect values for a and d for this mark.
4. [A1] 390√2-740 or 780/√2 - 740 ISW
5. [B1] r = sinθ/tanθ [= cosθ] Condone omission of θ.
6. [B1] S∞ = tanθ/(1-cosθ) or sinθ/(cosθ-cos²θ) or tan²θ/(tanθ-sinθ) ISW Do not allow fractions within fractions nor omission of θ.
7. [B1] a = √3 (1.73..) and r = 1/2 OE, SOI.
8. [M1] [S10 =]√3 (1-(1/2)¹⁰)/(1-1/2) This mark can be awarded for a correct formula with their values for a and r or a = tanθ and r = sinθ/tanθ or cosθ. Condone 1/2¹⁰.
9. [A1] = 3.46 AWRT Condone 1023√3/512