Cambridge A Level Mathematics 9709 — 2016 Oct/Nov Paper 1 · Variant 3

7 D C A B The diagram shows a triangular pyramid ABCD. It is given that −−→ −−→ −−→ AB = 3i + j + k, AC = i −2j −k and AD = i + 4j −7k. (i) Verify, showing all necessary working, that each of the angles DAB, DAC and CAB is 90Å. [3] (ii) Find the exact value of the area of the triangle ABC, and hence find the exact value of the volume of the pyramid. [4] [The volume V of a pyramid of base area A and vertical height h is given by V = 13Ah.]

Mark scheme: 7 (i) AB.AC = 3 − 2 −=1 0 hence perpendicular or 90˚ B1 3 ‒ 2 ‒ 1 or sum of prods etc must be seen AB.AD = 3 + 4 − 7 = 0 hence perpendicular or 90˚ B1 Or single statement: mutually AC.AD = 1 − 8 + 7 = 0 hence perpendicular or 90˚ AG B1 perpendicular or 90˚ seen at least [3] once . (ii) Area ABC = ( ½ ) 3 2 + 12 + 12 × 12 + ( − 2 ) 2 + ( − 1) 2 M1 = ½ 11 × 6 A1 Expect ½ 66 Vol. = ⅓× their ∆ ABC × 12 + 4 2 + ( − 7 ) 2 M1 1 = 66 × 66 = 11 A1 Not 11.0 6 [4] 2

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9 (a) Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and −12r respectively. Find the value of the common sum to infinity. [3] (b) The first term of an arithmetic progression is cos 1 and the second term is cos 1 + sin21, where 0 ≤1 ≤0. The sum of the first 13 terms is 52. Find the possible values of 1. [5] [Questions 10 and 11 are printed on the next page.]

Mark scheme: 6 12 9 (a) = M1 1 − r 1 + r 1 r = A1 3 S = 9 A1 [3] 13 2 (b)  2cosθ+ 12sin θ = 52 M1* Use of correct formula for sum of   2 AP 2cosθ+ 12(1 − cos²θ) = 8 → 6cos 2θ− cosθ− 2 ( = 0 ) DM1 Use s 2 = 1 − c 2 & simplify to 3- term quad cosθ = 2 / 3 or − 1/ 2 soi A1 Accept 0.268π, 2π/3. SRA1 for θ= 0.841 , 2.09 Dep on previous A1 A1A1 48.2˚, 120˚ Extra solutions in [5] range –1 d y 2 1  3  2 1 2 2

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