4754A A A * * MATHEMATICS (MEI) Applications of Advanced Mathematics (C4) Paper A ADVANCED GCE. Tuesday 13 January 2009 Morning

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1 ADVANCED GCE MATHEMATICS (MEI) Applications of Advanced Mathematics (C) Paper A 75A Candidates answer on the Answer Booklet OCR Supplied Materials: 8 page Answer Booklet Graph paper MEI Examination Formulae and Tables (MF2) Other Materials Required: None Tuesday 13 January 29 Morning Duration: 1 hour 3 minutes * * A A * * INSTRUCTIONS TO CANDIDATES Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet. Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully and make sure that you know what you have to do before starting your answer. Answer all the questions. Do not write in the bar codes. You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used. The total number of marks for this paper is 72. This document consists of pages. Any blank pages are indicated. NOTE This paper will be followed by Paper B: Comprehension. OCR 29 [T/12/2653] OCR is an exempt Charity 2R 8H16 Turn over

2 1 Express 2 Section A (36 marks) 3x + 2 x(x 2 in partial fractions. [6] + 1) 2 Show that (1 + 2x) x 9 x2 +..., and find the next term in the expansion. State the set of values of x for which the expansion is valid. [6] 3 Vectors a and b are given by a 2i + j k and b i 2j + k. Find constants λ and µ such that λa + µb j 3k. Prove that cot β cot α sin(α β) sin α sin β. [3] 5 (i) Write down normal vectors to the planes 2x y + 2 and x 1. Hence find the acute angle between the planes. (ii) Write down a vector equation of the line through (2,, 1) perpendicular to the plane 2x y + 2. Find the point of intersection of this line with the plane. 6 (i) Express cos θ + 3 sin θ in the form R cos(θ α), where R > and α is acute, expressing α in terms of π. (ii) Write down the derivative of tan θ. Hence show that 1 3 π 1 (cos θ + 3 sin θ) 2 dθ 3. OCR 29 75A Jan9

3 3 Section B (36 marks) 7 Scientists can estimate the time elapsed since an animal died by measuring its body temperature. (i) Assuming the temperature goes down at a constant rate of 1.5 degrees Fahrenheit per hour, estimate how long it will take for the temperature to drop (A) from 98 F to 89 F, (B) from 98 F to 8 F. [2] In practice, rate of temperature loss is not likely to be constant. A better model is provided by Newton s law of cooling, which states that the temperature θ in degrees Fahrenheit t hours after death is given by the differential equation dθ dt k(θ θ ), where θ F is the air temperature and k is a constant. (ii) Show by integration that the solution of this equation is θ θ + Ae kt, where A is a constant. The value of θ is 5, and the initial value of θ is 98. The initial rate of temperature loss is 1.5 F per hour. (iii) Find A, and show that k (iv) Use this model to calculate how long it will take for the temperature to drop (A) from 98 F to 89 F, (B) from 98 F to 8 F. (v) Comment on the results obtained in parts (i) and (iv). [1] [Question 8 is printed overleaf.] OCR 29 75A Jan9 Turn over

4 8 Fig. 8 illustrates a hot air balloon on its side. The balloon is modelled by the volume of revolution about the x-axis of the curve with parametric equations x sin θ, y 2 cos θ + sin 2θ, ( θ 2π). The curve crosses the x-axis at the point A (, ). B and C are maximum and minimum points on the curve. Units on the axes are metres. y B O A (, ) x C Fig. 8 (i) Find dy in terms of θ. dx (ii) Verify that dy dx when θ 1 π, and find the exact coordinates of B. 6 Hence find the maximum width BC of the balloon. (iii) (A) Show that y x cos θ. (B) Find sin θ in terms of x and show that cos 2 θ x 1 x2. (C) Hence show that the cartesian equation of the curve is y 2 x 3 1 x. [7] (iv) Find the volume of the balloon. [3] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. OCR 29 75A Jan9

5 75 Mark Scheme January (C) Applications of Advanced Mathematics Section A 1 3x + 2 A Bx+ C xx ( + 1) x ( x + 1) 3x + 2 A(x 2 + 1) + (Bx + C)x x 2 A coefft of x 2 : A + B B 2 coefft of x: 3 C correct partial fractions equating coefficients at least one of B,C correct 3x x + xx x x 2 2 ( + 1) ( + 1) [6] 1 2.( ) 1 (1+ 2 x) x+ 3 3 (2 x) ! x x x x +... * ( )( ) Next term (2 ) 3 x 3! 3 81 x 2(i) 1/3 2 binomial expansion correct unsimplified expression simplification www Valid for 1 < 2x < 1 ½ < x < ½ [6] 3 j 3k λ a + μ b λ(2i + j k) + μ(i 2j + k) 2λ + μ λ 2μ 3 λ + μ λ 2μ, 2λ λ 2, μ 1, equating components at least two correct equations LHS cot β cot α cos β cosα sin β sinα sinα cos β cosαsin β sinαsin β sin( α β) sinα sin β OR RHS sin( α β) sin α cos β cos α sin β sin α sin β sin α sin β sin α sin β cot β cot α [3] cot cos / sin combining fractions www using compound angle formula splitting fractions using cotcos/sin 13

6 75 Mark Scheme January (i) Normal vectors 1 and 1 1 Angle between planes is θ, where cosθ ( 1) + 1 ( 1) 2 + ( 1) ( 1) / 12 θ 73.2 or 1.28 rads scalar product finding invcos of scalar product divided by two modulae 2 2 (ii) r + λ λ λ 1+ λ 2(2 + 2λ) ( λ) + (1 + λ) λ 2 λ ½ So point of intersection is (1, ½, ½ ) 6(i) cosθ + 3 sinθ r cos( θ α) Rcosθ cosα + Rsinθsinα Rcosα 1, Rsinα 3 R ( 3) 2, R 2 tan α 3 α π/3 R 2 equating correct pairs tan α 3 o.e. (ii) derivative of tan θ is sec 2 θ π π sec ( ) (cos 3 sin ) d π θ θ 3 d θ θ + θ 1 π tan( θ ) 3 ¼ ( ( 3)) 3/ * π 3 ft their α 1 R 2 www [tan (θ π/3] ft their R,α(in radians) 1

7 75 Mark Scheme January 29 Section B 7(i) (A) 9 / hours (B) 18/ hours [2] (ii) dθ k( θ θ) dt dθ kdt θ θ θ θ ln( ) kt + c separating variables ln( θ θ) kt + c θ θ e kt+ c anti-logging correctly(with c) θ kt θ + Ae * Ae c (iii) Ae A 8 Initially d θ k(98 5) 8k 1.5 dt k.3125*.3125t (iv) (A) e 39/8 e.3125t t ln(39/8)/(.3125) 6.6 hours.3125t (B) e 3/8 e.3125t t ln(3/8)/(.3125) 15 hours equating taking lns correctly for either (v) Models disagree more for greater temperature loss [1] 15

8 75 Mark Scheme January 29 8(i) dy 2cos2θ 2sinθ dθ, dx 2cosθ dθ dy dy / dθ dx dx / dθ dy 2cos2θ 2sinθ cos 2θ sinθ dx 2cosθ cosθ, substituting for theirs oe (ii) When θ π/6, dy cos π / 3 sin π / 6 dx cos π / 6 1/2 1/2 3/2 Coords of B: x 2 + 2sin(π/6) 3 y 2cos(π/6) + sin(π/3) 3 3/2 BC 2 3 3/2 3 3, ft for either exact (iii) (A) y 2cosθ + sin2θ 2 cos θ + 2 sin θ cos θ 2cos θ (1 + sin θ) xcos θ * (B) sinθ ½ (x 2) cos 2 θ 1 sin 2 θ 1 ¼ (x 2) 2 1 ¼ x 2 + x 1 (x ¼ x 2 ) * (C) Cartesian equation is y 2 x 2 cos 2 θ x 2 (x ¼ x 2 ) x 3 ¼ x * [7] sin2θ 2 sin θ cos θ squaring and substituting for x (iv) 2 V π y dx 3 1 π ( x x ) dx π x x 2 π(6 51.2) 12.8π.2 (m 3 ) [3] need limits x x π or or better. 16

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