ADVANCED GCE UNIT MATHEMATICS (MEI) Applications of Advanced Mathematics (C4) Paper A THURSDAY 4 JUNE 7 Additional materials: Answer booklet (8 pages) Graph paper MEI Examination Formulae and Tables (MF) 4754(A)/ Afternoon Time: hour 3 minutes INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces provided on the answer booklet. Answer all the questions. 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. The total number of marks for this paper is 7. ADVICE TO CANDIDATES Read each question carefully and make sure you know what you have to do before starting your answer. 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. NOTE This paper will be followed by Paper B: Comprehension. This document consists of 6 printed pages and blank pages. HN/5 OCR 7 [T//653] OCR is an exempt Charity [Turn over
Section A (36 marks) Express sin q 3 cos q in the form R sin (q a), where R and a are constants to be determined, and a 9. Hence solve the equation sin q 3 cos q for [7] q 36. Write down normal vectors to the planes x 3y 4z and x y z 5. Hence show that these planes are perpendicular to each other. 3 Fig. 3 shows the curve y ln x and part of the line y. y O x Fig. 3 The shaded region is rotated through 36 about the y-axis. Û (i) Show that the volume of the solid of revolution formed is given by Ù pe y dy. ı (ii) Evaluate this, leaving your answer in an exact form. 4 A curve is defined by parametric equations x t, y t t. 3 x Show that the cartesian equation of the curve is y x. 5 Verify that the point (, 6, 5) lies on both the lines Ê ˆ Ê -ˆ r Á + l Á Ë - Ë 3 ʈ Ê ˆ and r 6 Á + m. Á Ë3 Ë - Find the acute angle between the lines. [7] OCR 7 4754A/ June 7
3 Û 6 Two students are trying to evaluate the integral Ù ı + e - x dx. Sarah uses the trapezium rule with strips, and starts by constructing the following table. x.5 + e -x.696.6.655 (i) Complete the calculation, giving your answer to 3 significant figures. [] Anish uses a binomial approximation for and then integrates this. (ii) Show that, provided e x is suitably small, + e e e + - e x ( ) ª + - -x -x - x. 8 Û - x (iii) Use this result to evaluate Ù + e dx approximately, giving your answer to 3 significant ı figures. OCR 7 4754A/ June 7 [Turn over
4 Section B (36 marks) 7 Data suggest that the number of cases of infection from a particular disease tends to oscillate between two values over a period of approximately 6 months. (a) Suppose that the number of cases, P thousand, after time t months is modelled by the equation P Thus, when t, P. sin t. (i) By considering the greatest and least values of sin t, write down the greatest and least values of P predicted by this model. [] dp (ii) Verify that P satisfies the differential equation [5] dt P cos t. (b) An alternative model is proposed, with differential equation As before, P when t. dp dt (P P) cos t. (*) (i) Express in partial fractions. P(P ) (ii) Solve the differential equation (*) to show that ln ÊP -ˆ Ë sin t. P [5] This equation can be rearranged to give P. t - e sin (iii) Find the greatest and least values of P predicted by this model. OCR 7 4754A/ June 7
8 y B 5 A C(x, y) O x Fig. 8 In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations x cos q 5 cos q, y sin q 5 sin q, ( q p), where x and y are in metres. dy (i) Show that dx cos q cos q. sin q sin q dy Verify that when dx Hence find the exact coordinates of the highest point A on the path of C. [6] q 3 p. (ii) Express x y in terms of q. Hence show that x y 5 cos q. (iii) Using this result, or otherwise, find the greatest and least distances of C from O. [] You are given that, at the point B on the path vertically above O, cos q cos q. (iv) Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. OCR 7 4754A/ June 7
4754 Mark Scheme June 7 Section A sin θ 3 cos θ R sin(θ α) R(sin θ cos α cos θ sin α) R cos α, R sin α 3 R + 3 R tan α 3 α 7.57 sin(θ 7.57 ) θ 7.57 sin (/ ) θ 7.57 8.43, 6.57 θ 9, 33. Normal vectors are and 3 4 3. 6+ 4 4 planes are perpendicular. 3 (i) y ln x x e y V (ii) π x dy y y π( e ) dy πe dy * y y πe dy π e ½ π(e 4 ) 4 x x t t + t x + + x x+ + x+ 3 y + + x+ + x+ x + [7] equating correct pairs oe ft www cao (7.6 or better) oe ft R, α www and no others in range (MR- for radians) ½ e y substituting limits in kπe or equivalent, but must be exact and evaluate e as. y Solving for t in terms of x or y Subst their t which must include a fraction, clearing subsidiary fractions/ changing the subject oe www or t 3+ 3+ x t + x t + t 3 t+ t t+ t t + y t + substituting for x or y in terms of t clearing subsidiary fractions/changing the subject 4
4754 Mark Scheme June 7 x λ 5 r + λ y + λ 3 z + 3λ When x, λ, λ y + λ 6, z + 3λ 5 point lies on first line x μ r 6 + μ y 6 3 z 3 μ When x, μ, y 6, z 3 μ 5 point lies on second line Angle between and is θ, where 3 + + 3 cosθ 4. 5 7 7 θ 46.8 acute angle is 33. cao [7] Finding λ or μ checking other two coordinates checking other two co-ordinates Finding angle between correct vectors use of formula 7 ± 7 Final answer must be acute angle 6(i) (.696 +.655 A.5[ +.6]. (3 s.f.) ( + e ). + e + ( e! ) +... x x + e e * 8 (ii) x / x x (iii) I 8 ( + x e x e ) dx x x x e + e 6 4 ( e e ) ( e + + e ) 6 6.9335.845. (3 s.f.) cao [] Correct expression for trapezium rule Binomial expansion with p ½ Correct coeffs integration substituting limits into correct expression 5
4754 Mark Scheme June 7 Section B 7 (a) (i) P max P min /3. + (ii) P ( sin t) sint dp ( sin t). cost dt cost ( sin t) 4 P cost cost ( sin t) cost dp ( sin t) dt (b)(i) A B + P(P ) P P A(P ) + BP P(P ) A(P ) + BP P A A P ½ A. + ½ B B So + P(P ) P P (ii) dp ( P P )cos t dt dp cos tdt P P ( ) dp costdt P P ln(p ) ln P ½ sin t + c When t, P ln ln ½ sin + c c P ln( ) sin t * P [] D [5] [5] chain rule ( ) soi (or quotient rule,numerator,denominator ) attempt to verify or by integration as in (b)(ii) correct partial fractions substituting values, equating coeffs or cover up rule A B separating variables ln(p ) ln P ft their A,B from (i) ½ sin t finding constant (iii) P max.847 / e P min.78 / e www www 6
4754 Mark Scheme June 7 dy cosθ + cos θ 8 (i) dx sinθ sin θ cosθ + cos θ * sinθ + sin θ When θ π/3, dy cos π / 3 + cos π / 3 dx sin π / 3+ sin π / 3 as cos π/3 ½, cos π/3 ½ At A x cos π/3 + 5 cos π/3 ½ y sin π/3 + 5 sin π/3 5 3/ [6] dy/dθ dx/dθ or solving cosθ+cosθ substituting π/3 into x or y ½ 5 3/ (condone 3 or better) (ii) x + y ( cosθ + 5cos θ) + (sinθ + 5sin θ) cos θ + cosθcos θ + 5cos θ + sin θ + sin θsin θ + 5sin θ + cos(θ θ) + 5 5 + cos θ * D expanding cos θ cos θ + sin θ sin θ cos(θ θ) or substituting for sin θ and cos θ (iii) Max 5 + 5 min 5 5 [] (iv) cos θ +cos θ cos θ ± ± 3 4 4 At B, cos θ + 3 OB 5 + 5( + 3) 75 + 5 3 6.6 OB 6.6.7 (m) quadratic formula or θ68.53 or.radians, correct root selected or OBsinθ+5sinθ ft their θ/cosθ oe cao 7