ADDITIONAL MATHEMATICS

Transcription

1 Oxford Cambridge and RSA Examinations Free Standing Mathematics Qualification (Advanced) ADDITIONAL MATHEMATICS Specimen Paper 699 Additional materials: Electronic calculator TIME hours Candidate Name Centre Number Candidate Number INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces above. Write your answers, in blue or black ink, in the spaces provided on the answer booklet. Answer all the questions. Read each question carefully and make sure you know what you have to do before starting your answer. You are expected to use an electronic calculator for this paper. 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 00. Question number TOTAL For examiner s use only OCR00

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3 Section A Express 4 x + 8x in the form a ( x + p) + q. Hence find the minimum value of 4x + 8x. [6] 5 Expand ( + x) in ascending powers of x, simplifying all the terms. [4] A triangle ABC is such that AB = 5 cm, BC = 8 cm and CA = 7 cm. Show that one angle is 60 o. [5] 4 Given that y = x + x 7 dy, find. dx Use your result to show that the graph of y = x + x 7 has no turning points. [5] 5 The line L goes through the points (, ) and (7, 6). (a) Find the equation of L. (b) Write down the equation of M, the line through the origin which is perpendicular to L. [5] 6 Find the value of ( x + ) dx. [4] Additional Mathematics OCR00 Specimen Question Paper

4 7 y θ O h x The London Eye can be considered to be a circular frame of radius 67.5m on the circumference of which are capsules carrying a number of people round the circle. Take a co-ordinate system where O is the base of the circle and Oy is a diameter. At any time after starting off round the frame, the capsule will be at height h metres when it has rotated θ. (a) Draw a graph of h against θ. [] (b) Give an expression for h in terms of θ. [] (c) Find values of θ when h = 00. [] x 6 8 (a) Simplify to a single fraction the expression. x + x x 6 (b) Hence use the quadratic formula to solve the equation = 4. [7] x + x 9 Show that ( x ) is a factor of x x 4x + = 0. Hence solve the equation x x 4x + = 0. [5] 0 Prove that if the sides of a triangle can be written as ( n ), n and ( n +) for n > then the triangle is right angled. [4] Additional Mathematics 4 OCR00 Specimen Question Paper

5 Section B The diagram shows a grass bank which was constructed as part of an assault course. The rectangle ABCD is horizontal and ABFE is a square inclined at 0 to the horizontal such that E is vertically above D and F is vertically above C. The area of the square ABFE is 600m. X is a point on AB such that AX = m. Calculate (a) the area of ABCD, [] (b) the length of the paths XE and XF, [] (c) the angle between the paths XE and XF, [] (d) the angle of slope of the path XE with the horizontal. [] Additional Mathematics 5 OCR00 Specimen Question Paper

6 A body is falling through a liquid, and the distance fallen is modelled by the formula s = 48t t t seconds is the time. until it comes to rest, where s centimetres is the distance fallen and (a) Find (i) the velocity when t =, [] (ii) the initial velocity, [] (iii) the acceleration when t =, [] (iv) the time when the body comes to rest, [] (v) the distance fallen when the body comes to rest. [] (b) Sketch the velocity/time graph for the period of time until the body comes to rest. [] China cups are packed in boxes of 0. It is known that in 8 are cracked. Find the probability that in a box of 0, chosen at random, (a) no cups are cracked, [] (b) exactly cup is cracked, [4] (c) exactly cups are cracked, [] (d) at least cups are cracked. [] Additional Mathematics 6 OCR00 Specimen Question Paper

7 4 A bicycle factory produces two models of bicycle, A and B. Model A requires 0 hours of unskilled and 0 hours of skilled labour. Model B requires 5 hours of unskilled and 5 hours of skilled labour. The factory employs 0 unskilled and 8 skilled labourers, each of whom work a 40 hour week. (a) Suppose the factory makes x model A and y model B per week. Show that the restriction of unskilled labour results in the inequality 4x + y 80 and find a similar inequality from the restriction on skilled labour. [5] (b) Draw graphs of two lines and shade the feasible region. [] (c) The factory makes a profit of 40 on model A and 60 on model B. Write down the objective function. [] (d) Show that making 5 model A bicycles and 5 model B bicycles is possible and find the number of each that should be made to maximise the profit. [] Additional Mathematics 7 OCR00 Specimen Question Paper

8 Additional Mathematics 8 OCR00 Specimen Question Paper

9 Oxford Cambridge and RSA Examinations Free Standing Mathematics Qualification (Advanced) ADDITIONAL MATHEMATICS MARK SCHEME 699 Note: In the following scheme, the following notations apply. M an M mark is earned for a correct method (or equivalent method) for that part of the question. A method may contain incorrect working, but there must be sufficient evidence that, if correct, it would have given the correct answer. A an A mark is earned for accuracy, but cannot be awarded if the corresponding M mark has not been earned. B a B mark is an accuracy mark awarded independently of any M mark. Note: B,, means that either, or B marks can be scored.

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11 Question Answer Mark Section A 4x + 8x 4( x + x) 4( x i.e. a = 4, p =, q = 5 Minimum value when x = ; + x + ) 4 4( x + ) 4x + 8x = 5 5 A M A ( + x ) = + 5. (x) + 0. (x) + 0. (x) + 5.( x) + (x) M (coeffs) A (coeffs) = x + 70x + 080x + 80x + 4x A Cosine rule applied to B: CosB = = B = (N.B. Calculation of approximate values for A and C with B = 80 - A - C will gain M for correct cosine rule only.) Selection of angle B M cos rule A A or any correct alternative method 4 dy y = x + x 7 = x dx + x + 0 for any x, so no turning points. A B B 5(a) y = 6 x 7 y = x y 6 = x y = x + 4 M A 5(b) Perpendicular line through origin is y + x = 0 6 A x 8 ( x + ) dx = + x = = 5 A 7(a) For graph B, 7(b) 7(c) h = 67.5( cosθ ) 00 Cosθ = = θ = and 4. B A Additional Mathematics OCR00 Specimen Mark Scheme

12 Question Answer Mark Section A 8(a) 8(b) x 6 x( x ) 6( x + ) x 7x x + x ( x + )( x ) ( x + )( x ) x 6 = 4 x x + x 7x = 4( x + x ) x + x + 4 = 0 ± 4..4 ± x = = = or =.57 or M A M A 9 f ( ) = = 0 x x 4x + = 0 ( x )( x x 6) = 0 B ( x )( x )( x + ) = 0 x =,, -. A A 0 If Pythagoras Theorem holds then the triangle is right-angled. 4 4 ( n ) + (n) n n + + 4n n + n + ( n + Since this is true for all n > the triangle is right-angled for n >. ) M A A Section A Total: 5 Additional Mathematics OCR00 Specimen Mark Scheme

13 Question Answer Mark Section B (a) (b) AB = AE = 40m AD = AEcos0 = 40cos0 = 7.59 Area ABCD = AB.AD = 40x7.59 = 504m By Pythagoras B M EX = 40 + = 744 = 4.76 m A Similarly FX = = 84 = 48.8 m A (c) (d) Cosine Rule: CosEXF = = EXF = The angle of slope requires an extra length, i.e. ED ED = 40sin 0 sin x 40sin 0 = = 0.76 x = M A A M A A (a)(i) s = 48t t v = 48 t v = 48 = 45 cms - (a)(ii) v0 = 48 0 = 48 cms - B (a)(iii) v = 48 t a = 6t a = 6 cms - A (a)(iv) v = 48 t v = 0 when t = 48 t = 4 seconds (a)(v) s = 48t t s 4 = = 8cm (b) B, Additional Mathematics OCR00 Specimen Mark Scheme

14 Question Answer Mark Section B (a) A (b) M(Coeff) A(Coeff) A A (c) A(Coeff) A (d) P(0) P() P() = (a) (b) (c) = [0.0 from exact answers to (a), (b), (c)] A Additional Mathematics 4 OCR00 Specimen Mark Scheme

15 Question Answer Mark Section B 4(a) 0 hrs of unskilled labour per Model A means 0x for x bikes. Likewise 5 hrs for Model B means 5y for y bikes giving 0x + 5y. Since the max possible is 0 x 40 hrs = 400 this gives 0x + 5y 400 i.e. 4x + y 80 For skilled labour 0x + 5y 8 x 40 i.e. x + 5y 64 B,, 4(b) B + B for lines B shading 4(c) 4(d) P = 40 x + 60y x y 4 x + y x + 5y P i.e. lies in feasible region so the best solution is 4 Model A and 7 Model B giving a profit of 980. B, Section B Total: 48 Total mark available: 00 Additional Mathematics 5 OCR00 Specimen Mark Scheme

16 Assessment Grid Question Marks AO AO AO AO4 AO5 Totals A M A 6 A 4 B M A A 5 4 A B B 5 5 M A 5 6 A A 4 4 B 7 B 7 A M A B A A 5 0 A 4 4 B A A A B A B A A A A A A B B 4 B Totals Additional Mathematics 6 OCR00 Specimen Mark Scheme