# Wednesday 30 May 2012 Afternoon

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2 2 Formulae Sheet: 6993 Additional Mathematics C In any triangle ABC Cosine rule a 2 = b 2 + c 2 2bc cos A A b c a B Binomial expansion When n is a positive integer (a + b) n = a n + ( n 1) an 1 b + ( n 2) an 2 b ( n r) an r b r + + b n where ( n r) = n C r = n! r!(n r)! 6993 Jun12

3 3 Section A 1 (i) Find the range of values of x satisfying x 2 4x [3] (ii) Show this range on the number line provided. [1] 2 A die has 6 faces numbered one to six. The die is biased so that when it is thrown the probability of obtaining a six is 1 5. The die is thrown 5 times. Find the probability of obtaining (i) at least 1 six, (ii) exactly 3 sixes. [2] [4] 3 The function f(x) = x 3 + ax + 6 is such that when f(x) is divided by (x 3) the remainder is 12. (i) Show that the value of a is 7. (ii) Factorise f(x). [2] [3] 4 A car moves from rest with constant acceleration on a straight road. When the car passes a point A it is travelling at 10 m s 1 and when it passes a point B further along the road it is travelling at 16 m s 1. The car takes 10 seconds to travel from A to B. Find the distance AB, the constant acceleration. [4] 6993 Jun12 Turn over

4 4 5 (i) Show that the equation 3cos 2 θ = sin θ + 1 can be written as 3sin 2 θ + sin θ 2 = 0. [2] (ii) Solve this equation to find values of θ in the range 0 < θ < 360 that satisfy 3cos 2 θ = sin θ + 1. [4] 6 The equation of a curve is y = 2x 3 9x x. (i) Show that the curve has a stationary point where x = 2. [4] (ii) Determine whether the stationary value where x = 2 is a maximum or minimum. [2] 7 A yachtsman wishes to sail from a port, A, to another port, B, which is 9 km due East of A. Because of the wind he is unable to sail directly East and sails 8 km on a bearing of 070 to point C. N C 8 km Not to scale 70 A 9 km B Calculate (i) the distance he is now from port B, [3] (ii) the angle ABC and hence the bearing on which he must sail to reach port B from point C, correct to the nearest degree. [4] 6993 Jun12

5 5 2 8 (i) Show that (x 2 + 2x 3) dx = 2 3. [3] 0 The diagram shows part of the curve y = x 2 + 2x y x (ii) Marc claims that the total area between the curve, the x-axis and the lines x = 0 and x = 2 is Explain why he is wrong. [1] (iii) Calculate the total area between the curve, the x-axis and the lines x = 0 and x = 2. [3] 9 The height above the ground of a seat on a fairground big wheel is h metres. At time t minutes after the wheel starts, h is given by h = 7 5cos(480t). (i) Write down the initial height above the ground of the seat (when t = 0). [1] (ii) Find the greatest height reached by the seat. [2] (iii) Calculate the time of the first occasion when the seat is 9 metres above the ground. Give your answer correct to the nearest second. [4] 6993 Jun12 Turn over

6 6 Section B 10 A (1, 10), B (8, 9) and C (7, 2) are three points. (i) Find the coordinates of the midpoint, M, of AC. [1] (ii) Find the equation of the circle with AC as diameter. [4] (iii) Show that B lies on this circle. [1] (iv) Prove that AM and BM are perpendicular. [3] (v) BD is a diameter of this circle. Find the coordinates of D. [3] 11 The shaded region in the diagram shows a wooden shape. The curve has equation y = 1 2 x2 and the coordinates of A are ( 2, 2). y B A O x The line AB is the normal to the curve at the point A. (i) Find the equation of the line AB. [5] (ii) Find the coordinates of the point B where the line AB meets the curve again. [3] (iii) Find the shaded area. [4] 6993 Jun12

7 7 12 The Highway Code gives a table of shortest stopping distances (d feet) for a vehicle travelling at v miles per hour. The formula used for this table is given by Two entries in the table are given below. d = av 2 + bv. v mph d feet (i) By forming and solving a pair of simultaneous equations in a and b, show that the formula is d = v2 + v. [5] 20 (ii) Find the difference between the stopping distances for a car travelling at 65 mph and a car travelling at 70 mph. [3] (iii) Many drivers maintain a distance of 50 feet or less when driving on a motorway. Use the formula in part (i) to find the speed at which the shortest stopping distance is 50 feet. [4] Question 13 is printed overleaf 6993 Jun12 Turn over

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