Partial preview of the text
Download A level statistics 2025 paper and more Exams Mathematics in PDF only on Docsity!
/ 5 Refer to the diagram for Question 1 in the Diagram Booklet. It shows a sketch of the graph with equation y=|x-4|+ 10 The point P, shown in the diagram, is the vertex of the graph. (a) State the coordinates of P (2 marks) (b) Use algebra to solve [x = 4| +10 = 3x (Solutions relying on calculator technology are not acceptable.) (2 marks) (Total for Question 1 is 4 marks) 6 (a) Sketch the curve with equation y = Q73x stating the coordinates of any points of intersection with the coordinate axes. There are blank axes on pages 34—45 in the Answer Booklet if you wish to use them. (2 marks) (b) By writing 52 as a power of 2, solve the equation -3x_ 1 2 = 3502 (2 marks) (Total for Question 2 is 4 marks) 8 Refer to the diagram for Question 4 in the Diagram Booklet. It shows a sketch of the curve with equation y = f(x), where f (x) = 2* — 10x The curve crosses the X—axis at X = &, &>1, as shown in the diagram. (a) Show that & lies in the interval [5, 6] (2 marks) Given that f(x) =p x 2*—10 WL (b) state the value of the constant p (1 mark) (continued on the next page) continued. (c) Taking Xq =6 as a first approximation to Q, apply the Newton-Raphson method once to f(x) to obtain a second approximation to Show your method and give your answer to 3 significant figures. (2 marks) The curve has a minimum turning point at Q, shown in the diagram. (d) Use the answer to part (b) to find the X coordinate of Q Show your working and give your answer to 3 significant figures. (2 marks) (Total for Question 4 is 7 marks) 11 continued. Using your answer to part (a) and making your method clear, estimate 115 (b) (i) e* dx -1°5 1°5 _ 2 (ii) (« x +7} 0 (3 marks) (Total for Question 5 is 6 marks) 12 (a) Find the first four terms, in ascending powers of X, of the binomial expansion of —_4 _ ~ (2+ 3x)? writing each term in simplest form. (4 marks) (b) Find the range of values of X for which this expansion is valid. (1 mark) (Total for Question 6 is 5 marks) 14 In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Refer to the diagram for Question 8 in the Diagram Booklet. It shows a sketch of part of the curve with equation y=j2x—-7 The region R, shown shaded in the diagram, is bounded by the curve, the X-axis, the y-axis and the line with equation y=5 Find the exact area of R (Total for Question 8 is 6 marks) 15 (a) Prove that the difference of the squares of two consecutive ODD numbers is always a multiple of 8 (3 marks) (b) Show that the difference of the squares of two consecutive EVEN numbers is NOT always a multiple of 8 (1 mark) (Total for Question 9 is 4 marks) 17 40. continued. Given that the equation f(x) = p, where Pp is a constant, has exactly two distinct roots, (c) find the range of possible values for P (3 marks) (Total for Question 10 is 8 marks) a 18 / 41. Given that 6 is measured in radians, prove, from first principles, that the derivative of sin 0 is cos 8 You may assume that as h —> 0, sinh cosh=—1 h — 1 and — 0 (Total for Question 11 is 5 marks) a } 20 42. continued. Given that —> — —> - OA=4i and OC =3j and OD = 2k * the points M and N are the midpoints of AB and CG respectively —> = « the point P lies on MN such that MP =3PN (a) show that = =j+21j43 OP =i + it res (4 marks) The straight line through O and P meets the face BFGC at the point Q (b) Find the coordinates of Q (2 marks) (Total for Question 12 is 6 marks) Turn over / / 21 f 13. In this question you must show all stages of your working. Given that the first three terms of a geometric sequence are 8sin8 3sin20 2+2cos20 and 0<0<5 (a) show that sin@= 8 (4 marks) (continued on the next page) Turn over 23 44. In this question you must show detailed reasoning. The height of the tide, H metres, in a harbour, is measured t hours after midnight. The rate of change of H is modelled by the differential equation gH _ dt = Acos(bt) where A and b are constants and t is measured in radians. (continued on the next page) 24 44. continued. Given that + the largest value of this rate of change is 2 metres per hour * the time between consecutive tides of maximum height, in the harbour, is 12 hours *¢ the maximum height of the tide, in the harbour, is 30 metres TT rashes PERE b find a complete equation for the model, giving H in terms of t (Total for Question 14 is 5 marks) 
