# Mathematics (JAN12MPC201) General Certificate of Education Advanced Subsidiary Examination January Unit Pure Core TOTAL

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1 Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Pure Core 2 Friday 13 January 2012 General Certificate of Education Advanced Subsidiary Examination January am to am For this paper you must have: the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator. Time allowed 1 hour 30 minutes MPC2 Instructions Use black ink or black ball-point pen. Pencil should only be used for drawing. Fill in the es at the top of this page. Answer all questions. Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. You must answer the questions in the spaces provided. around each page. Show all necessary working; otherwise marks for method may be lost. Do all rough work in this book. Cross through any work that you do not want to be marked. Information The marks for questions are shown in brackets. The maximum mark for this paper is 75. Advice Unless stated otherwise, you may quote formulae, without proof, from the booklet. You do not necessarily need to use all the space provided. Question TOTAL Mark (JAN12MPC201) 6/6/ MPC2

2 2 Answer all questions in the spaces provided. 1 The diagram shows a sector OAB of a circle with centre O and radius 6 cm. 6cm A O y 6cm B The angle between the radii OA and OB is y radians. The area of the sector OAB is 21.6 cm 2. (a) Find the value of y. (2 marks) (b) Find the length of the arc AB. (2 marks) (02)

3 3 Turn over s (03)

4 4 2 (a) Use the trapezium rule with five ordinates (four strips) to find an approximate value for ð x x þ 1 dx giving your answer to three significant figures. (4 marks) (b) State how you could obtain a better approximation to the value of the integral using the trapezium rule. (1 mark) (04)

5 5 Turn over s (05)

6 6 3 (a) Write pffiffiffiffiffi 4 x 3 in the form x k. (1 mark) (b) Write 1 x2 pffiffiffiffiffi in the form x p x q. (2 marks) 4 x 3 (06)

7 7 Turn over s (07)

8 8 4 The triangle ABC, shown in the diagram, is such that AB is 10 metres and angle BAC is 150. A 10 m 150 B C The area of triangle ABC is 40 m 2. (a) Show that the length of AC is 16 metres. (2 marks) (b) Calculate the length of BC, giving your answer, in metres, to two decimal places. (3 marks) (c) Calculate the smallest angle of triangle ABC, giving your answer to the nearest 0.1. (3 marks) (08)

9 9 Turn over s (09)

10 10 5 (a) (i) Describe the geometrical transformation that maps the graph of y ¼ 1 þ x 6 onto 3 the graph of y ¼ð1þ2xÞ 6. (2 marks) (ii) The curve y ¼ 1 þ x 6 3 is translated by the vector to give the curve y ¼ gðxþ. 3 0 Find an expression for gðxþ, simplifying your answer. (2 marks) (b) The first four terms in the binomial expansion of 1 þ x 6 are 1 þ ax þ bx 2 þ cx 3. 3 Find the values of the constants a, b and c, giving your answers in their simplest form. (4 marks) (10)

11 11 Turn over s (11)

12 12 6 An arithmetic series has first term a and common difference d. The sum of the first 25 terms of the series is (a) Show that a þ 12d ¼ 140. (3 marks) (b) The fifth term of this series is 100. Find the value of d and the value of a. (4 marks) (c) The nth term of this series is u n. Given that 33 X 25 u n X! k u n ¼ 67 X k u n n¼1 n¼1 n¼1 find the value of X k n¼1 u n. (3 marks) (12)

13 13 Turn over s (13)

14 14 7 (a) Sketch the graph of y ¼ 1 2x, indicating the value of the intercept on the y-axis. (2 marks) (b) Use logarithms to solve the equation 1 2 x ¼ 5, giving your answer to three significant 4 figures. (3 marks) (c) Given that log a ðb 2 y Þþ3log a y ¼ 3 þ 2 log a a express y in terms of a and b. Give your answer in a form not involving logarithms. (5 marks) (14)

15 15 Turn over s (15)

16 16 8 (a) Given that 2 sin y ¼ 7 cos y, find the value of tan y. (2 marks) (b) (i) Use an appropriate identity to show that the equation 6 sin 2 x ¼ 4 þ cos x can be written as 6 cos 2 x þ cos x 2 ¼ 0 (2 marks) (ii) Hence solve the equation 6 sin 2 x ¼ 4 þ cos x in the interval 0 < x < 360, giving your answers to the nearest degree. (6 marks) (16)

17 17 Turn over s (17)

18 18 9 The diagram shows part of a curve crossing the x-axis at the origin O and at the point Að8, 0Þ. Tangents to the curve at O and A meet at the point P, as shown in the diagram. y P O Að8, 0Þ x The curve has equation y ¼ 12x 3x 5 3 (a) (b) (i) Find dy dx. Find the value of dy dx at O. (2 marks) at the point O and hence write down an equation of the tangent (2 marks) (c) (ii) Show that the equation of the tangent at Að8, 0Þ is y þ 8x ¼ 64. (3 marks) Find ð 12x 3x 5 3 dx. (3 marks) (d) Calculate the area of the shaded region bounded by the curve from O to A and the tangents OP and AP. (7 marks) (18)

19 19 Turn over s (19)

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