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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level MATHEMATICS Paper 1 Pure Mathematics 1 (P1) 9709/12 May/June 2012 1 hour 45 minutes Additional Materials: Answer Booklet/Paper Graph Paper List of Formulae (MF9) *9656618107* READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all t
       *           9           6           5           6           6           1           8           1           0           7          *      UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSGeneral Certificate of Education Advanced Subsidiary Level and Advanced Level MATHEMATICS 9709/12 Paper 1 Pure Mathematics 1 (P1) May/June 20121 hour 45 minutes  Additional Materials: Answer Booklet/Paper Graph Paper List of Formulae (MF9) READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.Write your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use a soft pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid. Answer  all the questions.Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles indegrees, unless a different level of accuracy is specified in the question.The use of an electronic calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together.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 75.Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.This document consists of  4 printed pages. JC12 06_9709_12/RP© UCLES 2012 [Turn over   21 O x y 23  y =62 – 3  x The diagram shows the region enclosed by the curve y = 62  x  − 3, the x  -axis and the lines x  = 2 and  x  = 3. Find, in terms of  π  , the volume obtained when this region is rotated through 360 ◦ about the  x  -axis. [4] 2 The equation of a curve is y = 4 √   x  + 2 √   x  . (i) Obtain an expression ford  y d  x  . [3] (ii) A point is moving along the curve in such a way that the x  -coordinate is increasing at a constantrate of 0.12 units per second. Find the rate of change of the y -coordinate when x  = 4. [2] 3 The coefficient of  x  3 in the expansion of  ( a + x  ) 5 + ( 2 − x  ) 6 is 90. Find the value of the positiveconstant a . [5] 4 The point A has coordinates (− 1, − 5 ) and the point B has coordinates ( 7, 1 ) . The perpendicularbisector of  AB meets the x  -axis at C  and the y -axis at D . Calculate the length of  CD . [6] 5 (i) Prove the identity tan  x  + 1tan  x  ≡ 1sin  x  cos  x  . [2] (ii) Solve the equation2sin  x  cos  x  = 1 + 3tan  x  , for 0 ◦ ≤ x  ≤ 180 ◦ . [4] © UCLES 2012 9709/12/M/J/12  36  AO 8cm2.4rad  B The diagram shows a metal plate made by removing a segment from a circle with centre O and radius8cm. The line AB is a chord of the circle and angle AOB = 2.4 radians. Find (i) the length of  AB , [2] (ii) the perimeter of the plate, [3] (iii) the area of the plate. [3] 7 (a) In an arithmetic progression, the sum of the first n terms, denoted by S  n , is given by S  n = n 2 + 8 n .Find the first term and the common difference. [3] (b) In a geometric progression, the second term is 9 less than the first term. The sum of the secondand third terms is 30. Given that all the terms of the progression are positive, find the first term.[5] 8 (i) Find the angle between the vectors 3 i − 4 k and 2 i + 3  j − 6 k . [4]The vector −−→ OA has a magnitude of 15 units and is in the same direction as the vector 3 i − 4 k . Thevector −−→ OB has a magnitude of 14 units and is in the same direction as the vector 2 i + 3  j − 6 k . (ii) Express −−→ OA and −−→ OB in terms of  i , j and k . [3] (iii) Find the unit vector in the direction of  −−→  AB . [3] [Questions 9 and 10 are printed on the next page.] © UCLES 2012 9709/12/M/J/12 [Turn over  49  B A x yO y x x = – +8 –10 2 The diagram shows part of the curve y = −  x  2 + 8  x  − 10 which passes through the points A and B . Thecurve has a maximum point at A and the gradient of the line BA is 2. (i) Find the coordinates of  A and B . [7] (ii) Find   y d  x  and hence evaluate the area of the shaded region. [4] 10 Functions f and g are defined byf : x  → 2  x  + 5 for x  ∈  ,g : x  → 8  x  − 3for x  ∈  , x  ≠ 3. (i) Obtain expressions, in terms of  x  , for f  − 1 (  x  ) and g − 1 (  x  ) , stating the value of  x  for which g − 1 (  x  ) is not defined. [4] (ii) Sketch the graphs of  y = f  (  x  ) and y = f  − 1 (  x  ) on the same diagram, making clear the relationshipbetween the two graphs. [3] (iii) Given that the equation fg (  x  ) = 5 − kx  , where k  is a constant, has no solutions, find the set of possible values of  k  . [5] Permissiontoreproduceitemswherethird-partyowned material protectedbycopyrightisincludedhasbeen sought and clearedwhere possible. Everyreasonableeffort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher willbe pleased to make amends at the earliest possible opportunity.University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.© UCLES 2012 9709/12/M/J/12
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