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9231_s14_qp_11 | Eigenvalues And Eigenvectors | Geometry

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Further Maths 9231 June 2014
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       *                 2                 1                 9                 6                 9                 9                 6                 5                 5                 9                *            Cambridge International Examinations Cambridge International Advanced Level FURTHER MATHEMATICS  9231/11 Paper 1  May/June 20143 hours  Additional Materials: Answer Booklet/Paper Graph Paper List of Formulae (MF10) 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 an HB pencil for any diagrams or graphs.Do not use staples, paper clips, glue or correction fluid.DO  NOT  WRITE IN ANY BARCODES. 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 a calculator is expected, where appropriate.Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receivecredit.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.This document consists of   4  printed pages. JC14 06_9231_11/RP© UCLES 2014  [Turn over  www.onlineexamhelp.com     w    w    w .    o    n     l     i    n    e    e    x    a    m     h    e     l    p .    c    o    m  21  The equation  x  3 +  px   +  q  =  0, where  p  and  q  are constants, with  q  ≠  0, has one root which is thereciprocal of another root. Prove that  p + q 2 =  1. [5] 2  Expand and simplify   r   + 1  4 − r  4 . [1]Use the method of differences together with the standard results for n  r  = 1 r   and n  r  = 1 r  2 to show that n  r  = 1 r  3 =  14 n 2  n + 1  2 .   4  3  Prove by mathematical induction that, for all non-negative integers  n ,11 2 n + 25 n + 22is divisible by 24. [6] 4  Obtain the general solution of the differential equationd 2  x  d t  2  − 6d  x  d t   + 25  x   =  195sin2 t  .   6  5  The curve  C   has polar equation  r   =  a  1 + sin    , where  a  is a positive constant and 0  ≤  <  2   . Drawa sketch of   C  . [2]Find the exact value of the area of the region enclosed by  C   and the half-lines  =  13    and  =  23   . [4] 6  The linear transformation T :   4 →  4 is represented by the matrix  M , where M  =   2  − 1 1 32 0 0 56  − 2 2 1110  − 3 3 19  . (i)  Find the rank of   M  and state a basis for the range space of T. [4] (ii)  Obtain a basis for the null space of T. [4] 7  Use de Moivre’s theorem to show thattan5  =  5 t  − 10 t  3 + t  5 1 − 10 t  2 + 5 t  4 ,where  t   =  tan   . [4]Deduce that the roots of the equation  t  4 − 10 t  2 + 5  =  0 are  ± tan  15    and  ± tan  25   . [3]Hence show that tan  15   tan  25  =  5. [2] © UCLES 2014 9231/11/M/J/14  38  The curve  C   has parametric equations  x   =  t  2 ,  y  =  t  −  13 t  3 , for 0  ≤  t   ≤  1.Find (i)  the arc length of   C  , [5] (ii)  the surface area generated when  C   is rotated through 2    radians about the  x  -axis. [5] 9  The matrix  M , where M  =  − 2 2 22 1 2 − 3  − 6  − 7  ,has an eigenvector   01 − 1  . Find the corresponding eigenvalue. [2]It is given that if the eigenvalues of a general 3 × 3 matrix  A , where A  =  a b cd e f g h i  ,are   1 ,   2  and   3  then  1  + 2  + 3  =  a + e + i andthe determinant of   A  has the value   1  2  3 .Use these results to find the other two eigenvalues of the matrix  M , and find correspondingeigenvectors. [8] 10  It is given that  I  n  =    14   0 sin 2 n  x  cos  x   d  x  , where  n  ≥  0. Show that  I  n  −  I  n + 1  =  2 −  n + 12  2 n + 1.   5  Hence show that     14   0 sin 6  x  cos  x   d  x   =  ln  1 + 2  −  73120  2. [5] 11  The line  l 1  passes through the points  A  2, 3,  − 5   and  B  8, 7,  − 13  . The line  l 2  passes through thepoints  C   − 2, 1, 8   and  D  3,  − 1, 4  . Find the shortest distance between the lines  l 1  and  l 2 . [5]The plane    1  passes through the points  A ,  B  and  D . The plane    2  passes though the points  A ,  C  and  D . Find the acute angle between    1  and    2 , giving your answer in degrees. [6] © UCLES 2014 9231/11/M/J/14  [Turn over  412  Answer only  one  of the following two alternatives. EITHER The curve  C   has parametric equations  x   =  t  2 ,  y  =   2 − t   12 , for 0  ≤  t   ≤  2.Find (i)  d 2  y d  x  2  in terms of   t  , [5] (ii)  the mean value of   y  with respect to  x   over the interval 0  ≤  x   ≤  4, [6] (iii)  the  y -coordinate of the centroid of the region enclosed by  C  , the  x  -axis and the  y -axis. [3] OR The curve  C   has equation  y  =  ax  2 + bx  + c x  + d   ,where  a ,  b ,  c  and  d   are constants. The curve cuts the  y -axis at   0,  − 2   and has asymptotes  x   =  2 and  y  =  x  + 1. (i)  Write down the value of   d  . [1] (ii)  Determine the values of   a ,  b  and  c . [6] (iii)  Show that, at all points on  C  , either  y  ≤  3 − 2  6 or  y  ≥  3 + 2  6. [7] Permission to reproduce items where third-party owned material protected bycopyright isincluded has been sought and cleared where possible. Every reasonableeffort 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.Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge LocalExaminations Syndicate (UCLES), which is itself a department of the University of Cambridge.© UCLES 2014 9231/11/M/J/14 www.onlineexamhelp.com     w    w    w .    o    n     l     i    n    e    e    x    a    m     h    e     l    p .    c    o    m
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