A-level 进阶数学真题 06年 纯数学.pdf

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certifi cate of Education Advanced Level FURTHER MATHEMATICS 9231 01 Paper 1 May June 2006 3 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 on both sides of the paper You may use a soft pencil for any diagrams or graphs Do not use staples paper clips highlighters glue or correction fl uid Answer all the questions Give non exact numerical answers correct to 3 signifi cant fi gures or 1 decimal place in the case of angles in degrees unless a different level of accuracy is specifi ed 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 receive credit You are reminded of the need for clear presentation in your answers The number of marks is given in brackets at the end of each question or part question At the end of the examination fasten all your work securely together This document consists of 4 printed pages UCLES 2006 Turn over 2 1Express un 1 4n2 1 in partial fractions and hence fi nd N n 1 unin terms ofN 4 Deduce that the infi nite seriesu1 u2 u3 is convergent and state the sum to infi nity 2 2Draw a diagram to illustrate the regionRwhich is bounded by the curve whose polar equation is r cos2 and the lines 0 and 1 6 2 Determine the exact area ofR 4 3Prove by induction or otherwise that 232n 312n 46 is divisible by 48 for all integersn 0 6 4The linear transformation T 4 4is represented by the matrix A 1 123 2 345 5 61014 4 5811 Show that the dimension of the range space of T is 2 3 LetMbe a given 4 4 matrix and letSbe the vector space consisting of vectors of the formMAx wherex 4 Show that ifMis non singular then the dimension ofSis 2 4 5The curveChas equation y 2x 3 x 1 x 1 i Write down the equations of the asymptotes ofC 2 ii Find the set of values ofxfor whichCis above its oblique asymptote and the set of values ofx for whichCis below its oblique asymptote 3 iii Draw a sketch ofC stating the coordinates of the points of intersection ofCwith the coordinate axes 4 UCLES 20069231 01 M J 06 3 6 a The equation of a curve is y 2 3 3 x 3 2 Find the length of the arc of the curve from the origin to the point wherex 1 4 b The variablesxandyare such that y3 dy dx 3 x4 6 Given thaty 1 whenx 1 fi nd the value of d2y dx2 whenx 1 5 7Given that In 1 2 0 sinnxdx wheren 0 prove that In 2 n 1 n 2 In 4 The region bounded by thex axis and the arc of the curvey sin4xfromx 0 tox is denoted byR Determine they coordinate of the centroid ofR 5 8Obtain the general solution of the differential equation d2y dt2 6dy dt 25y 80e 3t 5 Given thaty 8 and dy dt 8 whent 0 show that 0 ye3t 10 for allt 5 9Given that ei andnis a positive integer show that n 1 n 2cosn and n 1 n 2isinn 2 Hence express cos7 in the form pcos7 qcos5 rcos3 scos where the constantsp q r sare to be determined 4 Find the mean value of cos72 with respect to over the interval 0 1 4 leaving your answer in terms of 5 UCLES 20069231 01 M J 06 Turn over 4 10The equation of the plane is 2x 3y 4 48 Obtain a vector equation of in the form r a b c wherea bandcare of the formpi qi rjandsi tkrespectively and wherep q r s tare integers 6 The linelhas vector equationr 29i 2j k 5i 6j 2k Show thatllies in 3 Find in the formax by c d the equation of the plane which containsland is perpendicular to 4 11Answer onlyoneof the following two alternatives EITHER Obtain the sum of the squares of the roots of the equation x4 3x3 5x2 12x 4 0 2 Deduce that this equation does not have more than 2 real roots 3 Show that in fact the equation has exactly 2 real roots in the interval 3 x 0 5 Denoting these roots by and and the other 2 roots by and show that 2 4 OR ThesquarematrixAhas asaneigenvaluewithcorrespondingeigenvectorx Thenon singularmatrix Mis of the same order asA Show thatMxis an eigenvector of the matrixB whereB MAM 1 and that is the corresponding eigenvalue 3 It is now given that A 100 a 30 bc 5 andM 101 010 001 i Write down the eigenvalues ofAand obtain corresponding eigenvectors in terms ofa b c 4 ii Find the eigenvalues and corresponding eigenvectors ofB 4 iii Hence fi nd a matrixQand a diagonal matrixDsuch thatBn QDQ 1 3 You are not required to fi ndQ 1 Permission to reproduce items where third party owned material protected by copyright is included has been sought and cleared where possible Every reasonable effort has been made by the publisher UCLES to trace copyright holders but if any items requiring clearance have unwittingly been included the publisher will be pleased to make