CAIE
Mathematics
(9709)
Questions & Answers
Sketch the graph of yx a2=- , where a is a positive constant.
Solve the inequality xa xa23 21-- . [2] 2 Express 25 12 69 16 in partial fractions.
The variables x and y satisfy the equation ab yx y21 , where a and b are constants.
Show that the graph of y against x is a straight line.
Given that ab =, state the equation of the straight line in the form ypxq=+ , where p and q are rational numbers in their simplest form.
The equation of a curve is eyy 6e xx22 += . Find the gradient of the curve at the point where y1=.
It is given that the equation ec osx53 =+ has only one root. Show by calculation that this root lies in the interval ..x07 0811 .
Show that if a sequence of values in the interval ..x07 0811 given by the iterative formula lncosxxnn1 converges then it converges to the root of the equation in part (a).
Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3] The diagram shows the curve eyx , where a is a positive constant, and its maximum point M.
Find the exact coordinates of M.
Find the exact value of edxxaxa y .
Show that cossin cos2 /ii i- .
Hence find the exact value of cossin sincosd4 44 22 ii ii i-+ `jy .
The points A, B and C have position vectors ijkOA 24=-++ , OB ij52=+ and ij kOC85 3=+ -, where O is the origin. The line l passes through B and C.
Find a vector equation for l The line l has equation ()ri jk ijk24 32n=-++ ++ -.
Find the coordinates of the point of intersection of l and l
The point D on l is such that ABBD=. Find the position vector of D.
The complex numbers z and ~ are defined by iz1=- and 333i~=-+ .
Express z~ in the form iab+, where a and b are real and in exact surd form.
Express z and ~ in the form er , where r02 and rr1Gi- . Give the exact values of r and i in each case.
On an Argand diagram, the points representing ~ and z~ are A and B respectively. Prove that OAB is an isosceles right-angled triangle, where O is the origin.
Using your answers to part
, prove that tan
By writing secy i= ascos , show that sinsec3ii=
The variables x and i satisfy the differential equation ()sin cosxx93 i+= +`j . It is given that x3= when ri=. Solve the differential equation to find the value of cosi when x0=. Give your answer correct to 3 significant figures. [8] Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. 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 amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.