CAIE Mathematics (9709)

Sketch the graph of yx a3=+ , where a is a positive constant.

Hence or otherwise solve the inequality xa ax322+- . [2] mTTTTuTTTTTTTim

Solve the equation 32 43 xx12 3##= . Give your answer correct to 3 significant figures. [4] mTTTTuTTTTTTTsm 3 Re2 The shaded region in the Argand diagram, bounded by a line and a circle, represents the complex numbers z satisfying Rez2G and ()iz32G-+ . The point P shown on the diagram is one of the points of intersection of the line and the circle.

Find the complex number represented by the point P. Give your answer in the form ixy+, where x and y are real and exact.

Find the greatest value of argz for points in the shaded region. [3] mTTTTuTTTTTTT m

Find the exact value of dtanxx x y . [6] mTTTTuTTTTTTTdm

It is given that ()() ()fgxx ax =- , where ()fx and ()gx are polynomials. Show that ()xa- is a factor of ()xfl.

It is given that ()x3 - is a factor of xx pxq24 -+ +, where p and q are constants. Find the values of p and q. [5] mTTTTuTTTTTTTom

By sketching a suitable pair of graphs, show that the equation cots inxx22 21=- has exactly one root in the interval rx0 11 .

Show by calculation that the root is in the interval ..x04 0611 .

Use the iterative formula tansin 22 1 eo to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] mTTTTuTTTTTTTcm

The equation of a curve is yx yx23 16 32 3 -- =.

Show thatxxy mTTTTuTTTTTTTum

Hence find the coordinates of the points on the curve at which the normal is parallel to the y-axis. [4] mTTTTuTTTTTTTmm

Prove the identity sins incosxx44 2 / cosxx-`j . [3] mTTTTuTTTTTThTm

Hence find the exact value of .cos dsinxx x4y [5] mTTTTuTTTTTThhm 9 Let () () () xa xa xaxa , where a is a positive constant.

Express ()fx in partial fractions. [5] mTTTTuTTTTTThim

Hence find the exact value of ()fdxx y . Give your answer in the form ()lnap q+ , where p and q are rational. [4] mTTTTuTTTTTThsm h cm 50 cm 50 cm The diagram shows a tank for holding water . The tank is in the shape of a cube of side 50 cm. At time t seconds, the depth of water in the tank is h cm. Water is poured into the tank at a rate of cms500031- Water pours out of the tank through a hole in the bottom at a rate proportional to h When h = 20, the depth of the water is increasing at a rate of .cms04

Show that mTTTTuTTTTTTh m

Given that h = 0 when t = 0, find the time taken for the depth of the water in the tank to reach 20 cm. [5] mTTTTuTTTTTThdm Relative to the origin O, the position vectors of the points A, B and C are ijOA42=- , ijkOB28 4=+ +, and ikOC 26=-+. The midpoint of AB is M, as shown in the diagram.

Find the vectors BM and MC.

Calculate the exact value of the cosine of angle CMB. [3] mTTTTuTTTTTThom

Hence or otherwise find the exact area of triangle ABC. [4] mTTTTuTTTTTThcm Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. mTTTTuTTTTTThum BLANK PAGE * 0000800000019 * mTTTTuTTTTTThmm Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable ef fort 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. BLANK PAGE * 0000800000020 * mTTTTuTTTTTTiTm