CAIE Mathematics (9709)

Express xx9368 -+ in the form ()pxqr ++ , where p, q and r are constants.

Hence find the set of values of the constant k for which the equation xx k9368 -+ = has no real roots.

Find the exact roots of the equation xx9368 15 -+ =-. [2] mTTTTuTTTTTTTim

Find the term independent of x in the expansion of x -bl . [3] mTTTTuTTTTTTTsm

The graph of ()yxf= is transformed to the graph of ()yx32f=+ . Describe fully the two transformations which have been combined to give the resulting graph.

A different graph has equation ()yxg= . This graph is stretched by scale factor 3 in the y-direction and then reflected in the y-axis. W rite down the equation of the transformed graph in terms of the function g. [2] mTTTTuTTTTTTT m

The equation of a curve is such that =+ , where k is a constant. The curve passes through the point S (2, 20) and the gradient of the curve at S is

Find the value of k.

The coordinates of a point T on the curve are (1, t). Find the value of t. [5] mTTTTuTTTTTTTdm a b The equation of a curve is yx x4 =- . The curve has a maximum point when xa= and crosses the x-axis at the point with coordinates (b, 0), where b02. The shaded region is bounded by the curve, the line xa= and the x-axis (see diagram).

Find the value of a. [3] mTTTTuTTTTTTTom

Find the exact area of the shaded region. [5] mTTTTuTTTTTTTcm

Sketch the graph of sinyx32=+ for rx02GG . [2] rr 2

Determine the number of solutions in the interval rx02GG of each of the following equations.

sinxx32+=

sinxx32 5+=- [1] mTTTTuTTTTTTTum

Solve the equation sinc osxx32 51 += - for rx02GG . [5] mTTTTuTTTTTTTmm

The coordinates of the points P and Q are (1, 1) and (7, 1 1) respectively. The line segment PQ forms a diameter of a circle.

Find the equation of the circle.

Find the equation of the tangent to the circle at the point Q. [3] mTTTTuTTTTTThTm

The other point on the circle with x-coordinate 7 is R. Find the coordinates of the point of intersection of the tangent at Q with the tangent at R. [4] mTTTTuTTTTTThhm

The first three terms of a geometric progression are a, b and c respectively, where a, b and c are positive constants. The first three terms of an arithmetic progression are a, b and -3c respectively .

Show that aa cc1090 -+ =. [3] It is now given that a9= and c takes the smaller of its two possible values.

(i) Find the sum to infinity of the geometric progression. [5] mTTTTuTTTTTThim

Find the sum of the first 20 terms of the arithmetic progression. [3] mTTTTuTTTTTThsm

The function f is defined by () () () xx36 for x22.

Find an expression for ()xfl and hence determine whether f is an increasing function, a decreasing function or neither.

State whether f exists. Give a reason for your answer. [1] mTTTTuTTTTTTh m The function g is defined by ()xx43g=- for xa2.

Find the range of g in terms of the constant a.

Find the set of values of a for which the composite function fg exists. [2] mTTTTuTTTTTThdm The diagram shows a circle with centre A and radius r passing through points B, C and D. A larger circle of radius s has centre C and passes through B and D. The length BD is also s.

Show that sr3= . [2] mTTTTuTTTTTThom

Find an expression for the area of the shaded region. Give your answer in the form rabr +`j , where a and b are constants to be found. [7] 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