CAIE Mathematics (9709)

Show that the equationcos tan7 += can be expressed as sins in12 1207 ii-- =.

Hence solve the equation cos tan7 += for 0 360GGicc .

The function f is defined as follows: ()fxx 1=- for x12.

Find an expression for ()fx The diagram shows the graph of ()gyx= where ()gx for xR!.

State the range of g and explain whether g exists. [2] The function h is defined by ()hx for x0H.

Solve the equation ()hf fx =eo. Give your answer in the form abc+ , where a, b and c are integers.

The first and second terms of an arithmetic progression are tani and sini respectively, where 11i .

Given that r i=, find the exact sum of the first 40 terms of the progression. [4] The first and second terms of a geometric progression are tani and sini respectively, where 11i .

(i) Find the sum to infinity of the progression in terms of i.

Given that r i=, find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures.

The curve with equation yx x28 =- has a minimum point at A and intersects the positive x-axis at B.

Find the coordinates of A and B.

yx x2828 The diagram shows the curve with equation yx x28 =- and the line AB. It is given that the equation of AB is y Find the area of the shaded region between the curve and the line.

The equation of a circle is () ()xy a61 8 -+ += . The line with equation ya x2=- is a tangent to the circle.

Find the two possible values of the constant a.

For the greater value of a, find the equation of the diameter which is perpendicular to the given tangent. [3] B C 2 cm rrad rrad The diagram shows a symmetrical plate ABCDEF. The line ABCD is straight and the length of BC is 2 cm. Each of the two sectors ABF and DCE is of radius r cm and each of the angles ABF and DCE is equal to r radians.

It is given that .cm.r04=

Show that the length .cm.EF24=

Find the area of the plate. Give your answer correct to 3 significant figures.

It is given instead that the perimeter of the plate is 6 cm. Find the value of r. Give your answer correct to 3 significant figures.

A function f is such that ()()fxx x6236 =- -l for xR!.

Determine the set of values of x for which ()fx is decreasing.

Given that ()f1 1=-, find ()fx.

The equation of a curve is ()yx52 5 =- + for x

A point P is moving along the curve in such a way that the y-coordinate of point P is decreasing at 5 units per second. Find the rate at which the x-coordinate of point P is increasing when y32=.

Point A on the curve has y-coordinate 32. Point B on the curve is such that the gradient of the curve at B is 3-. Find the equation of the perpendicular bisector of AB. Give your answer in the form axbyc0++ =, where a, b and c are integers. [6] 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. Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.