CAIE Mathematics (9709)

The first term of an arithmetic progression is -20 and the common dif ference is 5.

Find the sum of the first 20 terms of the progression. [2] It is given that the sum of the first 2k terms is 10 times the sum of the first k terms.

Find the value of k.

The equation of a curve is yx23 =- . Two points A and B with x-coordinates 2 and ()h2+ respectively lie on the curve.

Find and simplify an expression for the gradient of the chord AB in terms of h.

Explain how the gradient of the curve at the point A can be deduced from the answer to part

, and state the value of this gradient.

Find the term independent of x in the expansion of each of the following:

x +eo

()xx -+eo .

The function f is defined by ()fx for x

(i) State the value of ()f1-.

The diagram shows the graph of ()fyx= . Sketch the graph of ()fyx on this diagram. Show any relevant mirror line.

Find an expression for ()fx and state the domain of the function f The function g is defined by ()gxx32=+ for xRd.

Solve the equation ()fg fx =bl. [3] r cm r cm 2r cm2r cm 2rad radθ radB EC D The diagram shows a metal plate OABCDEF consisting of sectors of two circles, each with centre O. The radii of sectors AOB and EOF are r cm and the radius of sector COD is 2r cm. Angle AOB = angle EOF = i radians and angle COD = 2i radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is cm10 Given that r 2 and 1i, find the values of r and i.

By expressing xx28 11 -+ + in the form ()axbc -- +, where a, b and c are positive integers, find the coordinates of the vertex of the graph with equation yx x28 11 =- ++ .

The diagram shows part of the curve with equation yx x28 11 =- ++ and the line with equation yx89=+ . Find the area of the shaded region.

The equation of a circle is xypx yq20 ++++ =, where p and q are constants.

Express the equation in the form () ()xa yb r 22 2 -+ -= , where a is to be given in terms of p and is to be given in terms of p and q. [2] The line with equation xy210+= is the tangent to the circle at the point A (4, 3).

(i) Find the equation of the normal to the circle at the point A.

Find the values of p and q.

The equation of a curve is yk xkx22 =- + and the equation of a line is ykxp=+ , where k and p are constants with k0111 .

It is given that one of the points of intersection of the curve and the line has coordinates ,bl. Find the values of k and p, and find the coordinates of the other point of intersection.

It is given instead that the line and the curve do not intersect. Find the set of possible values of p.

A function f with domain x02 is such that ()()fxx x82310 =- -l . It is given that the curve with equation ()fyx= passes through the point (1, 0).

Find the equation of the normal to the curve at the point (1, 0).

Find ()fx. [4] It is given that the equation ()fx0=l can be expressed in the form xx1251281920 -+ =.

Determine, making your reasoning clear , whether f is an increasing function, a decreasing function or neither. [3] Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. BLANK PAGE * 0000800000019 * 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 *