CAIE Mathematics (9709)

Solve the equation . Give your answer correct to 3 decimal places.

Expand ()()xx61 2 in ascending powers of x, up to and including the term in x , simplifying the coefficients.

State the set of values of x for which the expansion is valid.

On an Argand diagram shade the region whose points represent complex numbers z which satisfy both the inequalities z32iG- and ()argz12i GGrr -- . [5] * 0000800000005 *

The square roots of 145i-- can be expressed in the Cartesian form xyi+, where x and y are real and exact. By first forming a quartic equation in x or y, find the square roots of 145i-- in exact Cartesian form.

By sketching a suitable pair of graphs, show that the equation sinxx22 has only one root in the interval x011 r.

Show by calculation that this root lies between 1 and 1.5.

Use the iterative formula sinxx 22 nn1 2 with an initial value of 1.03 to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

Express sinc os72 4ii+ in the form ()cosR ia-, where R02 and r0 11a . Give the value of a correct to 4 decimal places.

Hence solve the equation .sinc osxx72 42 45 += for x011 r.

The variables x and i satisfy the dif ferential equation ()sinc os x24 32 i=+ , and x0= when r i= . Solve the dif ferential equation and obtain an expression for x in terms of i.

With respect to the origin O, the points A, B and C have position vectors given by =- fp, OB fp and OC = fp.

Find a vector equation for the line through A and B.

Using a scalar product, find the exact value of cos BAC.

Hence find the exact area of triangle ABC.

Find the quotient and remainder when x is divided by x14 +.

Find the exact value of ()tanxx x2d ; . [6] The diagram shows the graph of sincosyx x52 = for x0 GG r and its maximum point M.

Find the exact x-coordinate of M.

By using the substitution cosux= , find the area of the region bounded by the curve, the x-axis between x0= and x r=, and the line x r=. [5] 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. Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.