Mathematics 2024 · Oct/Nov

By sketching a suitable pair of graphs, show that the equation cots ecxx2= has exactly one root in the interval rx0 11 .

Show that if a sequence of real values given by the iterative formula (tancosxxnn1 conver ges, then it converges to the root in part

. [1] mTTTTuTTTTTTTsm

The square roots of 68i- 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 68i- in exact Cartesian form. [5] mTTTTuTTTTTTT m

Solve the equation 55 10 xx 2 . Give your answer correct to 3 decimal places. [3] mTTTTuTTTTTTTdm

The complex number u is given by rr() cosi sin cosi sin Find the exact value of arg u.

The complex numbers u and u* are plotted on an Argand diagram. Describe the single geometrical transformation that maps u onto u* and state the exact value of arg u*. [2] mTTTTuTTTTTTTom ln y (0.50, 2.24) (3.40, 8.27) The variables x and y satisfy the equation ayb =, where a and b are constants. The graph of lny against x is a straight line passing through the points (0.50, 2.24) and (3.40, 8.27), as shown in the diagram. Find the values of a and b. Give each value correct to 1 significant figure. [4] mTTTTuTTTTTTTcm

Show that the equation tant antanxx x22 0 +- = may be expressed as tant anxx20 3-=- for tanx0!. [3] mTTTTuTTTTTTTum

Hence solve the equation tant an tan22 42 0 ii i+- = for r011i . Give your answers in exact form. [3] mTTTTuTTTTTTTmm

The parametric equations of a curve are tanxt 2= , costy 2= , for rt0 11 .

Show that cos =- . [4] mTTTTuTTTTTThTm

Hence find the equation of the normal to the curve at the point where rt =. Give your answer in the form ymxc=+ . [4] mTTTTuTTTTTThhm

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

The point D is such that ABCD is a trapezium with DC AB3= . Find the position vector of D.

The diagonals of the trapezium intersect at the point P. Find the position vector of P. [5] mTTTTuTTTTTThim

Using a scalar product, calculate angle ABC. [4] mTTTTuTTTTTThsm

A balloon in the shape of a sphere has volume V and radius r. Air is pumped into the balloon at a constant rate of r40 starting when time t0= and r0=. At the same time, air begins to flow out of the balloon at a rate of r.r08. The balloon remains a sphere at all times.

Show that r and t satisfy the dif ferential equation 50d

Find the quotient and remainder when r5 is divided by r50-. [3] mTTTTuTTTTTTh m

Solve the differential equation in part

, obtaining an expression for t in terms of r.

Find the value of t when the radius of the balloon is 12. [1] mTTTTuTTTTTThdm 11 Let ()x

Find ()fxl and hence find the exact coordinates of the stationary point of the curve with equation ()xyf= . [5] mTTTTuTTTTTThom

Use the substitution ue = and partial fractions to find the exact value of ()xxfdy . Give your answer in the form lna, where a is a rational number in its simplest form. [9] 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