STEP II - 91*4023334091 Sixth Term Examination Papers 9470...

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© UCLES 2010 91**4023334091* Sixth Term Examination Papers 9470 MATHEMATICS 2 Morning Wednesday 23 JUNE 2010 Time: 3 hours Additional Materials: Answer Paper Formulae Booklet Candidates may not use a calculator INSTRUCTIONS TO CANDIDATES Please read this page carefully, but do not open this question paper until you are told that you may do so. Write your name, centre number and candidate number in the spaces on the answer booklet. Begin each answer on a new page. INFORMATION FOR CANDIDATES Each question is marked out of 20. There is no restriction of choice. You will be assessed on the six questions for which you gain the highest marks. You are advised to concentrate on no more than six questions. Little credit will be given for fragmentary answers. You are provided with a Mathematical Formulae Booklet. Calculators are not permitted. Please wait to be told you may begin before turning this page. _____________________________________________________________________________ This question paper consists of 6 printed pages and 2 blank pages. [Turn over 
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Section A: Pure Mathematics 1 Let P be a given point on a given curve C . The osculating circle to C at P is defined to be the circle that satisfies the following two conditions at P : it touches C ; and the rate of change of its gradient is equal to the rate of change of the gradient of C . Find the centre and radius of the osculating circle to the curve y = 1 - x + tan x at the point on the curve with x -coordinate 1 4 π . 2 Prove that cos 3 x = 4 cos 3 x - 3 cos x . Find and prove a similar result for sin 3 x in terms of sin x . (i) Let I( α ) = Z α 0 ( 7 sin x - 8 sin 3 x ) d x . Show that I( α ) = - 8 3 c 3 + c + 5 3 , where c = cos α . Write down one value of c for which I( α ) = 0. (ii) Useless Eustace believes that Z sin n x d x = sin n +1 x n + 1 for n = 1 , 2 , 3 , . . . . Show that Eustace would obtain the correct value of I( β ) , where cos β = - 1 6 . Find all values of α for which he would obtain the correct value of I( α ).
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STEP II - 91*4023334091 Sixth Term Examination Papers 9470...

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