STEP III - 91*4023334091* Sixth Term Examination Papers...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
© UCLES 2010 91**4023334091* Sixth Term Examination Papers 9475 MATHEMATICS 3 Afternoon MONDAY 21 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 8 printed pages and 4 blank pages. [Turn over 
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Section A: Pure Mathematics 1 Let x 1 , x 2 , . . . , x n and x n +1 be any fixed real numbers. The numbers A and B are defined by A = 1 n n X k =1 x k , B = 1 n n X k =1 ( x k - A ) 2 , and the numbers C and D are defined by C = 1 n + 1 n +1 X k =1 x k , D = 1 n + 1 n +1 X k =1 ( x k - C ) 2 . (i) Express C in terms of A , x n +1 and n . (ii) Show that B = 1 n n X k =1 x 2 k - A 2 . (iii) Express D in terms of B , A , x n +1 and n . Hence show that ( n + 1) D > nB for all values of x n +1 , but that D < B if and only if A - r ( n + 1) B n < x n +1 < A + r ( n + 1) B n . 2 In this question, a is a positive constant. (i) Express cosh a in terms of exponentials. By using partial fractions, prove that Z 1 0 1 x 2 + 2 x cosh a + 1 d x = a 2 sinh a . (ii) Find, expressing your answers in terms of hyperbolic functions, Z 1 1 x 2 + 2 x sinh a - 1 d x and Z 0 1 x 4 + 2 x 2 cosh a + 1 d x . 2 2 9475 Jun10
Background image of page 2
3 For any given positive integer n , a number a (which may be complex) is said to be a primitive n th root of unity if a n = 1 and there is no integer m such that 0 < m < n and a m = 1. Write down the two primitive 4th roots of unity. Let C n ( x ) be the polynomial such that the roots of the equation C n ( x ) = 0 are the primitive n th roots of unity, the coefficient of the highest power of x is one and the equation has no repeated roots. Show that C 4 ( x ) = x 2 + 1 . (i) Find C 1 ( x ), C 2 ( x ), C 3 ( x ), C 5 ( x ) and C 6 ( x ), giving your answers as unfactorised poly- nomials.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 12

STEP III - 91*4023334091* Sixth Term Examination Papers...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online