General Certificate of Education January 2006 Advanced Level Examination

MATHEMATICS Unit Further Pure 2 Friday 27 January 2006

MFP2

1.30 pm to 3.00 pm

For this paper you must have: * an 8-page answer book * the blue AQA booklet of formulae and statistical tables You may use a graphics calculator.

Time allowed: 1 hour 30 minutes Instructions Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MFP2. Answer all questions. All necessary working should be shown; otherwise marks for method may be lost. * *

* *

Information The maximum mark for this paper is 75. The marks for questions are shown in brackets. * *

Advice Unless stated otherwise, formulae may be quoted, without proof, from the booklet. *

P80782/Jan06/MFP2 6/6/6/

MFP2

2

Answer all questions.

1

(a) Show that 1 r

1

2r ‡ 1 ˆ 2 …r ‡ 1† r …r ‡ 1†2

2

(2 marks)

2

(b) Hence find the sum of the first n terms of the series 12

3 5 7 ‡ 2 ‡ 2 ‡: : : 2 2 2 2 3 3  42

(4 marks)

2 The cubic equation x3 ‡ px 2 ‡ qx ‡ r ˆ 0 where p, q and r are real, has roots a, b and g. (a) Given that a‡b‡g ˆ4

and

a 2 ‡ b 2 ‡ g 2 ˆ 20

find the values of p and q.

(5 marks)

(b) Given further that one root is 3 ‡ i, find the value of r.

(5 marks)

3 The complex numbers z1 and z2 are given by 1‡i z1 ˆ 1 i

and

p 3 1 z2 ˆ ‡ i 2 2

(a) Show that z1 ˆ i .

(2 marks)

(b) Show that j z1j ˆ j z2 j .

(2 marks)

(c) Express both z1 and z2 in the form reiy , where r > 0 and

p