6667

Edexcel GCE

Further Pure Mathematics FP1 Advanced Level

Specimen Paper Time: 1 hour 30 minutes

Materials required for examination Answer Book (AB16) Graph Paper (ASG2) Mathematical Formulae (Lilac) Items included with question papers Nil

Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI-89, TI-92, Casio CFX-9970G, Hewlett Packard HP 48G.

Instructions to Candidates In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper reference (6667), your surname, initials and signature. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. Full marks may be obtained for answers to ALL questions. This paper has eight questions. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.

This publication may only be reproduced in accordance with London Qualifications Limited copyright policy. Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited

1.

Prove that

å (r

r =1

n

2

- r -1 =

)

1 (n - 2)n(n + 2) . 3

(5)

2.

1 f ( x ) = ln x - 1 - . x

(a) Show that the root a of the equation f(x) = 0 lies in the interval 3 < a < 4 .

(2)

(b) Taking 3.6 as your starting value, apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to a. Give your answer to 4 decimal places. (5) 3. Find the set of values of x for which 1 x > . x -3 x -2

(7)

4.

f ( x ) º 2 x 3 - 5 x 2 + px - 5, p Î ℝ. The equation f (x) = 0 has (1 – 2i) as a root. Solve the equation and determine the value of p. (7)

5.

(a) Obtain the general solution of the differential equation dS - 0.1S = t. dt

(6) (b) The differential equation in part (a) is used to model the assets, £S million, of a bank t years after it was set up. Given that the initial assets of the bank were £200 million, use your answer to part (a) to estimate, to the nearest £ million, the assets of the bank 10 years after it was set up. (4)

2

6.

The curve C has polar equation

r 2 = a 2 cos 2q , -p p £q £ . 4 4

(a) Sketch the curve C. (2) (b) Find the polar coordinates of the points where tangents to C are parallel to the initial line. (6) (c) Find the area of the region bounded by C. (4)

7.

Given that z = -3 + 4i and zw = -14 + 2i, find (a) w in the form p + iq where p and q are real, (4) (b) the modulus of z and the argument of z in radians to 2 decimal places (4) (c) the values of the real constants m and n such that mz + nzw = -10 - 20i .

(5)

3

Turn over

8.

(a) Given that x = e t , show that (i)

dy dy = e -t , dx dt

2 dy ö d2 y - 2t æ d y ç 2 - ÷. =e ç 2 dt ÷ dx ø è dt

(ii)

(5)

(b) Use you answers to part (a) to show that the substitution x = e t transforms the differential equation d2 y dy x 2 2 - 2x + 2y = x3 dx dx into d2 y dy - 3 + 2 y = e 3t . 2 dt dt (3) (c) Hence find the general solution of x2 d2 y dy - 2x + 2y = x3. 2 dx dx (6) END

4

Paper Reference(s)

6668

Edexcel GCE

Further Pure Mathematics FP2 Advanced Level

Specimen Paper Time: 1 hour 30 minutes

Materials required for examination Answer Book (AB16) Graph Paper (ASG2) Mathematical Formulae (Lilac) Items included with question papers Nil

Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX-9970G, Hewlett Packard HP 48G....