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Paper Reference(s)

6671

Edexcel GCE

Pure Mathematics P1

(New Syllabus)

Advanced/Advanced Subsidiary

Thursday 11 January 2001 ( Morning

Time: 1 hour 30 minutes

Materials required for examination Items included with question papers

Answer Book (ABO4) Nil

Graph Paper (GPO2)

Mathematical Formulae

Candidates may only use one of the following calculators:

Casio fx-83WA or fx-85WA

Texas Instruments TI-30s or TI-30 ecoRS

Texet Albert

Sharp EL-531 (irrespective of following letters) e.g. EL-531GH, EL-531LH

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 (Pure Mathematics P1), the paper reference (6671), 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 8 questions. Pages 6, 7 and 8 are blank.

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.

Printer’s s Log No

N6980

This publication may only be reproduced in accordance with Edexcel copyright policy.

Edexcel Foundation is a registered charity. ©2001 Edexcel

1. Given that (2 + (7)(4 ( (7) = a + b(7, where a and b are integers,

(a) find the value of a and the value of b. (2 marks)

Given that [pic]= c + d(7 where c and d are rational numbers,

(b) find the value of c and the value of d. (3 marks)

2. (a) Prove, by completing the square, that the roots of the equation x2 + 2kx + c = 0, where k and c are constants, are (k ± ((k2 ( c). (4 marks)

The equation x2 + 2kx +81 = 0 has equal roots.

(b) Find the possible values of k. (2 marks)

3. Find all values of ( in the interval 0 ( ( < 360 for which

(a) cos (( + 75)( = 0.5, (3 marks)

(b) sin 2( ( = 0.7, giving your answers to one decima1 place. (5 marks)

4.

Fig. 1

Figure 1 shows the curve with equation y = 5 + 2x ( x2 and the line with equation y = 2. The curve and the line intersect at the points A and B.

(a) Find the x-coordinates of A and B. (3 marks)

The shaded region R is bounded by the curve and the line.

(b) Find the area of R. (6 marks)

5. The third and fourth terms of a geometric series are 6.4 and 5.12 respectively.

Find

(a) the common ratio of the series, (2 marks)

(b) the first term of the series, (2 marks)

(c) the sum to infinity of the series. (2 marks)

(d) Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. (4 marks)

6.

Fig. 2

The points A (3, 0) and B (0, 4) are two vertices of the rectangle ABCD, as shown in Fig. 2.

(a) Write down the gradient of AB and hence the gradient of BC. (3 marks)

The point C has coordinates (8, k), where k is a positive constant.

(b) Find the length of BC in terms of k. (2 marks)

Given that the length of BC is 10 and using your answer to part (b),

(c) find the value of k, (4 marks)

(d) find the coordinates of D. (2 marks)

7.

Fig. 3

Triangle ABC has AB = 9 cm, BC 10 cm and CA = 5 cm.

A circle, centre A and radius 3 cm, intersects AB and AC at P and Q respectively, as shown in Fig. 3.

(a) Show that, to 3 decimal places, (BAC = 1.504 radians. (3 marks)

Calculate,

(b) the area, in cm2, of the sector APQ, (2 marks)

(c) the area, in cm2, of the shaded region BPQC, (3 marks)

(d) the perimeter, in cm, of the shaded region BPQC. (4 marks)

8.

Fig. 4

A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2x cm by x cm and height h cm, as shown in Fig. 4.

Given that the capacity of a carton has to be 1030 cm3,

(a) express h in terms of x, (2 marks)

(b) show that the surface area, A cm2, of a carton is given by

[pic] (3 marks)

The manufacturer needs to minimise the surface area of a carton.

(c) Use calculus to find the value of x for which A is a minimum. (5 marks)

(d) Calculate the minimum value of A. (2 marks)

(e) Prove that this value of A is a minimum. (2 marks)

END

-----------------------

[pic]

A

B

O

R

y

x

y = 2

y = 5 + 2x ( x2

O

A

C

D

B

y

x

x

2x

h

[pic]

C

B

A

Q

P

9 cm

5 cm

10 cm

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