Paper Reference(s)
6674
Edexcel GCE
Pure Mathematics P4
Advanced/Advanced Subsidiary
Thursday 20 June 2002 Morning
Time: 1 hour 30 minutes
Materials required for examination Items included with question papers
Answer Book (AB16) Nil
Graph Paper (ASG2)
Mathematical Formulae (Lilac)
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 (Pure Mathematics P4), the paper reference
(6674), your surname, other name 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.
N10595 This publication may only be reproduced in accordance with Edexcel copyright policy.
Edexcel Foundation is a registered charity. ©2002 Edexcel
1. Prove that
n
6(r
r 1
2
1) (n 1)n(2n + 5).
(4)
2. f(x) = e 2x – 15x – 2.
The equation f(x) = 0 has exactly one root between 1.5 and 1.7.
(a) Taking 1.6 as a first approximation to , apply the Newton-Raphson procedure
once to f(x) to find a second approximation to , giving your answer to 3 significant
figures.
(5)
(b) Show that your answer is the value of correct to 3 significant figures.
(2)
3. f(x) = x4 – 6x3 + 17x2 – 24x + 52.
(a) Show that 2i is a root of the equation f(x) = 0.
(1)
(b) Hence solve f(x) = 0 completely.
(6)
4. Using algebra, find the set of values of x for which
3
2x – 5 > .
x
(7)
N10592 2
5. Given that z = 3 + 4i and w = 1 + 7i,
(a) find w.
(1)
The complex numbers z and w are represented by the points A and B on an Argand
diagram.
(b) Show points A and B on an Argand diagram.
(1)
(c) Prove that △OAB is an isosceles right-angled triangle.
(5)
z
(d) Find the exact value of arg .
w
(3)
6. (a) Find the general solution of the differential equation
dy
cos x + (sin x)y = cos3 x.
dx
(6)
(b) Show that, for 0 x 2, there are two points on the x-axis through which all the
solution curves for this differential equation pass.
(2)
(c) Sketch the graph, for 0 x 2, of the particular solution for which y = 0 at x = 0.
(3)
7. (a) Find the general solution of the differential equation
d2 y dy
2 2
7 3 y = 3t2 + 11t.
dt dt
(8)
dy
(b) Find the particular solution of this differential equation for which y = 1 and =1
dt
when t = 0.
(5)
(c) For this particular solution, calculate the value of y when t = 1.
(1)
N10595 3 Turn over
8. Figure 1
The curve C shown in Fig. 1 has polar equation
r = a(3 + 5 cos ), < .
(a) Find the polar coordinates of the points P and Q where the tangents to C are
parallel to the initial line.
(6)
The curve C represents the perimeter of the surface of a swimming pool. The direct
distance from P to Q is 20 m.
(b) Calculate the value of a.
(3)
(c) Find the area of the surface of the pool.
(6)
END
N10595 4
