Write your name here

Surname Other names

Centre Number Candidate Number

Pearson Edexcel
International
Advanced Level

Further Pure
Mathematics F1
Advanced/Advanced Subsidiary
Monday 15 January 2018 – Afternoon Paper Reference

Time: 1 hour 30 minutes WFM01/01

You must have: Total Marks
Mathematical Formulae and Statistical Tables (Blue)

Candidates may use any calculator allowed by the regulations of the

Joint Council for Qualifications. Calculators must not have the facility
for symbolic algebra manipulation, differentiation and integration, or
have retrievable mathematical formulae stored in them.
Instructions
•• Use black ink or ball-point pen.
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
Coloured pencils and highlighter pens must not be used.
• centre
Fill in the boxes at the top of this page with your name,
number and candidate number.
• clearly
Answer all questions and ensure that your answers to parts of questions are
labelled.
• – there may
Answer the questions in the spaces provided
be more space than you need.
• You should show sufficient working to make your methods clear. Answers
without working may not gain full credit.
• When a calculator is used, the answer should be given to an appropriate
degree of accuracy.
Information
•• The total mark for this paper is 75.
The marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
Advice
•• Read each question carefully before you start to answer it.
Try to answer every question.
• Check your answers if you have time at the end. Turn over

P51511A
©2018 Pearson Education Ltd.

1/1/1/1/
*P51511A0132*
 Leave
blank
5
1. f ( x) = 3x 2 − − 6, x > 0
3 x

DO NOT WRITE IN THIS AREA

The single root Į of the equation f (x) = 0 lies in the interval [1.5, 1.6].

(a) Taking 1.5 as a first approximation to Į, apply the Newton-Raphson process once to
f (x) to obtain a second approximation to Į. Give your answer to 3 decimal places.
(4)

(b) Use linear interpolation once on the interval [1.5, 1.6] to find another approximation
to Į. Give your answer to 3 decimal places.
(3)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

2
*P51511A0232*
 Leave
blank
Question 1 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q1

(Total 7 marks)

3
*P51511A0332* Turn over
 Leave
blank
2. f ( z ) = z 4 − 6 z 3 + 38 z 2 − 94 z + 221

(a) Given that z = 2 + 3i is a root of the equation f (z) = 0, use algebra to find the three

DO NOT WRITE IN THIS AREA

other roots of f (z) = 0
(7)

(b) Show the four roots of f (z) = 0 on a single Argand diagram.

(2)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

4
*P51511A0432*
 Leave
blank
Question 2 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

5
*P51511A0532* Turn over
 Leave
blank
Question 2 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

6
*P51511A0632*
 Leave
blank
Question 2 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q2

(Total 9 marks)

7
*P51511A0732* Turn over
 Leave
blank
n n
3. (a) Use the standard results for ∑ r 2 and ∑r 3
to show that, for all positive integers n,
r =1 r =1

DO NOT WRITE IN THIS AREA

n
1
∑r 2
(r + 1) =
12
n(n + 1)(n + 2)(an + b)
r =1

where a and b are integers to be determined.

(4)

(b) Given that

25 k

∑ r 2 (r + 1) + ∑3 r
= 140543
r =5 r =1

find the value of the integer k.

(4)
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

8
*P51511A0832*
 Leave
blank
Question 3 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

9
*P51511A0932* Turn over
 Leave
blank
Question 3 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

10
*P51511A01032*
 Leave
blank
Question 3 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q3

(Total 8 marks)

11
*P51511A01132* Turn over
 Leave
blank
4. The quadratic equation

3x2 + 2x + 5 = 0

DO NOT WRITE IN THIS AREA

has roots Į and ȕ.

Without solving the equation,

(a) find the value of Į2 + ȕ 2

(2)
82
(b) show that Į3 + ȕ 3 =
27 (2)

(c) find a quadratic equation which has roots

⎛ α⎞ ⎛ β⎞
⎜⎝ α + and ⎜ β + ⎟
β ⎟⎠
2 ⎝ α2 ⎠

DO NOT WRITE IN THIS AREA

giving your answer in the form px 2 + qx + r = 0, where p, q and r are integers.
(4)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

12
*P51511A01232*
 Leave
blank
Question 4 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

13
*P51511A01332* Turn over
 Leave
blank
Question 4 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

14
*P51511A01432*
 Leave
blank
Question 4 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q4

(Total 8 marks)

15
*P51511A01532* Turn over
 Leave
blank
5. (i) Given that

2z + 3

DO NOT WRITE IN THIS AREA

= 1+ i
z + 5 − 2i

find z, giving your answer in the form a + E࣠i, where a and b are real constants.
(5)

(ii) Given that

w = (3 + Ȝi)(2 + i)

where Ȝ is a real constant, and that

_௘w௘_ 

find the possible values of Ȝ.

(4)
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

16
*P51511A01632*
 Leave
blank
Question 5 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

17
*P51511A01732* Turn over
 Leave
blank
Question 5 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

18
*P51511A01832*
 Leave
blank
Question 5 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q5

(Total 9 marks)

19
*P51511A01932* Turn over
 Leave
blank
6. The parabola C has equation \࣠2 ௘x and the point S is the focus of this parabola.
The point P(2, 8) lies on C and the point T lies on the directrix of C. The line segment
PT is parallel to the x-axis.

DO NOT WRITE IN THIS AREA

(a) Write down the coordinates of S.
(1)

(b) Find the length of PT.

(1)

(c) Using calculus, show that the tangent to C at the point P has equation

y = 2x + 4
(4)

The hyperbola H has equation [࣠\ = 4. The tangent to C at P meets H at the points L and M.

(d) Find the exact coordinates of the points L and M, giving your answers in their
simplest form.
(6)

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ DO NOT WRITE IN THIS AREA
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

20
*P51511A02032*
 Leave
blank
Question 6 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

21
*P51511A02132* Turn over
 Leave
blank
Question 6 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

22
*P51511A02232*
 Leave
blank
Question 6 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q6

(Total 12 marks)

23
*P51511A02332* Turn over
 Leave
blank
7. (i)
⎛ 6 k⎞
A= ⎜ , where k is a real constant, k z 8
⎝ −3 −4⎟⎠

DO NOT WRITE IN THIS AREA

Find, in terms of k,

(a) Aí
(3)

(b) A2
(1)
⎛ 5 9⎞
Given that A2 + 3Aí = ⎜
⎝ −3 −5⎟⎠

(c) find the value of k.

(3)

(ii) ⎛ 1 ⎞
− − 3

DO NOT WRITE IN THIS AREA

⎜ 2 ⎟
M= ⎜ ⎟
⎜ 3 −1 ⎟
⎝ 2 ⎠

The matrix M represents a one way stretch, parallel to the y-axis, scale factor p, where
p > 0, followed by a rotation anticlockwise through an angle ș about (0, 0).

(a) Find the value of p.

(2)

(b) Find the value of ș.

(2)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

24
*P51511A02432*
 Leave
blank
Question 7 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

25
*P51511A02532* Turn over
 Leave
blank
Question 7 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

26
*P51511A02632*
 Leave
blank
Question 7 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q7

(Total 11 marks)

27
*P51511A02732* Turn over
 Leave
blank
8. (i) A sequence of numbers is defined by

u1 = 3

DO NOT WRITE IN THIS AREA

un +1 = un + 3n − 2 n .1

Prove by induction that, for all positive integers n,

3 2 7
un = n − n+5
2 2 (5)

(ii) Prove by induction that, for all positive integers n,

f (n) = 3௘n௘௘௘níLVGLYLVLEOHE\
(6)
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

28
*P51511A02832*
 Leave
blank
Question 8 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

29
*P51511A02932* Turn over
 Leave
blank
Question 8 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

30
*P51511A03032*
 Leave
blank
Question 8 continued
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

31
*P51511A03132* Turn over
 Leave
blank
Question 8 continued
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________

DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DO NOT WRITE IN THIS AREA

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________ Q8

(Total 11 marks)
TOTAL FOR PAPER 75 MARKS
END

32
*P51511A03232*