Free Essay

Hey There Its Me

In:

Submitted By sickky123
Words 1192
Pages 5
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Question

General Certificate of Education
Advanced Level Examination
January 2012

Mark

1
2
3

Mathematics

MFP2

5

Unit Further Pure 2
Friday 20 January 2012

4

6

1.30 pm to 3.00 pm

7
For this paper you must have:
*
the blue AQA booklet of formulae and statistical tables.
You may use a graphics calculator.

8
TOTAL

Time allowed
*
1 hour 30 minutes
Instructions
*
Use black ink or black ball-point pen. Pencil should only be used for drawing. *
Fill in the boxes at the top of this page.
*
Answer all questions.
*
Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. *
You must answer the questions in the spaces provided. Do not write outside the box around each page.
*
Show all necessary working; otherwise marks for method may be lost. *
Do all rough work in this book. Cross through any work that you do not want to be marked.
Information
*
The marks for questions are shown in brackets.
*
The maximum mark for this paper is 75.
Advice
*
Unless stated otherwise, you may quote formulae, without proof, from the booklet.
*
You do not necessarily need to use all the space provided.

(JAN12MFP201)

P46339/Jan12/MFP2 6/6/6/

MFP2

Do not write outside the box 2

Answer all questions in the spaces provided.

1 (a)

Show, by means of a sketch, that the curves with equations y ¼ sinh x and y ¼ sech x

have exactly one point of intersection.
(b)

(4 marks)

Find the x-coordinate of this point of intersection, giving your answer in the form a ln b .
(4 marks)

QUESTION
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p
.
6
(1 mark)

Draw on an Argand diagram the locus L of points satisfying the equation arg z ¼

2 (a)

(b) (i)

A circle C, of radius 6, has its centre lying on L and touches the line ReðzÞ ¼ 0 .
(2 marks)
Draw C on your Argand diagram from part (a).

(ii) Find the equation of C, giving your answer in the form j z À z0 j ¼ k .

(3 marks)

(iii) The complex number z1 lies on C and is such that arg z1 has its least possible value.

Find arg z1 , giving your answer in the form pp, where À1 < p 4 1 .

(2 marks)

QUESTION
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A curve has cartesian equation

3

1

y ¼ 2 lnðtanh xÞ
(a)

Show that dy 1
¼
dx sinh 2x

(b)

(4 marks)

The points A and B on the curve have x-coordinates ln 2 and ln 4 respectively. Find the arc length AB , giving your answer in the form p ln q, where p and q are rational numbers. (8 marks)

QUESTION
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4

The sequence u1 , u2 , u3 , … is defined by u1 ¼

3
4

unþ1 ¼

3
4 À un

Prove by induction that, for all n 5 1 ,
3nþ1 À 3 un ¼ nþ1
3
À1

(6 marks)

QUESTION
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5

Find the smallest positive integer values of p and q for which


p p p cos þ i sin
8
8  ¼i

p p q cos À i sin
12
12

(7 marks)

QUESTION
PART
REFERENCE

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6 (a)

(b)

Express 7 þ 4x À 2x 2 in the form a À bðx À cÞ2 , where a, b and c are integers.
(2 marks)
By means of a suitable substitution, or otherwise, find the exact value of ð5 dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7 þ 4x À 2x 2
2

(6 marks)

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The numbers a , b and g satisfy the equations

7

a 2 þ b 2 þ g 2 ¼ À10 À 12i ab þ bg þ ga ¼ 5 þ 6i
(a)

Show that a þ b þ g ¼ 0 .

(b)

The numbers a , b and g are also the roots of the equation

(2 marks)

z 3 þ pz2 þ qz þ r ¼ 0
Write down the value of p and the value of q.

(2 marks)

It is also given that a ¼ 3i .

(c)
(i)

Find the value of r.

(3 marks)

(ii) Show that b and g are the roots of the equation

z 2 þ 3iz À 4 þ 6i ¼ 0

(2 marks)

(iii) Given that b is real, find the values of b and g .

(3 marks)

QUESTION
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8 (a)

(b)

Write down the five roots of the equation z5 ¼ 1 , giving your answers in the
(1 mark) form eiy , where Àp < y 4 p .
Hence find the four linear factors of z4 þ z3 þ z2 þ z þ 1

(c)

Deduce that
2

z þzþ1þz

(d)

(3 marks)

À1

Use the substitution

þz

À2


¼

z þ z À1

2p z À 2 cos þ z À1
5



4p z À 2 cos þ z À1
5

2p
¼
¼ w to show that cos
5

pffiffiffi
5À1
.
4


(4 marks)

(6 marks)

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END OF QUESTIONS

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There are no questions printed on this page

DO NOT WRITE ON THIS PAGE
ANSWER IN THE SPACES PROVIDED

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