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“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful: the ideas, like the colours
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              
 the
 words,
 fit 
must
or
together in a harmonious
way. Beauty is the first test.”
G H Hardy
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Further Maths A2 (M2FP2D1) Assignment  (upsilon)
PREPARATION Every week you will be required to do some preparation for future lessons,
to be advised by your teacher.
CURRENT WORK – MECHANICS
You need to complete the following past paper, timed and using good exam technique. Check the
mark scheme only after you have completed the paper: M2 Edexcel Summer 2006.
DECISION MATHS
Using the Decision Paper Solomon C select the questions you have covered in class and give them
in with this assignment
CONSOLIDATION – FP2
1.
Given that 3x sin 2 x is a particular integral of the differential equation
d2 y
 4 y  k cos 2 x ,
dx 2
where k is a constant,
a) calculate the value of k,
b) find the particular solution of the differential equation for which at x = 0, y = 2, and for


which at x  , y  .
4
2
2.
a) Use algebra to find the exact solutions of the equation 2 x 2  x  6  6  3 x .
b) On the same diagram, sketch the curve with equation y  2 x 2  x  6 and the line with
equation y = 6 – 3x.
c) Find the set of values of x for which 2 x 2  x  6  6  3 x .
3.
Obtain the general solution of the differential equation x
answer in the form y  f  x  .
dy
 2 y  cos x, x  0 giving your
dx
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Updated: 08/02/2016
4.
The figure above shows a sketch of the curve C with polar equation

r  4sin  cos 2  , 0    .
2
The tangent to C at the point P is perpendicular to the initial line.
3  
a) Show that P has polar coordinates  ,  .
2 6


The point Q on C has polar coordinates   2,  .
4

The shaded region R is bounded by OP, OQ and C, as shown in the figure above.

b) Show that the area of R is given by

  sin
4
6
2
1 1

2 cos 2   cos 4  d .
2 2

c) Hence, or otherwise, find the area of R, giving your answer in the form a  b , where a and
b are rational numbers.
5.
a) Given that z  cos  isin  , use de Moivere’s theorem to show that z n 
1
 2 cos n .
zn
b) Express 32 cos 6  in the form p cos 6  q cos 4  r cos 2  s , where p, q, r and s are
integers.
c) Hence find the exact value of


3
0
cos 6  d .
CHALLENGE QUESTION
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Updated: 08/02/2016
Answers:
Current work – M2 June 2006:
1)
6s
2a)
14.4 kW
2b)
0.4m s2
3a)
25.1 Ns
3b)
18.9 m s1
4a)
i)
4a)
ii)
4
a
3
4b)
= 14.9°
5b)
3.5 s
7a)
22.4 J
7b)
6.4 m s1
7c)
4.27 m s1
8a)
 1 e 

u ,
 5 
8c)
3
mu2
10
 4e  1 

u
 5 
5
a
2
Consolidation:
1a)
1b)
2a)
2b)
2c)
3)
4c)
5b)
5c)
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Updated: 08/02/2016
Challenge question: tau
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Updated: 08/02/2016
M THS
ASSIGNMENT
COVER SHEET
Name
Maths Teacher
Please tick honestly:
Yes
No - explain why.
Have you ticked/crossed your
answers using the answers
given?
Have you corrected all the
questions which were wrong?
How did you find this assignment?
Use this space to outline any problems you’ve had, how you overcame them as well as the things
which went well or which you enjoyed/learned from.
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Updated: 08/02/2016