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Sixth Term Examination Papers
9470
MATHEMATICS 2
Morning
Wednesday 23 JUNE 2010
Time: 3 hours
Additional Materials: Answer Paper
Formulae Booklet
Candidates may
not
use a calculator
INSTRUCTIONS TO CANDIDATES
Please read this page carefully, but do not open this question paper until you are
told that you may do so.
Write your name, centre number and candidate number in the spaces on the answer
booklet.
Begin each answer on a new page.
INFORMATION FOR CANDIDATES
Each question is marked out of 20. There is no restriction of choice.
You will be assessed on the
six
questions for which you gain the highest marks.
You are advised to concentrate on no more than
six
questions. Little credit will be given
for fragmentary answers.
You are provided with a Mathematical Formulae Booklet.
Calculators are not permitted.
Please wait to be told you may begin before turning this page.
_____________________________________________________________________________
This question paper consists of 6 printed pages and 2 blank pages.
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View Full DocumentSection A:
Pure Mathematics
1
Let
P
be a given point on a given curve
C
. The
osculating circle
to
C
at
P
is deﬁned to be
the circle that satisﬁes the following two conditions at
P
: it touches
C
; and the rate of change
of its gradient is equal to the rate of change of the gradient of
C
.
Find the centre and radius of the osculating circle to the curve
y
= 1

x
+ tan
x
at the point
on the curve with
x
coordinate
1
4
π
.
2
Prove that
cos 3
x
= 4 cos
3
x

3 cos
x .
Find and prove a similar result for sin 3
x
in terms of sin
x
.
(i)
Let
I(
α
) =
Z
α
0
(
7 sin
x

8 sin
3
x
)
d
x .
Show that
I(
α
) =

8
3
c
3
+
c
+
5
3
,
where
c
= cos
α
. Write down one value of
c
for which I(
α
) = 0.
(ii)
Useless Eustace believes that
Z
sin
n
x
d
x
=
sin
n
+1
x
n
+ 1
for
n
= 1
,
2
,
3
, . . .
. Show that Eustace would obtain the correct value of I(
β
) , where
cos
β
=

1
6
.
Find all values of
α
for which he would obtain the correct value of I(
α
).
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 Math, Addition, Sin, Quadratic equation, Mathematical Formulae Booklet, challenge match, Formulae Booklet

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