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STEP I, 2002
2
Section A:
Pure Mathematics
1
Show that the equation of any circle passing through the points of intersection of the ellipse
(
x
+ 2)
2
+ 2
y
2
= 18 and the ellipse 9(
x

1)
2
+ 16
y
2
= 25 can be written in the form
x
2

2
ax
+
y
2
= 5

4
a .
2
Let f (
x
) =
x
m
(
x

1)
n
, where
m
and
n
are both integers greater than 1. Show that the
curve
y
= f (
x
) has a stationary point with 0
< x <
1. By considering f
00
(
x
), show that this
stationary point is a maximum if
n
is even and a minimum if
n
is odd.
Sketch the graphs of f (
x
) in the four cases that arise according to the values of
m
and
n
.
3
Show that (
a
+
b
)
2
6
2
a
2
+ 2
b
2
.
Find the stationary points on the curve
y
= (
a
2
cos
2
θ
+
b
2
sin
2
θ
)
1
2
+ (
a
2
sin
2
θ
+
b
2
cos
2
θ
)
1
2
,
where
a
and
b
are constants. State, with brief reasons, which points are maxima and which
are minima. Hence prove that

a

+

b

6
(
a
2
cos
2
θ
+
b
2
sin
2
θ
)
1
2
+ (
a
2
sin
2
θ
+
b
2
cos
2
θ
)
1
2
6
(2
a
2
+ 2
b
2
)
1
2
.
4
Give a sketch of the curve
y
=
1
1 +
x
2
, for
x
>
0.
Find the equation of the line that intersects the curve at
x
= 0 and is tangent to the curve
at some point with
x >
0 . Prove that there are no further intersections between the line and
the curve. Draw the line on your sketch.
By considering the area under the curve for 0
6
x
6
1, show that
π >
3 .
Show also, by considering the volume formed by rotating the curve about the
y
axis, that
ln 2
>
2
/
3 .
[
Note
:
Z
1
0
1
1 +
x
2
d
x
=
π
4
.
]
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View Full DocumentSTEP I, 2002
3
5
Let
f (
x
) =
x
n
+
a
1
x
n

1
+
···
+
a
n
,
where
a
1
,
a
2
,
. . .
,
a
n
are given numbers. It is given that f (
x
) can be written in the form
f (
x
) = (
x
+
k
1
)(
x
+
k
2
)
···
(
x
+
k
n
)
.
By considering f (0), or otherwise, show that
k
1
k
2
. . . k
n
=
a
n
.
Show also that
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 Spring '12
 rotar
 Math

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