STEP III, 2003
2
Section A:
Pure Mathematics
1
Given that
x
+
a >
0 and
x
+
b >
0 , and that
b > a
, show that
d
d
x
arcsin
±
x
+
a
x
+
b
²
=
√
b

a
(
x
+
b
)
√
a
+
b
+ 2
x
and ﬁnd
d
d
x
arcosh
±
x
+
b
x
+
a
²
.
Hence, or otherwise, integrate, for
x >

1 ,
(i)
Z
1
(
x
+ 1)
√
x
+ 3
d
x
,
(ii)
Z
1
(
x
+ 3)
√
x
+ 1
d
x
.
[You may use the results
d
d
x
arcsin
x
=
1
√
1

x
2
and
d
d
x
arcosh
x
=
1
√
x
2

1
. ]
2
Show that
2
r
C
r
=
1
×
3
× ··· ×
(2
r

1)
r
!
×
2
r
,
for
r
>
1 .
(i)
Give the ﬁrst four terms of the binomial series for (1

p
)

1
2
.
By choosing a suitable value for
p
in this series, or otherwise, show that
∞
X
r
=0
2
r
C
r
8
r
=
√
2
.
(ii)
Show that
∞
X
r
=0
(2
r
+ 1)
2
r
C
r
5
r
=
(
√
5
)
3
.
[
Note:
n
C
r
is an alternative notation for
±
n
r
²
for
r
>
1 , and
0
C
0
= 1 .]
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View Full DocumentSTEP III, 2003
3
3
If
m
is a positive integer, show that (1 +
x
)
m
+ (1

x
)
m
6
= 0 for any real
x
.
The function f is deﬁned by
f (
x
) =
(1 +
x
)
m

(1

x
)
m
(1 +
x
)
m
+ (1

x
)
m
.
Find and simplify an expression for f
0
(
x
).
In the case
m
= 5 , sketch the curves
y
= f (
x
) and
y
=
1
f (
x
)
.
4
A curve is deﬁned parametrically by
x
=
t
2
,
y
=
t
(1 +
t
2
)
.
The tangent at the point with parameter
t
, where
t
6
= 0 , meets the curve again at the point
with parameter
T
, where
T
6
=
t
. Show that
T
=
1

t
2
2
t
and
3
t
2
6
= 1
.
Given a point
P
0
on the curve, with parameter
t
0
, a sequence of points
P
0
, P
1
, P
2
, . . .
on the
curve is constructed such that the tangent at
P
i
meets the curve again at
P
i
+1
. If
t
0
= tan
7
π
18
,
show that
P
3
=
P
0
but
P
1
6
=
P
0
. Find a second value of
t
0
, with
t
0
>
0 , for which
P
3
=
P
0
but
P
1
6
=
P
0
.
5
Find the coordinates of the turning point on the curve
y
=
x
2

2
bx
+
c
. Sketch the curve in
the case that the equation
x
2

2
bx
+
c
= 0 has two distinct real roots. Use your sketch to
determine necessary and suﬃcient conditions on
b
and
c
for the equation
x
2

2
bx
+
c
= 0 to
have two distinct real roots. Determine necessary and suﬃcient conditions on
b
and
c
for this
equation to have two distinct positive roots.
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 Spring '12
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 Math, Probability distribution, Probability theory, probability density function

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