Section A:
Pure Mathematics
1
How many integers between 10 000 and 100 000 (inclusive) contain exactly two different digits?
(23 332 contains exactly two different digits but neither of 33 333 and 12 331 does.)
2
Show, by means of a suitable change of variable, or otherwise, that
Z
∞
0
f((
x
2
+ 1)
1
/
2
+
x
) d
x
=
1
2
Z
∞
1
(1 +
t

2
)f(
t
) d
t.
Hence, or otherwise, show that
Z
∞
0
((
x
2
+ 1)
1
/
2
+
x
)

3
d
x
=
3
8
.
3
Which of the following statements are true and which are false? Justify your answers.
(i)
a
ln
b
=
b
ln
a
for all
a, b >
0.
(ii)
cos(sin
θ
) = sin(cos
θ
) for all real
θ
.
(iii)
There exists a polynomial P such that

P(
θ
)

cos
θ

6
10

6
for all real
θ
.
(iv)
x
4
+ 3 +
x

4
>
5 for all
x >
0.
4
Prove that the rectangle of greatest perimeter which can be inscribed in a given circle is a
square.
The result changes if, instead of maximising the sum of lengths of sides of the rectangle, we
seek to maximise the sum of
n
th powers of the lengths of those sides for
n
>
2. What happens
if
n
= 2? What happens if
n
= 3? Justify your answers.
1
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5
(i)
In the Argand diagram, the points
Q
and
A
represent the complex numbers 4 + 6
i
and
10 + 2
i
.
If
A
,
B
,
C
,
D
,
E
,
F
are the vertices, taken in clockwise order, of a regular
hexagon (regular sixsided polygon) with centre
Q
, find the complex number which
represents
B
.
(ii)
Let
a
,
b
and
c
be real numbers. Find a condition of the form
Aa
+
Bb
+
Cc
= 0, where
A
,
B
and
C
are integers, which ensures that
a
1 +
i
+
b
1 + 2
i
+
c
1 + 3
i
is real.
6
Let
a
1
= cos
x
with 0
< x < π/
2 and let
b
1
= 1. Given that
a
n
+1
=
1
2
(
a
n
+
b
n
)
,
b
n
+1
=
(
a
n
+1
b
n
)
1
/
2
,
find
a
2
and
b
2
and show that
a
3
= cos
x
2
cos
2
x
4
and
b
3
= cos
x
2
cos
x
4
.
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 Spring '12
 rotar
 Math, Integers, Probability, Complex number, different digits

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