Section A:
Pure Mathematics
1
A
proper factor
of an integer
N
is a positive integer, not 1 or
N
, that divides
N
.
(i)
Show that 3
2
×
5
3
has exactly 10 proper factors. Determine how many other integers of
the form 3
m
×
5
n
(where
m
and
n
are integers) have exactly 10 proper factors.
(ii)
Let
N
be the smallest positive integer that has exactly 426 proper factors. Determine
N
,
giving your answer in terms of its prime factors.
2
A curve has the equation
y
3
=
x
3
+
a
3
+
b
3
,
where
a
and
b
are positive constants. Show that the tangent to the curve at the point (

a, b
) is
b
2
y

a
2
x
=
a
3
+
b
3
.
In the case
a
= 1 and
b
= 2, show that the
x
coordinates of the points where the tangent
meets the curve satisfy
7
x
3

3
x
2

27
x

17 = 0
.
Hence ﬁnd positive integers
p
,
q
,
r
and
s
such that
p
3
=
q
3
+
r
3
+
s
3
.
3
(i)
By considering the equation
x
2
+
x

a
= 0 , show that the equation
x
= (
a

x
)
1
2
has
one real solution when
a
>
0 and no real solutions when
a <
0 .
Find the number of distinct real solutions of the equation
x
=
(
(1 +
a
)
x

a
)
1
3
in the cases that arise according to the value of
a
.
(ii)
Find the number of distinct real solutions of the equation
x
= (
b
+
x
)
1
2
in the cases that arise according to the value of
b
.
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The sides of a triangle have lengths
p

q
,
p
and
p
+
q
, where
p > q >
0 . The largest and
smallest angles of the triangle are
α
and
β
, respectively. Show by means of the cosine rule
that
4(1

cos
α
)(1

cos
β
) = cos
α
+ cos
β .
In the case
α
= 2
β
, show that cos
β
=
3
4
and hence ﬁnd the ratio of the lengths of the sides of
the triangle.
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 Spring '12
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 Math, Factors, Integers, Prime number, Sweets, red sweets, yellow sweets

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