STEP II, 2003
2
Section A:
Pure Mathematics
1
Consider the equations
ax

y

z
= 3
,
2
ax

y

3
z
= 7
,
3
ax

y

5
z
=
b ,
where
a
and
b
are given constants.
(i)
In the case
a
= 0 , show that the equations have a solution if and only if
b
= 11 .
(ii)
In the case
a
6
= 0 and
b
= 11 show that the equations have a solution with
z
=
λ
for
any given number
λ
.
(iii)
In the case
a
= 2 and
b
= 11 ﬁnd the solution for which
x
2
+
y
2
+
z
2
is least.
(iv)
Find a value for
a
for which there is a solution such that
x >
10
6
and
y
2
+
z
2
<
1 .
2
Write down a value of
θ
in the interval
π/
4
< θ < π/
2 that satisﬁes the equation
4 cos
θ
+ 2
√
3 sin
θ
= 5
.
Hence, or otherwise, show that
π
= 3 arccos(5
/
√
28) + 3 arctan(
√
3
/
2)
.
Show that
π
= 4 arcsin(7
√
2
/
10)

4 arctan(3
/
4)
.
3
Prove that the cube root of any irrational number is an irrational number.
Let
u
n
= 5
1
/
(3
n
)
. Given that
3
√
5 is an irrational number, prove by induction that
u
n
is an
irrational number for every positive integer
n
.
Hence, or otherwise, give an example of an inﬁnite sequence of irrational numbers which
converges to a given integer
m
.
[An irrational number is a number that cannot be expressed as the ratio of two integers.]
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3
4
The line
y
=
d
, where
d >
0 , intersects the circle
x
2
+
y
2
=
R
2
at
G
and
H
. Show that the
area of the minor segment
GH
is equal to
R
2
arccos
±
d
R
²

d
p
R
2

d
2
.
(
*
)
In the following cases, the given line intersects the given circle. Determine how, in each case,
the expression (
*
) should be modiﬁed to give the area of the minor segment.
(i)
Line:
y
=
c
;
circle: (
x

a
)
2
+ (
y

b
)
2
=
R
2
.
(ii)
Line:
y
=
mx
+
c
;
circle:
x
2
+
y
2
=
R
2
.
(iii)
Line:
y
=
mx
+
c
;
circle: (
x

a
)
2
+ (
y

b
)
2
=
R
2
.
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 Spring '12
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 Math, Equations, Probability distribution, Irrational number, 45◦

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