Anderson Junior College
Preliminary Examination 2007
H2 Mathematics Paper 2
Section A: Pure Mathematics (40 marks)
1
a)
Differentiate
13
tan (ln
)
x
−
with respect to
x.
[2]
b)
The curve
C
has parametric equations
22
2
,1
x
ty
t
t
t
=
+=
−
+
, where
t
is a nonzero
parameter.
(i)
Show that the gradient of the curve at any point (
x,y
) satisfies the equation
2
3
d(
2
1
)
d2
2
y
tt
xt
−
=
−
.
[2]
(ii)
The line
x = p
is a tangent to the curve
C.
By using the result in (i), find the exact
value of
p.
[2]
2
.
Express
(
in the form
)
23
r
+
2(
1)
(
rA
r
B
r
+
++
−
, where
A
and
B
are constants.
Using the method of difference, find
in terms of
n
.
[5]
()
1
2
n
r
r
r
=
+
∑
Hence, or otherwise, find an expression for
in terms of
n.
[2]
1
2
2
)
3
2
(
−
=
∑
+
r
n
n
r
r
3
.
In an Argand diagram, the point
A
represents the fixed complex number
a
, where
0a
r
g
(
)
2
a
π
<<
. The complex numbers
z
and
w
are such that
2i
za
a
−=
and
ww
i
a
=+
. Sketch, in a single diagram, the loci of the points representing
z
and
w
[3]
Find
a)
the minimum value of
zw
−
in terms of
a
,
[1]
b)
the range of values of
1
arg
z
⎛⎞
⎜⎟
⎝⎠
in terms of arg(
a
).
[3]
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a)
By completing the square, or otherwise, describe the geometrical transformation by
which the curve
can be obtained from the curve
.
[2]
22
45
xy
y
−−−
=
0
1
−=
b)
The diagram below shows the graph of
( )
f
y
=
x
⎟
⎟
with asymptotes
y
= 2 and
x
= 0. The
curve has turning points at (2, 2) and (3, 2).
y
)
2,2
−
5
2
On separate diagrams, sketch the graphs of
(i)
()
x
=−
[
3
]
2
f
y
(ii)
f'
x
=
[
3
]
y
Show all intercepts, asymptotes and turning points clearly on your diagrams if they can
be found.
5
.
Relative to an origin O, the point
A
has position vector
6
2
6
⎛⎞
⎜
⎜
⎜⎟
−
⎝⎠
, the line
has equation
,
1
l
51
81
⎛
⎜
=+
λ
⎜
⎜
−−
⎝
r
0
⎞
⎟
⎟
⎟
⎠
λ
∈
±
and the plane
1
Π
has Cartesian equation
54
.
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 Summer '07
 MOE
 Math, Equations, Standard Deviation, Parametric Equations, Complex number, Alice, red pens, Anderson Junior College

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