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赢渠梁澳洲数学(题目)

作者:高考题库网
来源:https://www.bjmy2z.cn/gaokao
2020-11-28 21:31
tags:高二数学, 数学, 高中教育

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2020年11月28日发(作者:衡位)
d#{tff'
*r
STELLA MARIS
COLLEGE
2005
TRIAL HIGHER
SCHOOL
CERTIFICATtr
Mathematics
Extension
2
General Instructions
Total marks
-
I20
o
Reading time
-
5 minutes
.
Attempt
-
8
.
All questions
Questions
1
Working time
-
3 hours
.
are of equal value.
.
Write using
black or blue
pen
.
Board- approved
calculators may be used
o
A table
of standard integrals is provided
at the
back of this
paper
.
All necessary working
should be shown
in every
question.
1- 15 Marks
Ouestion
(a)
(b)
(c)
(d)
(e)
I
Evaluare
I
0
By using
integration
by
parts,
find
Jr
ri2x dx
Use the substitution z
-
^[;
to evaluate
t.+
dx
,
expressing
your
answer
-t.f+VX
in simplest
exact form.
tl
t
Using
I
-
tanl
,
evaluate
dx
2
J
0
1+ sin x
(i)
trind real
constants
A,
B and C such that
x+4
:-aL
A
-
Bx +C
x(x2
+ 4)
x'
x2+4
(ii)
Hence find
J
l-:
x(x
J!-
+4 )
a*
Marks
2
-
Start
2
-
15 marks
on a new sheet of paper
Ouestion
(a)
(b)
(c)
The
diagram shows the
graph
of
y
=
f
(x)
On separate diagrams,
sketch the following,
showing essential features.
(i)
y-f(l
xl)
(u)
-
I
f(x)
(iii)
y-(fQ))'
(iv)
y'=
f
tx)
(v) y
=
ln(/(x))
The sketch
shows the
graph
of
y
=
f
(x),
where
f
(x)
=
r'
-3x,
.r
)
1.
Copy
the
diagram.
On
your
diagram, sketch the
graph
of the inverse
function
y
=
f
-1(x)
showing
any intercepts with the
coordinate axes, the
coordinates
of any endpoints and the
coordinates
of the
point
of intersection
of
J
=
f
Q)
and
y
=
7-t
{*)
Find the
equation of the tangent
to the
curv e ,t +
y3
-3xy
-3
atthe
point
(I,2)
on the
curve.
Marks
2
2
)
2
2
3
-
Start on
a new sheet of
paper
3
-
15
marks
Ouestion
(a)
(b)
(c)
(d)
Let a=I-Zi
andB-3+i
Find, in the
form x + iy
,
(i)
ap
(
/
1l
I
a
p
Let
z
-
-.6 -;
(i)
Express
e
in modulus- argument
form.
(ii)
Hence or otherwise,
find
ztO,

your answer in the
form x + iy
Let
z1
=
+@rr#+; sinfr)
and
z2
=
21cosy+; sinE)
(i)
On an
Argand
diagram,
draw the
vectors dn
,
O-B
and OC
representing
z1; z2
and
z1
+
z,
respectivelY.
(ii)
Hence find
I
zr
+
z.)l
rn simplest
exact form.
(i)
On an
Argand diagram
shade
the region where both
lr-l-;)l
<
JZ
and
0
<
arg(1) <;
rr
L
(ii)
Find the exact
area of
the shaded
region. Justify
your
answer.
Marks
/-
2
E___
-
Start
of
paper
sheet
a new
on
marks
4
-
15
Ouestion
(a)
s)
(b)
(c)
Sketch
the
graph of
the
eilipr.
*2
*+=t
,2
;
showing
the
l- ^ i-r^-^^6r^
intercepts
on
^h
the
rl-o
a
-voc
xes,
the
coordinates
of
the
focii and
the
equations
of the
directrices.
Thehyperbola
+
e.
a-
{--r,
a>b>0,haseccentricity
b'
(i)
Show
rhar
rhe
line
through
the
focus
S(ae,0)
that
is
perpendicular
to the
asymptote
y
=
lL
t'ru' equation
ax
+by
-a2e
=0
(ii)
Show
that
this
line
meets
the
asymptote
at
a
point on
the
coffesponding
directrix.
p(p,!)
p
and
e(q,l)
q
are
two
variable
points on
the
rectangular
hyperbola
x!
=
I
such
that
the
chord
PQ
passes
through
the
point A(0,2).
M is
the
midpoint
of
PQ.
(i)
Show
that
PQ
has
equation
x
+
pqy
-
(p+q)
=
0'
Hence
deduce
that
P
+
q
=
2Pq
(ii)
Deduce
that
the
tangent
drawn
from
the
point A
to the
rectangular
hyperbola
touches
the
curve
at
the
point
(
1' 1)
(iii)
Sketch
the
rectanguiar
hyperboia
showing
the
points
P,Q,A
and
M'
Find
the
equation
of the
locus
of M
and
state
any
restrictions
on
the
domain
of
this
locus.
Marks
4

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