QB_HL_CALCULUS_PAPER1_WS1
1.
[6
marks]
Paint
is
poured
into
a
tray
where
it
forms
a
circular
pool
with
a
uniform
thickness
of
0.5
cm.
If
the
paint
is
poured
at
a
constant
rate
of
,
find
the
rate
of
increase
of
the
radius
of
the
circle
when
the
radius
is
20
cm.
2.
[7
marks]
A
curve
is
defined
by
the
equation
.
Find
the
equation
of
the
tangent
to
the
curve
at
the
point
where
x
=
1
and
.
3a.
[2
marks]
Find
all
values
of
x
for
such
that
.
3b.
[3
marks]
Find
,
showing
that
it
takes
different
integer
values
when
n
is
even
and
when
n
is
odd.
3c.
[2
marks]
Evaluate
.
4a.
[2
marks]
Express
in
the
form
where
a,
h,
.
4b.
[3
marks]
The
graph
of
is
transformed
onto
the
graph
of
.
Describe
a
sequence
of
transformations
that
does
this,
making
the
order
of
transformations
clear.
4c.
[2
marks]
The
function
f
is
defined
by
.
Sketch
the
graph
of
.
1
4d.
[2
marks]
Find
the
range
of
f.
4e.
[3
marks]
By
using
a
suitable
substitution
show
that
.
4f.
[7
marks]
Prove
that
.
5.
[6
marks]
Find
the
exact
value
of
.
6a.
[2
marks]
Use
the
identity
to
prove
that
.
6b.
[2
marks]
Find
a
similar
expression
for
.
6c.
[4
marks]
Hence
find
the
value
of
.
7.
[6
marks]
The
first
set
of
axes
below
shows
the
graph
of
for
.
2
Let
for
.
(a)
State
the
value
of
x
at
which
is
a
minimum.
(b)
On
the
second
set
of
axes,
sketch
the
graph
of
.
8.
[6
marks]
A
body
is
moving
in
a
straight
line.
When
it
is
metres
from
a
fixed
point
O
on
the
line
its
velocity,
,
is
given
by
.
Find
the
acceleration
of
the
body
when
it
is
50
cm
from
O.
9.
[9
marks]
A
curve
has
equation
.
(a)
Find
in
terms
of
x
and
y.
(b)
Find
the
gradient
of
the
curve
at
the
point
where
and
.
10a.
[2
marks]
Consider
the
function
.
The
sketch
below
shows
the
graph
of
and
its
tangent
at
a
point
A.
3
Show
that
.
10b.
[3
marks]
Find
the
coordinates
of
B,
at
which
the
curve
reaches
its
maximum
value.
10c.
[5
marks]
Find
the
coordinates
of
C,
the
point
of
inflexion
on
the
curve.
10d.
[4
marks]
The
graph
of
crosses
the
-‐‑axis
at
the
point
A.
Find
the
equation
of
the
tangent
to
the
graph
of
at
the
point
A.
10e.
[7
marks]
The
graph
of
crosses
the
-‐‑axis
at
the
point
A.
Find
the
area
enclosed
by
the
curve
,
the
tangent
at
A,
and
the
line
.
4
11a.
[2
marks]
The
function
f
is
defined
by
Determine
whether
or
not
is
continuous.
11b.
[4
marks]
The
graph
of
the
function
is
obtained
by
applying
the
following
transformations
to
the
graph
of
:
a
reflection
in
the
–axis
followed
by
a
translation
by
the
vector
.
Find
.
12.
[7
marks]
Use
the
substitution
to
show
that
.
13a.
[2
marks]
The
graph
of
the
function
is
shown
below.
5
Find
.
13b.
[1
mark]
Hence
find
the
-‐‑coordinates
of
the
points
where
the
gradient
of
the
graph
of
is
zero.
13c.
[3
marks]
Find
expressing
your
answer
in
the
form
,
where
is
a
polynomial
of
degree
3.
13d.
[4
marks]
The
point
(1,
1)
is
a
point
of
inflexion.
There
are
two
other
points
of
inflexion.
Find
the
-‐‑coordinates
of
the
other
two
points
of
inflexion.
13e.
[6
marks]
Find
the
area
of
the
shaded
region.
Express
your
answer
in
the
form
,
where
and
are
integers.
14a.
[2
marks]
6
Consider
the
following
functions:
,
,
Sketch
the
graph
of
.
14b.
[2
marks]
Find
an
expression
for
the
composite
function
and
state
its
domain.
14c.
[7
marks]
Given
that
,
(i)
find
in
simplified
form;
(ii)
show
that
for
.
14d.
[3
marks]
Nigel
states
that
is
an
odd
function
and
Tom
argues
that
is
an
even
function.
(i)
State
who
is
correct
and
justify
your
answer.
(ii)
Hence
find
the
value
of
for
.
15.
[7
marks]
A
curve
has
equation
.
Find
the
coordinates
of
the
three
points
on
the
curve
where
.
16a.
[3
marks]
The
function
is
given
by
.
(i)
Find
an
expression
for
.
(ii)
Hence
determine
the
coordinates
of
the
point
A,
where
.
16b.
[3
marks]
Find
an
expression
for
and
hence
show
the
point
A
is
a
maximum.
7
16c.
[2
marks]
Find
the
coordinates
of
B,
the
point
of
inflexion.
16d.
[5
marks]
The
graph
of
the
function
is
obtained
from
the
graph
of
by
stretching
it
in
the
x-‐‑direction
by
a
scale
factor
2.
(i)
Write
down
an
expression
for
.
(ii)
State
the
coordinates
of
the
maximum
C
of
.
(iii)
Determine
the
x-‐‑coordinates
of
D
and
E,
the
two
points
where
.
16e.
[4
marks]
Sketch
the
graphs
of
and
on
the
same
axes,
showing
clearly
the
points
A,
B,
C,
D
and
E.
16f.
[3
marks]
Find
an
exact
value
for
the
area
of
the
region
bounded
by
the
curve
,
the
x-‐‑axis
and
the
line
.
17a.
[2
marks]
Consider
the
complex
number
.
Use
De
Moivre’s
theorem
to
show
that
.
17b.
[1
mark]
Expand
.
17c.
[4
marks]
Hence
show
that
,
where
and
are
constants
to
be
determined.
17d.
[3
marks]
Show
that
.
17e.
[3
marks]
8
Hence
find
the
value
of
.
17f.
[4
marks]
The
region
S
is
bounded
by
the
curve
and
the
x-‐‑axis
between
and
.
S
is
rotated
through
radians
about
the
x-‐‑axis.
Find
the
value
of
the
volume
generated.
17g.
[3
marks]
(i)
Write
down
an
expression
for
the
constant
term
in
the
expansion
of
,
.
(ii)
Hence
determine
an
expression
for
in
terms
of
k.
18a.
[4
marks]
The
graphs
of
and
are
shown
below.
9
Let
f
(x)
=
.
Draw
the
graph
of
y
=
f
(x)
on
the
blank
grid
below.
10
18b.
[4
marks]
Hence
state
the
value
of
(i)
;
(ii)
;
(iii)
.
19a.
[3
marks]
The
graph
below
shows
the
two
curves
and
,
where
.
11
Find
the
area
of
region
A
in
terms
of
k
.
19b.
[2
marks]
Find
the
area
of
region
B
in
terms
of
k
.
19c.
[3
marks]
Find
the
ratio
of
the
area
of
region
A
to
the
area
of
region
B
.
20.
[9
marks]
The
curve
C
has
equation
.
Determine
the
coordinates
of
the
four
points
on
C
at
which
the
normal
passes
through
the
point
(1,
0)
.
21a.
[5
marks]
Let
.
Show
that
and
deduce
that
f
is
an
increasing
function.
12
21b.
[6
marks]
Show
that
the
curve
has
one
point
of
inflexion,
and
find
its
coordinates.
21c.
[11
marks]
Use
the
substitution
to
show
that
.
22a.
[3
marks]
The
graph
of
is
shown
below,
where
A
is
a
local
maximum
point
and
D
is
a
local
minimum
point.
On
the
axes
below,
sketch
the
graph
of
,
clearly
showing
the
coordinates
of
the
images
of
the
points
A,
B
and
D,
labelling
them
,
,
and
respectively,
and
the
equations
of
any
vertical
asymptotes.
13
22b.
[3
marks]
On
the
axes
below,
sketch
the
graph
of
the
derivative
,
clearly
showing
the
coordinates
of
the
images
of
the
points
A
and
D,
labelling
them
and
respectively.
23.
[6
marks]
Let
.
Using
implicit
differentiation,
show
that
14
.
24a.
[1
mark]
The
function
f
is
defined
on
the
domain
by
.
State
the
two
zeros
of
f
.
24b.
[1
mark]
Sketch
the
graph
of
f
.
24c.
[7
marks]
The
region
bounded
by
the
graph,
the
x-‐‑axis
and
the
y-‐‑axis
is
denoted
by
A
and
the
region
bounded
by
the
graph
and
the
x-‐‑axis
is
denoted
by
B
.
Show
that
the
ratio
of
the
area
of
A
to
the
area
of
B
is
25a.
[4
marks]
Using
the
definition
of
a
derivative
as
,
show
that
the
derivative
of
.
25b.
[9
marks]
Prove
by
induction
that
the
derivative
of
is
.
26a.
[5
marks]
The
diagram
shows
the
graph
of
the
function
defined
by
.
15
The
function
has
a
local
maximum
at
the
point
A
and
a
local
minimum
at
the
point
B.
Find
the
coordinates
of
the
points
A
and
B.
26b.
[3
marks]
Given
that
the
graph
of
the
function
has
exactly
one
point
of
inflexion,
find
its
coordinates.
27a.
[6
marks]
Consider
the
curve
defined
by
the
equation
.
Find
the
gradient
of
the
tangent
to
the
curve
at
the
point
.
27b.
[3
marks]
Hence,
show
that
,
where
is
the
acute
angle
between
the
tangent
to
the
curve
at
and
the
line
y
=
x
.
28a.
[4
marks]
The
curve
C
is
given
by
,
for
.
Show
that
.
28b.
[3
marks]
16
Find
the
equation
of
the
tangent
to
C
at
the
point
.
29a.
[4
marks]
The
curve
C
is
given
implicitly
by
the
equation
for
.
Express
in
terms
of
x
and
y.
29b.
[2
marks]
Find
the
value
of
at
the
point
on
C
where
y
=
1
and
.
30a.
[2
marks]
The
function
f
is
defined
by
,
with
domain
.
Express
in
the
form
,
where
and
.
30b.
[2
marks]
Hence
show
that
on
D.
30c.
[2
marks]
State
the
range
of
f.
30d.
[8
marks]
(i)
Find
an
expression
for
.
(ii)
Sketch
the
graph
of
,
showing
the
points
of
intersection
with
both
axes.
(iii)
On
the
same
diagram,
sketch
the
graph
of
.
30e.
[7
marks]
(i)
On
a
different
diagram,
sketch
the
graph
of
where
.
(ii)
Find
all
solutions
of
the
equation
.
31a.
[1
mark]
Given
that
,
17
sketch
the
graph
of
;
31b.
[1
mark]
show
that
;
31c.
[4
marks]
find
the
volume
of
the
solid
formed
when
the
graph
of
f
is
rotated
through
radians
about
the
x-‐‑axis.
32.
[7
marks]
Given
that
,
use
mathematical
induction
to
prove
that
.
33.
[7
marks]
The
graphs
of
and
,
intersect
and
create
two
closed
regions.
Show
that
these
two
regions
have
equal
areas.
18
34a.
[3
marks]
The
diagram
below
shows
a
circular
lake
with
centre
O,
diameter
AB
and
radius
2
km.
19
Jorg
needs
to
get
from
A
to
B
as
quickly
as
possible.
He
considers
rowing
to
point
P
and
then
walking
to
point
B.
He
can
row
at
and
walk
at
.
Let
radians,
and
t
be
the
time
in
hours
taken
by
Jorg
to
travel
from
A
to
B.
Show
that
.
34b.
[2
marks]
Find
the
value
of
for
which
.
34c.
[3
marks]
What
route
should
Jorg
take
to
travel
from
A
to
B
in
the
least
amount
of
time?
Give
reasons
for
your
answer.
35a.
[8
marks]
At
12:00
a
boat
is
20
km
due
south
of
a
freighter.
The
boat
is
travelling
due
east
at
,
and
the
freighter
is
travelling
due
south
at
.
Determine
the
time
at
which
the
two
ships
are
closest
to
one
another,
and
justify
your
answer.
35b.
[3
marks]
20
If
the
visibility
at
sea
is
9
km,
determine
whether
or
not
the
captains
of
the
two
ships
can
ever
see
each
other’s
ship.
36a.
[3
marks]
The
curve
C
with
equation
satisfies
the
differential
equation
and
y
=
e
when
x
=
2.
Find
the
equation
of
the
tangent
to
C
at
the
point
(2,
e).
36b.
[11
marks]
Find
.
36c.
[6
marks]
Determine
the
largest
possible
domain
of
f.
36d.
[4
marks]
Show
that
the
equation
has
no
solution.
37.
[6
marks]
38.
[7
marks]
Show
that
the
points
(0,
0)
and
(
,
)
on
the
curve
have
a
common
tangent.
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2016
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