Download in solving problems

NAME:
TEACHER:
MATHEMATICS AND STATISTICS
2013
(MATHEMATICS with CALCULUS)
Standards:
AS 91577 Apply the algebra of complex numbers in solving problems.
Page
2
Page
13
Page
24
AS 91578 Apply differentiation methods in solving problems
AS 91579 Apply integration methods in solving problems
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Page 1
2013 EOY L3 M3 Mathematics
Writer: David Fortune
NAME:
TEACHER:
MATHEMATICS AND STATISTICS 2013
(MATHEMATICS)
91577 Apply the algebra of complex numbers
in solving problems
Credits: Five
QUESTION AND ANSWER BOOKLET
Make sure that you have a copy of the Formulae and Tables Booklet.
You should answer ALL the questions in this booklet.
Show ALL working for ALL questions.
The questions in this booklet are NOT in order of difficulty.
Attempt all questions otherwise you may not provide enough evidence to achieve the required standard.
If you need any more room for any answer, use the extra space provided on page 12 at the back of this
booklet and clearly number the question.
YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE
END OF THE EXAMINATION.
TOTAL
Produced by SINCOS
OVERALL
LEVEL OF
PERFORMANCE
Page 2
t
2013 EOY L3 M3 Mathematics
Writer: David Fortune
You are advised to spend 60 minutes answering the questions in this booklet.
Assessor’s
use only
QUESTION ONE
(a)
Solve the following equation to find an expression for x in terms of p:
log 5 ( x  p)  3 .
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(b)
Write
1 3
2 3 3
in the form a  b 3 where a and b are rational numbers.
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(c)
Solve for x:
x  3  3x .
You must show all your working steps.
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Page 3
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(d) (i) Sketch the locus of the complex number z satisfying the condition
z 1  1  z .
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Draw your answer on the grid.
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2013 EOY L3 M3 Mathematics
Writer: David Fortune
(ii) Find the complex number v that belongs to the locus in Q1d(i) above, with the
greatest positive argument  , ie where 0     .
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Page 5
2013 EOY L3 M3 Mathematics
Writer: David Fortune
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QUESTION TWO
(a)
u and v are complex numbers where u  2  3i and v  3  2i
Find:
(i) 3u -
expressing your answer in the form a  b i .
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(ii) u  v expressing your answer in polar form rcis .
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(b) One root of the equation 3x 2  qx  q  0 , q  0 , is three times the other root.
Find the value of q.
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Page 6
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(c)
Assessor’s
use only
Given that x and y are rational numbers solve the equation:
3 y  2i y  i
.

3x  2 y x  i
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Page 7
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(d)
Solve the equation for x :
x
1 x m


1 x
x
3
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x   1, 0 .
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2013 EOY L3 M3 Mathematics
Writer: David Fortune
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QUESTION THREE


Write  2cis  in the form a  b i where a and b are rational numbers.
3

6
(a)
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(b)
Solve for x:
x 3  3 x .
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Page 9
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(c)
Assessor’s
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Find all the solutions of z 3  1  3 i .
Write your solutions in polar form.
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(d)
Solve the equation for x : 3 x  2 p  2 x 1 .
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Page 10
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(e)
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z 1
 1 then x  0 .
Let z  x  iy . Show that if
z 1
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Page 11
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Assessor’s
use only
Extra paper for continuing your answers, if required.
Clearly number the question(s).
Question
Number
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Page 12
2013 EOY L3 M3 Mathematics
Writer: David Fortune
NAME:
TEACHER:
MATHEMATICS AND STATISTICS 2013
(MATHEMATICS)
91578 Apply differentiation methods in solving problems
Credits: Six
QUESTION AND ANSWER BOOKLET
Make sure that you have a copy of the Formulae and Tables Booklet.
You should answer ALL the questions in this booklet.
Show ALL working for ALL questions.
Show any derivative(s) that you need to find when solving the problems.
The questions in this booklet are NOT in order of difficulty.
Attempt all questions otherwise you may not provide enough evidence to achieve the required standard.
If you need any more room for any answer, use the extra space provided on page 23 at the back of this
booklet and clearly number the question.
YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE
END OF THE EXAMINATION.
TOTAL
Produced by SINCOS
OVERALL
LEVEL OF
PERFORMANCE
Page 13
t
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Assessor’s
use only
You are advised to spend 60 minutes answering the questions in this booklet.
Show all working.
QUESTION ONE
(a)
Differentiate y  3 2 x  4 .
You do not need to simplify your answer.
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(b) Find the equation of the tangent to the curve y 
2
at the point 0, 2 .
x 1
Show any derivatives that you need to find when solving this problem.
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(c)
A meteor enters the earth’s atmosphere and burns up at a rate that is always proportional
to its surface area. Assuming the meteor is always spherical, show that the
radius decreases at a constant rate.
Show any derivatives that you need to find when solving this problem.
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2013 EOY L3 M3 Mathematics
Writer: David Fortune
(d)
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The graph below defines the function y  f (x) .
For the function f (x) find:
(i)
lim f ( x) .
x 3
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(ii)
lim f ( x) .
x 5
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(iii) all value(s) of x where f (x) is not differentiable.
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(iv) all value(s) of x where f ( x)  0 .
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Page 15
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(e)
Assessor’s
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A triangular prism has ends which are equilateral triangles.
The total length of all the edges of the triangular prism is L cm.
Show that the maximum possible surface area of the prism is
L2
2(12  3 )
.
Show any derivatives that you need to find when solving this problem.
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Page 16
2013 EOY L3 M3 Mathematics
Writer: David Fortune
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QUESTION TWO
(a)
(a)
Differentiate
ex
.
y 2
4x  1
You do not need to simplify your answer.
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(b)
The graph shows the function y  f (x) .
On the axes below, sketch the graph of the derived function of f (x) .
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Page 17
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(c)
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An ellipse is defined by the parametric equations x  2 cos t and y  3 sin t .
Find the gradient of the tangent to the curve at t 

4
.
Show any derivatives that you need to find when solving this problem.
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2
d 2 y  dy 
2
   2
(d) Show that y  ln( x  5) is a solution of the differential equation
.
2
dx
x 5
 dx 
2
Show any derivatives that you need to find when solving this problem.
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Page 18
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(e)
3 -1
Coffee is poured at a uniform rate of 10 cm s into a cup whose inside shape is a
truncated cone. If the upper and lower radii of the coffee cup are 4 cm and 2 cm, and the
height of the cup is 6 cm, how fast will the level of the coffee be rising when the coffee
is half way up?
Assessor’s
use only
[It may be useful to extend the cup to form a cone.]
Give any derivative(s) you need to find when solving this problem.
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Page 19
2013 EOY L3 M3 Mathematics
Writer: David Fortune
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QUESTION THREE
(a)
Differentiate
y  e 4 x  ln( 3x  4) .
You do not need to simplify your answer.
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(b)
On Christmas Eve, in-store sales of Galaxy Tablets follow the model:
S  (20t  2t 2 )e 0.2t
0  t  10
where S is the number of sales (in hundreds)
and t is the time in hours after opening the shop doors.
Show any derivatives that you need to find when solving this problem.
(i) Find the rate of change of the number of sales 2 hours after opening the doors.
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(ii) At what time is the maximum rate of sales?
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Page 20
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(c)
A liquid is draining through a conical filter that is
16 cm high and with a radius of 4 cm at the top.
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The liquid is coming out of the cone at a constant rate
of 2 cm3/sec.
At what rate is the depth of liquid in the cone changing
when the liquid is 8 cm deep?
Show any derivatives that you need to find when solving this problem.
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Page 21
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(d)
Assessor’s
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A trapezium is inscribed inside a semicircle of radius
4 cm so that one side is along the diameter.
Find the maximum possible area for the trapezium.

[Hint: it may be useful to find the area in terms of angle a.]
Show any derivatives that you need to find when solving this problem.
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Page 22
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Assessor’s
use only
Extra paper for continuing your answers, if required.
Clearly number the question(s).
Question
Number
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Page 23
2013 EOY L3 M3 Mathematics
Writer: David Fortune
NAME:
TEACHER:
MATHEMATICS AND STATISTICS 2013
(MATHEMATICS)
91579 Apply integration methods in solving problems
Credits: Six
QUESTION AND ANSWER BOOKLET
Make sure that you have a copy of the Formulae and Tables Booklet.
You should answer ALL the questions in this booklet.
Show ALL working for ALL questions.
Show any integral(s) that you need to find when solving the problems.
The questions in this booklet are NOT in order of difficulty.
Attempt all questions otherwise you may not provide enough evidence to achieve the required standard.
If you need any more room for any answer, use the extra space provided on page 34 at the back of this
booklet and clearly number the question.
YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE
END OF THE EXAMINATION.
TOTAL
Produced by SINCOS
OVERALL
LEVEL OF
PERFORMANCE
Page 24
t
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Assessor’s
use only
You are advised to spend 60 minutes answering the questions in this booklet.
QUESTION ONE
(a) Find the integrals:
You do not need to simplify your answers. Do not forget the constant of integration.
(i)
 (cosec
2
4 x) dx .
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(ii)

5 x 4  3x
dx .
x2
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Page 25
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(b)
Calculate the enclosed area between the graphs of y  x 2  1 and y  x  1 .
Assessor’s
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Give the results of any integration needed to solve this problem.
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(c)
A particle moves with velocity function v(t )  cos  t m/s along a straight line where t is
the time in seconds from the start of the motion.
Assuming the distance after 1 second was 4 metres, find the distance function
for the particle.
Give the results of any integration needed to solve this problem.
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Page 26
2013 EOY L3 M3 Mathematics
Writer: David Fortune
(d)
The temperature, T, of a cake cools at a rate proportional to the difference between its
own temperature and the temperature Ts of the surroundings.
(i)
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Write a differential equation that expresses the rate at which the cake cools.
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(ii)
The hot cake had an initial temperature of 270 oC and cools down to 120 oC after
3 minutes. The surrounding room temperature is 20 oC.
Find the time that the hot cake takes to cool to 40 oC.
Give the results of any integration needed to solve this problem.
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Page 27
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Assessor’s
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QUESTION TWO
(a)
Find the integral
 p ( x  3)
3
where p is a constant.
dx
You do not need to simplify your answer.
Do not forget the constant of integration.
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1
(b)
Dierdre wanted to find

ex
2
2 x
dx .
0
She couldn’t integrate so decided to use Simpson’s Rule to find an approximation to
the integral.
She made up a table for the function as shown.
x
e
x2 2 x
0
1
0.25
0.646
0.5
0.472
0.75
0.392
1
0.368
1
Use these values and Simpson’s Rule to find an approximate value for

ex
2
2 x
dx .
0
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2013 EOY L3 M3 Mathematics
Writer: David Fortune
(c)
Find the integral:

4x  1
dx .
2x  1
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(d)
The graph shown is y  sin 2 x .
Given that the shaded area between the curve,
the x-axis and the line x = m is 0.2 find the
value of m.
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2013 EOY L3 M3 Mathematics
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(e)
The curves cy 2  x 3 and y 2  ax (where a > 0 and c > 0) intersect at the origin and at
the point P in the first quadrant.
The areas of the regions enclosed by the arcs OP, the x-axis and the vertical line through P
are A1 and A2 respectively.
Prove that
A1 3
 .
A2 5
Give the results of any integration needed to solve this problem.
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Page 30
2013 EOY L3 M3 Mathematics
Writer: David Fortune
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QUESTION THREE
(a)
Find the integrals:
You do not need to simplify your answers. Do not forget the constant of integration.
(i)
e
1
4x
dx .
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(ii)
 (16 cos 2x sin 2x) dx .
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(b)
Find the general solution of the differential equation
dy
 k ( x  2)( 2  y )
0  x  2 and 0  y  2 .
dx
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b
(c)
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 (2  7h( x)) dx  2 .
1
b
Find a simplified expression, in terms of b, for
 h( x) dx .
1
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(d)
A water trough for stock is in the shape of an isosceles triangular prism.
The top edges of the tank are parallel to the ground.
The stock trough is 40 cm deep, 60 cm wide and 2 m long.
Water evaporates from the trough in such a way that, at any time, the rate of change of
depth of water in the tank is proportional to the top surface area of the water in the tank
times half its depth.
Initially the water trough is full.
After 8 days the depth of water in the trough is 30 cm.
After how many days is the water in the trough 10 cm deep?
Give the results of any integration needed to solve this problem.
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2013 EOY L3 M3 Mathematics
Writer: David Fortune
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Page 33
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Extra paper for continuing your answers, if required.
Clearly number the question(s).
Assessor’s
use only
Question
Number
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Page 34
2013 EOY L3 M3 Mathematics
Writer: David Fortune
ASSESSMENT SCHEDULE
91577 Apply the algebra of complex numbers in solving problems
Achievement
Achievement with Merit
Achievement with Excellence
Apply the algebra of complex numbers in
solving problems involves:
Apply the algebra of complex numbers,
using relational thinking, in solving
problems involves one or more of:
Apply the algebra of complex numbers,
using extended abstract thinking, in solving
problems involves one or more of:
• selecting and using methods
• demonstrating knowledge of concepts
and terms
• communicating using appropriate
representations.
• selecting and carrying out a logical
sequence of steps
• connecting different concepts or
representations
• demonstrating understanding of
concepts
• forming and using a model;
• devising a strategy to investigate or solve
a problem
• identifying relevant concepts in context
• developing a chain of logical reasoning, or
proof
• forming a generalisation;
and also relating findings to a context, or
communicating thinking using
appropriate mathematical statements.
and also using correct mathematical
statements, or communicating mathematical
insight.
Sufficiency for each question:
N0: No response, no relevant evidence.
N1: Attempt at ONE question showing limited knowledge of algebra of complex numbers in solving
problems.
N2: 1 u
A3: 2 u
A4: 3 u
M5: 1 r
M6: 2 r
E7: 1 t with minor errors ignored
E8: 1 t with full excellence criteria
Judgement Statement
Score range
Produced by SINCOS
Not Achieved
Achievement
Achievement with
Merit
Achievement with
Excellence
0-6
7 - 13
14 - 18
19 -24
Schedule Page 35
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
ONE
Achievement (u)
Apply the algebra
of complex numbers
in solving
problems.
1a
x  p  53
1b
1
1c
1d(i)
[ = p  125]
1
3
3
x  3  9x 2
9x 2  x  3  0
x  0.636 valid
x  0.524 not valid
Merit (r)
Apply the algebra of
complex numbers,
using relational
thinking, in solving
problems.
Excellence (t)
Apply the algebra of
complex numbers,
using extended
abstract thinking, in
solving problems.
Correct
expression.
Correct solution.
2 solutions given.
Correct solution.
(one only).
Locus of z  1  1 is the shaded area
inside circle centre (1,0)
Locus of 1  z is the shaded area outside
circle centre (0,0)
Either correct
locus shown.
Correct solution
(correct locus
intersection shown
– dark shading in
right circle).
1d(ii)
( x  1) 2  y 2  x 2  y 2
 2x  1  0
3
1
y
ie x 
2
2
y
3
Argument  tan 1  tan 1
x
1


3
Meet at
Produced by SINCOS
Correct solution
showing correct
logical steps with
correct
mathematical
statements.
Schedule Page 36
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
TWO
Evidence
Achievement (u)
Apply the algebra
of complex numbers
in solving
problems.
2a(i)
2a(ii)
2b
3 7 i
26 cis 0.197 OR
Apply the algebra of
complex numbers,
using extended
abstract thinking, in
solving problems.
26 cis11.3o
If  and 3 are the two roots then
q
q
x   ( x   )( x  3 )
3
3
q
q
 4 and  3 2
3
3
4
  and thus q = 16
3
Correct line 2 or
equivalent.
Correct solution.
(3 y  2i)( x  i)  (3x  2 y )( y  i)
2  2 xi  2 y 2  (3x  2 y)i
Equating real and imaginary parts
y 2  1 and  2 x  3x  2 y
y  1 x 
and
Correct equating
real and
imaginary parts.
1
y
5
ie solutions are x 
2d
Apply the algebra of
complex numbers,
using relational
thinking, in solving
problems.
Excellence (t)
Correct
expression.
x2 
2c
Merit (r)
1
5
x
y 1
1
y  1
5
Correct solution.
x
1  x m2
2

1 x
x
9
2
2
9 x  18x(1  x)  9(1  x)  x(1  x)m 2
m2 x2  m2 x  9  0
Squaring
Correct line 3.
(or equivalent)
 m 2  m 4  36m 2
x
2m 2
1
m 2  36
=  
2
2m
Produced by SINCOS
Schedule Page 37
Correct solution
showing correct
logical steps with
correct
mathematical
statements.
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
THREE
Achievement (u)
Apply the algebra
of complex numbers
in solving
problems.
3a
64
Correct
expression.
3b
x396 x  x
x 1
Correct solution.
3c

 
z 3  2cis    2cis (  2k )
3
3

1
z1  2 3 cis ( )
9
1
7

z 2  2 3 cis ( )
9
1
1
13
5
z 3  2 3 cis (
) or z 3  2 3 cis ( )
9
9
3d
3e
( x  2 p) ln 3  ( x  1) ln 2
x(ln 3  ln 2)  ln 2  2 p ln 3
ln 2  2 p ln 3
x
ln 3  ln 2
x  1  yi
x  1  yi

( x  1) 2  y 2
( x  1) 2  y 2
Apply the algebra of
complex numbers,
using extended
abstract thinking, in
solving problems.
Correct solution
(all three) in either
real or degree
argument format.
Correct
expression
without exponents.
Correct solution.
1
Correct relationship
for x 2 and y 2
without modulus
signs. [line 2 or
equivalent]
positive
2x  2x
4x  0
x0
Produced by SINCOS
Apply the algebra of
complex numbers,
using relational
thinking, in solving
problems.
Excellence (t)
Correct polar
form for one
solution.
( x  1) 2  y 2  ( x  1) 2  y 2
as
Merit (r)
Schedule Page 38
Correct solution
showing correct
logical steps with
correct
mathematical
statements.
2013 EOY L3 M3 Mathematics
Writer: David Fortune
ASSESSMENT SCHEDULE
91578 Apply differentiation methods in solving problems
Achievement
Achievement with Merit
Achievement with Excellence
Apply differentiation methods in solving
problems involves:
Apply differentiation methods, using
relational thinking, in solving problems
involves one or more of:
Apply differentiation methods, using
extended abstract thinking, in solving
problems involves one or more of:
• selecting and using methods
• demonstrating knowledge of concepts
and terms
• communicating using appropriate
representations.
• selecting and carrying out a logical
sequence of steps
• connecting different concepts or
representations
• demonstrating understanding of
concepts
• forming and using a model;
• devising a strategy to investigate or solve
a problem
• identifying relevant concepts in context
• developing a chain of logical reasoning, or
proof
• forming a generalisation;
and also relating findings to a context, or
communicating thinking using
appropriate mathematical statements.
and also using correct mathematical
statements, or communicating mathematical
insight.
Sufficiency for each question:
N0: No response, no relevant evidence.
N1: Attempt at ONE question demonstrating limited knowledge of differentiation techniques
N2: 1 u
A3: 2 u
A4: 3 u
M5: 1 r
M6: 2 r
E7: 1 t with minor errors ignored
E8: 1 t with full excellence criteria
Judgement Statement
Score range
Produced by SINCOS
Not Achieved
Achievement
Achievement with
Merit
Achievement with
Excellence
0-6
7 - 13
14 - 18
19 -24
Schedule Page 39
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Achievement
(u)
Merit (r)
Excellence (t)
Apply
differentiation
methods in
solving problems.
Apply differentiation
methods, using
relational thinking, in
solving problems.
Apply differentiation
methods, using
extended abstract
thinking, in solving
problems.
Evidence
ONE
2
1a
1b
1c
1d

dy 1
 (2 x  4) 3 (2)
dx 3
Correct
derivative.
dy
dy
2
 2

. At x = 0
2
dx
dx ( x  1)
Equation of tangent y  2  2( x  0)
ie y  2 x  2
4
dV
V   r3
 4 r 2
A  4 r 2
3
dr
dV
 kA  4k r 2 k a constant
dt
dr dr dV 4k r 2



dt dV dt
4 r 2
 k a constant
Correct
derivative with
correct solution.
Correct
dV
.
dt
Correct derivative
with correct
solution.
.
(i) lim f ( x)  2
x 3
(ii) lim f ( x) does not exist
x 4
Correct solution in
at least 3
situations.
(iii) f (x) is not differentiable at
x  1, 3 and 5
(iv) x  5
1e
Let x be side of triangle and y be length of
prism.
L  6x
3
2
Area = 3xy  0.5x sin 60  2
6x  3y  L
y
 xL  6 x 2 
3 2
x
2
dA
dA
Correct
.
 L  12 x  3 x
dx
dx
At A  0 L  12 x  3x  0
L
x
12  3
3 2
x
Area  xL  6 x 2 
2
L2
6L2
3
L2




2 (12  3 ) 2
12  3 (12  3 ) 2

12L2  3L2
2 (12  3 ) 2
Produced by SINCOS

L2
2(12  3 )
Schedule Page 40
Correct maximum
for x.
Correct derivative
with correct
solution.
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
TWO
Achievement (u)
Apply
differentiation
methods in solving
problems.
2a
dy (4 x 2  1)e x  8 xe x

dx
(4 x 2  1) 2
Apply differentiation
methods, using
relational thinking, in
solving problems.
Apply differentiation
methods, using
extended abstract
thinking, in solving
problems.
Correct
derivative.
dx
dy
 2 sin t
 3 cos t
dt
dt
dy dy dt
3 cos t



dx dt dx  2 sin t
Correct

2d
Excellence (t)
Correct graph of
derivative.
[graph for x < 1
must be negative
but could be
different shape]
2b
2c
Merit (r)
3
At t 
Grad of tangent = 
4
2
dy
2x
r 2  32  w 2
 5
dx x  5
d 2 y d  dy  d  2 x 
   

dx 2 dx  dx  dx  x 2  5 
( x 2  5)  2  2 x  2 x  2 x 2  10


( x 2  5) 2
( x 2  5) 2
2
Correct
dy
.
dx
Correct derivative
with correct
solution.
dy
.
dx
4x
 dy 
   2
( x  5) 2
 dx 
2
d 2 y  dy 
2 x 2  10
2
   2
 2
2
2
dx
( x  5)
( x  5)
 dx 
12 h
1
dV

 10
ie r  h
4 r
3
dt
1
1
V   r2 h 
 h3
3
27
dV 1
  h2
dh 9
dh dh dV
90



dt dV dt
 h2
dh
90
10


At h = 9 then
dt 81 
9
Correct derivative
with correct
solution.
2
2e
Produced by SINCOS
Correct
dV
.
dh
[or equivalent
answer using diff
variables]
Schedule Page 41
Correct
dh
.
dt
Correct derivatives
with correct
solution.
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
THREE
Achievement (u)
Apply
differentiation
methods in solving
problems.
3a
3b(i)
3b(ii)
dy
3
 4e 4 x 
dx
3x  4
dS
dt
 (20  4t )e 0.2t  (20t  2t 2 )( 0.2e 0.2t )
dS
 3.75
At t = 2
dt
t = 2.93 hours after shop opens
3d
Excellence (t)
Apply differentiation
methods, using
relational thinking, in
solving problems.
Apply differentiation
methods, using
extended abstract
thinking, in solving
problems.
Correct
derivative.
Correct
derivative with
correct solution.
[Units not
required]
From (i) max when
(20  4t )  0.2(20t  2t 2 )  0
0.4t 2  8t  20  0
3c
Merit (r)
h 16
h

ie r 
r
4
4
dV
2
and
dt
1
1
V   r 2 h   h3
3
48
dV
1
dh dh dV
32
  h2



dh 16
dt dV dt  h 2
dh
1

At h = 8
[ = 0.159]
dt 2 
Two solution to
quadratic
equation given
[2.93 and 17.07]
Correct
dV
dh
Correct solution.
(Only one).
Correct derivative
with correct
solution.
1 2
1
4 sin a  2  4 2 sin(   2a)
2
2
= 16 sin a  8 sin( 2a)
Area=
dA
 16 cos a  16 cos( 2a)
da
= 0 when cos a   cos( 2a)
ie
Correct
2 cos 2 a  cos a  1  0
1
cos a 
or cos a  1
2
a

Correct derivative
with correct
solution.
3
ie maximum area  20.78 cm2
Produced by SINCOS
dA
.
da
Schedule Page 42
2013 EOY L3 M3 Mathematics
Writer: David Fortune
ASSESSMENT SCHEDULE
91579 Apply integration methods in solving problems
Achievement
Achievement with Merit
Achievement with Excellence
Apply integration methods in solving
problems involves:
Apply integration methods, using
relational thinking, in solving problems
involves one or more of:
Apply integration methods, using extended
abstract thinking, in solving problems
involves one or more of:
• selecting and using methods
• demonstrating knowledge of concepts
and terms
• communicating using appropriate
representations.
• selecting and carrying out a logical
sequence of steps
• connecting different concepts or
representations
• demonstrating understanding of
concepts
• forming and using a model;
• devising a strategy to investigate or solve
a problem
• identifying relevant concepts in context
• developing a chain of logical reasoning, or
proof
• forming a generalisation;
and also relating findings to a context, or
communicating thinking using
appropriate mathematical statements.
and also using correct mathematical
statements, or communicating mathematical
insight.
Sufficiency for each question:
N0: No response, no relevant evidence.
N1: Attempt at ONE question showing limited knowledge of integration techniques.
N2: 1 u
A3: 2 u
A4: 3 u
M5: 1 r
M6: 2 r
E7: 1 t with minor errors ignored
E8: 1 t with full excellence criteria
Judgement Statement
Score range
Produced by SINCOS
Not Achieved
Achievement
Achievement
with Merit
Achievement with
Excellence
0-6
7 - 13
14 - 18
19 -24
Schedule Page 43
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
ONE
Achievement (u)
Apply integration
methods in solving
problems.
1a(i)
1a(ii)
1b
1
 cot 4 x  C
4
Correct integration.
5 3
x  3 ln x  C
3
Correct integration.
not essential.

 x  1  (x
 1) dx

 ( x  x ) dx
2
0
1
(ii)
Apply integration
methods, using
extended abstract
thinking, in solving
problems.
1
2
0
1d(i)
Apply integration
methods, using
relational thinking,
in solving problems.
Excellence (t)
Meet at x 2  1  x  1
ie x = 0 and x = 1
Area required =
1
1c
Merit (r)
1
1 
1
  x 2  x3  =
6
3 0
2
sin  t
C
Distance =

At t = 1 4  C
sin  t
4
Distance =

dT
dT
 k (Ts  T ) OR
 k (T  Ts )
dt
dt
dT
 k (T  20)
dt
dT
 k dt
T  20
ln( T  20)  kt  C

Correct integration.
Correct
integration.
Correct integration
with correct
solution.
Correct DE.

T  20  Ae kt
At t = 0, T = 270
so T  20  250e kt
When t = 3 T = 120
100  250e 3k
ln 0.4
k
 0.3054
3
When T = 40 then
40  20  250e 0.3054t
 0.3054t  ln 0.08
t  8.28
ie temperature is 40 oC after 8.3
minutes
Produced by SINCOS
Correct
integration with
correct solution.
Correct integration
with correct
general solution
for T.
Correct solution
showing correct
integrations and
correct
mathematical logic
and statements.
Schedule Page 44
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
TWO
Achievement
(u)
Apply integration
methods in
solving problems.
2a
Merit (r)
Excellence (t)
Apply integration
methods, using
relational
thinking, in
solving problems.
Apply integration
methods, using
extended abstract
thinking, in
solving problems.
Correct
integration.
p
( x  3) 4  C
4
Integral =
2b
0.25
1  0.368  4(0.646  0.392)  2(0.472)
3
Correct
integration.
= 0.539
2c

4x  1
3
dx  (2 
)dx
2x  1
2x  1

= 2 x  1.5 ln 2 x  1  C
Correct
integration with
minor error in
division or
substitution.
Correct
integration with
correct solution.
not essential.
m
2d
 sin 2x. dx  0.2
Area 
0
m
1

  cos 2 x   0.2
2
0
1 1
 cos 2m  0.2
2 2
cos 2m  0.6
m  0.464
2e
Correct
integration.
Correct
integration with
correct solution.
Meet at cax  x 3 ie x = 0 and x  ac
ac
A1 

ac
ydx 
0
c 
1
3
x 2 dx
0
ac
5 3
2 4 4
2  

x  = a c
5
5 c  0
5
2
ac
A2 

ac
ydx  a
0

Set up one
correct
definitive
integral for area.
One correct
integration for
an area
completed.
1
x 2 dx
0
ac
5 3
 32 
2 4 4
a
c
=
x
 
3
 0
2
A
3
ie 1  5 
A2 2
5
3
2 a

3
Produced by SINCOS
Correct solution
showing correct
integrations and
correct
mathematical
statements.
Schedule Page 45
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
THREE
Achievement (u)
Apply integration
methods in solving
problems.
3a(i)
3a(ii)
1
 e 4 x  C
4
Merit (r)
Apply integration
methods, using
relational thinking, in
solving problems.
Excellence (t)
Apply integration
methods, using
extended abstract
thinking, in solving
problems.
Correct integration.
 (8 sin 4x) dx
Correct integration.
 2 cos 4x  C
3b




1
dy  k ( x  2)dx
2 y
1
dy  k (2  x)dx
y2
1
ln( y  2)  2kx  kx 2  C
2
2 kx 0.5 kx 2
OR
y  2  Ae
y  2  e 2kx0.5kx
2
c
Correct
integration.
Correct integration
with correct general
solution.
OR equivalent
b
3c
 (2  7h( x)) dx  2
1
b

ie 2 x  7 h( x) dx  2
b
1
Correct definite
integration.
1
b

ie 2b  2  7 h( x) dx  2
1
b

7 h( x) dx  2b ie
1
Produced by SINCOS
Correct integration
with correct
solution.
b

2b
h( x) dx 
7
1
Schedule Page 46
2013 EOY L3 M3 Mathematics
Writer: David Fortune
Question
Evidence
THREE
3d
Achievement (u)
Merit (r)
Excellence (t)
Let width of water at height h be w.
3h
h
40

ie w 
2
0.5 w 30
dh
h
h
 kA   k 200 w   150kh2
dt
2
2
1
dh  150k 1 dt
h2
1
  150kt  C
h
1
At t = 0 h = 40 C  
40
At t = 8 h = 30
1
1

 150k 8 
30
40
1
k 
144000
When h = 10
1
150t
1



10
144000 40


Correct integration.
Correct integration
with correct logical
steps and correct
solution.
t  72
ie 10 cm deep after 72 days
Produced by SINCOS
Schedule Page 47
2013 EOY L3 M3 Mathematics
Writer: David Fortune