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JAMES RUSE AGRICULTURAL HIGH SCHOOL
MATHEMATICS PROGRAMME - PRELIMINARY
MATHEMATICS EXTENSION 1 – 2006
Preliminary: Year 11 Terms 1,2,3
PREAMBLE
This program contains three major sections
A. Revision of year 9 and 10 work,
B. Preliminary Course to be completed in Year 11.
C. HSC Course to be completed in Year 12
Section A contains work on coordinate geometry, algebra, trigonometry, geometry, probability, logarithms
and use of the calculator. This should not be taught as a block at the commencement of the year but
should be revision progressively through the use of short questions at the start of periods, short diagnostic
tests and revision assignments which could be given as homework exercises.
The strand on functions and calculus has been designed so that a variety of functions have been
introduced before concepts and techniques involved in differentiation are met. This has been done to
ensure that initial practice of rules for differentiation is made more meaningful for pupils.
Teachers need to read carefully the appropriate pages of both syllabus and notes, to consult the support
notes for this program, and to peruse previous Higher School Certificate Examination Papers so that the
spirit and intentions of the syllabus are interpreted correctly.
Syllabus Summary – Mathematics (Ext 1)
Preliminary
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HSC
1
TOPIC SUMMARY – PRELIMINARY COURSE
Revision
(1) Algebra
(2) Absolute Value
(3) Co-ordinate Geometry
(4) Trigonometry
(5) Geometry Proofs
(6) Graphs
(7) Logarithms
Preliminary Course
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
Unit 11
Unit 12
Unit 13
Unit 14
Unit 15
Unit 16
Further Graphs and Locus
Quadratics
Trigonometry Expansions
Radian Measure
Limits and Calculus
Products, Quotients and Function of Function
Calculus of Trigonometric, Exponential and Logarithmic Functions
Applications of Differentiation
Geometry of the Circle
Inequalities
Further Trigonometry
Polynomials
Trigonometric Equations
Permutations and Combinations
Locus and The Parabola
Geometry of the Parabola
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2
A. Revision of Year 9 and 10 work.
Revision 1. Algebra
Student outcomes
(i)
(ii)
Student is able to:
Simplify expressions involving removing
parenthesis
Substitute:
Evaluate expressions involving the 4
operations, powers and roots.
Number substitution may involve integers,
fractions, decimals or surds
Implications, considerations and
implementations
Example: Expand and simplify:
4 x 2  5x  7  3 x 2  7 x  1 ,
2 x  3 x 2  5x  2 .

 



Example: Find value of t 4  t 2  1 when
t2 3
or
2
4
A4C
 2
 4
evaluate
when A    , B    ,
4
B
 3
 3
7
8
C   .
 3
Substitute into formulae
Find h if v = 10, r = 2 and v  r 2 h or find t
1
when a = 4, u = – 4, s = 6 and s  ut  at 2 .
2
(iii)
Factorise expressions containing
Common factors.
Differences of squares.
Differences and sum of cubes.
Trinomials.
Four terms by grouping in pairs.
e.g. Factor/factorise: 5 x 2  10 x , 16 x 2  1 ,
3 x 2  4 x  7 , 2 x 3  16 , x 2  y 2  3x  3 y .
(iv)
Simplify Algebraic Expressions:
(v)
Solve linear equations:
5t  3  21  t  ,
(vi)
Solve linear inequalities:
3x  4  2 12 , 3  2x  1 .
(vii)
Solve quadratic equations:
5 x 2  11x  2  0 , 8t 2  1  10t , y 2  6 y ,
3
a3
x 2  5x  6
 2
,
,
a2 a 4
x2
2m  n m  3n 2
3

,
.

3
6
x x x  2 
v  22
(viii) Solve equations reducible to quadratics:
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3x  4
3x  1 5 x  2
2 ,

.
x
3x  1 5 x  2
 16 .

 
2

x 4  9 x 2  8  0 , x 2  3x  x 2  3x  3  0 .
3
(ix)
Solve simultaneous equations:
Standard of difficulty only to the extent
required on later topics.
(x)
Perform arithmetic operations on Surds:
Simplification:
Solve: 3x  2 y  7 and 5x  y   2  18 .
50 , 4 18
Addition and subtraction:
2 3  27 , 3 2  18  3 .
Multiplication:

 3  1 3  1.
Rationalising denominators:
3
2
(xi)

 

2
2 3  3 6 , 2 3 1 3 3 1 , 2 3 1 ,
2
,
3 1
,
3 2
2 3 2
Simplify Indices:
Revision of index laws for positive indices.
Use in simplifying algebraic expressions
 
e.g: 3x
Meaning of negative indices.
Evaluation of numeric expressions.
 2
9 ,  
 3
Simplification of algebraic expressions.
3 x 1 2  x .4 2 x 1 a 1  b 1
,
,
ab
(2  x ) 3
y 2
Meaning of fractional index.
Evaluation of numeric expressions.
Writing expressions in index form
Writing expressions in surd form.
Solve Indicial equations:
Simultaneous indicial equations:
(xiii) Polynomials:
4 operations on polynomials.
Remainder and factor theorems.
Factoring cubic or quartic polynomials.
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1
2
2
3

x,
x
,
x
1
2

3
2
8 , 4 .
x , x
(xii)
2 x .4 x
,
(2 x ) 3
2 3
2
3
1
,
x2
, 2x

3
x2
1
2
4
3
Solve: 2  32 , y  81 , 9 n  27 .
n
Solve: 5 x  y  125 and 7 x  y  1 .
Exercises such as finding b and c if x 3  bx  c
is divisible by x  2 and x  3 .
4
Revision 2. Absolute Values
Students will have been introduced to this topic in the junior school. Therefore a large amount of time
should not be spent on this topic. Pre-testing should be used to identify areas of weakness that can be
addressed.
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
Give definition for absolute values:
a for a  0
(a) a  
 a for a  0
(b) a  a 2
(c) a considered in terms of the distance of
a from zero on a number line.
(d) a  b = the distance between a and b on
a number line
(ii)
Evaluate expressions containing absolute
values.
Evaluate: 4  5  13 , 9  3  5
(iii)
Graph simple functions involving absolute
values.
Graph: y  x  2 , y  3x  2 , y  3  x
(iv) Solve equations containing absolute values.
Solve: x  2  5 , 3x  2  1 ,
3x  2  5 x  4 , 3x  2  5 x  4
(v)
Solve inequalities containing absolute values. Solve: 3x  1  7 , 2  x  5 , 5x  1  8 ,
x  1  x 1 , 2 x 1  x  2  3
(Note: The first 3 could be solved by
considering distances on a number line)
(vi)
Sketch harder absolute value graphs
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Sketch: x  y  1 , x  y  a , y  2  x  1
5
Revision 3. Coordinate Geometry
Student outcomes
(i)
(ii)
Student is able to:
Find for a Straight line:
Slope of the line.
Intercepts.
Draw its graph.
Condition for a point to be on a line.
Distance between point.
Midpoint of interval.
Standard form for equations of lines.
Point–gradient form
Angle between line and positive direction of
the x-axis (Angle of inclination)
For Parallel and perpendicular lines:
(a) Find gradient of a given line.
Know condition for lines to be parallel
( m1  m2 ).
Implications, considerations and
implementations
e.g: Find k if A(2,3) lies on line 3 x  4 y  k .
e.g: y = 0 for x–axis, x = 0, for y–axis.
y = c for line parallel to x–axis, x = k for
line parallel to y–axis.
m  tan 
e.g: Find k if 3x  4 y  7 and kx  5 y  10 are
parallel.
Find the value of k which makes lines
parallel.
(b) Use the properties for parallel and
perpendicular lines to prove that a
particular figure is a parallelogram
e.g: Prove that the figure bounded by the lines
3x  2 y  8 , 4 x  3 y  7 , 3x  2 y  12
and 4 x  3 y  11 is a parallelogram.
(c) Find the equation of a line through a
given point parallel to a given line.
e.g: Find equation of line through A(2, 3) and
parallel to 3x  4 y  10 .
Derivation to be based on fact that equation is
of form 3x  4 y  c and A(2,3) lies on it.
An alternative equation for a line parallel to
ax  by  c and passing through x0 , y 0  is
ax  x0   b y  y0   0 .
(d) Know the condition for lines to be
perpendicular and prove that m1m2  1
Students should be able to find the slope of a
line perpendicular to a given line (ie use
1
m2  
) and also prove that lines are
m1
perpendicular (ie show that m1m2  1)
(e) Use in proving that a particular figure is a
rectangle (opposite sides equal, one angle
a right angle).
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(f) Find values of k which make a pair of
lines perpendicular
e.g.: Find k if 3x  4 y  7 is perpendicular to
kx  5 y  10 .
(g) Find the equation of a line through a
given point which is perpendicular to a
given line.
e.g.: Find equation of the line through A(4,–2)
and perpendicular to 3x  7 y  10 .
Derivation to be based on fact that equation is
of form 7 x  3 y  c and A(4,–2) lies on it.
An alternative equation for a line perpendicular
to ax  by  c and passing through x0 , y 0  is
bx  x0   a y  y0   0 .
(iii)
Find the equation of a line through the
intersection of two given lines.
Given lines l1  0 and l2  0 then the equation
of a line through the intersection point of these
lines has the form l1  kl2  0 .
Reference:
HSC Course in Mathematics Year 11 – 3 unit
Chapter 4 - Jones and Couchman
(iv)
Find perpendicular distance from a point to a
line.
(v)
Circles:
Find equation of a circle from its locus
definition.
Write down the equation of a circle given its
centre and radius.
Find the centre and the radius of a circle
from an equation given in central form or
general form.
(vi)
e.g.1: Write down the centre and radius of the
circle with equation
x  32   y  42  25
e.g.2: Find the centre and radius of circle with
equation x 2  y 2  6 x  8 y  1  0 .
Test if a line is a tangent or a chord to a
circle through
(a) solution of simultaneous equations and
(b) use of the perpendicular distance formula.
e.g. Find the point(s) of intersection of the line
x  y  2 and the circle x 2  y 2  5 .
e.g. Find R if the line x  y  4 is tangent to the
circle x  1   y  2  R 2 .
2
(vii)
3U
Divide a line into a particular ratio m:n, both
internally and externally.
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2
e.g. Use similar triangles to derive the ratio
formula
7
Revision 4. Trigonometry
Students will have been introduced to this topic in the junior school. Therefore a large amount of time
should not be spent on this topic. Pre-testing should be used to identify areas of weakness that can be
addressed.
Student outcomes
(i)
Student is able to:
Redefine trig. ratios in terms of unit circle
incorporating exact values.
Evaluate trig. Ratios for angles in all
quadrants (ASTC).
Implications, considerations and
implementations
Value of expressions such as sin 210 ,
cos 225 , tan 240 .
Simplifying expressions such as sin 180    ,
cos180    , tan 360    .
Find exact values for trig. Ratios of angles
associated with 30, 45, 60 and also
0, 90, 180, 270, 360
(ii)
Sketch graphs of y  sin x , y  cos x ,
y  tan x .
(iii)
Solve simple equations of form sin x  c ,
cos x  c , tan x  c .
(iv)
Solve Problems involving right angled
triangles including those in which:
(a) a diagram is not given
e.g, A ladder 20m long rests against a vertical
wall. The ladder makes an angle of 60o with
the ground. Find the distance that the ladder
will reach up the wall.
(b) angles of elevation and depression are
involved
(c) simple bearings are used
(v)
Use the sine rule to calculate the length of a Application to problems involving angles of
side given 2 angles and a side or calculate an elevation and depression, bearings and
angle given 2 sides and an angle.
geometrical figures.
(See Year 10 program and syllabus for guidance
on standards).
(vi)
Use the cosine rule to obtain sides and angles
and apply to bearings and geometrical
figures.
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(vii) Solve cases where two triangles have a
common side and do not overlap and cases
where triangles overlap. Questions should
include problems on angles of elevation and
depression as well as problems with
bearings.
(Note: The method to be adopted at all times
requires students to obtain a general
expression for the length and angle required
and then to carry out a single calculation, or
find calculation as a final step).
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Revision 5. Geometry
Revision of Years 7–10 geometry. The syllabus notes and sample questions should be closely
examined in order to decide on the standard and explanations sequence. Practice should be given in
choosing correct strategies and techniques for solutions of problems.
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
Define triangles and special quadrilaterals.
(ii)
Define properties of angles on a straight line,
vertically opposite angles, angles at a point
and use these properties in a variety of
problems.
(iii)
Identify parallel lines and angles on parallel
lines and use tests for parallel lines.
(iv)
Identify and calculate:
(a) exterior angles angle sum of a triangle.
(b) angle sum of quadrilaterals and general
polygons.
(c) sum of external angles of general
polygon.
(v)
Test for congruent triangles
(vi)
(a) Recognise properties of parallelograms,
rhombus, rectangle and square and
(b) use sufficiency conditions to test for these
quadrilaterals.
(vii)
Test for similar triangles and calculate sides
in similar triangles.
(viii) State and use Pythagoras' Theorem and its
converse.
(ix)
Derive and use area formulae for
parallelogram, triangle, trapezium and
rhombus.
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10
(x)
Prove the properties:
(a) A line parallel to one side of a triangle
divides the other two sides in the same
proportion.
(b) A line joining the midpoints of two sides
of a triangle is parallel to the third side
and half its length.
(c) Parallel lines preserve ratios of intercepts
on transversals.
(xi)
Complete simple numerical exercises of a
deductive nature, on the above properties.
e.g.
(a) In the figure AE = 15cm , EC = 21cm,
DB = 24 cm , and DE||BC. Find the length of
AD.
C
21
B
E
15
x
D
A
(b) D, E, F are the midpoints of the sides of a
triangle ABC. Prove that triangles ABC and
DEF are similar.
(xii)
Solve problems involving all of the above
properties with or without diagrams, using
numerical or general cases.
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For examples see Syllabus pages 17 - 20
11
Revision 6. Graphs
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
Graph straight lines using intercepts.
(ii)
Graph parabola by using axes of
symmetry/vertex approach with points of
intersection with axes where appropriate.
(iii)
Graph polynomial functions:
(iv)
Graph circles of form
x  a 2   y  b 2  R 2 and semi-circles of
Graph: y  xx  2x  3 , y  x  1x  1 ,
y  x 3  3x 2  4 x , y  x 3  6 x 2  11x  6 .
2
the form y  a 2  x 2 or y   a 2  x 2 .
(v)
Graph hyperbolae:
(vi)
Graph examples involving asymptotes:
E
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1
2
( x  2)
, y
, y
,
x3
x
( x  3)
1
1
y  2  , y  1 .
x
x
Graph: y 
y
x
x
( x  1)
, y
, y 2
.
( x  2)( x  2)
x 1
( x  1)
2
12
Revision 7. Logarithmic and Exponential Functions
Students will have been introduced to this topic in the junior school. Therefore a large amount of time
should not be spent on this topic. Pre-testing should be used to identify areas of weakness that can be
addressed.
Student outcomes
Implications, considerations and
implementations
Student is able to:
(i)
12.3
Find values of 2 x ,3 x ,4 x , .... using a hand
calculator.
Draw graphs of y  2 x ,3 x ,4 x , ....
Graph y  e x .
(ii) Define log a x, log e x .
12.1
N  a x  x  log a N , N  0, a  0, x  
(iii) Evaluate simple logarithmic expressions
12.2 from this definition.
1
eg: log28, log93, log 2   .
 4
(iv)
Sketch log. graphs
Sketch y  log a x, a  2, 3, 4,, 10, e
(v)
Use index laws to prove rules for logarithms:
(for a, x, y > 0)
Verify the results: log a a x   x and a loga x  x .
12.2
(a) log a a  1
(b) log a 1  0
(c) log x  log y  log xy
 
(d) log x n  n log x
x
(e) log x  log y  log  
 y
12.3
(f) log a b 
log c b
log c a
(vi) Simplify numerical expressions
12.3
log 2 20  log 2 5 ,
(vii) Solve simple equations.
eg: log 7 x  2 , 3 x  10 , log 2 x  8 .
log 2 8
.
log 2 4
12.2 Practice in writing a logarithm statement as
in index form and vice versa.
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(viii) Simplify algebraic expressions
12.3
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 x2 y 
 in terms of a, b and c if
Express log 
z


log x  a , log y  b and log z  c .
14
Prelim. UNIT 1.
FURTHER GRAPHS and LOCUS
Student outcomes
(i)
(ii)
Implications, considerations and
implementations
Student is able to:
Understand the concept of a function.
Interpret function notation.
State domain and range.
State the natural domain of a function, as
well as the range corresponding to this
domain, as each function is graphed.
x : x  3,
x : 2  x  2, x : x  4, y : y  2 is to be
Appropriate set notation e.g.
introduced.
For a real domain that excludes a point (say
x  3 ) use x : x  , x  3 or
x : x  3 or x  3 but not x  3.
Discussion of a curve such as x 2  y 2  r 2 as
two functions y  r 2  x 2 and y   r 2  x 2
Also consider the semi-circles derived from the
2
2
circle x  a    y  b   r 2 .
i.e. y  b  r 2  x  a 
2
(iii)
Graph simple cases where the function rule
varies for different parts of the domain
0 if x  0
e.g. f ( x)  
2 x if x  0
and functions where domain is restricted
e.g. graph y  4  x 2 for x  1 .
(iv)
Identify and use the symmetry properties of
odd and even functions.
Define even functions as functions such that
f(x) = f(–x) and give a geometric
interpretation ( = graph is invariant under a
reflection in y-axis)
Define odd functions as functions such that
f(–x) = –f(x) and give a geometric
interpretation ( = graph is invariant under a
180 rotation about the origin)
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Use of symmetry properties of odd and even
functions in graphing functions such as:
y  2x 3 , y 
x2 1
.
x2 1
Find the axis of symmetry of graphs of even
functions.
Understand point symmetry of graphs of odd
functions.
15
(v)
Determine the set of points which satisfy a
Examples:
given set of conditions either algebraically or (1) The locus whose equation is x 2  y 2  0
geometrically.
consists of the points lying on either of the
straight lines y  x , and y   x .
(2) The locus of points P equidistant from two
distinct points A and B. ie PA=PB [Locus is
the perpendicular bisector of the segment
AB].
(3) The locus of a point P such that its distance
from a point A is k times its distance from
point B. ie AP=kBP [Locus is a circle]
(4) Locus of a point P and two fixed points A
and B such that PA+PB = constant. [Locus
is an ellipse]
(5) Locus of points equidistant from a fixed
point and a fixed line. [Locus is a parabola –
will be treated in more depth in Unit 15]
Circle already covered in Revision Unit 3
(vi)
Graph inequalities with at most one nonlinear inequality.
Examples:
(1) Indicate the region determined by the
inequality: 2 x  3 y  6 .
(2) Indicate the region determined by the
2
inequality:  x  3  y 2  1 .
(3) Shade the region that satisfies all three
inequalities: x 2  y 2  1 , y  2 x , x  0 .
(vii)
Sketch graphs by addition or subtraction of
ordinates
Graphs could include: y  x 
2
1
, y  x 2 ,
x
x
E
y  x  2 x , y  sin x  cos x
(viii) Sketch further graphs by:
deciding if the function is even or odd,
finding points of intersection with the axes,
E
finding asymptotes, plotting additional
points.
Examples could include:
x2  2
x2
, y
y  x4 , y  2
( x  1)( x  3)
x 2
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16
Prelim. UNIT 2
QUADRATICS
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
Solve quadratic equations using
(a) Factorising
(b) Completion of square
Solve equations of the type:
(a) x 2  7 x  12  0 , 4 x 2  7 x  3  0 ,
x 2  4x .
(b) 4 x 2  7 x  3  0 , x 2  6 x  4  0 .
(ii)
Finding roots of quadratic equation using the Prove the quadratic formula by completing the
quadratic formula.
square.
(iii)
Solve quadratic inequalities by the use of an
appropriate graph.
(iv)
Establish the relation between roots and
coefficients.
(v)
Find the discriminant.
Use the discriminant to determine types of
roots.
Solve x 2  4 x  3  0 , x 2  16 , x 2  3x .
(1) Finding values of k that give (a) real,
(b) unequal and (c) unreal roots for
equations such as x 2  kx  4  0 .
(2) Find the value(s) of m so that the line
y  mx  m is a tangent to the parabola
y  x2 .
(vi)
Identify types of quadratic expressions:
Find value(s) of k for which an expression of
positive definite, negative definite, indefinite. the form kx2  3x  4 is positive definite or
negative definite.
(vii)
Write identical quadratic expressions.
Examples should include the expression of a
quadratic polynomial ax 2  bx  c in the form
Axx 1  Bx  C .
(viii) Equations reducible to quadratic equations:
Equations should be of the order of difficulty:
R
(a) x 4  9 x 2  20  0 .
(a) From the 2 unit syllabus
3U
2
(b) From the 3 unit syllabus
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1
1


(b)  x    3 x    2  0 ,
x
x


2n
n
2  9.2  8  0 , 9 n  7.3 n  18  0
17
Prelim. UNIT 3
TRIGONOMETRY EXPANSIONS
This topic is revised and extended in Prelim. Unit 10
Student outcomes
(i)
(ii)
Student is able to:
Write down basic identities:
Implications, considerations and
implementations
sin 2  A  cos A , cos2  A  sin A ,
sin  A   sin A , cos A  cos A ,
tan  A   tan A
Prove that
cos A  B  cos A cos B  sin A sin B .
Derive results for cos A  B, sin  A  B  ,
tan  A  B .
(iii)
Use formulae in both "directions".
(iv)
Derive the results for sin 2 A , cos 2 A , tan 2 A
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e.g. Write an equivalent expression for
sin  A  Bsin  A  B  cos A  Bcos A  B
and an expression for tan 2 X  Y  .
18
Prelim. UNIT 4 RADIAN MEASURE
This topic is encountered again in the HSC course , Unit 6
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
(a) Define a radian as an angle subtended at
the centre of a circle by an arc equal to a
radius.
Convert degrees radians using a calculator
and the relationship   180 0 .
(b) Evaluate expressions involving radians
(ii)
(iii)
(b) sin

4
, sin 2
Find the length of an arc: l  R .
Calculate arc length given R and  and
calculating  given l and R.
1 2
R.
2
Find the area of a segment as the difference
between the areas of a triangle and sector.
1
A  R 2   sin   .
2
Find the area of sector: A 
(iv)
Apply the formula for the length of an arc
and the area of a sector to solving problems.
e.g. Finding the volume of a cone formed from
a sector of given dimensions.
(v)
Graph trigonometric functions using radians
e.g. y  3 cos 2 x , y  sin x , y  1  cos x
(vi)
Solve simple equations using graphs
e.g. Graph y  sin 2 x and y  12 x , hence find an
approximation to the solution of
sin 2 x  12 x .
(vii)
Prove that sin x < x < tan x for 0  x 

2
.
This can be done by comparing areas
T
B
rtan
r

O
r
A
area OAB < area sector OAB < area OAT
1 2
1
1

r sin   r 2  r 2 tan 
2
2
2
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(viii) Prove that for small values of x,
sin x  x  tan x .
(ix)
lim sin x
,
x0 x
lim tan x
lim 1  cos x
,
x0 x
x0
x
Deduce results for
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20
Prelim. UNIT 5
LIMITS AND CALCULUS
Student outcomes
(i)
8.3
Student is able to:
Develop the notion of a gradient function by
measurement and tabulation of gradient at a
number of points on curves.
Implications, considerations and
implementations
e.g. y  x 2 , y  x 3
Predict a gradient function for these
functions proposed from tabulated values.
(ii)
Develop an informal idea of a limit.
8.4
Limits of expressions such as
x2  a2
as x a
xa
2 xh  h 2
and
as h0.
h
No extensive work with evaluating limits is
required in the Mathematics (2u) syllabus
Not to be examined
(iii)
Define continuity
8.2
(a) A function is continuous at a point x  c
if
  f x is defined at x  c
  the limit of f x  as x  c (from
above and below) exists
and
  f c equals this limit
No formal work is required.
Not to be examined
(b) A function is continuous for a  x  b if
it is continuous for all point within the
domain a  x  b .
(iv)
Define differentiability
(a) A function is differentiable at a point
x  c if
  f x is defined at x  c
  the limit of f x  as x  c (from
above and below) exists
and
  f (c) equals this limit
Graphs such a those involving absolute values
could be used to demonstrate that the gradients
before and after a point may not be the same.
Not to be examined
(b) A function is differentiable for a  x  b
if it is differentiable for all point within
the domain a  x  b .
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(v)
8.5
State the formal definition of derivative as:
f '  x   lim
x 0
(vi)
f ( x  x)  f ( x)
x
Practise finding derivatives of quadratic and
cubic expressions from first principles.
8.7
Find derivative of xn for positive integer n.
Alternative notation could include
f ( x  h)  f ( x )
f '  x   lim
or
h 0
h
f ( x )  f (c )
f '  c   lim
x c
xc
Use of various notations should be introduced:
dy
d 2
x  3x 
, y , f  x , f 3 ,
dx
dx
(vii) Find, by first principles, the derivative of
1
8.8 y  x and y  x .
(viii) Prove from the definition of the rules, the
8.8 derivatives for cf(x), f(x)±g(x).
(ix)
Complete simple exercises.
Examples:
(a) Differentiate: x 3  2 x  1 , 3 x 
8.8
x
x
(b) Differentiate expressions that have been
2
simplified by expanding: y  2 x  3 ,
y  4x  53x  2 .
(c) If f x   x 3  4 x 2  6 x  2 find f 2 .
(x)
Use correct notation:
Derivatives of q  f  p , s  g t  .
Derivatives with variables other than x and y:
ds
dV
s  3t 2  4t  5 find
, V  4r 2 find
.
dt
dr
(xi)
Find equation of tangent and normal.
e.g. Find the area of the triangles bounded by
the coordinate axis and the tangent to
y  x 2  x at A(1,2).
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22
Prelim. UNIT 6 PRODUCTS, FUNCTION OF FUNCTIONS, QUOTIENTS
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
Prove that if y = u(x).v(x) then
dy
dv
du
dy
u v
 uv   u v
or
dx
dx
dx
dx
Simple exercises such as finding derivatives of
2x  13x  4.
Further practise can be included with exercises
using the function of function rule
Find equations of tangent and normals.
See Fitzpatrick Ch 14
(ii)
Understand the concept of a function of a
function introduced through calculators.
Derive function of function rule.
If y  F (u ) where u  f ( x) then
dy dy du
=
.
dx du dx
(iii)
Complete exercises using the function of
function rule.
Differentiate:
3 x  47 , 6
x2 1 ,
x
5
2
4

3
, etc.
Exercises could include questions that reinforce
the product rule
Example: Find the derivative of
differentiate x 2 4 x  1
(iv)
Find the derivative of y 
y  v x  .
4 x  1 hence
1
considered as
v( x)
1
Prove the quotient rule:
du
dv
v
u
d u
dx
dx or dy  vu   uv 
 =
2
dx
dx  v 
v2
v
(v)
Apply these rules to finding equations of
tangents and normals.
Find derivatives of functions such as:
3
x 1
,
, etc.
2x  1 x  1
The first may be more easily done using the
function of function rule.
e.g. (i) Find the equation of the tangent to
x 1
 1
y
at  2,  .
x 1
 3
(ii) Find the equation of the normal to
y  2 x  3 at the point where x  6 .
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Prelim. UNIT 7
CALCULUS of TRIGONOMETRIC, EXPONENTIAL and LOGARITHMIC FUNCTIONS
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
Prove the derivative of sin x .
Find the derivatives of the other trig.
functions and their reciprocals using trig.
identities and rules of differentiation.
(ii)
Find simple derivatives of expressions
containing trig. functions.
(iii)
Complete exercises using the product,
quotient and function of function rules for
expressions involving trigonometric
expressions.
(iv)
Find the equation of the tangent or normal to Find the equation of the normal to y  sin 2 x at
a curve

the point where x 
8
(v)
Find derivative of 10x from
10 x  x  10 x
.
x  0
x
lim
5x  sin x , tan x 
x 2 sin x ,
lim
1
, cos x  x .
x
sin x
, cos3x  4 .
sin x  1
10
x
1
(= ln10) can be estimated by
x  0 x
considering successively smaller values for x.
Generalise for y = ax.
Find derivative of ex, e f  x  , a x .
(vi)
Investigate the relation
dy dx

 1.
dx dy
This could be demonstrated by considering the
functions such as y  x 3 and y  3 x .
(vii) Find the derivative of log e x and log e  f x . Differentiate expressions in bases other than “e”
Use of change of base theorem.
(viii) Find simple derivatives of functions
containing log. functions.
(ix)
Solve simple problems involving log.
differentiation.
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Differentiate:
3
 log e x , x 2 ln x
2
x
e.g. Find the equation of the tangent at the point
where y  log e x cuts the x–axis
24
Prelim. UNIT 8
APPLICATIONS OF CALCULUS
This topic, even though it contains material from the HSC course, is included here to give a better overall
view of differentiation and its uses and also to further reinforce the various methods of differentiation.
The various sections in the topic will be revised and extended in the appropriate units of the HSC course.
Student outcomes
Implications, considerations and
implementations
Student is able to:
CURVE SKETCHING
(i)
(a) State the significance of the sign of the
first derivative.
(b) Identify monotonic increasing and
decreasing functions.
(ii)
(a) Find stationary points on a curve.
dy
(A definition can be found in the syllabus For a stationary point at x0 , y 0  , dx  0 . For
note 10.2).
a max/min turning point, finding values of x
dy
 0 is sufficient for sketching
for which
(b) Identify local maxima and minima.
dx
dy
(c) Distinguish between local maximum or
 0 does not always
most curves. However
dx
minimum and absolute maximum or
imply that there is a turning point; but in all
minimum value of a function over a
dy
given domain.
cases of a turning point
must change sign
dx
for points before and after x0 , y 0  , while for
dy
some curves
may not exist at : x0 , y 0  yet
dx
the curve changes direction. Consider curves
such as y  3 x 2 and y  x for which the
gradient functions do not exist at their max/min
turning points.
y
5
y
4
5
3
4
2
3
1
2
x
-10
-5
1
5
x
-1
-10
-5
5
-1
(iii)
Define the second derivative
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introduce appropriate notation –
d  dy  d 2 y
, y , f x , g 4 , etc
 ,
dx  dx  dx 2
25
(iv)
(a) Find second derivative and use it to
determine concavity – concave up and
concave down.
NOTE: Stress that a point of inflexion is a point
about which the concavity of the curve
changes.
2
d y
 0 is not a sufficient test for inflexion
(b) Investigate the geometrical significance
dx 2
of the sign of the second derivative.
d2y
points, since
 0 when x = 0 for both
dx 2
(c) Examine inflexional tangents – these may
be horizontal, oblique or vertical.
y  x 2 and y  x 3 but y  x 2 has a minimum
turning point at 0,0 while y  x 3 has a
(d) Find points of inflexion and horizontal
horizontal inflexion point at 0,0 .
points of inflexion.
Curves may also have inflexion points where
the tangent is vertical – consider y  3 x . There
is still a change in y" before and after 0,0 .
1
x3
hx =
2
y
1.5
1
0.5
-3
-2
-1
x
1
2
3
-0.5
-1
-1.5
-2
d2y
for
dx 2
values before and after x0 , y 0  .
In all cases students must test
Students should be made aware of the
geometrical significance of the inflection point it is the only point where the tangent at a point
crosses the curve at that point.
(v)
Sketch simple polynomial curves and
rational functions using calculus techniques
combined with work from Years 9 and 10.
(Polynomials of degree 3 and higher.)
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e.g. sketch y  x 3  3x 2  9 x  2 , y  x 2 
2
.
x
26
MAXIMUM and MINIMUM VALUE PROBLEMS
(vi)
Construct the function to be analysed from
data given in words or on a diagram.
Geometrical and practical problems are to be
stressed.
e.g.1: Constructing various containers or
enclosures to maximise/minimise areas,
volumes, costs etc. given fixed
perimeters, surface areas etc.
Prove that a closed cylinder of fixed
surface area has maximum volume when
its diameter equals its height.
e.g.2: Given the hourly running cost of a ship as
a function of its velocity, find the most
economical running speed.
Students need to pay particular attention to
restrictions on variables and their explanation
of why there is a local as well as an absolute
max./min. for the values under consideration.
* Consider problems in which more than one
case needs to be analysed – HSC 1988 Q7b
** Consider problems in which the required
solution is at an endpoint – Fitzpatrick (2u)
Ex15e Q5
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27
Prelim. UNIT 9
GEOMETRY OF CIRCLE
Except for the assumption "equal arcs subtend equal angles at the circumference or the centre in circles of
equal radius ", students should be reasonably familiar with this work from the junior school. The
emphasis in this topic is to be on:
(i) a logical development of the theorems based on the definitions and assumption,
(ii) solution of problems giving reasons for each step.
Students should be asked to prove the preliminary theorems and the theorems for which proof may be
examined themselves. The chief thrust of the work in exercises should be centred on developing the
ability to choose the appropriate facts for the solution of problems.
All circle geometry proofs are 3 unit work only.
Student outcomes
(i)
Implications, considerations and
implementations
Student is able to:
Understand and use definitions, assumptions
and theorems within circle geometry.
ALL
3U
(a) Define a circle, centre, radius, diameter,
arc, sector, segment, chord, tangent,
concyclic points, cyclic quadrilateral, an
angle subtended by an arc or chord at the
centre and at the circumference, and of
an arc subtended by an angle should be
given.
Assumptions:
(1) Two circles touch if they
have a common tangent at the point of
contact.
(2) The tangent to a circle is
perpendicular to the radius drawn to the point
of contact. Converse.
(ii)
Discuss and prove the following results.
Reproduction of memorised proofs will not
be required.
Prove any of the following results using
properties obtained
(a) Equal angles at the centre stand on
equal chords. Converse.
(b) The perpendicular from the centre of a
circle to a chord bisects the chord.
Converse
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28
(c) Equal chords in equal circles are
equidistant from the centres. Converse
(d) Any 3 non-collinear points lie on a
unique circle whose centre is the point
of concurrency of the perpendicular
bisectors of the intervals joining pairs
of non-collinear points.
(e) The angle at the centre is twice the
angle at the circumference subtended
by the same or equal arcs.
(f) Angles at the circumference in the
same segment are equal.
(g) The angle in a semi-circle is a right
angle.
(h) Opposite angles of a cyclic
quadrilateral are supplementary.
(i) The exterior angle at a vertex of a
cyclic quadrilateral equals the interior
opposite angle.
(j) If the opposite angles in a quadrilateral
are supplementary then the
quadrilateral is cyclic (also a test for 4
points to be concyclic).
(k) If the exterior angle of a quadrilateral
equals it opposite interior angle then
the quadrilateral is cyclic.
(l) If an interval subtends equal angles at 2
points on the same side of it then the
endpoints of the interval and the 2
points are concyclic.
(m) The angle between a tangent and a
chord through the point of contact is
equal to the angle in the alternate
segment.
(n) Tangents to a circle from an external
point are equal.
(o) When circles touch, the line of centres
passes through their point of contact.
(p) The products of the intercepts of 2
intersecting chords/secants are equal.
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29
(q) The square of the length of the tangent
from an external point is equal to the
product of the intercepts on the
secant.
(iii)
Solve problems.
The syllabus supplies a number of numerical
and deductive exercises that can be used as a
guide to the standard to be encouraged.
(a)
Find the value of x.
xo
o
100
(b)
A
B
>>
30
130
o
o
O
>>
C
E
D
AB||CD. O is centre of circle.
Find size of angle BDE.
(c)
D
C
O
150
75
o
o
B
A
O is the centre.
Prove that AC||BD
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30
(d)
T
P
6 cm
10 cm
O
TP is a tangent. O is the centre.
Find the length of TP
(e)
B
>>
A
^
^
>>
D
C
ABCD is a parallelogram
inscribed in a circle. Prove
that ABCD is also a
rectangle.
(e)
A
Q
P
B
AP and AQ are diameters of
two circles that intersect at A
and B. Prove that P, B and Q
are collinear.
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31
A
(f)
D
E
O
B
C
AB is a chord of lenght 8 cm. CD is a
diameter of length 10 cm. AB and CD
intersect at E and AE = 3 cm. Find the
length of OE.
(iv)
Problem Areas:
Have adequate practice in the type of
situations that invariably cause problems.
These include:
(a)
126
(a) failure to discern between cyclic
quadrilaterals and quadrilaterals which are
not cyclic.
o
o
126
xo
xo
Find the value of x.
(b)
(b) failure to recognise all relationships
existing in a figure associated with angles
between a tangent and a chord and the
angles in the alternate segments.
o
63
156
o
x
o
B
T
A
ATB is a tangent.
Find the value of x.
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32
(c) failure to use the fact that the external
angle of a cyclic quadrilateral is equal to
a remote interior angle.
(c)
D
A
F
B
E
C
G
ABCD and ABEF are cyclic quadrilaterals. AB is
a common chord. DC and FE intersect at G.
(i) Prove that GCBE is a cyclic quadrilateral.
(ii) Hence prove that GCE = GBE.
(d)
C
(d) failure to recognise that a quadrilateral is
cyclic.
D
A
E
B
ABC is an acute angled triangle. BD and CE are
altitudes. Prove that AED = ACB.
Sources of additional exercises are:
JENKS "Geometrical Exercises"
FORDER "School Geometry"
HALL & STEPHENS "Geometry"
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Prelim. UNIT 10
INEQUALITIES
Student outcomes
(i)
3U
(ii)
3U
(iii)
3U
Student is able to:
Define a > b , a < b.
Implications, considerations and
implementations
a > b if a–b > 0
Understand the theorems on inequalities :
If a > b then
(a) a+c > b+c
(b) ac > bc if c > 0
(c) ac < bc if c < 0
Know the special rule for inequalities
involving only positive numbers.
(a) If a  b , c  d  ac  bd
(b) If a  b  a 2  b 2
1 1

(c) If a  b 
a b
(iv)
3U
Solve simple inequalities.
(v)
3U
Graph inequalities on the number plane:
Intersections and unions of regions.
e.g ax  by  c  0 , x  a    y  b   r 2 ,
(vi)
Prove simple inequalities
e.g. x 2  y 2  2 xy for all x and y with equality
when x = y. (Consider x 2  y 2  2 xy )
or
x 4  y 4  2 x 2 y 2 for all x and y with
equality when x = y
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(Stress ax  b  x 
b
when a  0 ).
a
2
2
y  x 2 etc.
Exercises relying on harder AM/GM
relationships are covered in Ext 2 course
(vii)
3U
Solve inequalities
JRAHS Ext I (16.12.05) - Prelim 2006
e.g.
2t  1
x2 1
1
0 ;
t 1
x
34
Prelim. UNIT 11
FURTHER TRIGONOMETRY
Student outcomes
(i)
Student is able to:
(a) Write down basic identities:
Complete simple proofs of identities.
(b) Find the solution of simple linear
equations
(ii)
3U
Implications, considerations and
implementations
sin 2  A  cos A , cos2  A  sin A ,
sin  A   sin A , cos A  cos A ,
sin A
,
tan  A   tan A , tan A 
cos A
sin 2 A  cos 2 A  1 , 1  tan 2 A  sec 2 A ,
cos   sin  2  cos   sin  2  2 ,
1
1

 2 sec 2 B , etc.
1  sin B 1  sin B
2 sin A  1, 3 cos A  1  0 , sin A  2
Write down expansions for cos A  B,
sin  A  B , tan  A  B .
(iii)
3U
Find the values of sin105o, tan15o etc.
expressed as surds.
(iv)
Practice using formulae in both "directions".
3U
(v)
Find the acute angle between two lines.
3U
Treat case where one line is parallel to the
y-axis as a special case.
(vi)
(a) Write down the expansions for sin 2 A ,
cos 2 A , tan 2 A
e.g. Write an equivalent expression for
sin  A  Bsin  A  B  cos A  Bcos A  B
and an expression for cos3x  5 y  .
3U
(b) Simplify expressions involving the above
results.
( c) Calculate exact values using the above
results.
(vii)
State the sine and cosine rule.
Complete calculations in triangles with
common sides.
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35
(viii) Complete more difficult
(a) The elevation of a hill at a place P due East of it is 48o, and
questions involving simple 3at a place Q due South of P the elevation is 30o. If the
dimensional problems and/or
3U
distance from P to Q is 500 metres, find the height of the
the necessity to select a
hill.
particular triangle (or
triangles) from a figure:
(b) From a point A the bearings of two points B and C are
found to be 333oT and 013oT respectively. From a point D,
5km due north of A, the bearings are 301oT and 021oT
respectively. By considering the triangle ABC, show that if
the distance between B and C is d km, then
2
2


 sin 210 
sin 59 0 sin 210
 sin 59 0 
2
0




d  25


2
cos
40

 sin 8 0 
sin 32 0 
sin 32 0 sin 8 0







JRAHS Ext I (16.12.05) - Prelim 2006
36
Prelim. UNIT 12
POLYNOMIALS
Student outcomes
(i)
3U
(ii)
Implications, considerations and
implementations
Student is able to:
Define polynomial of degree n.
Coefficient of polynomials.
Monic polynomials.
Zeros of polynomials.
Graph polynomial: Linear, quadratic, cubic,
quartic polynomials.
3U
Graph of functions y   x  2   x  4  ,
2
2
y  x  1 x  5 .
3
Know the significance of multiple roots of a
polynomial.
(It should be noted that a polynomial of odd
degree always has at least one zero.)
2
Prove that Px   x  a  Qx  is tangent to
x-axis at x  a .
(iii)
3U
Divide polynomials.
Understand the division transformations:
Px  Qxx  a  r .
(iv)
3U
Graph rational functions.
Discuss natural domain.
JRAHS Ext I (16.12.05) - Prelim 2006
Graph of y 
x 1
x2  4
,
y

x 1
x2  4
37
(v)
Prove and use
Examples:
3U
(a) The Remainder Theorem.
Find k if x 3  kx  4 is divisible by x  2 .
(b) The Factor Theorem.
Factor polynomials such as x 3  6 x 2  11x  6 .
Solve polynomial equations.
Solve equations such as x 3  x 2  5 x  2  0 .
Graph polynomial equations
Px   x 3  6 x 2  11x  6 developed from the
graphs of their linear factors.
Deduce from the factor theorem:
(a) If Px has zeros a1 , a2 , a3 ,, an then
x  a1 x  a2 x  a3 x  an  is
factor of Px .
(b) If Px is of degree n and has n distinct
zeros a1 , a2 , a3 ,, an then
Px   P0 x  a1 x  a2 x  a3 x  an .
(c) A polynomial of degree n cannot have
more that n zeros.
(d) If Px vanishes for more than n values
of x it vanishes for all values of x.
If x 3  9 x  8  px  2  qx  1 x  2  r
find p, q, r.
3
2
(e) If Ax   Bx  for more than n values of x
they are equal for all values of x.
(vi)
(a) Consider a polynomial as a product of
linear and quadratic factors.
e.g. a polynomial which is monic, of degree 4
and with double root at x  3 is of form:
x  32 x 2  bx  c  .
(b) Construct polynomials from given set of
data.
e.g. P(x) has degree 3, with zeros at
x  0, 1  2 , 1  2 and P1  3 . What is
P(x)?
3U
JRAHS Ext I (16.12.05) - Prelim 2006
38
(vii)
3U
Recognize the relationship between roots and Finding expressions such as 2 + 2 + 2
coefficients for quadratic , cubic and quartic
polynomials.
(a) By use of algebraic identities:
(++)2 = 2+2+2 + 2(++)
(Note: x 3  px  q  0 is often used in
H.S.C. exams as all cubic equations are
reducible to this form by a translation of the
origin to the inflexion point.)
(b) By substitution into the polynomial.
If  ,  ,  are roots of x 3  px  q  0 ,
show that  3   p  q  and  2   p 

        p       3q  3q and
 1 1 1
 2   2   2  3 p  q     2 p 
   
3
JRAHS Ext I (16.12.05) - Prelim 2006
q
3
3
39
Prelim. UNIT 13
TRIGONOMETRIC EQUATIONS and IDENTITIES
This unit deals with solutions expressed in degrees and minutes. HSC Unit 6 and HSC Unit 10 extend the
solutions using radians and inverse trigonometry
Student outcomes
(i)
3U
Implications, considerations and
implementations
Student is able to:
(a) Write down the expansions for sin 2 A ,
cos 2 A , tan 2 A .
Numerical exercises involving 2A , 3A , t .
(b) Prove the results for sin 3A , cos 3 A ,
tan 3A .
(c)
(ii)
Know the t result for sin A , cos A ,
tan A .
2t
.
1 t2
hence deduce that tan 150  2  3 .
e.g prove that if t = tan then tan 2 
(d) Simplfy identities involving sin x  y  ,
2A , 3A , t.
e.g. cot Z  tan Z  2cosec2Z ,
sin 3B cos 3B 2 sin 5B


,
sin 2 B cos 2 B
sin 4 B
1  cos 2 A

 tan A for 0  A  .
1  cos 2 A
2
(e) Eliminate parameters for simple cases.
e.g. find y in terms of x if x  sec , y  cos 2
(Note restrictions)
Solve equations (expessing the solution in a
general formula with degrees):
3U
Derive and use the general solution:
tan x  c  x  180m    , where
tan   c ,  90 0    90 0 and m is an
arbitrary integer.
cos x  c  x  360m    , where
cos  c , 0    180 0 and m is an
arbitrary integer.
e.g. cos x  0.5  x  360m  120 and m is
an arbitrary integer.
sin x  c  180m   1    , where
NOTE:
Equations of the form tan Ax  tan Bx etc. will
be covered in the HSC section involving Radian
Measure
m
sin   c ,  90 0    90 0 and m is an
arbitrary integer.
(iii)
Application to solving equations such as
sin x  0.2 , 3 cos 2x  1 , tan 4x  3  0 in
cases where the general solution and/or
solutions from a given finite domain are
required.
Solve equations of the type Asin x  B cos x
B
which can be rewritten as tan x  .
A
JRAHS Ext I (16.12.05) - Prelim 2006
e.g. 2 sin x  3 cos x .
40
(iv)
(a) Solve quadratic equations involving one
trig. function.
(a) e.g. Solve 2 sin 2 x  3 sin x  2  0 ,
2 sin 2 x  3 sin x  3  0 .
(b) Solve equations in which trig. identities
are used.
(b) e.g. sec x  2 cos x , 3 sin 2 x  2  cos 2 x ,
2 cos 2 A  7 cos A .
3U
(c) Solve quadratic equations of the type:
A sin 2 x  B sin x cos x  C cos 2 x  0 .
(v)
3U
(c) e.g. 2 sin 2 x  3 sin x cos x  2 cos 2 x  0 .
Dividing by cos 2 x  equation in tan 2 x .
(a) Use auxilliary angle to write expressions
of the form Asin x  B cos x in the form
Rsin x    or R cosx   
(b) Solve equations of the form
Asin x  B cos x  C .
(vi)
Solving equations of the form
Asin x  B cos x  C using the t method.
3U
Discuss the problem of the failure to find the
solution x  180 which arises when the
original equation reduces to a linear
equation, rather than a quadratic equation,
after substitution of the t results.
JRAHS Ext I (16.12.05) - Prelim 2006
41
Prelim. UNIT 14
PERMUTATIONS AND COMBINATIONS
Student outcomes
(i)
Student is able to:
determine the number of arrangements of
unlike elements in a line.
3U
Implications, considerations and
implementations
e.g.
(a) Find the way of arranging 6 cards each of
which is a different colour. Factional
notations
(b) Find the number of 4 letter words that can be
formed from the letters of PROBLEM.
(c) Find the number of ways of sitting 8 people
at a rectangular table with 4 seats on each
side if one side faces a painting and the other
side does not.
(The technique to be used in each case is simply
by filling spaces. However students are to be
instructed that they must write a DETAILED
explanation of the reason for each numerical
entry into a space).
(ii)
determine the number of arrangements of
unlike elements in a line and involving
special conditions.
e.g. Find the number of ways of positioning A,
B, C, D, E, F in 6 seats in line if A and B
are to fill the end positions.
e.g. Find the number of ways of positioning A,
B, C, D, E, F if A, B, C are to sit together.
3U
determine the number of arrangements of
unlike elements in a line and involving
groups.
(iv)
use complementary events in space filling.
e.g. Find the number of ways of positioning A,
B, C, D, E, F in 6 seats in line if A and B
are together but E and F are not.
determine the number of arrangements of
unlike elements in a circle.
e.g. Find the number of arrangements of 7
people around a circular table. (= 6!)
Since we can place any person in any position at
the start and then there are 6 people to place in
the remaining 6 positions = 6  5  4  3  2 1 .
3U
(iii)
3U
(v)
3U
In general n objects can be arranged in n  1!
ways around a circle.
(vi)
3U
determine the number of arrangements of
unlike elements in a circle and involving
special conditions.
JRAHS Ext I (16.12.05) - Prelim 2006
42
(vii)
3U
determine the number of arrangements of
unlike elements in a circle and involving
groups.
(viii) Arrangements of objects some of which are
to be separated .
3U
e.g. Find the number of 7 letter words which can
be formed from the letters of PROBING if
no 2 vowels are to be together.
(Note: Decide on the number of ways of
arranging the consonants P R B N G. There are
5! such arrangements. There are 6 possible
places for the first vowel positioned and 5
places for the second vowel. There are 3600
such words).
(ix)
Arrangment of elements some of which are
alike.
e.g. Find the number of ways of arranging the
letter of the word CALL.
3U
(Note: The student should be led by carefully
considering suitable examples to a general
method for arranging n objects of which p are of
one type, q are of a second type and r are of a
third type.)
Examples using this method:
e.g. Find the number of ways of arranging the
letters of WOOLLOOMOOLOO if all
letters are used in the arrangement.
e.g. 3 green, 4 blue and 5 red cards are placed
alongside one another. Find the number of
colour patterns that can be formed.
(x)
3U
(xi)
3U
Introduction of the symbol for placement:
n!
n
Pr = (n–r)!
Selection of r objects from a group
developed by realising that order of choice is
irrelevant and that the r objects can be
considered to be identical.
Proof that the number of selections is
n!
n
r!(n–r)! , which is given the symbol r or
n
C r (Read as "n choose r")

JRAHS Ext I (16.12.05) - Prelim 2006

The result for nr should be initially developed
through a series of exercises of the following
type: Find the number of ways of selecting 3
boys from 7 boys to form a committee.
Students should also see the relation between
n
and the number of branches on a tree
r
diagram.

43
(xii)
Exercises involving use of
n
C r result.
3U
e.g.
(a) Find the number of different committees of 4
that can be selected from 10 people.
(b) Find the number of different committees
containing 2 boys and 3 girls that can be
selected from 5 boys and 6 girls.
(c) Find the number of different committees of 5
that can be selected from 5 boys and 5 girls
if a definite boy Tom and a girl Maree are to
be included.
(xiii) Selection of groups of identical sizes.
3U
JRAHS Ext I (16.12.05) - Prelim 2006
e.g. 22 students are divided into 2 hockey teams
each containing 11 players? How many
team combinations are possible?
22

C1111C11 
 Note this result is

2
!


44
Prelim. UNIT 15
LOCUS and THE PARABOLA.
Student outcomes
(i)
Student is able to:
Find the perpendicular distance from a point
to a line.
(ii)
Define the parabola as a locus
(iii)
Use the locus definition to obtain the
equation of a parabola.
Implications, considerations and
implementations
e.g. Find the equation of the locus of a point
equidistant from (3,4) and the line y  1 .
e.g. Find the equation of the locus of a point
equidistant from (5,12) and the line
3 x  4 y  12  0 .
(iv)
Sketch a parabola given in one of the
following forms:
x 2  4ay , y 2  4ax ,
x  h 2  4a y  k  or
 y  k 2  4ax  h 
(v)
Write down the equation of a parabola given
two of the following: focus, vertex or
directrix.
(vi) Sketch parabolas
R
(vii) Complete the square to find the vertex, the
focus and the directrix from the equation of a
parabola.
(viii) Solve inequalities involving quadratic
E functions.
JRAHS Ext I (16.12.05) - Prelim 2006
Note: A sketch in these types of questions is
essential.
y  x 2  4 , y  x  2x  3 , y  2 x 2  5x  2
y  2 x 2  6 x  3  x  1 12   418  y  34 
2
Note: A sketch in these types of questions is
essential.
e.g. Find the inequality in x and y such that the
point P(x,y) is closer to A( –1, –2) than it
is to the line y = 2.
45
Prelim. UNIT 16
GEOMETRY OF THE PARABOLA.
Student outcomes
(i)
3U
Student is able to:
Convert parametric equations to a cartesian
equation.
(ii)
Find the gradient of a curve for curves given
in parametric form.
3U
Find the equation of a tangent/normal to a
curve given in parametric form.
(iii)
3U
Write down the parametric co-ordinates
2ap, ap 2 of a point on x 2  4ay .
(iv)
derive the equation of a chord:
1
( p  q) x  y  apq .
2
3U
(v)
3U
(vi)
3U
(vii)
3U
Implications, considerations and
implementations

Given x  x (t ) and y  y (t ) ,
dy dy dt
 
dx dt dx

derive the equation of the tangent to the
parabola:
px  y  ap 2 .
derive the equation of the normal to the
parabola:
x  py  2ap  ap 3 .
derive the equation of the chord of contact
drawn from the external point x0 , y 0  :
xx0  2a y  y0 
Student should also recognise that this equation
also represents the equation of the tangent at the
point x0 , y 0 
(viii) prove simple geometrical properties such as:
3U
(a) the tangent to a parabola at a given point
is equally inclined to the axis and the
focal chord through the point. Practical
applications.
(b) the tangents at the extremities of a focal
chord intersect at right angles on the
directrix.
(ix)
3U
solve exercises on simple geometrical
properties.
JRAHS Ext I (16.12.05) - Prelim 2006
46
(x)
Find the locus of a point where:
3U
(a) only one parameter is involved
Students should be encouraged to check that all
points of the equation derived from the
elimination of the parameter in the equations are
on the locus.
(b) two parameters are involved and either
one parameter becomes a constant or
there is a relationship between the two
parameters and they can be eliminated
through the use of the identity:
 p  q 2  p 2  q 2  2 pq
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47
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END OF PRELIMINARY COURSE
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JRAHS Ext I (16.12.05) - Prelim 2006
48