Contents
Section 1 Quadratic Equations in One Unknown
1.1 Solve Quadratic Equations ... 1
1.2 Form Quadratic Equations with Given Roots... 4
1.3 Relations between Roots and Coefficients ... 5
1.4 Complex Number ... 7
Section 2 Functions and Graphs 2.1 Intuitive Concept and Notation ... 8
2.2 Graphs of Quadratic Functions ... 9
2.3 Find the Maximum or Minimum Value of a Quadratic Function by Algebraic Method ... 12
Section 3 Exponential and Logarithmic Functions 3.1 Definition of Rational Indices... 14
3.2 Laws of Rational Indices ... 14
3.3 Definition of Logarithm... 16
3.4 Properties of Logarithm ... 16
3.5 Solve Exponential and Logarithm Equations ... 18
3.6 Graphs of Exponential and Logarithmic Functions ... 19
3.7 Applications of Logarithm in Daily Life .. 23
Section 4 More about Polynomials
4.1 Division of Polynomials... 25 4.2 Remainder Theorem and Factor
Theorem... 27 4.3 Greatest Common Divisor (GCD) and
Least Common Multiple (LCM) of
polynomials... 29 4.4 Arithmetic of Rational Functions... 30
Section 5 More about Equations
5.1 Solve Simultaneous Equations in Two Unknowns, One Linear and One
Quadratic... 31 5.2 Solve Equations which can be
Transformed into Quadratic Equations. 37
Section 6 Variations
6.1 Direct and Inverse Variations... 40 6.2 Joint and Partial Variations... 42
Section 7 Arithmetic and Geometric Sequences and Their Summations 7.1 Arithmetic Sequence... 46 7.2 Geometric Sequence... 49 7.3 Application to Daily-life Problems... 52
Section 8 Inequalities and Linear Programming 8.1 Solve Compound Linear Inequalities in
One Unknown... 54 8.2 Solve Quadratic Inequalities in One
Unknown... 56 8.3 Represent the Graphs of Linear
Inequalities in Two Unknowns on a Plane 62 8.4 Solve Systems of Linear Inequalities in
Two Unknowns... 64 8.5 Solve Linear Programming Problems... 66
Section 9 More about Graphs of Functions 9.1 Sketch and Compare Graphs of
Various Types of Functions... 68 9.2 Solve the Equation f(x) = k using the
Graph of y = f(x) ... 70 9.3 Solve the Inequalities f(x) > k, f(x) < k, f(x) ≥ k
and f(x) ≤ k using the Graph of y = f(x)... 71 9.4 Understand the Transformation of the
Function f(x) including f(x) + k, f(x + k), kf(x) and f(kx) ... 73
Section 10 Basic Properties of Circles
10.1 Basic Terminology... 76 10.2 Properties of Chords... 77 10.3 Properties of Angles... 79 10.4 Relation between Angles, Arcs and Chords 81 10.5 Properties of Cyclic Quadrilaterals... 85 10.6 Tests for Concyclic Points... 86 10.7 Properties of and Tests for Tangents... 87
Section 11 Locus
11.1 Intuitive Concept... 90
11.2 Describe and Sketch the Locus of Points Satisfying Given Conditions... 90
11.3 Describe the Locus of Points with Algebraic Equations... 93
Section 12 Equations of Straight Lines and Circles 12.1 Equations of Straight Lines... 97
12.2 Possible Intersection of Two Straight Lines... 101
12.3 Equations of Circles... 103
12.4 Intersection of a Straight Line and a Circle... 106
Section 13 More about Trigonometry 13.1 Trigonometric Ratios of Any Angle... 111
13.2 Trigonometric Identities... 113
13.3 Graphs of Trigonometric Functions... 117
13.4 Trigonometric Equations... 119
13.5 Sine and Cosine Formulae... 122
13.6 Area of Triangle... 125
13.7 Applications to 2D and 3D Problems.... 127
Section 14 Permutation and Combination 14.1 Addition Rule and Multiplication Rule of Counting... 130
14.2 Factorial Notation... 131
14.3 Permutation... 132
14.4 Combination... 134
Section 15 More about Probability
15.1 Set Language and Notation... 137
15.2 Operations on Sets... 139
15.3 Venn Diagram... 140
15.4 Probability in terms of Set Notation... 142
15.5 Addition Law of Probability... 142
15.6 Multiplication Law of Probability... 144
15.7 Conditional Probability... 145
15.8 Use Permutation and Combination to Solve Problems relating to Probability... 148
Section 16 Measures of Dispersion 16.1 Intuitive Concept of Dispersion... 153
16.2 Range and Inter-quartile Range... 154
16.3 Box-and-Whisker Diagram... 157
16.4 Standard Deviation... 161
16.5 Comparison on the Measures of Dispersion... 164
16.6 Effects of Modification of Data on Dispersion... 165
16.7 Standard Score and Normal Distribution... 167
Section 17 Uses and Abuses of Statistics 17.1 Survey Sampling... 170
17.2 Questionnaire Construction... 173 17.3 Uses and Abuses of Statistics Methods. 173
LAST MINUTE MATHEMATICS – COMPULSORY PART (HKDSE)
90 Section 11 Locus
11.1 Intuitive Concept
When a point moves under some specified conditions, the path traced out is called the locus of the point.
Example 11.1
Consider a wheel rolling on a horizontal surface.
The locus of the centre of the wheel is a straight line.
The locus of a point on the circumference of the wheel is a cycloid.
11.2 Describe and Sketch the Locus of Points Satisfying Given Conditions
Conditions and locus Sketch of the locus The point P maintains a
fixed distance r from a fixed point O.
The locus of P is a circle centred at O with radius r.
The point P maintains equal distances from two given points A and B.
The locus of P is the perpendicular bisector of the line segment AB.
cycloid straight line
O r P
A P
B
SECTION 11 ‐ LOCUS
91
Conditions and locus Sketch of the locus The point P maintains a
fixed distance d from a given line L.
The locus of P is a pair of parallel lines parallel to L, one on either side of L, each at a distance d from L.
The point P maintains a fixed distance d from a given line segment AB.
The locus of P is a closed curve around AB consisted of two line segments and two semi-circles.
i. The two line segments are parallel to AB, one on either side of AB, each at a distance d from AB.
ii. The two semi-circles are centred at A and B respectively. Their radii are both d.
The point P maintains equal distances from a pair of parallel lines L1 and L2. The locus of P is a straight line parallel to L1 and L2 equidistant between them.
The point P maintains equal distances from a pair of intersecting lines L1 and L2. The locus of P is the pair of bisectors of the two angles between L1 and L2.
P
L P
d d
P
P B d d A d d
P P
P L1
L2
L1
P
L2
P
LAST MINUTE MATHEMATICS – COMPULSORY PART (HKDSE)
92 Example 11.2
A and B are two fixed points on a plane. P is a point moving on the same plane such that P, A and B are not collinear and PAB PBA. Describe and sketch the locus of P.
Solution
Consider PAB, PAB PBA (given)
PA PB (sides opp., equal s)
P is a point equidistant from A and B.
The locus of P is the perpendicular bisector of the line segment AB, excluding the mid-point of AB.
Example 11.3
A and B are two fixed points on a plane. P is a point moving on the same plane such that APB 55.
Describe and sketch the locus of P.
Solution
Suppose P1 and P2 are two points lying on the locus of P and on the same side of the line segment AB.
Since AP1B AP2B ( 55),
A, P1, P2 and B are concyclic.
(converse of s in the same segment)
The locus of P consists of two major arcs with end-points A and B, excluding A and B.
A
P2
B 55
P1
A
P 55 B
P locus
A
P
B locus
LAST MINUTE MATHEMATICS – COMPULSORY PART (HKDSE)
130
Section 14 Permutation and Combination 14.1 Addition Rule and Multiplication Rule of
Counting
Let n(E) be the number of ways that event E happens.
(1) Addition Rule
For any two events A and B,
n(A or B) n(A) n(B) n(A and B) Moreover, if A and B cannot happen at the same time,
i.e. n(A and B) 0, then A and B are said to be mutually exclusive. In this case,
n(A or B) n(A) n(B) Example 14.1
Among the positive integers up to 1250, how many are either a multiple of 2 or a multiple of 5?
Solution
Since 1250 is a common multiple of 2 and 5, which are relatively prime,
Number of multiples of 2 1250 2 625 Number of multiples of 5 1250
5 250 Number of common multiples of 2 and 5 1250
2 5 125
Required number 625 250 125 750
SECTION 14 ‐ PERMUTATION AND COMBINATION
131 (2) Multiplication Rule
For a 2-step process in which event A may happen in n1(A) ways in the 1st step while event B may happen in n2(B) ways in the 2nd step, if the way in which A happens does not affect the value of n2(B), then
nprocess(A and B) n1(A) n2(B)
Example 14.2
(a) Express 2592 as a product of prime factors, using index notation.
(b) Hence find the number of factors of 2592.
Solution (a) 2592 25 34
(b) Note that every factor of 2592 can be written as 2x 3y where x 0, 1, 2, 3, 4 or 5
y 0, 1, 2, 3 or 4
Thus there are 6 possible values for x and 5 possible values for y. Also the value of x does not affect the number of possible values of y.
Number of factors of 2592 6 5 30 14.2 Factorial Notation
For any positive integer n, n factorial is defined as the product of the first n positive integers.
n! n (n 1) (n 2) … 3 2 1 For n 0, it is defined as 0! 1.
Thus for any positive integer n, n! n (n 1)!
