Yuen Long Merchants Association Secondary School
Teaching Programme for Mathematics (Module 1 – Calculus and Statistics)
Duration 
Lessons 
ctep 
real 
Topic 
S.4 Term 2 
5 
3 
4 
1. Binomial Theorem 
7 
5 
2 
2. Limits and Derivatives 

9 
8 
7 
3. Differentiation 

11 
6 
12 
4. Applications of Differentiation 

10 
8 
6 
5. Exponential and Logarithmic Functions 

(32) 
(32) 
(31) 
S.4 Second Term Examination 

Summer 

7 

6. Derivative of Exp. and Log. Functions and their Applications 
S.5 Term 1 
10 
10 

7. Indefinite Integration and its Applications 
7 
9 

8. Definite Integration 

11 
10 

9. Applications of Definite Integration 

(28) 
(29) 

S.5 First Term Examination 

S.5 Term 2 
15 


10. Further Probability 
16 


11. Discrete Random Variables 

(31) 


S.5 Second Term Examination 

Summer & S.6 
14 


12. Discrete Probability Distributions 
11 


13. Continuous Random Variables and Normal Distribution 

17 


14. Parametric Estimation 

(42) 


S.6 Mock HKDSE Examination 
YUEN LONG MERCHANTS ASSOCIATION SECONDARY SCHOOL
Teaching Programme for S.4 Mathematics Module 2 (Algebra and Calculus)
[The course M2 will starts from second term of S.4.]
15  16 Chapter 1 Surds and Rationalization of Denominators 3 periods
17  19 Chapter 2 Mathematical Induction 10 periods
l Understand the principle of mathematical induction (Only the First Principle of Mathematical Induction is required. Applications to proving propositions relating to summation of series and divisibility are required. Proving propositions involving inequalities is not required.)
20  22 Chapter 3 Binomial Theorem for Positive Integral Indices 6 periods
l Expand binomials with positive integral indices using the Binomial Theorem (Proving the Binomial Theorem is required. The use of the summation notation (å) should be introduced. The following contents are not required:
expansion of trinomials
knowledge of the greatest coefficient, the greatest term and the properties of binomial coefficients
applications to numerical approximation )
23  25 Chapter 4 Trigonometry (I) 12 periods
l Understand the concept of radian measure
l Find arc lengths and areas of sectors through radian measure
l Understand the functions cosecant, secant and cotangent and their graphs
l Understand the identities 1 + tan^{2} q = sec^{2} q and 1 + cot^{2} q = cosec^{2} q (Simplifying trigonometric expressions by identities are required.)
26  29 Chapter 5 Trigonometry (II) 10 periods
l Understand compound angle formulae and double angle formulae for the functions sine, cosine and tangent, and producttosum and sumtoproduct formulae for the functions sine and cosine (The following formulae are required:
n sin (A ± B) = sin A cos B ± cos A sin B
n cos (A ± B) = cos A cos B_{}sin A sin B
n tan (A ± B) = (tan A ± tan B)/(1_{}tan A tan B)
n sin 2A = 2 sin A cos A
n cos 2A = cos^{2} A – sin^{2} A = 1 – sin^{2} A = 2 cos^{2} A – 1
n tan 2A = 2 tan A/(1 – tan^{2} A)
n sin^{2} A = ½ (1 – cos 2A)
n cos^{2} A = ½ (1 + cos 2A)
n 2 sin A cos B = sin(A + B) + sin(A – B)
n 2 cos A cos B = cos(A + B) + cos(A – B)
n 2 sin A sin B = cos(A – B) – cos(A + B)
n sin A + sin B = 2 sin(A+B/2)cos(AB/2)
n sin A – sin B = 2 cos(A+B/2)sin(AB/2)
n cos A + cos B = 2 cos(A+B/2)cos(AB/2)
n cos A – cos B = 2 sin(A+B/2)sin(AB/2)
"Subsidiary angle form" is not required.
sin^{2}A = ½ (1  cos 2A) and cos^{2}A = ½ (1 + cos 2A) can be considered as formulae derived from the double angle formulae. )
S.4 Second Term Examination
YUEN LONG MERCHANTS ASSOCIATION SECONDARY SCHOOL
Teaching Programme for S.5 Mathematics Module 2 (Algebra and Calculus)
1 – 4 Chapter 6 Limits and the number e 13 periods
Introduction of the number e
l Recognize the definitions and the notations of the number e and the natural logarithm
Limits
l Understand the intuitive concept of the limit of function (Students should be able to distinguish "continuous functions" and "discontinuous functions" from their graphs. The absolute value function x, signum function sgn(x), ceiling function éxù (or ]x[) and floor function ëxû (or [x]) are examples of continuous functions and discontinuous functions. Theorems on the limits of sum, difference, product, quotient, scalar multiple and composite functions should be introduced but the proofs are not required.)
l Find the limit of a function (Finding the limits of a rational function at infinity is required.)
5  8 Chapter 7 Differentiation 28 periods
l Understand the concept of derivatives of a function (Students should be able to find the derivatives of elementary functions, such as C, x^{n} (n is a positive integer), Öx, sin x, cos x, e^{x}, ln x from first principles. Notations including y', f '(x) should be introduced. Test for differentiability of functions is not required.)
l Understand the addition rule, product rule, quotient rule and chain rule of differentiation
l Find the derivatives of functions involving algebraic functions, trigonometric functions, exponential functions and logarithmic functions (The following formulae are required:
n (C)' = 0
n (x^{n})' = nx^{n}^{1}
n (sin x)' = cos x
n (cos x)' = sin x
n (tan x)' = sec^{2}x
n (cot x)' = csc^{2}x
n (sec x)' = sec x tan x
n (cosec x)' = cosec x cot x
n (e^{x})' = e^{x}
n (ln x)' = 1/x
The following types of algebraic functions are required:
n polynomial functions
n rational functions
n power functions x^{α}
n functions derived from the above ones through addition, subtraction, multiplication, division and composition)
l Find derivatives by implicit differentiation (Logarithmic differentiation is required.)
l Find the second derivatives of an explicit function (Notations such as y'', f ''(x) should be introduced. Third and higher order derivatives are not required.)
9  12 Chapter 8 Applications of Differentiation 28 periods
l Find the equations of tangents and normals to a curve
l Find maxima and minima (Local and global extrema are required.)
l Sketch curves of polynomial functions and rational functions (The following points are noteworthy in curve sketching:
n symmetry of the curve
n limitations on the values of x and y
n intercepts with the axes
n maximum and minimum points
n points of inflexion
n vertical, horizontal and oblique asymptotes to the curve
Students may deduce the equation of the oblique asymptote to the curve of a rational function by division.)
l Solve the problems related to rate of change, maximum and minimum
S.5 First Term Examination
15 – 19 Chapter 9 Indefinite Integration and its Applications 32 periods
l Recognise the concept of indefinite integration (Indefinite integration as the reverse process of differentiation should be introduced.)
l Understand the properties of indefinite integrals and use of the integration formulae of algebraic functions, trigonometric functions and exponential functions to find indefinite integrals (The following formulae are required:
n ò k dx = kx + C
n ò x^{n} dx = x^{n}^{+1}/(n + 1) + C where n ¹ 1
n ò 1/x dx = ln x + C
n ò e^{x} dx = e^{x} + C
n ò sin x dx = cos x + C
n ò cos x dx = sin x + C
n ò sec^{2}x dx = tan x + C
n ò cosec^{2}x dx = cot x + C
n ò sec x tan x dx = sec x + C
n ò cosec x cot x dx = cosec x + C )
l Understand the applications of indefinite integrals in reallife or mathematical context (Applications of indefinite integrals in some fields such as geometry and physics are required.)
Use integration by substitution find indefinite integrals
l Use trigonometric substitution to find the indefinite integrals involving Ö(a^{2} – x^{2}), Ö(x^{2} – a^{2}) or Ö(a^{2} + x^{2}) (Notations including sin^{1} x, cos^{1} x and tan^{1} x and their related principal values should be introduced.)
l Use integration by parts to find indefinite integrals ( ò ln x dx = x ln x – x + C can be used as an example to illustrate the method of integration by parts. The use of integration by parts is limited to at most two times in finding an integral.)
20 – 24 Chapter 10 Definite Integration 22 periods
25 – 29 Chapter 11 Applications of Definite Integration 14 periods
l Understand the application of definite integrals in finding the area of a plane figure
l Understand the application of definite integrals in finding the volume of a solid of revolution about a coordinate axis or a line parallel to a coordinate axis (Both "disc method" and "shell method" are required. Finding the volume of a hollow solid is required.)
S.5 Second Term Examination
YUEN LONG MERCHANTS ASSOCIATION SECONDARY SCHOOL
Teaching Programme for S.6 Mathematics Module 2 (Algebra and Calculus)
1 – 3 Chapter 12 Determinants and Systems of Linear Equations 9 periods
4 – 6 Chapter 13 Matrices and Systems of Linear Equations 27 periods
Matrices
l Understand the concept, operations and properties of matrices (The addition, scalar multiplication and multiplication of matrices are required. The following properties are required:
n A + B = B + A
n A + (B + C) = (A + B) + C
n (l + m)A = lA + mA
n l(A + B) = lA + lB
n A(BC) = (AB)C
n A(B + C) = AB + BC
n (A + B)C = AC + BC
n (lA)(mB) = (lm)AB
n AB = AB )
l Understand the concept, operations and properties of inverses of square matrices of order 2 and order 3 (The following properties are required:
n the inverse of A is unique
n (A^{1})^{1} = A
n (lA)^{1} = l^{1}A^{1}
n (A^{n})^{1} = (A^{1})^{n}
n (A^{t})^{1} = (A^{1})^{t}
n A^{1} = A^{1}
n (AB)^{1} = B^{1}A^{1} where A, B are invertible matrices and l is a nonzero scalar. )
Systems of linear equations
l Solve systems of linear equations of order 2 and order 3 by Cramer's rule, inverse matrices and Gaussian elimination (The following theorem is required:
n A system of homogeneous linear equations in three unknowns has nontrivial solutions if and only if the coefficient matrix is singular
The wording "necessary and sufficient conditions" could be introduced to students.)
7 – 9 Chapter 14 Introduction to Vectors 13 periods
l Understand the operations and properties of vectors (The addition, subtraction, and scalar multiplication of vectors are required. The following properties are required:
n a + b = b + a
n a + (b + c) = (a + b) + c
n a + 0 = a
n 0a = 0
n l(ma) = (lm)a
n (l + m)a = la + ma
n l(a + b) = la + lb
n If aa + bb = a_{1}a + b_{1}b (a and b are nonzero and are not // to each other), then a = a_{1} and b = b_{1}. )
10 – 13 Chapter 15 Multiplications of Vectors 23 periods
Scalar product and vector product
l Understand the definition and properties of the scalar products (dot product) of vectors (The following properties are required:
n a × b = b × a
n a × (lb) = l(a × b)
n a × (b + c) = a × b + a × c
n a × a = a^{2} ³ 0
n a × a = 0 if and only if a = 0
n a b ³ a × b
n a – b^{2} = a^{2} + b^{2} – 2(a × b) )
l Understand the definition and properties of the vector products (cross product) of vectors in R^{3} (The following properties are required:
n a × a = 0
n b × a = (a × b)
n (a + b) × c = a × c + b × c
n a × (b + c) = a × b + a × c
n (la) × b = a × (lb) = l(a × b)
n a × b^{2} = a^{2} b^{2} – (a × b)^{2}
The following properties of scalar triple products should also be introduced:
n (a × b) × c = a × (b × c)
n (a × b) × c = (b × c) × a = (c × a) × b )
Applications of vectors
l Understand the applications of vectors (Division of a line segment, parallelism and orthogonality are required. Finding angles between two vectors, the projection of a vector onto another vector, the volume of a parallelepiped and the area of a triangle are required.)
Further Learning Unit
l Inquiry and Investigation
14 – 16 Revision
Whole HKDSE Syllabus, Past Examination Papers and PreMock Exam.
S.6 Mock HKDSE Examination