For those going to take A levels this year, I have compiled a list of topics you can use to help yourself prepare for your prelims or A levels.
J1s can use it as well. Just blocked off those not taught in your first year.
And to JC Maths or Physics tutors, feel free to download to use as well.
Anyone wants to help contribute for O levels or other subject for A levels can email me the checklist as well.
http://examworld.blogspot.com/2009/04/compiled-topical-revision-checklist.html
Thanks! :D
wow thanks a lot.
Hi,
I will attempt to do a drill-down to sub-topics and skills tested, so that students can focus on what to study for examinations. This is part 1.
Topic: Equations and inequalities
- Formulate and solve a system of linear equations
- Solve inequalities via algebraic method
- Solve inequalities via graphical method
- Solve a new inequality that looks similar to the previously solved one
Topic: Differentiation and its applications
- Differentiation techniques
- Equations of tangents and normals of curves defined in cartesian form, in parametric form or in implicit form
1. find equations of tangents and normals
2. find point where tangent/normal meets curve again
3. find area formed by tangent/normal
4. find equations/points where tangent/normal is parallel to x-/y-axis
- Rates of change (via chain rule)
- Maxima and minima
1. Formulate an expression, then find max/min
2. Graph sketching with stationary points
- Maclaurin's expansion
1. Find expansion via standard series in formulae booklet
2. Find expansion via repeated differentiation
3. Small angle trigonometric approximations
4. Use series to approximate values, to find definite integrals, to find limits, to find equation of tangent at the origin
5. Binomial expansion (focus on these skills: give the expansion in ascending/descending powers, state validity range, use suitable substitution to find an approximate value, find coefficient of x^n term)
Topic: Functions
- Determine whether composite/inverse function exists
- Find composite/inverse function
- Find restricted domain for composite/inverse function to exist
- Find domain/range for a function
- Sketch graphs of functions, their inverses and reflection lines
Thanks!
Cheers,
Wen Shih
wen shih r u a lecturer ?
Hi,
This is part 2.
Topic: Integration and its applications
- Standard integration techniques
- Integration by partial fractions
- Integration by given substitutions
- Integration by parts
- Approximate areas via sum of rectangles, then find limit as number of rectangles approaches infinity
- Find areas of curves defined parametrically
- Find areas/volumes of curves defined in cartesian form or in implicit form
Topic: Differential equations
- Solve DE via direct integration
- Solve DE via variable separable
- Solve DE via substitution, followed by any of the above
- Formulate DE based on a given problem description
1. Newton's cooling model
2. In-out rate flow model
3. Birth-death model
4. Money interest model
- Find particular solutions
- Sketch solution curves
- Comment on appropriateness of model and interpret solutions (e.g. long-term behaviour)
Topic: Sequences and series
- AP/GP
1. Show series is AP/GP
2. Solve a pure AP/GP problem (e.g. find term, sum, common difference, common ratio, sum to infinity)
3. Solve a problem where AP/GP are related (e.g. terms of AP are terms of GP)
4. Solve monetary problems (compound interest)
5. Solve problems involving patterns, e.g. find first term in the nth bracket given that {1}, {3, 5}, {7, 9, 11}, ...
- Summation
1. Sum of r, r^2, r^3, a^r and the like
- Method of differences
1. Use of partial fractions, trigonometric identities or appropriate algebraic manipulations to obtain difference of two similar expressions
2. Cancellation of terms
3. Find expression in terms of n or N
4. Find the sum as n or N approaches infinity (concept of convergence/divergence)
5. Find an inequality for a summation of a similar form, based on the previous one
- Recurrences
1. Find the limit as n approaches infinity
2. Prove some results or inequalities, e.g. x_n > x_(n + 1)
3. Relate to graphs when proving, for instance, x_n > x_(n + 1) when x_n < alpha
- Mathematical induction
1. Prove a result that involves summation, then at times use the result to find sums
2. Prove a result that involves recurrence
3. Prove a conjecture, obtained by observing a pattern that comes from guided steps
Thanks!
Cheers,
Wen Shih
wow!
thanks wen shih!
will use it to make sure my students score As :D
Hi,
This is part 3.
Topic: Graphing techniques
- Sketch basic graphs of parabolas, cubic/quartic equations, rectangular hyperbolas, hyperbolas, ellipses, circles
- Sketch graphs of rational functions with emphasis on skills like:
1. Find axial intercepts
2. Find equations of asymptotes
3. Find stationary points
4. Find range of values for which curve does not lie
5. Find conditions for curve to have stationary values
6. Find unknown variables given certain properties of curve
- Describe linear transformations (of reflections, translations, scalings) in words
- Sketch curves after being transformed by reflections, translations, scalings or their combinations
- Sketch graphs of y = f'(x) and appreciate concepts like concave up/down
- Sketch graphs of the forms y = 1 / f(x), y^2 = f(x), y = f(|x|), y = |f(x)| with combinations of reflections, translations, scalings
- Given a transformed graph, sketch the original y = f(x)
Topic: Vectors
- Use of ratio theorem and mid-point theorem to find position vectors
- Find lengths of projections (e.g. vector onto vector, vector onto line, vector onto plane, vector onto normal of plane)
- Find shortest distances (e.g. point & line, point & plane)
- Find areas (via vector product or standard formula like 1/2 bh)
- Find intersections (e.g. line & line, line & plane, plane & plane, 3 planes)
- Find angles (e.g. vector & vector, line & line, line & plane, plane & plane)
- Find equations of lines, planes
- Solve vector problems that involve diagrams
Topic: Complex numbers
- Find moduli and arguments of complex numbers
- Convert between different forms i.e. cartesian, polar, exponential
- Represent complex numbers as points on argand diagram and prove geometrical properties
- Solve simple or simultaneous complex equations
- Solve polynomial equations
- Solve z^n = c (c is any complex number) type of equations
- Factorise the equation based on roots found
- Sketch loci (e.g. circles, perpendicular bisectors, half-lines) with or without inequalities
- Find max/min moduli/arguments based on loci sketched
- Find intersections of loci sketched
All topics for Pure Mathematics have been covered, I hope ;)
Thanks!
Cheers,
Wen Shih
Hi,
The diligent student could use the lists I've written to track progress made in revision or to select appropriate questions from his/her school revision package to practise.
The committed teacher could use the lists to cover all areas when preparing his/her students.
The statistics component will come soon, if I find time to write ;)
Thanks!
Cheers,
Wen Shih
Hi,
A useful tip that I'd like to share: have a look at Cambridge exam papers of the last 2 years, in case schools have the habit of setting questions of the same flavour for mid-year exams.
Thanks!
Cheers,
Wen Shih
Hi,
Here is the checklist for Probability and Statistics, as promised.
Topic 1: Permutations and combinations
- Application of multiplication and addition principles.
- Solving problems involving arrangements related to lines, circles and beads.
- Solving problems involving restrictions and repetitions.
Topic 2: Probability
- Application of multiplication and addition principles.
- Appreciation of mutually exclusive and independent events.
- Solving problems involving conditional probabilities.
- Solving problems involving the use of tables of outcomes (e.g. sum of two die scores), Venn diagrams and tree diagrams.
- Solving problems involving the use of an infinite geometric progression.
For example: A and B play a game in which they each throw a die in turn until someone throws a six. Find the probability that A wins if he starts the game.
- Application of permutations and combinations in finding probabilities.
Topic 3: Binomial and Poisson distributions
- Finding Binomial and Poisson probabilities.
- Solving problems involving conditional probabilities.
- Commenting on the appropriateness of using Binomial/Poisson distribution for a random variable.
- Finding unknowns (n, p for Binomial or mean for Poisson) based on given probabilities.
- Finding probabilities involving the sum of Poisson variables.
- Solving problems involving a combination of Binomial and Poisson distributions.
- Using the Poisson distribution to approximate the Binomial distribution.
Topic 4: Normal distribution
- Finding Normal probabilities.
- Solving problems involving conditional probabilities.
- Commenting on the appropriateness of using Normal distribution for a random variable.
- Finding unknowns (mean, standard deviation or variance) based on given probabilities.
- Finding unknowns like the value a that satisfies P(X > a) = given probability.
- Solving problems involving the linear combination of independent Normal variables, i.e. aX + bY.
- Appreciating the difference between nX and X_1 + X_2 + ... + X_n.
- Using the Normal distribution to approximate the Binomial distribution with continuity correction.
- Using the Normal distribution to approximate the Poisson distribution with continuity correction.
Topic 5: Sampling
- Commenting on the appropriateness of sampling methods (random, systematic, stratified, quota) for given contexts.
- Solving problems involving the sampling distribution from a normal distribution.
- Application of the Central Limit Theorem.
Topic 6: Hypothesis testing
- Finding of unbiased estimates of the population mean and variance from a sample.
- Appreciation of the meaning of p-value and level of significance.
- Solving problems involving the t-test and z-test.
- Obtaining inequalities involving the level of significance, sample size or population mean.
Topic 7: Regression and correlation
- Sketching a scatter diagram and using it to comment on the relationship between variables or to identify a data pair which should be regarded as suspect.
- Appreciation of the cases where the product moment correlation coefficient (r) is negative, positive, zero.
- Finding the equations of regression lines and r from the given data.
- Finding unknowns, e.g. pair of x, y values, given the data.
- Using equations of regression lines to estimate values and commenting on their reliability.
- Application of a square, reciprocal or logarithmic transformation to achieve linearity for non-linear relationships between variables.
Thanks!
Cheers,
Wen Shih
Hi Mr Wee,
Well done.
Thanks.
It is good that now lecturers and "A" level H2 Maths students will have access to a quick and excellent reference on the patterns of questions asked for each topic.
May I recommend that this thread be made sticky in the forum ?
Thank you for your kind attention.
Regards,
ahm97sic
PS : Will you publish it ? I look forward to the publication of this formula, quick
reference and essential notes guide.
Hi Ahm97sic,
Thanks for your encouragement!
I may consider publication, if there is time for me to organise properly the contents :P
Cheers,
Wen Shih
Hi,
Now that you have a comprehensive checklist to know what to focus on when you revise, it is also important to know something about effective problem-solving.
I will describe Polya's problem-solving approach and illustrate its application with an example. The approach consists of four steps with some further points under each step.
Step 1: Understand the problem
What am I given?
What do I need to solve?
Step 2: Devise or make a plan
What approaches are available for me to solve?
Have I solved similar problems before?
Are there smaller problems for me to solve?
Step 3: Carry out the plan
I will write out the steps of my solution.
Step 4: Look back at the completed solution
Is my final answer a reasonable one?
Are my steps clear, logical and error-proof?
Can I check the answer with calculator?
Can I substitute back?
Can I obtain the same answer via another approach?
Now let's look at an example involving differential equations:
The curve C satisfies the equation y '' = sin^3 x. It is given that C cuts the y-axis at -1 and passes through the point (4/pi, 2/9). Find y in terms of x.
Step 1: Understand the problem
What am I given?
1. y '' = sin^3 x
2. Points (0, -1) and (4/pi, 2/9) lie on C.
What do I need to solve?
1. Find the particular solution, because I am given particular points.
2. Express y in terms of x, because it is required by the question.
Step 2: Devise or make a plan
What approaches are available for me to solve?
They are direct integration, variable separable, substitution which I have been exposed to.
Have I solved similar problems before?
Perhaps, I may have seen it before in lectures, tutorials, tests, exams, etc.
Are there smaller problems for me to solve?
1. Yes, I need to find dy/dx before I can find y in terms of x.
2. The expression sin^3 x may need to be simplified by applying a suitable trigonometric identity before I can integrate it.
3. Since I am given two points that lie on C, I will need to find constants in the particular solution.
Step 3: Carry out the plan
I will write out the steps of the solution as follow.
y '' = sin^3 x
= (sin x)(sin^2 x), by Pythagorean identity
= (sin x)(1 - cos^2 x)
= sin x - (sin x)(cos^2 x)
I recognise that -(sin x)(cos^2 x) looks like f '(x) . {f(x)}^n, so I can integrate the expression directly.
Integrating both sides wrt x,
y ' = -cos x + 1/3 cos^3 x + c, so I have solved a smaller problem.
y ' = -cos x + 1/3 (cos x)(1 - sin^2 x) + c, following the same approach earlier.
= -cos x + 1/3 cos x - 1/3 (cos x)(sin^2 x) + c
= -2/3 cos x - 1/3 (cos x)(sin^2 x) + c
Integrating both sides wrt x again,
y = -2/3 sin x - 1/3 (1/3 sin^3 x) + cx + d
= -2/3 sin x - 1/9 sin^3 x + cx + d, so I have obtained a general solution and another smaller problem has been tackled.
Since (0, -1) lies on C, d = -1.
Now y = -2/3 sin x - 1/9 sin^3 x + cx - 1.
Since (4/pi, 2/9) lies on C, -2/3 - 1/9 + (4/pi) c - 1 = 2/9
=> c = pi/2
Finally, the particular solution is y = -2/3 sin x - 1/9 sin^3 x + (pi/2) x - 1.
Step 4: Look back at the completed solution
Is my final answer a reasonable one?
It is, because the expression is trigonometric given that the original expression is trigonometric.
Are my steps clear, logical and error-proof?
Yes, though I must be careful with minus signs.
Can I substitute back?
Yes, I can check using the two points that lie on C.
Can I obtain the same answer via another approach?
No, but I can differentiate y twice and see if I could obtain the expression
y '' = sin^3 x,
though it may be time-consuming in the exam. I may do it if it is a tutorial question or a question that I'm doing during my revision.
I hope this example convinces students that the Polya's method is effective, especially in its steps 1 & 2. By carrying out the first two steps rigorously, it will bring about the solution naturally.
Thanks for reading!
Cheers,
Wen Shih
this is one rare useful forum haha (= thankss
Hi,
Do share with your classmates and friends who may be struggling with H2 maths :)
Thanks in advance!
Cheers,
Wen Shih
Hi,
It is important for students to read the H2 maths syllabus carefully to have a good feel of what's in and out of the content assessed.
It is usual for one to focus on what is in the syllabus and to neglect finer points.
I'll highlight these points below.
1.1 Functions, inverse functions and composite functions
- Exclude the use of the relation (fg)^(-1) = g^(-1) f^(-1).
- Restriction of domain to obtain a composite function is not required.
1.2 Graphing techniques
- y = f(kx + a) is a composition of y = f(x + a) followed by y = f(kx); or a composition of y = f(kx) followed by y = f(x + a/k).
2.1 Summation of series
- Students are not required to know the tests for convergence.
3.2 Scalar and vector products of vectors
- Exclude triple products a.b x c and a x b x c.
3.3 Three-dimensional geometry
- Exclude finding the shortest distance between two skew lines.
- Exclude finding an equation for the common perpendicular to two skew lines.
4.2 Complex numbers expressed in polar form
- Exclude loci such as |z - a| = k|z- b|, where k is not equal to 1 and
arg(z - a) - arg(z - b) = alpha.
- Exclude properties and geometrical representation of the nth roots of unity.
- Exclude use of de Moivre’s theorem to derive trigonometric identities.
5.1 Differentiation
- Exclude finding non-stationary points of inflexion.
- Exclude problems involving small increments and approximation.
- Differentiation from first principles is optional and will not be tested in the examination.
- Determining the maximum and minimum points using the second derivative test, is assumed knowledge covered in O-level Additional Maths.
5.2 Maclaurin's series
- Exclude derivation of the general term of the series.
5.3 Integration techniques
- Exclude reduction formulae.
- Knowledge of t-formula is not required.
5.4 Definite integrals
- Exclude approximation of area under a curve using the trapezium rule.
- Finding the volume of revolution of a curve defined parametrically is not required.
5.5 Differential equations
- Students are not expected to solve differential equations of the form dy/dx = f(x, y).
7.1 Binomial and Poisson distributions
- Exclude calculation of mean and variance for other probability distributions.
- Goodness of fit tests are not required.
7.2 Normal distribution
- Exclude finding probability density functions and distribution functions.
- Exclude calculation of E(X) and Var(X) from other probability density functions.
- The normal distribution is defined by a probability density function in terms of mu and sigma. The introduction of the probability density function is not required.
8.1 Sampling
- ‘Large’ samples will usually be of size at least 50, but students should know that using the approximation of normality can sometimes be useful with samples that are smaller than this.
8.2 Hypothesis testing
- Exclude testing the difference between two population means.
- The Central Limit Theorem should be applied if the population is not normal.
- Knowledge of Type I and Type II errors is not expected.
9.1 Correlation coefficient and linear regression
- Exclude derivation of formulae.
- Exclude hypothesis tests.
- Zero correlation does not necessary imply ‘no relationship’, but rather ‘no linear relationship’.
- The sum of squares of the errors can be introduced to explain how the linear regression line is obtained.
- A different line of regression will be obtained if we interchange the independent and dependent variables. Students are not expected to know the relationship r^2 = (b_1)(b_2), where b_1 and b_2 are regression coefficients.
Thank you.
Cheers,
Wen Shih
2009 A level H2 Syllabus
http://www.seab.gov.sg/SEAB/aLevel/syllabus/2009_GCE_A_Level_Syllabuses/9740_2009.pdf
Some A level books.
Some written by wee_ws
http://www.a-level-books.com/alevel/tenyearseries.asp?currentpage=1&rowcount=0&sid=21&cat=3
Originally posted by wee_ws:Hi,
It is important for students to read the H2 maths syllabus carefully to have a good feel of what's in and out of the content assessed.
It is usual for one to focus on what is in the syllabus and to neglect finer points.
I'll highlight these points below.
1.1 Functions, inverse functions and composite functions
- Exclude the use of the relation (fg)^(-1) = g^(-1) f^(-1).- Restriction of domain to obtain a composite function is not required.
1.2 Graphing techniques
- y = f(kx + a) is a composition of y = f(x + a) followed by y = f(kx); or a composition of y = f(kx) followed by y = f(x + a/k).2.1 Summation of series
- Students are not required to know the tests for convergence.3.2 Scalar and vector products of vectors
- Exclude triple products a.b x c and a x b x c.3.3 Three-dimensional geometry
- Exclude finding the shortest distance between two skew lines.- Exclude finding an equation for the common perpendicular to two skew lines.
4.2 Complex numbers expressed in polar form
- Exclude loci such as |z - a| = k|z- b|, where k is not equal to 1 and
arg(z - a) - arg(z - b) = alpha.- Exclude properties and geometrical representation of the nth roots of unity.
- Exclude use of de Moivre’s theorem to derive trigonometric identities.
5.1 Differentiation
- Exclude finding non-stationary points of inflexion.- Exclude problems involving small increments and approximation.
- Differentiation from first principles is optional and will not be tested in the examination.
- Determining the maximum and minimum points using the second derivative test, is assumed knowledge covered in O-level Additional Maths.
5.2 Maclaurin's series
- Exclude derivation of the general term of the series.5.3 Integration techniques
- Exclude reduction formulae.- Knowledge of t-formula is not required.
5.4 Definite integrals
- Exclude approximation of area under a curve using the trapezium rule.- Finding the volume of revolution of a curve defined parametrically is not required.
5.5 Differential equations
- Students are not expected to solve differential equations of the form dy/dx = f(x, y).7.1 Binomial and Poisson distributions
- Exclude calculation of mean and variance for other probability distributions.- Goodness of fit tests are not required.
7.2 Normal distribution
- Exclude finding probability density functions and distribution functions.- Exclude calculation of E(X) and Var(X) from other probability density functions.
- The normal distribution is defined by a probability density function in terms of mu and sigma. The introduction of the probability density function is not required.
8.1 Sampling
- ‘Large’ samples will usually be of size at least 50, but students should know that using the approximation of normality can sometimes be useful with samples that are smaller than this.8.2 Hypothesis testing
- Exclude testing the difference between two population means.- The Central Limit Theorem should be applied if the population is not normal.
- Knowledge of Type I and Type II errors is not expected.
9.1 Correlation coefficient and linear regression
- Exclude derivation of formulae.- Exclude hypothesis tests.
- Zero correlation does not necessary imply ‘no relationship’, but rather ‘no linear relationship’.
- The sum of squares of the errors can be introduced to explain how the linear regression line is obtained.
- A different line of regression will be obtained if we interchange the independent and dependent variables. Students are not expected to know the relationship r^2 = (b_1)(b_2), where b_1 and b_2 are regression coefficients.
Thank you.
Cheers,
Wen Shih
Hi Mr Wee,
I have seen your 2009 E.Maths and Add.Maths TYS by Dyna in Big bookshop but not yet available in Popular Bookshop.
I have not seen your 2009 H1 and H2 Maths TYS by Dyna in both bookshops yet.
I like your revision notes for H2 Maths as discussed in homework forum. Will the essential notes and formula guide be published at the same time ? This useful guide will fill the void that the other H2 Maths guides have not covered. It will be a very handy and useful guide for H2 Maths students.
Originally posted by Lee012lee:Hi Mr Wee,
I have seen your 2009 E.Maths and Add.Maths TYS by Dyna in Big bookshop but not yet available in Popular Bookshop.
I have not seen your 2009 H1 and H2 Maths TYS by Dyna in both bookshops yet.
I like your revision notes for H2 Maths as discussed in homework forum. Will the essential notes and formula guide be published at the same time ? This useful guide will fill the void that the other H2 Maths guides have not covered. It will be a very handy and useful guide for H2 Maths students.
Hi,
Thanks for your support!
Dyna is still rushing the 2009 H1 Maths TYS. H2 Maths TYS is ready, if you wish to purchase directly from Dyna.
The checklist and formula guide are made freely available on my website at this moment to reach out to students directly. Publishing and making students purchase them may not be advisable in this difficult time, I feel.
Thanks again.
Cheers,
Wen Shih
Originally posted by wee_ws:Hi,
Thanks for your support!
Dyna is still rushing the 2009 H1 Maths TYS. H2 Maths TYS is ready, if you wish to purchase directly from Dyna.
The checklist and formula guide are made freely available on my website at this moment to reach out to students directly. Publishing and making students purchase them may not be advisable in this difficult time, I feel.
Thanks again.
Cheers,
Wen Shih
Hi Mr Wee,
Thanks for the reply.
Your H2 maths revision notes are excellent.
H2 Maths students who do not visit homework forum or your website will be unable to benefit from your excellent notes.
May I suggest that you incorporate these notes into your H1 and H2 Maths TYS ie revision notes and examples for each topic before the past years examination questions ?
Originally posted by Lee012lee:Hi Mr Wee,
Thanks for the reply.
Your H2 maths revision notes are excellent.
H2 Maths students who do not visit homework forum or your website will be unable to benefit from your excellent notes.
May I suggest that you incorporate these notes into your H1 and H2 Maths TYS ie revision notes and examples for each topic before the past years examination questions ?
Hi,
Thanks for your encouragement! You are probably the first person to commend my work :D
FYI, I have included my website in my TYS and Solutions books as the first weblink, so that students will be able to visit my website and benefit from the learning resources.
I will take your suggestion to incorporate the notes next time if the publisher is agreeable, because it will make the books become more expensive and schools will complain even louder than before :P
Thanks again!
P.S. Dyna Publisher already sets TYS books at extremely reasonable prices to discourage illegal photocopying by students who want to avoid buying. Take the 2007 edition of H2 Maths TYS for instance, each page costs less than 4 cents.
Cheers,
Wen Shih
Mr wee,
In your year 2007 H2 Maths worked solution book, topic 1 Function Q1 got mistake. I am not sure if you have already edit out the error in the later edition or not.
Hi,
Thanks! If you are referring to N98/I/5, there is a typographical error in which I missed out the square root signs :P
I have just highlighted to the publisher via email.
Cheers,
Wen Shih