Mastering Mathematics Smartly
by Wee Wen Shih

All thanks to students who attended the course. You have shown a positive learning attitude and the commitment to want to succeed. To excel in mathematics, it is necessary to know yourself and to know the examination proper.

I mentioned during the course the problem-solving approach of Polya. Here is a nice description of the approach on a website, which I would encourage you to apply.

As promised, I have put up the materials you have requested:

• Worked solutions to N'07 papers.
• Worked solutions to school prelim statistics questions.
• Worked example on binomial expansion related to approximations.

If you wish to purchase the solutions book I have written, you may do so at large Popular bookstore branches (e.g. Bishan, Brash Basah, Jurong Point, etc.). Alternatively, you may call the publisher at 62730783 directly and buy the book (quoting my name for a possible discount subject to the publisher's generosity).

If you have further questions to ask, feel free to drop me an email or post your enquiries on the site's forum.

Finally, share this site and its resources with your friends and teachers, so that everyone will benefit :)

Thank you for your keen attention during session 1. Some of you have given feedback and I'd like to thank you.

As promised, here are the worked solutions to the unresolved questions.

As promised, these are the steps for calculating (with a Casio GC) the terms produced by a recurrence relation.

To prepare for final examination, I suggest that you attempt 2007 VJC, YJC prelim papers and 2008 JJC, IJC, SRJC prelim papers as well as specimen and 2007 exam papers. Go for timed practice of at least 2 hours by focusing on selected questions, if you cannot spare the time to do the entire paper.

For questions we have attempted in class, let me provide some key points below, for your reflection.

On Functions:

• Composite function exists if the Range of the first function is smaller than or equal to (set-wise) the Domain of the second function.
  
  
• The range of the composite function can be obtained by using the range produced by the first function. We place this set of values on the x-axis of the graph of the second function and obtain the y-values.

On Transformations:

• To sketch the graph of y = f '(x), we look at the slope at any point on the graph of y = f(x). If vertical asymptotes exist in the graph of y = f(x), they will remain in the graph of y = f '(x). If turning points exist in the graph of y = f(x), they become x-intercepts in the graph of y = f '(x). For any point that lies close to a horizontal asymptote, the slope at that point will be close to positive 0 (i.e. just above the x-axis) or negative 0 (just below the x-axis). Thus, the x-axis becomes the horizontal asymptote.
  
  
• The sequence of transformations must be carefully considered. For example, sketching y = f(-|x|) will require us to carry out this sequence:
  1. start with y = f(x);
  2. then we obtain y = f(-x), i.e. replace x by -x;
  3. then we obtain y = f(-|x|), i.e. replace x by |x|.
  
  If we interchange steps 2 and 3, we do not get the graph of y = f(-|x|).

On Applications of integration, integration techniques:

• Integration via substitution requires you to do the following: (1) differentiate the given substitution, (2) find the new limits, (3) use the substitution in the expression to be integrated, (4) apply chain rule.
  
  
• To integrate a modulus function, we need to know when the function is positive or negative. If f(x) > 0, then |f(x)| = f(x); otherwise |f(x)| = -f(x).

On Sequences:

• A.P. in the recurrence representation:
  x_(n+1) = d + x_n, x_1 = a, d is the common difference and n >= 1.
  
  
• G.P. in the recurrence representation:
  x_(n+1) = r . x_n, x_1 = a, r is the common ratio and n >= 1.
  
  
• A sequence either converges to a limit l or diverges. For example, any A.P. diverges while some G.P. converge. When a G.P. converges, then its sum to infinity exists.
  
  
• In Question 11(iv), we used a different approach below (following the style of 2007 exam):
  
  - In (i), we proved that the sequence converges to -1, 0 or 1.
  
  - In (iii), we deduced that x_(n+1) > x_n if 0 < x_n < 1.
  
  - To explain the behaviour we witnessed in (ii), we note that x_1 = 0.5, so x_1 lies between 0 and 1.
  
  - By the deduction in (iii) and the proof in (i), x_1 = 0.5 < x_2 < x_3 < ... < 1, i.e. the behaviour that the sequence converges to 1 with x_1 = 0.5.

On Applications of differentiation:

• In Question 5(b), we are unable to compute d/dt (xy) directly. Since xy is an expression involving theta, we need to use connected rates, i.e. d/d theta (xy) . dx/dt . d theta/dx to compute the value. dx/dt is already given, so it must be used in our connected rates. d theta/dx is the reciprocal of dx/d theta, which can be found by substituting theta with pi/6.

Good luck for your coming exams!

Solutions to the unanswered questions are here :)

Key ideas on Series:

• Cancellation of terms results in 2 cases: top-left, bottom-right terms remain or top-right, bottom-left terms stay.
  
  
• It is common for a question to ask the student to compare a new series with the old one, e.g. Q1(iv).

Key ideas on Vectors:

• 2 important results are: a . b = |a| |b| cos theta, |a x b| = |a| |b| sin theta. Many vector results come from them.
  
  
• Draw vector diagrams to visualise the problem better.
  
  
• We discussed a little bit on the reason for the line of intersection L to have a direction vector d = n1 x n2, where n1 and n2 are normal vectors of planes 1 and 2.
  - L lies on plane 1 and on plane 2.
  - So d is perpendicular to n1 and is perpendicular to n2.
  - We know that a x b is a vector that is perpendicular to a and is perpendicular to b, by definition.
  - So d = n1 x n2.
  
  This type of reasoning can also be applied to Q4(iii) and all other similar exam questions you will encounter.
  - Line 1 lies on the plane, so the plane's normal n is perpendicular to d1, the direction vector of line 1.
  - The plane is parallel to line 2, so the plane's normal n is perpendicular to d2, the direction vector of line 2.
  - By our understanding of the definition of vector product, n = d1 x d2.

Key ideas on P&C:

• Addition principle considers mutually exclusive cases. Multiplication principle considers tasks carried out in sequence.
  
  
• Permutation is made up of selecting items and ordering them.

Key ideas on Probability:

• This topic may make use of P&C concepts, e.g. P(girls are apart) = (number of ways girls are apart) / (number of ways without restriction). Another example: P(1 win, 1 lose, 2 draws) = P(win) . P(lose) . {P(draw)}^2 . (4!)/(2!). Here we see 4 outcomes of which 2 of them are identical, i.e. draw.
  
  
• P(A | B) = {P (A and B)} / P(B). P(A and B) typically requires critical thinking of what it means to combine two events, as we saw in Q7(iii).

Key ideas on Sampling:

• Random sampling, systematic sampling, stratified sampling and quota sampling are methods you need to recall on the approach, advantages and disadvantages.

Key ideas on Hypothesis Testing:

• The procedure is fairly standard. However, a question may be asked for a student to give an inequality of alpha (i.e. significance level), x bar (i.e. sample mean) or miu (population mean) so that the null hypothesis or alternative hypothesis can be concluded. Study these carefully (please refer to the "Hypothesis testing" page) when you revise.

Key ideas on Regression & Correlation:

• Regression line of y (dependent variable) on x (independent variable): y = a + bx.
  
  
• Regression line of x (dependent variable) on y (independent variable): x = c + dy.
  
  
• The pair x-bar, y-bar lies on both lines.
  
  
• r^2 = bd. r = sqrt(bd) if b, d > 0 or r = -sqrt(bd) if b, d < 0. r = 0 means lines are perpendicular. r = 1 means lines coincide.
  
  
• Linearisation of non-linear relationships makes use of prior knowledge of O-level, i.e. equations of straight lines.
  
  
• The scatter diagram helps us to see if there is a linear or non-linear relationship between 2 variables.
  
  
• Estimation is reliable if interpolation is carried out within the data range. Extrapolation of data should be avoided, as we cannot be sure of the relationship between variables outside the data range.