Mastering Mathematics Smartly
by Wee Wen Shih

• Topic 1: Functions and graphs
• Topic 2: Sequences and series
• Topic 4: Complex numbers
• Topic 5: Calculus

More coming soon...

This page highlights key skills for various topics that students should take note of.

Focus on the four key skills when you study for this topic:

  1. Find modulus and argument of complex numbers, e.g. z(w^2). Remember results like |z1.z2| = |z1||z2|, arg (z1.z2) = arg (z1) + arg (z2), |z1/z2| = |z1|/|z2|, arg (z1/z2) = arg (z1) - arg (z2), etc.

  2. Solve equations involving complex numbers. This typically requires us to let z = x +yi and then we compare real and imaginary parts.

  3. Solve equations of the form z^n = 1 or z^n = c (where c is a complex number). Remember the formula to help you find z.

  4. Draw argand diagrams of standard types (i.e. circle, perpendicular bisector and half-line). Remember the standard equations of circle (i.e. |z - a| = r), perpendicular bisector (i.e. |z - a| = |z - b|), half-line (i.e. arg (z - a) = theta). Find shaded region, intersection points, maximum/minimum modulus and maximum/minimum argument.

This topic typically covers the following:

• Equations of tangents and normals. Curves may be given in several forms: (1) y = f(x) (i.e. y in terms of x), (2) an expression in terms of both x and y, (3) a pair of parametric equations x = f(t), y = g(t). Questions tend to focus on the these skills: (a) find equations of tangents and normals at a general point (x, y) or at a particular point by direct differentiation, implicit differentiation or parametric differentiation, (b) find point(s) on the curve where the tangent and/or normal is parallel to the x- or y-axis, (c) check whether the tangent or normal meets the curve again, (d) find the area of the triangle formed by tangent and normal to the curve.
  
  
• Connected rates of change. Questions tend to require students to form meaningful relationships between variables and to apply chain rule, e.g. dA/dt = (dA/dr)(dr/dt).
  
  
• Maxima and minima problems. Questions often ask students to establish a relationship and to use the first derivative or second derivative test to determine the nature of a stationary value. The first derivative test is practical for expressions which are difficult to differentiate another time. The second derivative test is good for expressions which are easy to differentiate the second time.
  
  
• Curve sketching. Determining the nature of stationary points via first or second derivative test is an important skill. On top of that, students will have to carry out the usual steps of finding axial intercepts and asymptotes.

This topic offers many possibilities:

• Direct integration by using standard results which students should recall from memory or by referring to the formulae booklet or by making use of suggestions given in the question.
  
  
• Integration by a given substitution. Students should carry out these steps: (1) determine the new lower and upper limits if the integral is definite, (2) differentiate the given substitution, (3) apply the substitution onto the original expression and use the chain rule, (4) evaluate the integral in terms of the new variable.
  
  
• Integration by partial fractions.
  
  
• Integration by parts. Some questions may require students to carry out the process twice.

It is common to see techniques of integration well-connected with differential equations and areas & volumes.