# A Level Math Preparation Tip #3: Complex numbers

Complex numbers usually come as 2 questions, 1 on the loci and 1 on algebraic properties involving Cartesian, Polar or Exponential form.

One common type of question is on the properties of argument and modulus when we multiply, divide or raise a complex number to a certain power. Recall:

$\displaystyle \arg ({{z}_{1}}{{z}_{2}})=\arg ({{z}_{1}})+\arg ({{z}_{2}})$

$\displaystyle \arg \left( {\frac{{{{z}_{1}}}}{{{{z}_{2}}}}} \right)=\arg ({{z}_{1}})-\arg ({{z}_{2}})$

$\displaystyle \arg ({{z}_{1}}^{n})=n\arg ({{z}_{1}})$

$\displaystyle \left| {{{z}_{1}}{{z}_{2}}} \right|=\left| {{{z}_{1}}} \right|\left| {{{z}_{2}}} \right|$

$\displaystyle \frac{{\left| {{{z}_{1}}} \right|}}{{\left| {{{z}_{2}}} \right|}}=\frac{{\left| {{{z}_{1}}} \right|}}{{\left| {{{z}_{2}}} \right|}}$

$\displaystyle \left| {{{z}_{1}}^{n}} \right|={{\left| {{{z}_{1}}} \right|}^{n}}$

Next is the finding roots of a polynomial with unknown coefficients.

One common “not smart” approach student is to write “since -2+i is a root, the conjugate -2-i is also a root”. Followed by factorising the polynomial into linear factors and compare coefficient to find “a” and “b” and then the roots.

It is easier to follow the instructions given the question. A root means when you sub z = -2+i into the polynomial you will get 0. Then compare coefficient (real part = 0, imaginary part = 0) to get “a” and “b”. This method is faster as once I know the values of “a” and “b”, I can use GC “polynomial root finder” to search for the 3 remaining roots (usually it is quite nice).

Also if the question did not mention “a” and “b” are real, it is not right to say -2-i is a root.

# A level H2 Math preparation tips #1: Differentiation

## Calculus (Differentiation, Integration and applications)

• Calculus is the heaviest weightage among all the pure math “modules” in the exam. Over the years the total marks for this module is between 50 to 60. Hence I always emphasise to my students that this module is a MUST to master.
• A level exams will not test directly on techniques of differentiation. They will ask application questions only. So far over the years they have been testing on tangents / normals and maximum / minimum problems. No rate of change questions (yet).
• Tangents / Normals

For the past 8 years, 7 years have questions on tangents / normals asked in the form of parametric equation (since implict differentiation can be asked through Maclaurin’s series). Furthermore, the recent years’ questions has been algebraic intensive. E.g. they ask you to find equation of a tangent at a point with parameter “p” instead of say “2” to make the workings nicer. Also it is quite common for them to merge graphing techniques (sketch parametric curves) and integration (area under curve through parametric form) as 1 big question.

• Maximum / minimum problem

It is quite common that they will give you an object such as a box of various shapes and find the value of one of the lengths for maximum volume or surface area. Students may find the first part on formulating the equation to be harder than the differentiation itself. My suggestion is: if the equation is mentioned in the question as a “show” question, just use the answer to continue with the maximum / minimum computation which will usually have more marks. DO NOT give up the entire say 9 marks question just because you could not form the equation which is just approx 3 marks.

• Maclaurin’s series
For Macluarin’s series there are two main approaches: successive differentiation and using standard series from MF 15. Over the years, binomial expansion has been tested quite frequently. Also if you see the phrase “sufficiently small” and the question involves a triangle and angle –> small angle approx (using sine or cosine rule). Usually small angle approx questions is followed by binomial expansion too.