# A. Math Reflections & Study Tips

## (1) Simultaneous Equations

Generally an easy topic, but beware of certain kinds of questions such as:

(a) Questions involving reciprocals of x and y, such as

Solve  3(1/x) + 1/y = 1;  1/(x^2) + 1/(y^2) = 5

Ans: x = 1, y = -1/2 or x = -5/2, y = 5/11

Do not make common denominators and cross-multiply; the better technique is to let p = 1/x and q = 1/y.

(b) Questions involving coefficient matrices that are singular (determinant = 0), such as

Given that x and y satisfy the simultaneous equations mx + (m-1)y = 10 and (m-2)x + 3my = 20,

(i) if the equations have no unique solution, find the values of m; (ans: 1/2, -2)

(ii) if the equations have no solution, find the value of m.(ans: 1/2)

## (2) Indices, Surds & Logarithms

Generally an easy topic, except for students who do not understand or memorize the laws of indices and logarithms properly. I find that quite a number of weak students do not know the meaning of logarithm, thus not appreciating and enjoying the topic. Weak students do not realize that while Indices is concerned with the answer or expression obtained when a power is applied to a base number, Logarithms is about the power itself; the power that is needed to be applied to a base to give a certain number.

For eg, why is lg1000 = 3? ‘Evaluate lg1000’ or ‘What is lg1000?’ is the same as asking, “What is the Power that must be applied to the number 10 (the base) to obtain 1000?” Since 10^3 = 1000, therefore the answer is 3.

Weak students almost always make one or more of the following mistakes, thinking that (i) logA x logB = logA + logB, (ii) log(A + B) = logA + logB, (iii) logAB = logA x logB, (iv) (logA)^n = nlogA, (v) logAB^n = nlogAB, (vi) (a^m)^n = a^(m+n). All these are wrong.

How to be good in Indices and Logarithms? READ AND UNDERSTAND THE LAWS OF INDICES AND LOGARITHMS CAREFULLY. Yes, there is such a thing as READING MATH, not just practising Math.

## (3) Quadratic Functions & Equations

Some of the points to note are:

(1) Understand that alpha and beta by themselves also satisfy the quadratic eqn because they are the roots of the equation! So it’s not just abt finding the sum and product of roots;

You must be able to solve questions like:

If α is the root of the equation x^2 = 2x – 1, show that α^4 – α^2 = 2α – 2.

(2) Understand that the discriminant (b^2 – 4ac) is less than or equal to zero when the question involves the phrase “for which the function is never positive or never negative”;

(3) Understand that sometimes you are required to solve an inequality involving the discriminant but at other times the inequality involves the function itself;

(4) Understand how to complete the square to determine the maximum or minimum value of a quadratic function;

(5) Recognise that “real and distinct roots”, “real and equal roots”, and “no real roots” have other names that express the same meaning.

TO BE CONTINUED ……….. (by Mr Ilyasa)

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