Mastering Mathematics Smartly II - Recently Updated Pages

Roots of Quadratic Equations

weews — Tue, 17 Nov 2009 06:00:51 CST

We recall that for whose roots are and , sum of roots and product of roots .

It is a common difficulty for students to form quadratic equations whose roots are more complicated, e.g. , .

The strategy is to find the sum and product of the new roots and make use of the original values of , and a very useful identity .

Let's apply the above method for our example: The equation has roots and . Form an equation whose roots are and .

Step 1: Find and based on the original equation.

Step 2: Find sum and product of new roots.

Step 3: Form the new equation.

The required equation is .

Sketching modulus graphs

weews — Sun, 25 Oct 2009 00:35:55 CDT

Consider the question:

A function g(x) = 3 - l2x - 4l is defined for the domain 0 <= x <= 5.
Sketch the graph of g(x) and state the range of g corresponding to this domain.

Approach: The most direct way is to remove the modulus sign. Let's see how we do that.

By the definition of modulus, |2x - 4| = 2x - 4 when 2x - 4 >= 0, i.e. x >= 2.
So when x >= 2, g(x) = 3 - (2x - 4) = 7 - 2x.

By the definition of modulus, |2x - 4| = -(2x - 4) when x < 2.
So when x < 2, g(x) = 3 + (2x - 4) = 2x - 1.

To sketch the final graph of g(x), we do these:
1. Between 0<= x < 2, sketch g(x) = 2x - 1.

2. Between 2 <= x <= 5, sketch g(x) = 7 - 2x.

From the sketch (go draw!), we obtain the range of g as [-3, 3].
The value of g(x) = -3 lies on g(x) = 7 - 2x when x = 5.
The value of g(x) = 3 is the location where both graphs meet when x = 2.

Finally, we note that g is a piecewise function, since it is made up of several pieces depending on the set of values that x takes.

Checklist of concepts and skills

weews — Wed, 05 Aug 2009 00:30:38 CDT

Every student should have a strategic broad picture of the concepts and skills assessed in the Additional Mathematics syllabus. Such knowledge will enable the student to study smartly and to anticipate likely questions that may appear in any examination. The following checklist has been compiled with the aim to help the student learn more effectively and efficiently (minus all the hassles, I hope).

1. Trigonometry
- Prove identities.
- Solve equations (sometimes after proving an identity).
- Find min/max of an expression using the R-sine or R-cosine result.
- Find values of trigonometric expressions, given trigonometric ratios.
- Sketch trigonometric graphs (directly or after some transformations) and show appreciation of period and amplitude.
- Transformations include these forms: y = f(ax), y = f(x) + a, y = |f(x)| and combinations.
- Determine the number of roots of given equations, where their graphs need to be sketched and checked for the number of intersections.
- Find trigonometric ratios of specific angles (30, 45, 60 degrees) without using calculator.
- Know about radian and degree modes.

2. Surds and indices
- Form expressions using surds.
- Rationalise surds.
- Solve equations involving indices.

3. Logarithmic and exponential functions
- Solve logarithmic equations.
- Solve real-life problems involving exponential functions (e.g. radioactive decay of a substance).
- Simplify and evaluate, without using calculator, using laws of logarithm.
- Sketch logarithmic and exponential curves.

4. Modulus functions
- Sketch graphs of modulus functions.
- Solve equations and/or inequalities (by graphical approach) involving modulus functions.

5. Circles
- Find equations of circles and show appreciation of radius and centre.
- Find new equations given some transformation (e.g. reflection about x-axis).

6. Quadratic equations
- Appreciate the nature of roots (real, equal, imaginary) for graphs which are always above/below the x-axis, for graphs that do not intersect, for graphs that are tangent to each other and the like.
- Form new quadratic equations via sum and product of roots.
- Solve quadratic inequalities.

7. Binomial expansion
- Expand.
- Find terms independent of x or terms with certain powers.
- Approximate values by suitable substitutions.
- Know how to expand nCr where n is an unknown.

8. Linearisation of non-linear expressions
- Express a non-linear equation by manipulation to become Y = mX + c.
- Find slopes and intercepts in order to find unknowns in the original non-linear form.
- Find values from graphs.
- Interpretations of linear graphs.
- Draw another graph to find the number of roots of an equation.

9. Differentiation and its applications
- Differentiation techniques, especially the use of product rule, quotient rule and differentiation of composite functions like cos (3/x).

- Tangents and normals
(i) find gradients of tangents and normals;
(ii) find equations of tangents and normals;
(iii) find axial intercepts of tangents and normals.

- Rates of change via the chain rule.

- Maxima and minima
(i) form an expression based on the given information;
(ii) carry out differentiation, set the derivative to zero, find the stationary value(s);
(iii) determine the nature of stationary value via first or second derivative test.

- Curve sketching
(i) find axial intercepts.
(ii) carry out differentiation, set the derivative to zero, find the stationary values;
(iii) determine the nature of stationary points via first or second derivative test.

10. Integration and its applications
- Integration of standard functions.
- Integration via partial fractions.
- Find areas of regions via integration.
- Given known definite integrals, evaluate some other integrals.

11. Applications involving both differentiation and integration
- Kinematics (involving displacement, velocity, acceleration).
- Applying the derivative of a given expression to find the integral of a related one, e.g. differentiate x ln x, then find the definite integral of ln x.

12. Coordinate geometry
- Find gradients or equations of lines (parallel to or perpendicular to given lines).
- Find points of intersections.
- Find areas.
- Apply known geometrical properties of quadrilaterals.
- Find mid-points.
- Prove collinearity of points.

13. Remainder and factor theorems
- Factorise completely some expression and find its solutions.
- Form simultaneous equations based on given information and solve for unknowns.
- Use the fact that polynomial = quotient * divisor + remainder to solve problems.
- Carry out long division of polynomials.
- Three types of partial fractions (linear, repeated linear, non-factorisable quadratic factors).

14. Geometrical proofs
- Prove congruency and similarity.
- Prove results by applying mid-point and intercept theorems for triangles.
- Prove results by applying tangent-chord, intersecting-chord and tangent-secant theorems for circles.
- Know and apply angle properties and symmetric properties of circles.
- Know and apply the alternate segment theorem.

15. Simultaneous equations in two unknowns
- Solve both linear equations.
- Solve one linear, the other non-linear equations.
- Solve via substitution or inverse matrix method.

Geometrical Proofs

weews — Sat, 06 Jun 2009 08:41:05 CDT

Approach to part ii:

1. Angle ADB = 90 degrees (from part i).

2. AB = AC (given).

3. Triangle ABC is isoceles (by 2).

4. BD = DC (by 1 & 3).

5. EG = GB (given).

6. DG // CE (by mid-point theorem).

Approach to part iii:

1. CE // DG (from part ii).

2. FE // DG (by 1).

3. AE = EG (given).

4. AF = FD (by intercept theorem).

So AF = 1/2 AD.

Sketching Trigonometric Graphs

weews — Sat, 06 Jun 2009 08:15:54 CDT

Question: The function f is defined, for 0 <= x <= pi radians, as f(x) = 3 sin (x/2) - 2. Sketch the graph of y = f(x).

Approach:
The usual sine curve has an amplitude of 1 unit and a period of 2 pi radians, so we have

0 <= x/2 <= 2 pi, which leads to 0 <= x <= 4 pi.

Note that we are stretching the curve horizontally so that it is twice as long as before.

When we sketch y = f(x) for the interval [0, pi], it'll be a quarter of the original sine curve.

Next, we stretch the sine curve vertically so that its amplitude is 3 units (i.e. thrice as tall as before).

Finally, we shift the entire curve down by 2 units.

The result looks like this:

Prime Factorisation

weews — Thu, 04 Jun 2009 09:39:54 CDT

Question:
Find the smallest integer n such that 840n is a perfect square.

Approach:
First, express 840 as a product of prime factors, i.e. (2^3)(3)(5)(7).

Next, we must consider two facts:
1. Any perfect square must have powers which are divisible by 2.
2. 840n must have prime factors 2, 3, 5 and 7.

Mastering Mathematics Smartly II Home

weews — Fri, 10 Apr 2009 09:08:55 CDT

This site is set up with the aim to help secondary school students (preparing for O-level) learn mathematics effectively and efficiently. It is hoped that the articles (in the form of questions and solving approaches) will address the learning difficulties that students face.

This site is an off-shoot from the original "Mastering Mathematics Smartly".

The webmaster, Wee Wen Shih, is a mathematics educator and A-level mathematics book writer. He is passionate about mathematics and mathematics education. In his free time, he reads and runs.

Trigonometric Proofs

weews — Fri, 10 Apr 2009 09:07:31 CDT

We want to prove that this identity holds:

Polynomials

weews — Thu, 09 Apr 2009 03:51:07 CDT

This is a question that requires one to apply the factor theorem:

The approach to solve it is discussed below:

Kinematics

weews — Wed, 08 Apr 2009 21:27:42 CDT

This interesting question was asked by a forumer:

This was my response, describing the solving approach:

Second Derivative Test

weews — Wed, 25 Mar 2009 00:25:08 CDT

This is the response I gave to answer a question that was posed on a discussion forum regarding the ambiguity of the second derivative test when it yields the value of zero.