Mathematical Warehouse

2004-01-07T00:02:00.000+08:00

2004-01-05T02:17:00.000+08:00

4/6 Additional Mathematics Class Test

Date: 02/01/2004

1. Explain the meaning of {f}

[Solution]
A set with an empty set as an element.

[Comment]
- Many students gave answer as an empty set.
- Poor understanding of symbols used in Set Theory.

2. List all the 9 Laws of Indices.

[Solution]
-

[Comment]
- It is very shocking to know that none of the students is able to list out all the 9 laws.
- Some of the students used specific numbers in the laws, failing to know that laws require expressions to be in generic form (algebraic form).
- Some students resort to explain in words, failing to know that Mathematics is already a form of language (An Universal Language).
- Some students are confused with logarithmic expressions and exponential expressions, writing their laws in logarithmic forms.
- ALL students failed to show the conditions of a ¹ 0 for both a⁰=1 and a^-n= 1 / (aⁿ).

3. Solve Ö[ x / (x - 1) ] + Ö[ (x - 1) / x ] = 5 / 2.
Hence, find the range of values of x which satisfy Ö[ x / (x - 1) ] + Ö[ (x - 1) / x ] - 5 / 2 ³0

[Solution]

[Comment]
- Many students showed the WRONG understanding of (a+b)² = a²+b²
- Some showed WRONG understanding of Ö(a / b) + Ö(c / d) = Ö[ (a / b) + (c / d) ].
Simple Counterexample: Ö4 + Ö9 ¹ Ö13.

4. Given the equation cx² - dx - e = 0 , c ¹ 0 and c, d, e Î Â .
Prove that x = [ d ± Ö(d² + 4ce) ] / 2c

[Solution]

[Comment]
- No students showed the ability of completing the square.
- Many students used the general solution directly, changing the variable. That is NOT the proof.
- Specific instruction was mentioned during the test that substitution of variables in the general solution is not acceptable and many failed to listen.

5. Given 2 points (a, b) and (c, d), a, b, c, d Î Â.
Find the equation of the straight line that passes through the 2 points, having y
as the subject.

[Solution]

[Comment]
- ALL students lacked understanding of necessary condition (c ¹ a)for general case.
- Many students simply used symbol "c" in the equation y = mx + c without realising that "c" is already a constant given in one of the points. Thus, the confusion of symbol used.
- There are cases of WRONG understaning: ( a / b )(c) = ( ac / bc ). This is a very serious mistake, indicating that the students have very weak concept in fundamental algebra.

6. Express the simultaneous equations in matrix form. 2y + 5x = 37 & 2x - y = 4.
Hence, solve for x and y.
[Solution]

[Comment]
- Many students failed to express the sim equations into matrix form. They have no knowledge of the pre-requirement to arrange the x and y variables before changing into matrix form.
- Many students just solve the sim equation using their faimilar ways, failing to realise that the examiner wanted them to solve using the matrix approach. (The indication by the word "Hence" in the question.)

2003-12-15T11:52:00.000+08:00

Additional Mathematics 4018

Vectors in 2 Dimensions Lesson 1 (refer to syllabus)

Definition

A vector is a quantity with direction as well as magnitude. We normally denote a vector in bold, eg. a (printed form) or underlined, eg. a (written form).

The displacement from one point A to another point B written as AB, represents a vector pointing from A to B.

Position Vectors: If the starting point of reference happens to be the origin, O, then OA, where A is another point is the position vector of A.
Equal Vectors: When 2 vectors AB and CD have both the same direction and magnitude, ie. they have the same length and are parallel, they are said to be equal.; ie. AB = CD
Free Vectors: Vectors that are not restricted with the origin as the starting point.
Vector Addition: When we add vector AB to vector BC, we have vector AC.; ie. AB + BC = AC; Vector AC is also called the resultant vector of AB and BC.
Negative Vectors: A negative vector -AB has the same magnitude as AB but is opposite in direction.; Therefore we can say that -AB = BA.
Vector Subtraction: The subtraction of vector CB by vector AB is the same as the addition of vector AB with vector BC.; ie. AB - CB = AB + BC; Therefore, the resultant vector is AC.
Zero Vector: A vector with 0 magnitude, denoted by 0.; The addition of vector AB to its own negative is equal to 0.
Scalar Multiplication Of Vectors: When a vector is multiplied by a positive scalar, the length of the vector is multiplied by the scalar but the direction remains unchanged. If a vector is multiplied by a negative scalar, the length of the vector is multiplied by the scalar and the direction is reversed.
Parallel Vectors: When 2 non-zero vectors that are parallel to each other, they are scalar multiples of each other.; ie. If a and b are parallel, then a = a b, a Î Â.

coming up next........ Vectors in 2 Dimensions Lesson 2 ....

want to learn more? let me know

2003-12-14T04:42:00.000+08:00

Some Definitions

1. Integer: a number that may be expressed as the sum or difference of 2 natural numbers.
2. Irrational number: any real number that cannot be expressed as the ratio of 2 integers. ie. pi, square root of a prime number, ...
3. Natural number: one of the counting numbers; a number that can represent the cardinality of a finite set of objects, usually identified with the positive integers. ie. 1, 2, 3, ...
4. Prime number: a natural number divisible by no integers other than 1 and itself. ie. 2, 3, 5, 7, 11, ...
5. Rational number: any number that can be expressed as a ratio, a/b, of 2 integers, a and b, of which the latter cannot be 0.
6. Real number: any rational or irrational number.
7. Whole number: another term for a natural number, usually including zero. ie. 0, 1, 2, 3, ...