Be Awesome With Math

Form 4 Chapter 5

Saturday, February 22, 2014 | 12:49 AM | 0 Rain[s]

Gradient of a Straight Line

*Vertical distance and horizontal distance1. The diagram below show a straight line AB.

OA is known as the horizontal distance and OB is known as the vertical distance.
2.Vertical distance and horizontal distance are perpendicular to each other.

*Ratio of vertical distance to horizontal distance1.The gradient of a straight line is the ratio of the vertical distance to the horizontal distance between
two points on the straight line.

For example:

5.2 Gradient of a Straight Line in Cartesian Coordinates

*Formula for gradient of straight line The gradient,m,of a straight line passing through point P (x1,y1) and point Q (x1,y2)

5.3 Intercepts

*The x-intercept and the y-intercept of a straight line1.The x-intercept is the x-coordinate of the intersection point between a straight line and the x-axis.

2.The y-intercept is the y-coordinate of the intersection point between a straight line and the y-axis.

*Intercepts and gradient of a straight lineGiven a straight line with x-intercept = a and y-intercept =b,

5.4 Equation of a Straight Line

*Drawing the graph given an equation y = mx + c 1.The graph of the linear equation u = mx + c is a straight line.To draw the graph,follow the steps below :

STEP 1 :Construct a table of values using any two values of x
STEP 2 :Plot the two points on a Cartesian plane
STEP 3 :Draw a straight line through these two points

*Determining whether a given points lies on a straight line1.If a point lies on a specific straight line y = mx + c,then the coordinates of the point satisfy the equation
  of the straight line.
2.If a point does not lie on a specific straight line y = mx + c,then coordinates of the point does not
  satisfy the equation of the line.
3.To determine whether a given points lies on specific straight line :STEP 1 :Substitute the value of x-coordinate and the value of y-coordinate into the equation
STEP 2 :Compare the values obtained on LHS AND RHS.
             (a) If LHS = RHS, then the point lies on the straight line.
             (b) If LHS ≠  RHS, then the point does not lie on the straight line.

*Writing the equation of a straight lineTo write the equation of a straight line,the values of m and c need to be identified.

*Determining the gradient and the y-intercept of a straight lineWhen the equation of a straight line is given in the from y = mx + c,then the gradient of the straight line is m and its y-intercept is c.

*Finding the equation of a straight lineI. The equation of a straight line that is parallel the x-axis or the y-axis

Form 4 Chapter 4

| 12:40 AM | 0 Rain[s]

Mathematical Reasoning

4.1 : Quantifiers "All" and  "Some"

* Constructing statements using the quantifiers "all" and "some"A quantifiers denotes the number of objects or cases involved in a statement.
    (a) "All" refers to each and every object or case that satisfies a certain condition.
    (b) "Some" refers to several and not every object or case that satisfies a certain condition.*Determining the truth value of statements that contain the quantifier "all"In statement that contains the quantifier "all",each and every objects is being considered in
the statement.If  there is one object (or more) that contradicts the statement,then the statement is false.

*Generalising statements using the quantifier "all"Sometimes a statement can be generalised to cover all cases using the quantifier "all" without changing its truth value.

*Constructing true statements using the quantifier "all" or "some"To construct a true statement based on given objects and their properties :
    (a)Use the quantifier "all" if each and every object satisfies the given property.
    (b)Use the quantifier  "some" if there is one or more objects that contradicts with the given property.

4.2 Operations on Statements

*Changing the truth value of statements using the word "not" or "no"1.The word  "not" or  "no" can be used to change the truth value of a statement.
2.The process of changing the truth value of a statement using the word "not" or "no" is known as negation.
3.~p represent the negation for statement p.
4.Example:

*Identifying two statement from a compound statement that contains the word "and"1.In compound statement containing the word "and" we can identify two statement.
2.For example,7 is an odd number and 14 is an even number.Is a compound  statement that is made up from the following two statement.
         statement 1 : 7 is an odd number.           statement 2 : 14 is an even number.

*Forming compound statement using the word "and"W can use the word "and" to from a compound statement from two statement.

*Identifying two statement from a compound statement that contains the word "or"  In a compound statement containing the word "or" we can identify two statement.
        statement 1 : -5 < -2        statement 2 :½  = 0.5

*Forming compound statements using the word "or"We can from compound statement from two statement by using the word "or"

*Truth value of compound statement that contains the word "and"1.When two statements are combined  with the word "and" the compound statement formed is :

  (a) True,if both statement are true.
  (b) False, if one of the statement or both the statement are false.

2.The truth value are summarised in the truth table

*Truth value of a compound statement that contains the word "or"1.when two statements are combined with the word "or" the compound statement formed is :
(a) true,if one of the statement or both the statement are truth.
(b) false,if both the statement are false.
2.The truth values are summarised in the truth table

4.4 Implication

*Antecedent and consequent of an implicationStatement in the form "if p,then q"is known as an implication.p is the antecedent and q is the consequent.

*Writing two implications from a compound statement containing "if and only if"A compound statement in the form "p if and only if q"is a combination of two implications.
    Implication 1 : "if p, then q"
    Implication 2 : "if q, then p"

*Constructing implications "if p,then q" and  "p if and only if q"Based on the given antecedent and consequent,we can construct a mathematical statement in the form :
    (a) "if p,then q"
    (b) " p if and only if q"

*The converse of implicationFor the implication "if p,then q",the converse of the implication is "if q,then p".

*Truth value of the converse of an implicationThe converse of an implication is not necessarily true.

4.5 Arguments
*Premises and conclusion of an argument1. An argument consist of collection of statements,which are the premises,followed by another statement,which is the
   conclusion of the argument.

*Making a conclusion based on the given premisesArgument Form IPremise I : All A re B
Premise II : C is A
Conclusion : C is B

For example,Premise I : All multiples of 10 has the unit digit 0
Premise II : M is a multiple of 10
Conclusion : M has the unit digit 10

4.6 Deduction and induction
*Reasoning by deduction and induction1.Deduction is the process of making a specific conclusion  based on a general statement.
2.Induction is the process of making a general conclusion based on specific cases.

*Making conclusion by deductionThrough reasoning by deduction ,we can make conclusion for a specific case based on a general statement.

*Making generalisations by induction.Through reasoning by induction,we can make generalisation based on the pattern of a numerical sequence.

Form 4 Chapter 3

Friday, February 21, 2014 | 12:23 AM | 0 Rain[s]

Sets

A set is a collection of objects with common characteristics.

*Identifying elements of a set

1.The objects in a set are known as elements.
2.For example,if A is a set of even numbers,then 2 is an element of set A.
   We write 2 ∈  A an 5 ∉   A.
3.The symbol ∈ is used to denote the phrase 'is an element of' or' is a member of'.

*Representing sets using Venn diagrams

1.Sets can be represented using Venn diagrams.
2.A Venn diagram is an enclosed geometrical diagram in the shape of a circle,ellipse,triangle,square or
  rectangle.
3.For example,A = {1,2,3,4} can be represented in the Venn diagram below.

4.Notice that there are 4 elements in set A.Set A can also be represented in a Venn diagram as follows.
*Listing elements and stating the number of element in a setWe use the notation  n(A) to represent the number of elements in set A.

*Empty sets

1.An empty set is a set  with elements.
2.We use { } or   to represent empty sets.
3.For example,M = {x : x < 0 and x is a positive integer} is an empty set as M contains no elements.We write M =   or M = {  }

*Equal set

1.Two sets,A and B are equal if both have the same elements.
2.For example,A = { 1,2,3,4} and B = {2,4,3,1} are equal sets.It is written as A = B.
3.If set M is not equal to set N,then it is  denoted as M ≠ N

3.2 Subset,Universal Set and the Complement of a set

*Subsets

1.Element of set A is also an element of set B,then A is a subset of B.
2.if not all the elements of set M are the elements of set N,then M is not the subset of N.The relationship is
   written as M ¢  N

*Representing subsets using Venn diagrams

Subsets can be represented using Venn

*Listing the subsets of a specific set
Given that A = {1,2,3}.The possible subsets of A are  ,{1},{2},{3},{1,2},{1,3},{2,3} and {1,2,3}.

*Universal set

1.Universal set is a set consisting of all the elements under discussion.
2.the symbol    is used to denote a universal set.
3.All the sets under discussion are subsets of the universal set.

*The complement of a set

1.The complement of set A is a set consisting of all the elements in  ,which are not the elements of set A
2.The symbol A' denotes the complement of set A.
3.In the Venn,diagram below,the shaded region represents

3.3 Operation on Sets

*Intersection of sets

1.Intersection of set A and B is a set of elements which common to both sets A and B.
2.The intersection of set A and B is denoted by A B.
3.The shaded region in Venn diagram represents A B.

*The complement of the intersection of sets

1.The complement of A ∩ B is a set containing all the elements which are not the elements of set A ∩ B.
This is denoted by (A ∩ B)'.

2.The shaded region in the Venn diagram below represents (A ∩ B)'.

*Solving problems involving the intersection of sets
We can solve some problems in our daily life by applying the concepts of the intersection of sets.

*Union of sets

1.The union of sets A and B is a set of elements belonging to either of the sets or both.
2.The symbol A B denotes the union of set A and B.

*The complement of the union of sets

1.The complement of A U B is a set containing all the elements in the universal set, ξ which are not elements
of the set A U B.This is denoted by (A U B)'.
2.The shaded region in Venn diagram represent (A U B)'.

*Solving problems involving the union of sets
Venn diagram is very useful when solving problems involving the union of sets.

*Combined operations on setsWhen combined operations are involved,carry,out the operations in the brackets first.
*Solving problems involving combined operations on sets
Venn diagram is very useful when solving problems involving the union of sets

Form 4 Chapter 2

| 12:09 AM | 0 Rain[s]

2.1 Quadratic expressions

* Identifying quadratic expressions

1. A quadratic expression in the form ax2+bx+c,where a,b and c are constants,a≠0 and × is an unknown.
For example:
                  (a)  3x2 - 4x + 5
                  (b)  2x2 + 6x
                  (c)   x2-9

In a quadratic equations :

there is only one unknown
the highest of the power unknown is 2 .

* Forming quadratic expressions by multiplying two linear expressions1. When two linear expressions with the same unknown are multiplied,the product is a quadratic expression.
2. The multiplication process is known as expansion* Forming quadratic expressions based on specific situationsTo form quadratic expressions based on specific situations :

1.Choose a letter to represent the unknown.2.Form a quadratic expressions based on the information given.

2.2  Factorisation of quadratic expressions
*Factorising quadratic expressions of the form ax2 + bx + c,where b =0 or c =01.When b = 0, ax2 + c can be factorised by finding the highest common factor (HCF) of the coefficients a and c.

2.When c=0,ax2x< + bx can be factorised by finding the highest common factor of the coefficients a and b.x is also a common factor of the two              terms.
*Factorising quadratic expressions of the form px2 - q,where p and q are perfect squaresLet p = a² and q = b²

px² - q  = (ax)² - b²
            = (ax + b) (ax - b)

*Factorising quadratic expressions of the form ax2 + bx + c, where a ≠ 0, b ≠ 0 and c ≠ 01.We can use the inspection method and cross method to factorise quadratic equations of this form.
2.To factorise quadratic expressions :

(a) ax² + bx + c, where a = 1

             x² + bx + c =(x + p) (x + p)
                                =x² + qx + px + q²                         =x² + (p + q) x + pqIn comparison,
            b = p + q ,c = pq
- Find the combination of two numbers whose product is c.
- Choose the number combination from step 1 whose sum is b.

Quadratic Expressions and Equations

(b) ax² + bx + c, where a > 1

     ax² + bx + c =(mx + p) (nx + q)
                        =mnx² + mqx + npx + pq
                        =mnx² + (mq + np) x + pq

In comparison,
          a = mn  ,  b = mq + np  ,  c = pq

- Find the combination of two numbers whose product is a.
- Find the combination of two numbers whose product is c.
- Choose the number combination from step 1 and step 2 whose sum is b.

*Factorising quadratic expressions containing coefficients with common factors For quadratic expressions containing coefficients with common factor first before
carrying out the factorisation of the expressions.

2.3 Quadratic Equations

* Indentifying quadratic equations with one unknown1. Quadratic equations with one unknown are equations involving quadratic expressions.
2. In a quadratic equation :

there is an equal sign "="
there is only one unknown
the highest power of the unknown is 2

2.4 Roots of Quadratic Equations

* Determining the roots of a specific quadratic equationThe roots of a quadratic equation are the values of the unknown which satisfy the quadratic equation.
* Determining the solutions for quadratic equations1.The solutions for a quadratic equation can be determined by :

trial and improvement method
factorisation

2.To determine the solutions for ax2 + bx + c = 0 using trial and improvement method :

try use the factors and the last term , c

Form 4 Chapter 1

Thursday, February 20, 2014 | 11:54 PM | 0 Rain[s]

Standard Form

Significant figures

Rounding off positive numbers to a given number of significant figures

Form 2 Chapter 6

Tuesday, February 18, 2014 | 9:15 AM | 0 Rain[s]

Line bisector

Angle Bisector