A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor

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Binomial Expansion

(1+x)^n = 1 + nx + [n(n-1)]/2!(x^2) + [n(n-1)(n-2)]/3!(x^3)……

This formula is good to memorise as it very useful for all value of x.

Note:
1. n is a negative or rational number (not positive integer). For such n, the expansion must be of the form (1+x )^n
2. The expansion has an infinite number of terms.
3. The condition lxl < 1 is very important. This ensures that the expansion is valid.

Binomial Expansion has two cases

Case 1: Expansion in terms of ascending powers of x.

Case 2: Expansion in terms of descending powers of x.

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Arithmetic Progression

Definition

An arithmetic progression (A.P.) is a sequence of numbers in which each term other than the first term is obtained from the preceding one by the addition of a constant number called the common difference.

Therefore, if we let a be the first term and d be the common
difference of the sequence, then the sequence is an A.P. of the form

a, a + d, a + 2d, a + 3d, ……

general nth term = Un = a + ( n – 1)d

Sum of AP

Sn = (n/2)[2a+(n-1)d]

Sn = (n/2){First term + last term)

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Geometric Progression

A geometric progression is a sequence such that the ratio between every pair of consecutive terms is a constant number i.e. a sequence of the form

a, ar, ar2, ar3, ……

The constant r is called the common ratio.

General nth term of a geometric progression

Un = ar^(n-1)

Sum of GP

Sn = [a(1-r^n]/(1-r) lrl 1

Sum to infinity = a/(1-r)

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Arithmetic Progression – Practice Questions

Question 1
The sum of the first nine terms of an arithmetic progression is 75 and the twenty-fifth term is also 75. Find the common difference and the sum of the first hundred terms. [10/3,16000]

Question 2
The sum of the first n terms of a series is given by Sn = n^2 – 3n, n > 1. Show that the series is an arithmetic series. Hence, find the first term, a and the common difference, d. [-2,-2]

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Problem Students Encounter

1. Do not understand the concept at all.
2. Understand the concept but do not know how to apply especially complex questions
3. Understand Lecture Notes and can do the tutorials but do badly in the exam
4. How to memorized all formulae or concepts especially Vectors, Complex Number & Integration and Applications

If these are the problems you encounter. Math tutor in One.Tution Place can help you. Please call Mr Ong @9863 9633

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J1 – Teaching Binomial Expansion and exam practice

J2 – Teaching Permutations and Combinations and exam practice

If you have problems in your tutorial. Math tutor in One.Tution Place can help you. Please call Mr Ong @9863 9633

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Permutations and Combinations

Fundamental Principles of Counting

Addition Principle
If there are r choices for performing a particular task, and the number of ways to carry out the kth choice is ,nk for k = 1, 2, 3, …, r, then the total number of ways of performing the task is equal to the entire sum of the number of ways for all the r different choices i.e.

n1+n2+n3…………nk

Multiplication Principle
If one task can be performed in m ways, and following this, a second task can be performed in n ways (regardless of which way the first task was performed), then the number of ways of performing the 2 tasks in succession is m × n.

Permutations
A permutation is an ordered arrangement of objects.

Combinations
A combination is a selection of objects in which the order of selection does not matter.

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Examination Skills

DO
1. Plan your timetable well so that you are able to allocate time to revise all the topics.

2. Revise all tutorials, lecture tests, short quizzes given by your tutors, Common Test, and Past Years GCE ‘A’ Level Questions

3. Optimise the use of Graphic Calculator whenever possible.

4. Practice time management.

DO NOT
1. Practice questions without understanding – must grasp correct concepts
learnt from every question.

2. Skip topic(s).

3. Be complacent.

Important notes on accuracy of answers

In general,
1. Only if the final answer is not exact, then leave your answer to 3 significant figures unless otherwise stated.

2. For angles, leave your answer to 1 decimal place(in degree) or
3 significant figures(in radian) unless otherwise stated.

3. In Statistics, intermediate values should be stated fully if it is exact. If not, an accuracy of 5 significant figures is recommended unless otherwise stated.

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Skill Set – Permutations and Combinations

Skill Set 1

Consider consecutive operations one after another and use multiplication principle

Example
A man has 10 different pieces of jewellery and wishes to give his
three daughters one piece each as dowry. In how many ways can he
do this?

Consider each of the three daughters in sequence:
1st daughter / 2nd daughter / 3rd daughter
No. of ways = 10 x 9 x 8 = 720

Skill Set 2

Consider mutually exclusive cases and use addition principle

A bag contains five red and three blue tokens, all of which are
identical, apart from colour. Four tokens are taken out of the bag at
random without replacement and arranged in a row. In how many ways can this be done?

There are 4 possible cases:

Tokens No. of ways
Case 1 3 blue, 1 red
No of ways = 4!/3!

Case 2 2 blue, 2 red
No of ways = 4!/(2!2!)

Case 3 1 blue, 3 red
No of ways = 4!/3!

Case 4 4 red
No of ways = 4!/4!

Total = 15

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DO

1. Plan your timetable well so that you are able to allocate time to revise all the topics.

2. An overview of every topic is provided in the revision package, which
highlights some of the important concepts. However, this is not to replace
the lecture notes that contain more examples.

4. Revise all lessons notes and examples given by One.Tuition Place, other JCs Prelim Papers and Past Years GCE ‘A’ Level Questions

5. Optimise the use of Graphic Calculator whenever possible.

6. Practice time management.

If you have problems in your tutorial. Math tutor in One.Tution Place can help you. Please call Mr Ong @9863 9633

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Vectors 1

Unit Vector
A unit vector is a vector of magnitude 1.

The unit vector in the direction of a^ is given by a^ = a/lal

Parallel Vectors
Two non zero vectors a and b are parallel if and only if a = kb for some k is element of real number.

Collinear Points
Three points A, B and C are collinear if and only if AB = k x BC for some k is element of real number

Length of Projection
Length of projection of a vector a on a vector b
= | a . ^b |

Angle Between 2 Vectors
cos x = (a.b)/(lal x lbl)
x = the angle between the 2 vectors

Area of Parallelogram and Triangle

Area of parallelogram = l a x b l
Area of triangle = 1/2 l a x bl

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Vectors 2

Scalar Product

The scalar (or dot) product of two vectors a and b (denoted by a.b ) is defined as

a.b = |a||b| cos ,

where  is the angle between a and b.

Vector Product

The vector (or cross) product of two vectors a and b is defined by
a x b = lal lbl sin nˆ
where  is the angle between a and b, nˆ is the unit vector perpendicular to plane containing a and b

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Probability Part 1

BASIC PROBABILITY CONCEPTS

a) An experiment is a situation involving chance or probability that leads to results called outcomes.

Tossing a fair die once.

b) An outcome (sample point) of an experiment is the result of a single trial of the experiment.

{1} or {2} or . . . or {6}

c) The sample space, S, (or probability space) of an experiment is the set of all possible outcomes.

S = {1, 2, 3, 4, 5, 6}

d) An event, E, is one or more outcomes of an experiment and it is a subset of the sample space S. E = {set of even numbers}
= {2, 4, 6}

e) Probability is the measure of how likely an event is.

P(E) = 3/6 = 1/2

Experiment A: Tossing a fair coin and noting the outcome.
Sample Space, = {H, T}
P({H}) = P({T}) = 1/2 .

Experiment B: Tossing a fair die and noting the outcome.
Sample Space, = {1, 2, 3, 4, 5, 6}, and
P({1}) = P({2}) = P({3}) = P({4}) = P({5}) = P({6}) = 1/6

Experiments A and B each have a Uniform Space where each outcome is equally likely. A uniform space is a finite sample space with all outcomes equally likely, i.e. the probability of each outcome is 1/n where n is the number of outcomes.

Experiment C: Selecting a coloured ball, at random, from a box containing 3 white balls, 2 black balls and 4 blue balls.

Sample Space, = {white, black, blue}
P({white}) = 3/9 = 1/3
P({black}) = 2/9
P({blue}) = 4/9

As we can see, the outcomes in Experiment C are NOT equally likely to occur. We are more likely to choose a blue ball than any other colour. We are least likely to choose a black ball.

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Binomial Distribution

1. A Binomial situation arises when
• there is a fixed number, n, of repeated & independent trials
• each trial has only two mutually exclusive outcomes, namely ‘success’ or ‘failure’
• the probability of success, p, is the same for each trial

2. The binomial random variable X is the number of successes in the n trials carried out.
It is denoted by X ~ B(n, p).

3. If X ~ B(n, p), then
• P(X = x) = nCx p^x (1-p)^n-x , where x = 0, 1, 2, … , n
• use binompdf (n, p, x) to compute P(X = x)
• use binomcdf (n, p, x) to compute P(X ≤ x)

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Poisson Distribution

A Poisson distribution is another special case of a discrete distribution.

Characteristics / Assumptions of a Poisson Distribution:

 The outcomes occur singly and randomly.
 Whether an event occurs in a particular point in time or space is independent of what happens elsewhere.
 At all points in time (or space), the probability of one event occurring within a small fixed interval of time (or space) is the same.
 There is no (or negligible) chance of 2 events occurring simultaneously at precisely the same point in time (or space).
 The distribution is commonly used to model rare events (discrete).

In a Poisson distribution, the random variable is a count of the number of occurrences of a random event in a given region of time and space when a mean number of occurrences, , is given for a particular interval of time and space.

Real-Life examples in which the Poisson distribution is a good model: The number of
 particles emitted in a minute by a radioactive substance.
 typographical errors in a randomly chosen page of a book.
 accidents in a factory in a randomly chosen day.

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