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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Hi A-Level/H2 Math Students
Normal Approximation to the Binomial Distribution
X ~ B (n, p) can be approximated by a normal distribution with μ = np and σ2 = np(1 – p)
i.e. X ~ N (np,np(1- p))approximately if
(i) n is large, such that n > 30 ;
(ii) p is large enough, such that np > 5 and n(1- p) > 5 .
Note that X ~ B(n, p) has a discrete distribution
whereas X ~ N(np,np(1-p)has a continuous distribution.
Therefore, we need to adjust the values of X by using a continuity correction
Normal Approximation to the Poisson Distribution
X ~ Po(λ) can be approximated by a normal distribution with μ = λ and σ2 = λ
i.e. X ~ N(λ,λ) approximately if λ is large (λ > 10).
Note that Po(λ) has a discrete distribution whereas N(λ,λ) has a continuous distribution.
Therefore, we need to adjust the values of X by using a continuity correction
A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Hi A-Level/H2 Math Students
Graphing Techniques & Curve Sketching
1. y = f(x) —> f(x + a)
Translation
The graph of y = f (x) is translated -a units parallel to the x-axis.
2. y = f(x) —> f(x – a)
Translation
Translation The graph of y = f (x) is translated a units parallel to the y-axis.
3. y = f(x)—> f(ax)
Scaling
The graph of y = f(x) is scaled 1/a units parallel to the x-axis.
3. y = f(x)—> af(x)
Scaling
The graph of y = f(x) is scaled a units parallel to the y-axis.
4. y = f(x) —>f(ax+b)
Translation then scaling
The graph of y = f (x) is translated -b units parallel to the x-axis and then scaled 1/a units parallel to the x-axis.
5. y = f(x) —> af(x) + b
Scaling then translation
The graph of y = f (x) is scaled a units parallel to the y-axis and then
translated b units parallel to the y-axis.
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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Mathematical Induction
Proof by Induction involves the following steps:
STEP 1: Let Pn be the statement
STEP 2: Prove P1 is true [Note that for some cases,
(By substituting into LHS and RHS of the statement to prove that P1 is true)
STEP 3: Assume Pk is true for a
STEP 4: Prove “ Pk+1 is true also true”
STEP 5: Conclude the proof:
If Pk is true, then Pk+1 is also true
By mathematical induction,Pn is true for x is element Z+
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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
2014 Aug A level Math Intensive Revision
28-Jul Mon 7.30pm to 9.30pm 2 hrs Differentiation & Applications
29-Jul Tue 5pm to 7pm 2 hrs Inequalities/Conics/Differentiation/Integration
2-Aug Sat 11am to 1pm 2 hrs Inequalities/Conics/Differentiation/Integration
2-Aug Sat 5.30pm to 7.30pm 2 hrs Differentiation & Applications
3-Aug Sun 10am to 12pm 2 hrs Differentiation & Applications
3-Aug Sun 4pm to 6pm 2 hrs Inequalities/Conics/Differentiation/Integration
4-Aug Mon 7.30pm to 9.30pm 2 hrs Differentiation & Applications Exam FAQ
5-Aug Tue 5pm to 7pm 2 hrs Complex Numbers
9-Aug Sat 11am to 1pm 2 hrs Complex Numbers
9-Aug Sat 5.30pm to 7.30pm 2 hrs Differentiation & Applications Exam FAQ
10-Aug Sun 10am to 12pm 2 hrs Differentiation & Applications Exam FAQ
10-Aug Sun 4pm to 6pm 2 hrs Complex Numbers
11-Aug Mon 7.30pm to 9.30pm 2 hrs AP & GP
12-Aug Tue 5pm to 7pm 2 hrs Complex Numbers Exam FAQ
16-Aug Sat 11am to 1pm 2 hrs Complex Numbers Exam FAQ
16-Aug Sat 5.30pm to 7.30pm 2 hrs AP & GP
17-Aug Sun 10am to 12pm 2 hrs AP & GP
17-Aug Sun 4pm to 6pm 2 hrs Complex Numbers Exam FAQ
18-Aug Mon 7.30pm to 9.30pm 2 hrs Series and Sequences
19-Aug Tue 5pm to 7pm 2 hrs Permutations and Combinations
23-Aug Sat 11am to 1pm 2 hrs Permutations and Combinations
23-Aug Sat 5.30pm to 7.30pm 2 hrs Series and Sequences
24-Aug Sun 10am to 12pm 2 hrs Series and Sequences
24-Aug Sun 4pm to 6pm 2 hrs Permutations and Combinations
25-Aug Mon 7.30pm to 9.30pm 2 hrs Graphing Techniques
26-Aug Tue 5pm to 7pm 2 hrs Probalility
30-Aug Sat 11am to 1pm 2 hrs Probalility
30-Aug Sat 5.30pm to 7.30pm 2 hrs Graphing Techniques
31-Aug Sun 10am to 12pm 2 hrs Graphing Techniques
31-Aug Sun 4pm to 6pm 2 hrs Probalility
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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Binomial Expansion Concept
Concept 1 : Partial fractions
Use of partial fractions to provide simplification of expression before expansion.
The partial fractions decomposition can be found in MF15.
Concept 2 : Recognise the difference between the Binomial Theorem & Binomial
Series.
Binomial Theorem formula can only be used for expansion of (a + b)^n , where n is positive integer. The expansion works for any value of a and b
Knowing the range of values where Binomial Series is valid Expansion lxl < 1
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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
2015 A level Mathematics Schedule
J1 H2 Thu 5.30pm – 7.30pm
J1 H1 FRI 5.30pm – 7.30pm
J1 H2 SAT 11am – 1pm
J1 H2 SUN 4pm – 6pm
J2 H2 MON 7.30pm – 9.30pm
J2 H1 MON 5.30pm – 7.30pm
J2 H2 SAT 5.30pm – 7.30pm
J2 H2 SUN 10am – 12pm
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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
ARITHMETIC & GEOMETRIC SERIES
Question 1
A set consists of four integers {24 , a , b , 108} such that 24 < a < b < 108. The first three integers are consecutive terms of an arithmetic progression and the last three integers are consecutive terms of a geometric progression. Find the integers a and b. Ans a=48 b=72 Question 2 The fifth, tenth and twentieth terms of a convergent geometric progression, G, are the first three consecutive terms of an arithmetic progression, A. (i) Determine the common ratio of G. [0.908] Given that the first term of G is 2, (ii) evaluate the sum to infinity for G. [21.8] (iii) find the sum of the first 10 odd-numbered terms in A. [- 33.2] Question 3 The second, fifth and tenth terms of an arithmetic progression are consecutive terms of a geometric progression. The eighth term of the arithmetic progression is 6. Find the sum of the first 12 terms of the arithmetic progression. [432/7] Question 4 A geometric progression and an arithmetic progression has the common first term, a. The sum of the first three terms of the geometric progression is 19/27 of the sum to infinity. Find the common ratio, r. [2/3] The second and third terms of the geometric progression are increasing consecutive terms of the arithmetic progression respectively. Find the sum of the 1st 55 terms of the arithmetic progression in terms of a. [ −275a ] Question 5 An arithmetic series A has first term a and a geometric series G has first term b. The common difference of A is four times the first term of G and the common ratio of G is twice the first term of A. Each term of A is added to the corresponding term of G to form the terms of a third series S. Given that the first two terms of S are 1/2 and 0 respectively, find the value of a.[-1] Please contact Angie @96790479 or Mr Ong @98639633 if you need help in Mathematics
A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Revision Notes on Recurrence Relation
1. A recurrence relation is an equation that relates the general term of a sequence to one or more of its preceding terms.
For example,
xn+1 = 7 + xn , x1 = 0 ,
un = un−1 + 2 , u1 =1 , and
Fn+2 = Fn + Fn+1 , F1 = F2 =1 are recurrence relations.
These equations have the common property that if one term of the sequence is known, then we can find the next term by recursive substitution of terms. In this way, we can obtain the entire sequence of numbers. The subscript n
gives the position of the terms of the sequence; eg x1 is the first term and x2 the second term etc.
The general term n x of the sequence can take any real value; eg x1 = 0 , x2 =1 and x2 = −0.5 etc.
2. We say that a sequence of numbers xn converges to the limit l if xn ®l as n®¥ where l is a finite number. If no such number l exists, then we say that the sequence diverges.
3. A sequence is said to be increasing if the terms are increasing. Mathematically, this means xn+1 > xn for all n = 1, 2, 3, ….
The general approach in proving that a sequence is increasing is thus to show that xn+1 − xn > 0 for all n = 1, 2, 3, ….
4. A sequence is said to be decreasing if the terms are decreasing. Mathematically, this means xn+1 < xn for all n = 1, 2, 3, …. The general approach in proving that a sequence is decreasing is thus to show that xn+1 − xn < 0 for all n = 1, 2, 3, …. 5. A sequence is said to be constant if all the terms have the same value. Mathematically, this means xn+1 = xn for all n = 1, 2, 3, … Please contact Angie @96790479 or Mr Ong @98639633 if you need help in Mathematics
A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Permutations and Combinations – Skill Set
1. Box Method
Consider consecutive operations one after another and use multiplication
principle
2. Cases Method
Consider mutually exclusive cases and use addition principle
3. Restrictions Method
Take note of restrictions or constraints and arrange the others
4. Complementation Method
Apply method of ‘Complementation’
5. Grouping Method
Apply method of ‘Grouping’
6. Slotting Method
Apply method of ‘Slotting’
7. Polygons Method
Arrangements around polygons
8. Identical Group Size Method
Groups with identical size
Please contact Angie @96790479 or Mr Ong @98639633 if you need help in Mathematics
A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Probability – Skill Set
1. Recognize equally likely outcomes and use
P(A) = n(a) / n(s)
2. Identify regions in Venn diagram and use suitable additions/subtractions of
regions
3. Identify conditional probability (through key phrases like “given that”,
“it is now known that”, etc.) and apply formula
P(A/B) = P (A intersect B) / P(B)
4. Construct a tree diagram with suitable number of branches and use tree to
visualize cases
5. Identifying non-ending trees or trees with unknown branch length as G.P. and using G.P. formulas
6. Using P&C techniques with skill 1
Please contact Angie @96790479 or Mr Ong @98639633 if you need help in Mathematics
A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
JC1 Promo Exams and JC2 Prelim Exams Preparatory Classes
for Mathematics.
Open for Registration Now!
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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Introduction
In general, Mathematics is a common pre-requisite for many courses in the university
H1 Mathematics is designed to provide students with a foundation in mathematics and statistics that will support their business or social sciences studies at the university. The syllabus aims to equip students with the skills to analyse and interpret data and to make informed decisions.
H2 Mathematics is designed to prepare students for a range of university courses, including mathematics, sciences and related courses, where a good foundation in mathematics is required. It develops mathematical thinking and reasoning sk9lls that are essential for further learning of mathematics. Through the applications of mathematics, students also develop an appreciation of mathematics and its connections to other disciplines and to the real world.
H2 Further Mathematics is designed for students who are mathematically inclined and who wish to further expand and deepen their knowledge of mathematics and its applications. Students will develop advanced mathematical thinking and reasoning skills and learn a wider range of mathematics methods and tools. This will give a head-start to students who plan to study mathematics or mathematics-related university courses such as science and engineering in the form of a stronger and richer foundation in mathematics. H2 Further Mathematics is to be taken with H2 Mathematics as ‘Double Mathematics.
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A-Level Mathematics Tuition Singapore/JC Maths/H2 Math Tuition and Tutor
Math Tuition Center Singapore
Arithmetic Progression & Geometric Progress
The sum of the first 100 terms of an arithmetic progression is 10,000. The first, second and fifth terms of this progression are three consecutive terms of a geometric progression. Find the first term and the non-zero common difference of the arithmetic progression.
Answer
First Term = 1
Common difference = 2
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