## Secondary 4 O Level E Maths and A Maths Group Tuition, Bishan

Maths Tuition @ Bishan starting in 2014.

Secondary 4 O Level E Maths and A Maths.

Patient and Dedicated Maths Tutor (NUS Maths Major 1st Class Honours, Dean’s List, RI Alumni)

Email: mathtuition88@gmail.com

## What is the Difference between H1 Mathematics, H2 Mathematics and H3 Mathematics?

Note: Additional Mathematics is very helpful to take H2 Mathematics in JC!

## Curriculum

There are three mathematics syllabi, namely H1 Mathematics, H2 Mathematics and H3 Mathematics.

Students who offered Additional Mathematics and passed the subject at the GCE ‘O’ level examination may take up H2 Mathematics. Students posted to the Arts stream and did not offer Additional Mathematics at the GCE ‘O’ level examination are not allowed to take H2 Mathematics but may consider taking up H1 Mathematics. However, students who are posted to the Science stream but did not offer Additional Mathematics at the GCE ‘O’ level examination are advised to offer H2 Mathematics if they intend to pursue Science or Engineering courses at a university. Students who wish to offer H3 Mathematics must offer H2 Mathematics as well.

The use of a Graphing Calculator (GC) without a computer algebra system is expected for these Mathematics syllabi. The examination papers will be set with the assumption that candidates will have access to GCs.

#### H1 Mathematics

H1 Mathematics provides a foundation in mathematics for students who intend to enrol in university courses such as business, economics and social sciences. The topics covered include Graphs, Calculus and Statistics. A major focus of the syllabus would be the understanding and application of basic concepts and techniques of statistics. This would equip students with the skills to analyse and interpret data, and to make informed decisions.

#### H2 Mathematics

H2 Mathematics prepares students adequately for university courses including mathematics, physics and engineering, where more mathematics content is required. The topics covered are Functions and Graphs, Sequences and Series, Vectors, Complex Numbers, Calculus, Permutations and Combinations, Probability, Probability Distributions, Sampling, Hypothesis Testing, and Correlation and Regression. Students would learn to analyse, formulate and solve different kinds of problems. They would also learn to work with data and perform statistical analysis.

#### H3 Mathematics

H3 Mathematics offers students who have a strong aptitude for and are passionate about mathematics a chance to further develop their mathematical modeling and reasoning skills. Opportunities abound for students to explore various theorems, and to read and write mathematical proofs. Students would learn the process of mathematical modeling for real-world problems, which involves making informed assumptions, validation and prediction. Students may choose from the three H3 Mathematics modules, namely the MOE-UCLES module, the NTU Numbers and Matrices module and the NUS Linear Algebra module.

The MOE-UCLES module is conducted by tutors from our Mathematics Department. The three main topics to be investigated are Graph Theory, Combinatorics and Differential Equations. This module would be mounted only if there’s demand.

The NTU Numbers and Matrices module is conducted by lecturers from the Nanyang Technological University (NTU). Students would have to travel to Hwa Chong Institution to attend this module.

The NUS Linear Algebra module is conducted by lecturers at the National University of Singapore (NUS). Students who offer this module would have to attend lessons together with the undergraduates at the university.

## H2 Maths A Level 2012 Solution, Paper 2 Q5; H2 Maths Tuition

5(i)(a)

$P(\text{patient has the disease and test positive})=0.001(0.995)=9.95\times 10^{-4}$

$P(\text{patient does not have the disease and he tests positive})=(1-0.001)(1-0.995)=4.995\times 10^{-3}$

$P(\text{result of the test is positive})=9.95\times 10^{-4}+4.995\times 10^{-3}=5.99\times 10^{-3}$

(b)

Let A=patient has disease

Let B=result of test is positive

$\displaystyle\begin{array}{rcl}P(A|B)&=&\frac{P(A\cap B)}{P(B)}\\ &=&\frac{(0.001)(0.995)}{5.99\times 10^{-3}}\\ &=&0.166 \end{array}$

Note that the probability is surprisingly quite low! (This is called the False positive paradox, a statistical result where false positive tests are more probable than true positive tests, occurring when the overall population has a low incidence of a condition and the incidence rate is lower than the false positive rate. See http://en.wikipedia.org/wiki/False_positive_paradox)

(ii)

$\displaystyle P(A|B)=\frac{(0.001)p}{(0.001)p+(1-0.001)(1-p)}=0.75$

By GC, $p=0.999666$ (6 d.p.)

# A Level H2 Maths 2012 Paper 2 Q3 Solution

(i)

(The graph above is drawn using the Geogebra software 🙂 )

(ii)

$x^3+x^2-2x-4=4$

$x^3+x^2-2x-8=0$

By GC, $x=2$

By long division, $x^3+x^2-2x-8=(x-2)(x^2+3x+4)$

The discriminant of $x^2+3x+4$ is

$D=b^2-4ac=3^2-4(1)(4)=-7<0$

Hence, there are no other real solutions (proven).

(iii) $x+3=2$

$x=-1$

(iv)

(v)

$|x^3+x^2-2x-4|=4$

$x^3+x^2-2x-4=4$ or $x^3+x^2-2x-4=-4$

$x^3+x^2-2x-8=0$ or $x^3+x^2-2x=0$

$x^3+x^2-2x-8=0 \implies x=2$ (from part ii)

$x^3+x^2-2x=x(x^2+x-2)=x(x-1)(x+2)=0$

$x=0,1,-2$

In summary, the roots are $-2,0,1,2$