O Level – Additional Mathematics

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O-Level Additional Mathematics Tuition Singapore

Quadratic Inequalities

1. A quadratic expression is an expression of the form ax^2 + bx + c, where a, b and c are constants and a # 0.
For example: 4x^2 — 2x + 1

2. A quadratic function is a function with rule being a quadratic expression. For example: y = x^2 — 3x + 1 or f(x) = x^2 — 3x + 1

3. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b and c are constants and a # 0.
For example: 2x^2 + 5x — 10 = 0

4. A quadratic inequality is an inequality that can be expressed in one variable on one side and zero on the other such that
ax^2 + bx + c < 0, where a, b and c are constants and a # 0.

For example: 4x^2 — 2x + 1 < 0 and (x — 1)(x + 3)< 0

5. Steps to solve a quadratic inequality:
(a) Make the right-hand side (RHS) of the inequality zero by bringing all the terms to the left-hand side (LHS).
(b) Factorise the expression on the LHS.
(c) Use one of the following methods to find the solution.
(i) Sketch the quadratic curve, OR
(ii) Use the sign test

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O-Level Additional Mathematics Tuition Singapore

Partial fractions

1/(X+a)(X+b) = A/(X+a) + B/(X+b)

1/(X+a)^2 = A/(X+a)^2 + B/(X+b)

1/(aX^2+b)(X+c) = (AX + B)/(aX^2+b) + C/(X+c)

Note
a,b,c,A.B,C, are constant

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O-Level Additional Mathematics Tuition Singapore

2014 Aug O level Additional Math Intensive Revision

28-Jul Mon 4pm to 5.30pm 1.5 hrs Surds and Indices and Logarithms

30-Jul Wed 7pm to 8.30pm 1.5 hrs Surds and Indices and Logarithms

31-Jul Thu 5.30pm to 7pm 1.5 hrs Surds and Indices and Logarithms

3-Aug Sun 6pm to 7.30pm 1.5 hrs Surds and Indices and Logarithms

4-Aug Mon 4pm to 5.30pm 1.5 hrs Co-ordinate Geometry & P2 Exam Practice

6-Aug Wed 7pm to 8.30pm 1.5 hrs Co-ordinate Geometry & P2 Exam Practice

7-Aug Thu 5.30pm to 7pm 1.5 hrs Co-ordinate Geometry & P2 Exam Practice

10-Aug Sun 6pm to 7.30pm 1.5 hrs Co-ordinate Geometry & P2 Exam Practice

11-Aug Mon 4pm to 5.30pm 1.5 hrs Trigonometric Identities and Eqns & P1 Exam Practice

13-Aug Wed 7pm to 8.30pm 1.5 hrs Trigonometric Identities and Eqns & P1 Exam Practice

14-Aug Thu 5.30pm to 7pm 1.5 hrs Trigonometric Identities and Eqns & P1 Exam Practice

17-Aug Sun 6pm to 7.30pm 1.5 hrs Trigonometric Identities and Eqns & P1 Exam Practice

18-Aug Mon 4pm to 5.30pm 1.5 hrs Differentiation and Application & P2 Exam Practice

20-Aug Wed 7pm to 8.30pm 1.5 hrs Differentiation and Application & P2 Exam Practice

21-Aug Thu 5.30pm to 7pm 1.5 hrs Differentiation and Application & P2 Exam Practice

24-Aug Sun 6pm to 7.30pm 1.5 hrs Differentiation and Application & P2 Exam Practice

25-Aug Mon 4pm to 5.30pm 1.5 hrs Integration and Application & P1 Exam Practice

27-Aug Wed 7pm to 8.30pm 1.5 hrs Integration and Application & P1 Exam Practice

28-Aug Thu 5.30pm to 7pm 1.5 hrs Integration and Application & P1 Exam Practice

31-Aug Sun 6pm to 7.30pm 1.5 hrs Integration and Application & P1 Exam Practice

For exam based question with full worked solution, please contact
@9863 9633

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O-Level Additional Mathematics Tuition Singapore

TOPIC 1 SIMULTANEOUS LINEAR EQUATIONS

Exam Tip 1
When we solve 2 simultaneous equations, the solutions are actually the intersection points (x and y coordinates) of the 2 intersecting curves (or straight lines) represented by the simultaneous equations.

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O-Level Additional Mathematics Tuition Singapore

2015 Schedule Additional Math

S3 1.5 hrs WED 4pm – 5.30pm

S4 2 hrs MON 4pm – 6pm

S4 2 hrs SUN 6pm – 8pm

Please contact Angie @ 96790479 or Mr Ong @ 98639633

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O-Level Additional Mathematics Tuition Singapore

Exam Questions

Question 1

Find the range of values of x for which 1+ x^2 > – (7x – x^2)/3 .
Hence or otherwise, find the minimum value of y if
y = 1 + x^2 + (7x – x^2)/3

Question 2

A polynomial, f(x), of degree 4 has a quadratic factor of x^2 — x+ 2 and a linear factor of (x + 3). Given that f(x) leaves a remainder of 8 and —24 when divided by (x — 1) and (x+ 1) respectively, find

(i) the remaining factor of f(x),

(ii) the number of real roots of the equation f(x) = 0, justifying
your answer,

(iii) the remainder when f(x) is divided by x.

For exam based question with full worked solution, please contact Mr Ong
@98639633 0r Angie @96790479

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O-Level Additional Mathematics Tuition Singapore

Exam Question 1

Find the values of k, for which -2x^2+kx-8 is always negative. [3 marks]

Ans -8 < k < 8 Exam Question 2 Given that 4x^2 - 6x + 9 = A(x - 1)(2x + 1)  B(x - 1) + C for all values of x, find the values of A, B and C. [3 marks] Ans A = 5, B = −4, C = 7 For exam based question with full worked solution, please contact Mr Ong @98639633 0r Angie @96790479

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O-Level Additional Mathematics Tuition Singapore

Question 1

A rhombus PQRS has coordinates P(-3, 12) and Q(9,11). The diagonals of the
rhombus intersect at the point M(−1, 7).
(i) Find the coordinates of R.
(ii) Find the equation of the diagonal QS.

Ans : (i) R (1, 2) (ii) 5y = 2x + 37

Question 2

(i) Expand (1 – 2x)^4 in ascending powers of x, up to and including the term
in x2.

(ii) Find the value of a such that the coefficient of x is zero in the expansion of (2 + ax)^2 * (1 – 2x)^4

Ans : (i) 2 – 8x + 24x^2…….. (ii) a = 8

For exam based question with full worked solution, please contact Mr Ong
@98639633 0r Angie @96790479

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O-Level Additional Mathematics Tuition Singapore

Question 1

(i) Show that x^2 -3x – 54 = 0 has real and distinct roots.

(ii) Solve the inequality x^2 -3x – 54 > and equal 0 and represent the solution set on the number line.

Ans :

(i) Show D > 0 . Conclusion.
(ii) x equal and < -6 , x equal and > 9

Question 2

Given the polynomial f(x) = 6x^3 – 25x^2 + 34x – 15
(i) find the remainder when f(x) is divided by x – 2,
(ii) show that 3x – 5 is a factor of f(x),
(iii) solve the equation 6x^3 – 25x^2 + 34x – 15 = 0

Ans

(i) 15
(ii)Show that f(5/3) = 0
(iii) x = 5/3 or or 1 or 3/2

For exam based question with full worked solution, please contact Mr Ong
@98639633 0r Angie @96790479

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O-Level Additional Mathematics Tuition Singapore

Secondary 3 Year End Exams and Secondary 4 Prelim Preparatory Classes

for Additional Math.

Open for Registration Now!

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O-Level Additional Mathematics Tuition Singapore

Question

(a) Find the smallest value of the integer a for which ax^2-8x+3 is positive for all values of x.

(b) The equation of a curve is y = x^3 + 5/2x^2 -2x +1
Find the set of values of x for which y is a decreasing function.

Solution

(a)
ax^2-8x+3>0
(-8)^2-12a<0
12a>64
a>5 1/3
The smallest integer a is 6.

(b)
y = x^3 + 5/2x^2 -2x +1
dy/dx = 3x^2 + 5x – 2
For decreasing function, dy/dx < 0 3x^2 + 5x - 2 < 0 (3x - 1)(x + 2) < 0 Draw number line -2 < x < 1/3 Please contact Angie @ 96790470 or Mr Ong @ 98639633 if you need help in Elementary Math

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O-Level Additional Mathematics Tuition Singapore

Question

Find the range of values of k for which the line x + 3y = k intersects the curve
y^2 = 2x + 3 at 2 distinct points.

(ii) State the value of k if the line x + 3y = k is tangent to the curve
y^2 = 2x + 3.

Answer

i) k > -6
ii) k = -6

Please contact Angie @ 96790479 or Mr Ong @ 98639633 if you need Add Math Tuition

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