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O-Level Additional Mathematics Tuition Singapore
The Factorial n!
The notation n! is read as ‘n factorial’. It is defined for a positive integer n as the product of the first n positive integers.
Examples
(a) 3! = 3 x 2 x 1 = 6
(b) 4!= 4 x 3 x 2 x 1 = 24
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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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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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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
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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
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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.
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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
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
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@98639633 0r Angie @96790479
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
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@98639633 0r Angie @96790479
O-Level Additional Mathematics Tuition Singapore
Secondary 3 Year End Exams and Secondary 4 Prelim Preparatory Classes
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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
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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
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