GENERAL AND SPECIFIC COMPETENCIES IN MATHEMATICS IV
(Advanced Algebra, Trigonometry and Statistics)

A. Functions
1. Demonstrate knowledge and skill related to functions in general 1.1 Define a function
1.2 Differentiate a function from a mere relation
* real life relationships
* set of ordered pairs
* graph of a given set of ordered pairs
* vertical line test
* given equation
1.3 Illustrate the meaning of the functional notation f(x) 1.4 Determine the value of f(x) given a value for x
B. Linear Functions
1. Demonstrate knowledge and skill related to linear functions and apply in solving problems
1. Define the linear function f(x) = mx + b
2. Given a linear function Ax + By = C, rewrite in the form f(x) = mx + b and vice versa 3. Draw the graph of a linear function given the following: * any two points

* x and y intercepts
* slope and one point
* slope and the y-intercept
4. Given f(x) = mx + b, determine the following: * slope
* trend: increasing or decreasing
* x and y intercepts
* some points
5. Determine f(x) = mx + b given:
* slope and y-intercept
* x and y intercepts
* slope and one point
* any two points
6. Solve problems involving linear functions
C. Quadratic Functions
1. Demonstrate knowledge and skill related to quadratic functions and apply these in solving problems
1.1 Identify quadratic functions f(x) = ax2 + bx + c
1.2 Rewrite a quadratic function ax2 + bx + c in the form f(x) = a(x-h)2 + k and vice versa
1.3 Given a...

...Level 1/Level 2 Certificate
Mathematics
Specification
Edexcel Level 1/Level 2 Certificate in Mathematics
(KMAO)
First examination June 2012
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Acknowledgements
This specification has been produced by Edexcel on the basis of consultation with
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thank all those who contributed their time and expertise to its development.
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...Name _______________________________________ Score ______________________
Section __________________________
REVISING IN MATHEMATICS IV
A.Y. 2014 - 2015
(Science Class)
General Directions: Analyze the following statements then write the correct answer.
I. MULTIPLE CHOICES
Directions: Read the following test items carefully. Write the letter of the correct answer.
1. Which of the following is a polynomial function?
a. P(x) = 3x-3 – 8x2 + 3x + 2 c. P(x) = 2x4 + x3 + 2x + 1
b. P(x) = x3 + 4x2 + – 6 d. G(x) = 4x3 – + 2x + 1
2. What is the degree of the polynomial function f(x) = 5x – 3x4 + 1?
a. 2 c. 4
b. 3 d. 5
3. What will be the quotient and the remainder when y = 2x3 – 3x2 – 8x + 4 is divided by
(x +2)?
a. q(x) = 2x2 – 7x + 6 , R = -8
b. q(x) = 2x2 – 7x + 6, R = 8
c. q(x) = 2x2 – 7x – 6, R = -8
d. q(x) = 2x2 – 7x - 6 , R = 8
4. If f(a) = 2a3 + a2 + 12, what will be the value of f(a) at a = -2?
a. 1
b. -1
c. 0
d. 2
5. What must be the value of k so that when f(x) = kx2 - x + 3 divide by (x + 1) and the remainder is 5?
a. 2
b. -2
c. 0
d. 1
6. What must be the value of k in the function f(x) = x4 + x3 – kx2 – 25x – 12 so
that (x – 4) is a factor?
a. -12
b. -13
a. 13
b. 12
7. What is the remainder when f(x) = x4 + 3x2 + 4x – 1 divided by (x – 1)?
a. 7
b. -7
c. 6
d. 5
8. Which of the following binomial is a factor of f(x) = x3 – x2 – 5x – 3?
a. x - 1
b. x + 2
c. x - 3...

...values of t.
s is a complex algebra variable defined by: s = a +jω where j = sqrt(-1), so you will be partly using imaginary numbers.
The symbol i (j in electrical engineering) is used to represent √ -1. Therefore, for example, √(-4) = 2i. The number called i, or 1i , or xi are called purely imaginary numbers.
One use of the complex plane is known as the s-plane. It is used to visualize the roots of the equation describing a system's behavior (the characteristic equation) graphically. The equation is normally expressed as a polynomial in the parameter 's' of the Laplace transform, hence the name 's' plane.[1]
Complex plane using Argand diagrams show the z-plane, where z = x + iy and may use z-Transforms as well as the Laplace. In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation. It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time scale calculus. Through bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform).
Z = a+ ib, = r e^itheta, a = real part of z, b = imaginary part of z, r = modulus of z, theta = argument of z, a & b are real numbers. While this mapping is (necessarily) nonlinear, it is useful in that it maps the entire jΩ axis of the s-plane onto the unit circle in the...

...Semi-Detailed Lesson Plan in Mathematics (Transformations)
Level: First Year High School
Subjects: Mathematics, Geometry, Transformations
I. Objectives:
A. To recognize Euclidean transformations.
B. To recognize reflections, translations, and rotations.
C. To prove theorems related to transformations.
D. To solve problems involving transformations.
E. To apply transformations to real-world situations.
F. To create designs using transformations.
II. Materials:
papers, protractor, ruler
tangram puzzle
worksheets
III. Procedure:
A. Presentation
Activity - Folding of Paper
The teacher will give an activity that involves the folding of paper and tracing of shapes.
B. Discussion
From the activity, the teacher will point out that geometry is not only the
study of figures but is also the study of the movement of figures.
Is the original figure congruent to the other figures?
How does the second image compare to the original figure?
C. Input
Definitions:
Transformations
Reflection
Rotation
Translation
Dilation
Rigid Motion
Theorems:
Theorem 18-1
Theorem 18-2
Theorem 18-3
Theorem 18-4
C. Discussion
The above definitions and theorems will be discussed and proved. The teacher will ask the student to give examples of transformations.
D. Activity
Tangram Puzzle
The students will form six groups. Each group is going to make images of animals using tangram puzzle and they will identify the...

...Contribution of India in mathematics
The most fundamental contribution of India in mathematics is the invention of decimal system of enumeration, including the invention of zero. The decimal system uses nine digits (1 to 9) and the symbol zero (for nothing) to denote all natural numbers by assigning a place value to the digits. The Arabs carried this system to Africa and Europe.
1) Aryabhata is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya, he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of as 3.1416, claiming, for the first time, that it was an approximation. He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhatasiddhanta. Even today, this data is used in preparing Hindu calendars. In recognition to his contributions to astronomy and mathematics, India's first satellite was named Aryabhata.
2) Brahmagupta is renowned for introduction of negative numbers and operations on zero into arithmetic. His main work was Brahmasphutasiddhanta, which was a corrected version of old...

...Zero in Mathematics
Zero as a number is incredibly tricky to deal with. Though zero provides us with some useful mathematical tools, such as calculus, it presents some problems that if approached incorrectly, lead to a breakdown of mathematics as we know it.
Adding, subtracting and multiplying by zero are straightforward.
If c is a real number,
c+0=c
c-0=c
c x 0=0
These facts are widely known and regarded to hold true in every situation.
However, division by zero is a far more complicated matter. With most divisions, for example,
10/5=2
We can infer that
2 x 5=10
But if we try to do this with zero,
10/0=a
0 x a=10
Can you think of a number that, when multiplied by 0, equals 10? There is no such number that we have ever encountered that will satisfy this equation.
Another example will emphasise the mysteriousness of dividing by zero.
One may assume that
(c x 0)∕0=c
The zeroes should cancel, as would be done with any other number. But since we know that
c x 0=0
it follows that
(c x 0)/0=0/0=c
This does not seem to make sense. This also means that
1=0/0=2
1=2
since 1 and 2 are both real numbers. Actually, this means that 0/0 is equal to every real number!
In effect, there is no real answer to a division by zero. It cannot be done.
In fact, if we could divide by zero, it would be possible to prove anything that we could dream of. For example, imagine a student trying to prove to his teacher that he...

...Introduction
Mathematics is an indispensable subject of study. It plays an important role in forming the basis of all other sciences which deal with the material substance of space and time.
What is Mathematics?
Mathematics may be described as the fundamental science. It may be broadly described as the science of space, time and number. The universe exists in space and time, and is constituted of units of matter. To calculate the extension or composition of matter in space and time and to compute the units that make up the total mass of the material universe is the object of Mathematics. For the space-time quantum is everywhere full of matter and we have to know matter mathematically in the first instance.
Importance of Mathematics
Knowledge of Mathematics is absolutely necessary for the study of the physical sciences.
Computation and calculation are the bases of all studies that deal with matter in any form.
Even the physician who has to study biological cells and bacilli need to have a knowledge of Mathematics, if he means to reduce the margin of error which alone can make his diagnosis dependable.
To the mechanic and the engineer it is a constant guide and help, and without exact knowledge of Mathematics, they cannot proceed one step in coming to grips with any complicated problem.
Be it the airplane or the atom bomb,...