Notes, summaries, assignments, exams, and exercises for Mathematics for Secondary education

Comparison of Accounting Statements

Measuring Angles

An angle can be measured in three ways:

1. In degrees: The circumference is divided into 360 equal parts. Each part represents an angle, with the apex at the center of the circle (sexagesimal measure), and is indicated by placing 1°.
2. In grads: The circumference is divided into 400 equal parts. Each part represents an angle, with the apex at the center of the circle, measuring a grad, and is indicated by placing 1^g.
3. In radians: An angle of one radian is an angle whose arc on the circumference has the same length as the radius of the circle, and is indicated by placing 1 rad.

Definition of Sine, Cosine, and Tangent

For an acute angle β in a right triangle:

• Sine (sin β): Defined as the ratio of the opposite leg to the hypotenuse.
• Cosine

Understanding Mathematical Functions

A relationship based on two numerical variables, usually called X and Y. X is the independent variable, and Y is the dependent variable. The function is usually denoted by y = f(x), associating each value of x with a single value of y: x → y = f(x).

Domain and Image of a Function

Domain: The domain of a function f(x) consists of all values of X for which the function is defined.

Image (Range): Im(f) is the set of values that the function takes. That is, the set of values y for which there exists an x such that f(x) = y.

Methods of Representing Functions

Ways to represent functions:

• Through its graphical expression
• Through a statement
• By a table of values
• By an analytical expression or formula

Calculating Different

Linguistic Sign Features

Root: The root is the carpenter-joiner and wood SIGIF.

Morpheme: It is the -er meaning 'person who works with' and the -o meaning 'male', while -s refers to the plural.

Characteristics of the Linguistic Sign

• Biplane: It consists of two levels. The signifier (expression level) is the string of sounds or spellings (e.g., the signifier of 'niño' is /n/ /i/ /ñ/ /o/). The signified (content level) is the concept in our minds associated with the signifier (e.g., 'niño' means 'young person').
• Arbitrary: The meaning of 'young person' varies by language: 'niño' in Spanish, 'warm' in Quechua, and 'boy' in English.
• Articulate: It is composed of smaller parts.

Monemes: Units of Meaning

The first articulation consists of units that...

Transposed and Symmetric Matrices

When we swap rows and columns, we obtain a transposed matrix.

A symmetric square matrix is a specific case where the matrix equals its transpose. For example: F (121/224/146) where F^t = (121/224/146).

Systems of Linear Equations

A way to represent a system of linear equations with unknowns is a System:

• ax + by = c
• dx + ey = f

Opposite Matrices and Sum Properties

The opposite of a given matrix is the result of replacing each element by its opposite. The opposite of A is -A.

The properties of the matrix sum include:

• Associative Property
• Commutative Property
• Null Element
• Matrix Opposed

Properties of Determinants

The properties of determinants and their examples are as follows:

• The determinant of a square matrix equals the determinant

Triangles

• Median: The segment connecting a vertex to the midpoint of the opposite side.
• Centroid (G): The intersection point of the three medians. It divides each median into two segments.
• Height (Altitude): The perpendicular segment from a vertex to the opposite side or its extension.
• Orthocenter: The point where the three altitudes intersect.
• Perpendicular Bisector: The line perpendicular to a segment at its midpoint.
• Angle Bisector: The ray that divides an angle into two equal angles.
• Circumcenter: The intersection point of the three perpendicular bisectors. It is equidistant from the three vertices.
• Incenter: The intersection point of the three angle bisectors. It is equidistant from the sides of the triangle.

Polygons

• Polygon: A plane region bounded

Mathematics Term 3 Reference

Chapter 2

Divisibility Tests

• 1: All numbers are divisible by 1.
• 2: All even numbers are divisible by 2. Last digit must be 0, 2, 4, 6, or 8.
• 3: The sum of the digits must be divisible by 3.
• 4: The number formed from the last two digits must be divisible by 4.
• 5: The last digit must be 0 or 5.
• 6: Must pass the divisibility tests for both 2 and 3.
• 7: (There is no simple test for divisibility by 7).
• 8: The number formed from the last three digits must be divisible by 8.
• 9: The sum of the digits must be divisible by 9.
• 10: The last digit must be 0.

Terms for Division

• Dividend: The starting number; the total; the amount you have.
• Divisor: The number doing the dividing; the number of groups.
• Quotient: The number of times the divisor

Current Ratio

The current ratio measures a company's ability to pay short-term obligations with its current assets.

• Positive (>1): Current assets can meet current liabilities. Specifically, assets cover [X]% of the liabilities.
• Negative (<1): Current assets cannot meet all current liabilities. They only cover [X]%, indicating a risk of short-term default.

Acid Test (Quick Ratio)

This metric determines if a company can meet its short-term debts without relying on inventory sales.

• Positive (0.8–1.2): With bank cash and accounts receivable, the company can satisfy [X]% of current liabilities without selling stock.
• Negative (<0.8): The company can only satisfy [X]% of liabilities without stock, making it too dependent on inventory sales to

Properties of Sets

Important Set Formulas

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• If A and B are disjoint: n(A ∪ B) = n(A) + n(B)
• Number of subsets of a set with n elements = 2ⁿ
• Number of proper subsets = 2ⁿ – 1
• n(P(A)) = 2ⁿ

1. Identity Function

f(x) = x

• Graph: Straight line through origin
• Line: y = x
• Domain & Range: ℝ
• Passes through: (0, 0), (1, 1), (-1, -1)

2. Constant Function

f(...

Adding Like Terms in Algebra

When working with algebraic expressions, like terms are those that have the same pronumeral (letter) and the same powers. To add like terms, you combine their coefficients while keeping the pronumeral and its power unchanged.

For example:

5x + 3x

Because 'x' is the common pronumeral in this expression, you can simply add the coefficients (5 and 3) together. The sum is 8. Since 'x' is present in both terms, it must be included in the final answer. Therefore, the result is:

8x

Subtracting Like Terms in Algebra

Similar to addition, subtracting like terms involves combining terms that share the same pronumeral and powers. You subtract their coefficients while maintaining the pronumeral and its power.

For example:

5x - 3x

Since...