Notes, summaries, assignments, exams, and problems for Mathematics

1. Simplifying Algebraic Expressions

How to Simplify Algebraic Expressions

• Multiply numbers and letters together.
• Combine like terms (terms with the same letters and powers).

Example:
−2ac × 4bd = −8abcd

Example:
5ab − 8b²a + ba = 5ab − 8ab² + ab = −8ab² + 6ab

2. Expanding Brackets (Distributive Law)

How to Expand Brackets

• Multiply everything inside the bracket by what is outside.

Example:
−4(x + 7) = −4x − 28

Example:
5(x − 3) + 2(6 − x) = 5x − 15 + 12 − 2x = 3x − 3

3. Solving Equations

How to Solve Equations

• Get all variables (e.g., x's) on one side and numbers on the other.
• Perform the same operation on both sides of the equation.

Example:
3x + 2 = 17
3x = 15
x = 5

Example:
9x − 8 = 4x + 7
5x = 15
x = 3

4. Solving Inequalities

How

Understanding Inheritance in Python

Inheritance allows a class to use and extend the properties and methods of another class. It promotes code reusability and reduces duplication. Think of it as “child learns from parent.” 👨‍👦‍💻

Code Duplication: Program Without Inheritance

When classes share common attributes or methods (such as first_name, last_name, and get_age), implementing them separately leads to redundant code, as shown below:

import datetime

class TennisPlayer:
    def __init__(self, fname, lname, birth_year):
        self.first_name = fname
        self.last_name = lname
        self.birth_year = birth_year
        self.aces = []

    def get_age(self):
        now = datetime.datetime.now()
        return now.year -

System Types
The systems of equations can be classified by the number of solutions that can arise. According to that case may have the following cases:
· Incompatible system if it has no solution.
· Compatible system if you have any solution in this case can also distinguish between:
or compatible system determined when it has a finite number of solutions.
indeterminate or compatible system when it admits an infinite set of solutions.
Fitting and classification:

Calculating the rank of a matrix for determining

1. We can rule a line if:.
· All the coefficients are zeros.
· There are two equal lines.
• A line is proportional to another.
• A line is a linear combination of others.
Delete the third column because it is a linear... Continue reading "Matrices: multiplication, rank, determinant, inverse and Rouche-Frobenius theorem" »

Part I: Complete the Following Table

Part II: Complete the Following Sentences

1. She is from the United States. She is American.
2. She is from Nigeria. She is Nigerian.
3. She is from Germany. She is German.
4. They are from Egypt. They are Egyptian.
5. We are from Canada. We are Canadian.

Part III: Select the Correct Sentence

1. Select the correct sentence.
   • a) She lives with Carlos, and she works on Saturdays.
2. Select the correct sentence.
   • c) They are running in the park.
3. Select the correct sentence.
   • b) She is 20 years old.
4. Select the correct sentence.
   • a) We always

Fundamental Accounting Principles and Practices

Core Accounting Definitions

Assets = Liabilities + Owner’s Equity: This is the fundamental accounting equation, representing the balance of a company's financial position.

Double-Entry Accounting

The recording of both debit and credit parts of every transaction, ensuring the accounting equation remains balanced.

Objective Evidence Principle

Requires that a source document (e.g., invoice, receipt) be prepared for every entry in a journal, providing verifiable proof of a transaction.

File Maintenance

The process of arranging accounts in a general ledger, assigning account numbers, and keeping records current.

Opening an Account

Writing an account title and number on the heading of an account in the ledger.... Continue reading "Essential Accounting Concepts and Financial Reporting" »

Chapter 1: Core Accounting Fundamentals

Key Financial Statements Equations

• Balance Sheet (BS) Equation: Assets = Liabilities + Stockholders' Equity (SHE)
• Income Statement (IS) Equation: Revenues - Expenses = Net Income (NI)
• Retained Earnings (RE) Equation: Beginning RE + Net Income - Dividends = Ending RE

Debit and Credit Rules

• A Debit increases: Expenses, Assets, Dividends (DEAD)
• A Credit increases: Liabilities, Revenues, Equity (CLEAR)

Objective of Financial Reporting

1. Provide information useful to equity investors and creditors.
2. Maintain the Entity Perspective (separating the company from its owners/people).
3. Ensure Decision Usefulness.

Adjusting Entries: Deferrals and Accruals

Deferrals (Cash Paid/Received Before Recognition)

Prepaid Expenses

Meaning: Paid... Continue reading "Core Financial Accounting Concepts and Principles" »

Metallurgical sample preparation for microscopy is a crucial step in analyzing the microstructure and properties of metallic materials. Proper sample preparation is essential to obtain accurate and meaningful results. The process involves several key steps:

1. Sample Selection for Microscopy

Choose a representative portion of the material to be analyzed. Ensure that the sample is free from external contaminants and has a flat surface for preparation.

2. Precision Cutting Techniques

Use a precision cutting method to obtain a small section of the material for analysis. Common cutting techniques include abrasive cutting (using a saw with abrasive blades), wire cutting, or electrical discharge machining (EDM) for hard materials.

3. Sample Mounting for

Bond Characteristics

• Coupon: The interest payment made by the bond issuer, usually expressed as an annual percentage of the bond's face value.
• Par (Face Value): The amount the bondholder receives when the bond matures, typically $1,000.
• Term to Maturity: The time remaining until the bond's maturity date when the issuer must repay the bond's par value.
• Denomination: The face value of the bond, usually in increments of $1,000.
• Quotation: Bonds are quoted as a percentage of their face value (e.g., a bond quoted at 95 is selling for 95% of $1,000, or $950).

Bond Prices, Yield to Maturity (YTM), Current Yield, and Rate of Return (HPR)

• Bond Prices: The market price of a bond depends on interest rates. Prices and interest rates have an inverse relationship.

Essential Financial Statement Formulas

• Stockholder Equity: Total Assets – Total Liabilities
• Retained Earnings: Net Income – Dividends Declared
• Net Income: Sales Revenue – Expenses
• Gross Profit: Sales Revenue – Cost of Goods Sold (COGS)
• Cost of Goods Sold (COGS) based on Rate: (1 – Gross Profit Rate) × Net Sales

Inventory Accounting Adjustments (LIFO and FIFO)

• Calculating LIFO Reserve: FIFO Ending Inventory Cost – LIFO Ending Inventory Cost
• Adjusting Balance Sheet from LIFO to FIFO Inventory: LIFO Reserve + LIFO Inventory

Accounting Estimates and Depreciation

Changing Accounting Estimates

• Book Value at Date of Change: Costs – Accumulated Depreciation
• New Remaining Useful Life: Original Life – Years Depreciated + Additional Years
• Depreciation

Discrete Random Variables

Discrete random variables are variables that can take on a finite number of distinct values. In simpler terms, a discrete random variable is a set of possible outcomes that is countable.

Continuous Random Variables

Continuous random variables are random variables that take an infinitely uncountable number of potential values, typically measurable amounts.

Example

1. List the sample space in the given experiment. How many outcomes are possible?

The sample space is: S = {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD}

2. Count the number of defective keyboards in each outcome in the sample space and assign this number to the outcome. For instance, if you list NND, then the number of defective keyboards is 1.

The possible values of X are 0,... Continue reading "Introduction to Statistics: Discrete and Continuous Random Variables, Probability Distributions, and Sampling Techniques" »