General Mathematics Overview

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Questions and Answers

Explain how the order of operations (PEMDAS/BODMAS) is crucial in solving the following arithmetic expression: $2 + 3 \times (6 - 4) \div 2$.

PEMDAS/BODMAS dictates the sequence: Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right). Following this order ensures we subtract within the parentheses first, then multiply, then divide, and finally add, leading to the correct result.

How can you determine if the number 457 is a prime or composite number without performing full division for all numbers less than 457?

Check for divisibility by prime numbers less than the square root of 457 (approximately 21.4). If 457 is not divisible by any of these primes (2, 3, 5, 7, 11, 13, 17, 19), then it is a prime number.

Describe two different methods to solve the following system of equations:

$x + y = 5$

$2x - y = 1$

Substitution: Solve one equation for one variable (e.g., $x = 5 - y$) and substitute into the other equation.
Elimination: Add or subtract the equations to eliminate one variable (in this case, adding the equations directly eliminates y).

Explain how the Pythagorean theorem can be used to find the height of an isosceles triangle with base 10 and two equal sides of length 13.

The height bisects the base, creating two right triangles with hypotenuse 13 and one side 5 (half of the base). Using the Pythagorean theorem ($a^2 + b^2 = c^2$), we can solve for the height ($a$): $a^2 + 5^2 = 13^2$. Therefore, $a = \sqrt{13^2 - 5^2} = 12$. Signup and view all the answers

Describe the difference between differential and integral calculus, and provide an example of a practical application for each.

Differential calculus deals with rates of change and slopes (e.g., finding the velocity of an object at a specific time), while integral calculus deals with accumulation and areas (e.g., calculating the area under a curve representing the total distance traveled). Signup and view all the answers

Explain the difference between descriptive and inferential statistics. Provide an example of when each type of statistic would be used.

Descriptive statistics summarize data (e.g., finding the average test score of a class), while inferential statistics uses sample data to make predictions about a population (e.g., predicting election outcomes based on a poll). Signup and view all the answers

Briefly explain the Fundamental Theorem of Arithmetic and provide an example of how it applies to the number 60.

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely expressed as a product of prime numbers. For example, $60 = 2^2 \times 3 \times 5$, and this prime factorization is unique. Signup and view all the answers

Explain the difference between deductive and inductive reasoning. Provide a simple example of each.

Deductive reasoning starts with general premises and draws specific conclusions (e.g., All men are mortal, Socrates is a man, therefore, Socrates is mortal). Inductive reasoning starts with specific observations and makes general conclusions (e.g., Every swan I have seen is white, therefore, all swans are white). Signup and view all the answers

Explain the difference between accuracy and precision in measurement, and provide an example to illustrate the difference.

Accuracy refers to how close a measurement is to the true value, while precision refers to the repeatability of a measurement. For example, a scale that consistently measures a 10 kg weight as 12 kg is precise but not accurate, while a scale that measures weights randomly around 10 kg is accurate but not precise. Signup and view all the answers

Explain the concept of compound interest and contrast it with simple interest. Why is compound interest more beneficial over long investment periods?

Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal and accumulated interest. Compound interest is more beneficial over long periods because interest earns interest, leading to exponential growth. Signup and view all the answers

Flashcards

Arithmetic

Basic operations including addition, subtraction, multiplication, and division.

Order of Operations (PEMDAS/BODMAS)

The order in which mathematical operations should be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).