"Comprehensive Guide to Year 12 Mathematics: From Algebra to Practical Applications"
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Course
Mathematics
Institution
12th Grade
The document is a comprehensive guide to Year 12 mathematics, covering a wide range of topics essential for practical applications in everyday life and future endeavors. It begins with algebraic mathematics, exploring techniques for manipulating expressions, solving equations and inequalities, and ...
Introduction: Year 12 General Mathematics offers a diverse range of topics
essential for practical applications in everyday life and future endeavors.
This guide provides comprehensive explanations and examples of each
topic, including relevant formulas and step-by-step instructions to
facilitate understanding and mastery.
1. Algebra and Number: Algebraic Techniques: Algebraic techniques
involve manipulating algebraic expressions to simplify or solve
equations. For example, consider the expression
3𝑥2+5𝑥−23x2+5x−2. To simplify it, you would combine like terms
(terms with the same variable and exponent) by adding or
subtracting their coefficients: 3𝑥2+5𝑥−23x2+5x−2.
Solving Equations and Inequalities: Equations and inequalities involve
finding the value(s) of the variable that satisfy the given conditions. For
instance, to solve the equation 2𝑥+3=92x+3=9, you would isolate the
variable by performing inverse operations: 2𝑥=9−32x=9−3 ⇒2𝑥=6⇒2x=6
⇒𝑥=62⇒x=26 ⇒𝑥=3⇒x=3.
Indices and Surds: Indices and surds deal with expressions involving
powers and roots. For example, to simplify 23×2423×24, you would add
the exponents: 23×24=23+4=2723×24=23+4=27.
Polynomials: Polynomials are expressions with one or more terms
involving variables raised to non-negative integer powers. To multiply two
polynomials, such as 2𝑥+32x+3 and 𝑥−4x−4, you would use the
distributive property: (2𝑥+3)(𝑥−4)=2𝑥2−8𝑥+3𝑥−12(2x+3)
(x−4)=2x2−8x+3x−12.
Algebraic Fractions: Algebraic fractions involve fractions containing
algebraic expressions in the numerator, denominator, or both. To add or
subtract algebraic fractions, find a common denominator and perform the
operation on the numerators, such as 𝑎𝑏+𝑐𝑏=𝑎+𝑐𝑏ba+bc=ba+c.
2. Financial Mathematics: Simple and Compound Interest: Simple
interest is calculated using the formula 𝐼=𝑃𝑅𝑇I=PRT, where 𝐼I is the
interest, 𝑃P is the principal amount, 𝑅R is the interest rate per
period, and 𝑇T is the time period. Compound interest is calculated
using 𝐴=𝑃(1+𝑟/𝑛)𝑛𝑡A=P(1+r/n)nt, where 𝐴A is the future value, 𝑟r is
the annual interest rate, 𝑛n is the number of compounding periods
per year, and 𝑡t is the time in years.
Depreciation: Depreciation refers to the decrease in value of an asset over
time. The straight-line method divides the initial cost by the useful life to
determine the annual depreciation, while the reducing-balance method
applies a constant depreciation rate to the diminishing value of the asset
each year.
Annuities: Annuities involve regular payments or investments over a
specified period. The present value of an annuity is calculated using the
formula 𝑃𝑉=𝑃𝑀𝑇×[(1−(1+𝑟)−𝑛)/𝑟]PV=PMT×[(1−(1+r)−n)/r], where 𝑃𝑉PV is
the present value, 𝑃𝑀𝑇PMT is the annuity payment, 𝑟r is the interest rate
per period, and 𝑛n is the number of periods.
Loans and Borrowing: Loan calculations involve determining the total
repayment amount over the loan term, including principal and interest
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