Grade 12
  1. Algebra  1.1. Matrices  1.1.1. Definition and Types of Matrices  1.1.2. Operations on Matrices  1.1.3. Transpose of a matrix  1.1.4. Determinants and their properties  1.1.5. Inverse of a Matrix  1.1.6. Solving Linear Equations Using Matrices  1.1.7. Applications of Matrices  1.2. Complex Numbers  1.2.1. Definition and representation  1.2.2. Algebra of complex numbers  1.2.3. Modulus and argument  1.2.4. Polar and Exponential Forms  1.2.5. De Moivre's Theorem  1.2.6. Roots of complex numbers  1.2.7. Applications in Geometry  1.3. Vectors and 3D Geometry  1.3.1. Vector Algebra  1.3.2. Scalar and Vector Products  1.3.3. Equation of a Line in Space  1.3.4. Equation of a Plane  1.3.5. Distance between lines and planes  1.3.6. Angle between Lines and Planes
  2. Calculus  2.1. Limits and Continuity  2.1.1. Concept of Limits  2.1.2. Continuity of Functions  2.1.3. Types of Discontinuities  2.1.4. Left-hand and Right-hand Limits  2.2. Differentiation  2.2.1. Derivative as a Rate of Change  2.2.2. Techniques of Differentiation  2.2.3. Higher Order Derivatives  2.2.4. Applications of Derivatives  2.2.5. Tangents and Normals  2.2.6. Increasing and Decreasing Functions  2.3. Integration  2.3.1. Indefinite Integrals  2.3.2. Definite Integrals  2.3.3. Integration Techniques  2.3.4. Area under a curve  2.3.5. Applications of Integrals in Geometry  2.4. Differential Equations  2.4.1. Formation of Differential Equations  2.4.2. Order and Degree of Differential Equations  2.4.3. Solutions of Differential Equations  2.4.4. Applications of Differential Equations  2.4.5. Solving First-Order Linear Differential Equations
  3. Probability and Statistics  3.1. Probability  3.1.1. Conditional Probability  3.1.2. Bayes' Theorem  3.1.3. Random Variables  3.1.4. Probability Distributions  3.1.5. Binomial and Normal Distributions  3.2. Statistics  3.2.1. Mean and Standard Deviation  3.2.2. Variance and Covariance  3.2.3. Correlation and Regression  3.2.4. Hypothesis Testing
  4. Linear Programming  4.1. Graphical Method  4.1.1. Feasible and Infeasible Regions  4.1.2. Optimal solutions  4.1.3. Sensitivity Analysis in Graphical Method  4.2. Simplex Method  4.2.1. Formulation of Problems  4.2.2. Solving linear programming problems  4.2.3. Dual Simplex Method
  5. Relations and Functions  5.1. Types of Relations  5.1.1. Reflexive Relations  5.1.2. Symmetric Relations  5.1.3. Transitive Relations  5.1.4. Equivalence Relations  5.2. Types of Functions  5.2.1. Onto Functions  5.2.2. One-to-One Functions  5.2.3. Inverse Functions  5.2.4. Composite Functions  5.3. Inverse Trigonometric Functions  5.3.1. Principal Values in Inverse Trigonometric Functions  5.3.2. Properties and Graphs of Inverse Trigonometric Functions  5.3.3. Applications in Calculus
  6. Advanced Trigonometry  6.1. Trigonometric Equations  6.1.1. General solutions of equations  6.1.2. Transformation formulae  6.2. Hyperbolic Functions  6.2.1. Definitions and properties  6.2.2. Relationships between hyperbolic and trigonometric functions
  7. Mathematical Reasoning  7.1. Logical Reasoning  7.1.1. Statements and Logical Connectives  7.1.2. Tautologies and Contradictions  7.2. Proofs  7.2.1. Direct proof  7.2.2. Indirect Proof  7.2.3. Proof by contradiction  7.2.4. Proof by Mathematical Induction
  8. Applications of Mathematics  8.1. Applications of Calculus  8.1.1. Maxima and Minima Problems  8.1.2. Economics and Biology Applications  8.1.3. Rate of Change in Physics  8.2. Applications of Probability  8.2.1. Risk Analysis  8.2.2. Decision Making  8.2.3. Game Theory

Grade 12 ↓


Applications of Mathematics


Mathematics is not just abstract numbers and theories. It is a powerful tool that applies to many aspects of our daily lives, scientific advancements, and engineering marvels. When you understand how to apply mathematics to real-world problems, you discover the language that helps you communicate and solve essential problems in a variety of fields.

1. Finance and economics

In the world of finance and economics, mathematics plays an important role in almost everything. Concepts such as interest rates, loan calculations, investment strategies, and risk assessment are all based on mathematics.

Compound interest

Compound interest is one of the most powerful concepts in finance. The idea is that you earn interest not only on the initial amount of money but also on the interest that accumulates over time.

A = P(1 + r/n)^(nt)

Where:

  • A = future value of the investment/loan, including interest
  • P = original investment amount (initial deposit or loan amount)
  • r = annual interest rate (decimal)
  • n = number of times the interest is compounded per year
  • t = number of years for which the money is invested or borrowed

For example, if you invest $1,000 at a 5% annual compound interest rate, how much money will you have after 3 years?

A = 1000(1 + 0.05/1)^(1*3) = 1157.63

2. Engineering

Engineering is full of mathematical applications, ranging from simple calculations to complex algorithms. Mathematics helps create precise designs and ensure the structural integrity of buildings and bridges.

Structural analysis

An example of this is the use of mathematics in structural analysis, where engineers have to determine the strength and durability of a construction.

Consider a simple beam that can support a weight. The maximum load it can support without failure can be calculated using the stress and strain formulas:

Stress (σ) = Force (F) / Area (A) Strain (ε) = Change in Length (ΔL) / Original Length (L)
beam

3. Medicine

In medicine, math helps with everything from calculating drug dosages to interpreting statistical data in clinical trials. It ensures accuracy in diagnosis and the effectiveness of treatment plans.

Calculation of dosage

Calculating medication dosage is important for patient safety. Doctors and pharmacists use a formula to determine how much medication a patient should take:

Dose = (Desired Concentration × Volume) / Potency

For example, if a doctor prescribes a medication with a concentration of 50 mg/mL and the volume needed is 5 mL, the potency of the medication may mean that the patient needs more or less than initially thought.

4. Environmental science

Mathematics is an important part of environmental science. It is used to model and analyze environmental systems and the interactions between ecosystem components.

Population dynamics

Mathematical models can predict how a population changes over time. For example, the logistic growth model can simulate population growth:

P(t) = K / (1 + (K-P₀)/P₀)e^(-rt)

Where:

  • P(t) = population at time t
  • K = carrying capacity, or maximum population size
  • P₀ = initial population size
  • r = growth rate
  • t = time

This model takes into account the slowing down of growth when resources become limited.

5. Computer science

Computers are based on mathematical principles. Algorithms, data structures, and encryption all require a deep understanding of mathematics.

Encryption

Modern encryption methods use complex mathematical algorithms to secure data. A common method is RSA encryption which is based on the difficulty of factoring large numbers.

The RSA algorithm consists of keys - a public key for encrypting messages and a private key for decrypting them:

public key: (e, n) private key: (d, n)

6. Statistics in daily life

We encounter statistics in our daily lives, from the probability of rain to election predictions. Understanding how to interpret statistical data helps make informed decisions.

Understanding probability

Probability can predict the likelihood of an event occurring. For example, the probability of throwing a six-sided dice is:

P(rolling a 3) = 1/6

Understanding probability helps in assessing risks and making prospective decisions.

Conclusion

The applications of mathematics are far-reaching and deeply embedded in many fields. Whether designing bridges, calculating financial interest, or securing data through encryption, mathematics provides the tools and language necessary to understand and shape the world around us. The versatility of mathematics enables it to make profound contributions to our scientific understanding, technological advancement, and everyday decision-making.


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