Welcome to the ultimate revision guide for SS3 Mathematics, focusing on the key topics covered in the second term. This article provides a thorough and easy-to-understand breakdown of the major concepts, techniques, and applications to ensure you’re well-prepared for your examination. Whether you’re revisiting material or learning it for the first time, this guide will give you the clarity needed to master the subject.
Week 1: Review of First Term Work
1. Bonds and Debentures: Bonds and debentures are types of financial instruments used by companies and governments to raise capital.
- Bonds: Issued by governments or corporations and typically have a fixed interest rate and maturity date.
- Debentures: Similar to bonds but are unsecured, meaning they are not backed by any asset.
Key Points to Note:
- Interest Rate: The percentage paid by the issuer to the bondholder.
- Maturity: The date when the bond is due to be repaid.
2. Shares: Shares represent ownership in a company. When you buy shares, you become a shareholder and have a claim on the company’s profits.
Types of Shares:
- Ordinary Shares: Gives voting rights and dividends.
- Preference Shares: Pay fixed dividends but don’t offer voting rights.
3. Rates: This refers to the percentage used to calculate the tax payable or other financial metrics such as interest on loans.
4. Income Tax: Income tax is a tax levied on an individual’s or entity’s earnings. The rate at which income tax is charged depends on the income bracket you fall under.
5. Value Added Tax (VAT): VAT is a tax levied on goods and services at each stage of production and distribution.
Week 2: Coordinate Geometry of a Straight Line
1. Cartesian Coordinates: A coordinate system that specifies each point uniquely in a plane using a pair of numerical coordinates.
2. Plotting Linear Graphs: To plot a linear graph, you need at least two points. Once you have these, you can draw a straight line through them. The graph of a linear equation is always a straight line.
3. Determining the Distance Between Two Coordinate Points: To find the distance between two points (x₁, y₁) and (x₂, y₂) on a Cartesian plane, use the distance formula:
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x₂ – x₁)² + (y₂ – y₁)²}
4. Finding the Midpoint of the Line Joining Two Points: The midpoint formula helps find the point that lies exactly halfway between two coordinates (x₁, y₁) and (x₂, y₂):
Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2} \right)
5. Practical Application of Coordinate Geometry: Coordinate geometry is used in real-world problems such as finding distances, determining locations, or calculating areas.
6. Gradient and Intercept of a Straight Line:
- Gradient: The slope of the line, showing how steep the line is.
Gradient=y2−y1x2−x1\text{Gradient} = \frac{y₂ – y₁}{x₂ – x₁}
- Intercept: The point where the line crosses the y-axis. This can be found using the equation y=mx+cy = mx + c, where cc is the intercept.
Week 3: Coordinate Geometry of a Straight Line (Continued)
1. Gradient and Intercepts: Understanding the gradient and y-intercept of a straight line is crucial for analyzing the behavior of graphs.
2. Angle Between Two Intersecting Lines: The angle between two lines can be calculated using the formula:
tanθ=∣m1−m21+m1m2∣\tan \theta = \left| \frac{m₁ – m₂}{1 + m₁ m₂} \right|
Where m1m₁ and m2m₂ are the gradients of the two lines.
3. Application of Linear Graphs to Real-Life Situations: Linear graphs are used in various fields, such as economics, physics, and engineering, to represent relationships between variables.
Week 4: Differentiation of Algebraic Functions
1. Meaning of Differentiation/Derived Function: Differentiation is the process of finding the rate of change of a function with respect to a variable. The result is called the derivative.
2. Differentiation from First Principles: The first principle of differentiation is based on the limit process. It’s used to define the derivative of a function:
f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}
3. Standard Derivatives of Basic Functions:
- The derivative of xnx^n is nxn−1nx^{n-1}.
- The derivative of exe^x is exe^x.
- The derivative of sin(x)\sin(x) is cos(x)\cos(x).
Week 5: Differentiation of Algebraic Functions (Continued)
1. Rules of Differentiation:
- Sum and Difference Rule: The derivative of the sum (or difference) of two functions is the sum (or difference) of their derivatives.
ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)
- Product Rule: The derivative of the product of two functions is:
ddx[f(x)⋅g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)
- Quotient Rule: The derivative of the quotient of two functions is:
ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}
2. Applications of Differentiation:
- Maxima and Minima: Used to find the maximum or minimum values of functions, often in optimization problems.
- Velocity and Acceleration: The derivative of position with respect to time gives velocity, and the derivative of velocity gives acceleration.
- Rate of Change: Differentiation is often used to find rates of change in various contexts.
Week 6: Integration of Simple Algebraic Functions
1. Definition of Integration: Integration is the reverse process of differentiation. It is used to find the area under a curve.
2. Methods of Integration:
- Substitution Method: Used when an integral contains a function and its derivative.
- Partial Fraction Method: Used to integrate rational functions by expressing them as the sum of simpler fractions.
- Integration by Parts: Based on the product rule of differentiation.
3. Application of Integration in Calculating Area Under the Curve: To find the area under a curve, integrate the function that represents the curve between the given limits.
4. Simpson’s Rule for Estimating the Area: Simpson’s Rule is a numerical method for estimating the area under a curve. It is used when exact integration is difficult or impossible.
Likely Questions for Each Topic:
Topic 1: Bonds and Debentures
- What is the difference between bonds and debentures?
- How is the interest rate on bonds calculated?
- What are the advantages of issuing debentures over shares?
- What does a fixed maturity date imply for bonds?
- Explain the concept of a preference share.
- What is income tax and how is it calculated?
- Define VAT and explain how it works.
- How are shares traded on the stock market?
- What factors influence the rates of income tax?
- Describe a scenario where bonds would be more beneficial than shares.
Topic 2: Coordinate Geometry of a Straight Line
- What is the Cartesian coordinate system?
- How do you plot a linear graph?
- Explain how to calculate the distance between two points.
- What is the midpoint formula and how is it used?
- Define the gradient of a straight line.
- How do you calculate the intercept of a line?
- What is the formula for finding the angle between two intersecting lines?
- Provide a real-life example of coordinate geometry.
- How do you calculate the gradient between two points?
- Explain the concept of the practical application of coordinate geometry.
Topic 3: Differentiation of Algebraic Functions
- What is the derivative of a function?
- How do you differentiate using the first principle?
- What is the standard derivative of xnx^n?
- What is the product rule in differentiation?
- How do you apply the quotient rule in differentiation?
- What are the applications of differentiation in real-life situations?
- Define maxima and minima in the context of differentiation.
- How do you differentiate trigonometric functions like sin(x)\sin(x)?
- What is the meaning of velocity and acceleration in terms of differentiation?
- How is differentiation used to find rates of change?
Topic 4: Integration of Algebraic Functions
- What is integration and how is it related to differentiation?
- Explain the substitution method in integration.
- What is partial fraction decomposition in integration?
- How do you use Simpson’s rule to estimate areas under curves?
- What are the different methods of solving integrals?
- How do you calculate the area under a curve?
- What is the application of integration in physics?
- Define integration by parts and give an example.
- How is integration used in calculating the volume of solids?
- Provide an example of a real-life problem solved using integration.
Conclusion:
This comprehensive guide should provide clarity on key mathematical concepts for SS3 students preparing for their second term examination. Practice the questions provided, and ensure you understand the methods of solving each problem. With consistent effort and the right approach, you’ll be well-equipped to tackle your exams.