Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 ↓


Coordinate Geometry


Coordinate geometry, also known as analytical geometry, is a fascinating branch of mathematics that connects algebra with geometry. It allows us to use algebra to solve geometric problems and vice versa. This field uses a coordinate system to represent geometric shapes and enables us to study their properties using algebraic methods.

Cartesian coordinate system

The Cartesian coordinate system is the foundation of coordinate geometry. It is named after the French mathematician René Descartes. This system consists of two number lines that cross each other at right angles. These lines are called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. The point where they intersect is called the origin, denoted by (0, 0).

(0, 0) (11) X-axis Shaft

Point in the coordinate plane

Every point in the plane can be represented by an ordered pair of numbers (x, y). x value represents the horizontal position of the point, while the y value represents its vertical position. For example, the point (3, 4) is 3 units to the right of the origin and 4 units above the origin.

Distance between two points

The distance between two points in a plane can be calculated using the distance formula. If you have two points A (x1, y1) and B (x2, y2), then the distance d between them is:

d = √((x2 - x1)² + (y2 - y1)²)

Suppose you want to find the distance between the points (3, 4) and (7, 1). Applying the formula:

d = √((7 - 3)² + (1 - 4)²) = √(16 + 9) = √25 = 5

Hence, the distance between the two points is 5 units.

Midpoint of a line segment

The midpoint of a line segment is the point that divides the line segment into two equal parts. If you have a line segment connecting points A (x1, y1) and B (x2, y2), then the midpoint M (x, y) is given by:

x = (x1 + x2) / 2 y = (y1 + y2) / 2

For example, the midpoint of the line segment connecting the points (2, 3) and (4, 7) is:

x = (2 + 4) / 2 = 3 y = (3 + 7) / 2 = 5

Hence the midpoint is (3, 5).

Slope of the line

The slope of a line measures its steepness and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Given two points A (x1, y1) and B (x2, y2), the slope of the line connecting them is m :

m = (y2 - y1) / (x2 - x1)

Consider the points (1, 2) and (3, 6), the slope of the line is:

m = (6 - 2) / (3 - 1) = 4 / 2 = 2

This means that the line rises 2 units for every 1 unit it moves to the right.

Equation of a straight line

The equation of a straight line can take many forms, but the most familiar form is the slope-intercept form, which is expressed as:

y = mx + b

Here, m is the slope of the line, and b is the y-intercept, which represents the point where the line crosses the y-axis.

For a given slope m = 2 and y-intercept b = -3, the equation of the line is:

y = 2x - 3
(0, 0) y = 2x – 3

Forms of linear equations

Besides the slope-intercept form, there are other forms of the linear equation:

1. Point-slope form

This form is useful when you know a point on the line and its slope. Its formula is:

y - y1 = m(x - x1)

2. Standard form

The standard form of the line equation is:

Ax + By = C

Here, A, B and C are integers, and A ≥ 0.

Parallel and perpendicular lines

In coordinate geometry, the concepts of parallel and perpendicular lines are important:

Parallel lines

Parallel lines have the same slope. So, if two lines are parallel, their equations will be as follows:

Line 1: y = m1x + b1 Line 2: y = m2x + b2

where m1 = m2.

Perpendicular lines

The slopes of perpendicular lines are the negative reciprocals of each other. If the slope of a line is m, then the slope of a line perpendicular to it will be -1/m.

For example, if the slope of a line is 2, then the slope of a line perpendicular to it will be -1/2.

Circles in the coordinate plane

The equation of a circle in the coordinate plane is given by its center and radius. With center at (h, k) and radius r, the general equation of a circle is:

(x - h)² + (y - k)² = r²

The equation for a circle with centre (2, 3) and radius 5 becomes:

(x - 2)² + (y - 3)² = 25

Applications of coordinate geometry

Coordinate geometry is used extensively in various fields such as physics, engineering, computer graphics, and game development. Its ability to provide a clear link between algebra and geometry makes it extremely useful in solving complex mathematical problems.

For example, in physics, it is used to determine the trajectories of moving objects. In computer graphics, coordinate geometry allows the rendering of visual scenes on a computer screen, helping to create realistic graphics in video games and simulations.

In addition, coordinate geometry is essential in navigation systems, such as GPS, where it is used to calculate distances between geographic locations, and in architecture and city planning, where accurate measurements and location determination are important.

Conclusion

Coordinate geometry provides a powerful framework for analyzing and solving geometric problems with algebraic methods. Its fundamental concepts, such as points, lines, slopes, distances, and properties of geometric shapes such as circles, provide a thorough understanding of spatial relationships and mathematical calculations needed to excel in mathematics.

By mastering coordinate geometry, you gain valuable skills applicable in both academic and real-world contexts, increasing your ability to solve practical problems with confidence and accuracy.


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