Basic Mathematics Form 4

Indice

1. 1. Coordinate Geometry

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   • Gradient and the distance between two points
   • The midpoint of a line segment
   • Equations of straight lines
   • Parallel and perpendicular lines
   • Collinear points, intersections and shapes
   • Summary

   Review In Form One you learned to plot points, find gradients and write the equation of a straight line in the form y = mx + c. In Form Four we add the distance formula , the midpoint formula , and the conditions for lines to be parallel or perpendicular . Distance between two points The distance between A(x 1 , y 1 )…

2. 2. Areas and Volumes

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   • Area of a triangle using sine
   • Area of regular polygons
   • Prisms and cylinders
   • Pyramids and cones
   • Spheres, composite solids and recasting
   • Summary

   Area of a triangle using sine When two sides a and b and the angle C between them are known, the area of a triangle is Area = ½ ab sin C For example, with sides 10 cm and 12 cm and an included angle of 30°: area = ½ × 10 × 12 × 0.5 = 30 cm 2 . A regular polygon with n sides can be split into n congruent isosceles…

3. 3. Three-Dimensional Figures

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   • Solids, Euler's formula and nets
   • Lengths inside solids
   • The angle between a line and a plane
   • The angle between two planes
   • Summary

   Solids and their parts A three-dimensional (3-D) figure has length, width and height. A solid bounded by flat polygon faces is a polyhedron ; examples are the cube, cuboid, prism and pyramid. Cylinders, cones and spheres have curved surfaces. Each polyhedron has faces (F), edges (E) where faces meet, and vertices (V)…

4. 4. Probability

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   • The language and calculation of probability
   • Experimental probability and expected numbers
   • Combined events: mutually exclusive and independent
   • Tree diagrams
   • Summary

   What is probability? Probability measures how likely an event is to happen. It is a number from 0 to 1. An impossible event has probability 0, a certain event has probability 1, and an event that is equally likely to happen or not has probability 1/2. Key terms Experiment : an action with uncertain results, such as…

5. 5. Trigonometry

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   • Trigonometric ratios of angles from 0° to 360°
   • Graphs of sine and cosine; simple equations
   • The sine rule
   • The cosine rule, bearings and areas
   • Compound angle formulas
   • Summary

   Angles beyond 90° Using a unit circle, sine, cosine and tangent can be defined for any angle from 0° to 360°. The signs depend on the quadrant : 1st quadrant (0°–90°): all ratios positive. 2nd quadrant (90°–180°): only sin positive. sin θ = sin (180° − θ), cos θ = −cos (180° − θ). 3rd quadrant (180°–270°): only tan…

6. 6. Vectors

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   • Scalars, vectors and their notation
   • Magnitude, direction and unit vectors
   • Adding, subtracting and multiplying vectors
   • Position vectors, midpoints and proofs
   • Applications: velocities and forces
   • Summary

   Scalars and vectors A scalar is a quantity that has size (magnitude) only, such as mass, time, temperature or distance. A vector has both magnitude and direction , such as displacement, velocity, force and acceleration. "5 km" is a distance (scalar); "5 km on a bearing of 060°" is a displacement (vector). Writing…

7. 7. Matrices and Transformations

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   • Matrices: order, addition and scalar multiplication
   • Multiplying matrices
   • The determinant and the inverse
   • Solving simultaneous equations with matrices
   • Matrices and transformations
   • Summary

   What is a matrix? A matrix is a rectangular arrangement of numbers in rows and columns. A matrix with m rows and n columns has order m × n. In this book a 2 × 2 matrix is written on one line as [a b; c d], where the semicolon separates the first row (a b) from the second row (c d). For example, [2 1; 3 4] has first row…

8. 8. Linear Programming

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   • Linear programming: key ideas and forming constraints
   • Graphing constraints: the feasible region
   • Finding the optimum: vertices and search lines
   • Solving linear programming problems
   • Summary

   What is linear programming? Linear programming is a method for finding the best possible result, such as the maximum profit or the minimum cost , when there are limits (constraints) on resources like money, time, land or raw materials. The limits and the quantity to be optimised are all linear (no squares or products…