Formula Sheet For Igcse Maths

The Ultimate IGCSE Maths Formula Sheet: Your Guide to Success

The IGCSE Maths exam can feel daunting, but with the right preparation and resources, you can conquer it! One crucial element of success is mastering the key formulas and knowing when to apply them. This comprehensive formula sheet covers the essential formulas for IGCSE Maths, providing explanations and examples to help you understand and apply them effectively. This article aims to be your ultimate guide, equipping you with not just a list of formulas, but also a deeper understanding to boost your confidence and improve your problem-solving skills.

Number and Algebra

This section covers the core algebraic concepts and numerical manipulations crucial for IGCSE Maths.

1. Number Properties & Operations

• Order of Operations (BODMAS/PEMDAS): Remember the order of operations: Brackets, Orders (exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This ensures consistent calculation results. For example, 2 + 3 × 4 = 14, not 20.

• Prime Factorization: Expressing a number as a product of its prime factors. For example, the prime factorization of 12 is 2 x 2 x 3 (or 2² x 3). This is essential for finding the highest common factor (HCF) and lowest common multiple (LCM).

• HCF and LCM: The highest common factor is the largest number that divides two or more numbers without a remainder. The lowest common multiple is the smallest number that is a multiple of two or more numbers. Methods for finding these include prime factorization or listing multiples.

• Indices (Exponents): Understanding rules for manipulating indices is crucial:

  • a^m × aⁿ = a^m+n
  • a^m ÷ aⁿ = a^m-n
  • (a^m)ⁿ = a^mn
  • a⁰ = 1
  • a^-n = 1/aⁿ
  • a^1/n = ⁿ√a

• Standard Form (Scientific Notation): Expressing numbers in the form a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This is useful for handling very large or very small numbers.

• Surds: Numbers that cannot be expressed as a simple fraction. Simplifying surds involves removing perfect square factors from under the square root. For example, √12 = √(4 x 3) = 2√3.

  • √a × √b = √(ab)
  • √a / √b = √(a/b)
  • (√a)² = a

2. Algebraic Manipulation

• Expanding Brackets: Removing brackets by multiplying each term inside the bracket by the term outside. For example, 3(x + 2) = 3x + 6.

• Factorization: Expressing an algebraic expression as a product of simpler expressions. Common methods include taking out a common factor, difference of two squares (a² - b² = (a + b)(a - b)), and quadratic factorization.

• Solving Linear Equations: Finding the value of the unknown variable that makes the equation true. This involves using inverse operations to isolate the variable.

• Simultaneous Equations: Solving two or more equations simultaneously to find the values of two or more unknowns. Methods include elimination and substitution.

• Quadratic Equations: Equations of the form ax² + bx + c = 0. Solutions can be found using factorization, completing the square, or the quadratic formula:

  • x = [-b ± √(b² - 4ac)] / 2a

• Inequalities: Expressions that show the relative size of two values using symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to). Solving inequalities involves similar techniques to solving equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.

Geometry and Measurement

This section deals with shapes, their properties, and measurements.

1. Mensuration

• Area of a rectangle: Area = length × width
• Area of a triangle: Area = ½ × base × height
• Area of a parallelogram: Area = base × height
• Area of a trapezium: Area = ½ × (sum of parallel sides) × height
• Area of a circle: Area = πr²
• Circumference of a circle: Circumference = 2πr or Circumference = πd
• Volume of a cuboid: Volume = length × width × height
• Volume of a cylinder: Volume = πr²h
• Volume of a sphere: Volume = (4/3)πr³
• Surface area of a cuboid: Surface area = 2(lw + lh + wh)
• Surface area of a cylinder: Surface area = 2πr² + 2πrh
• Surface area of a sphere: Surface area = 4πr²

2. Geometry

• Pythagoras' Theorem: In a right-angled triangle, a² + b² = c², where a and b are the lengths of the shorter sides (legs) and c is the length of the hypotenuse (longest side).

• Trigonometry (SOH CAH TOA):

  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent

• Similar Triangles: Triangles with the same angles but different sizes. Corresponding sides are proportional.

• Congruent Triangles: Triangles that are identical in size and shape.

• Properties of Shapes: Understand the properties of various shapes, including angles, sides, and symmetry (e.g., squares, rectangles, triangles, circles, parallelograms, etc.).

Vectors and Matrices

This section deals with vector quantities and matrix operations.

1. Vectors

• Vector Addition: Add vectors by adding their components.
• Vector Subtraction: Subtract vectors by subtracting their components.
• Scalar Multiplication: Multiply a vector by a scalar by multiplying each component by the scalar.
• Magnitude of a Vector: The length of the vector, calculated using Pythagoras' theorem.
• Unit Vectors: Vectors with a magnitude of 1.

2. Matrices

• Matrix Addition and Subtraction: Add or subtract matrices by adding or subtracting corresponding elements.
• Scalar Multiplication: Multiply a matrix by a scalar by multiplying each element by the scalar.
• Matrix Multiplication: Multiplying matrices involves a specific rule for multiplying rows and columns. The number of columns in the first matrix must equal the number of rows in the second matrix.
• Determinant of a 2x2 Matrix: For matrix [[a, b], [c, d]], the determinant is ad - bc.
• Inverse of a 2x2 Matrix: The inverse of a matrix A is denoted by A^-1. For a 2x2 matrix, the inverse can be calculated using the determinant.

Statistics and Probability

This section involves analyzing data and calculating probabilities.

1. Statistics

• Mean: The average of a set of numbers.
• Median: The middle value in a set of numbers when they are arranged in order.
• Mode: The most frequent value in a set of numbers.
• Range: The difference between the largest and smallest values in a set of numbers.
• Frequency Tables: Tables that show how often each value occurs.
• Histograms: Graphical representations of frequency distributions.
• Cumulative Frequency: The running total of frequencies.
• Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1).

2. Probability

• Probability: The likelihood of an event occurring. Probability = (Number of favorable outcomes) / (Total number of possible outcomes).
• Independent Events: Events where the outcome of one event does not affect the outcome of another.
• Dependent Events: Events where the outcome of one event does affect the outcome of another.
• Mutually Exclusive Events: Events that cannot occur at the same time.

Calculus (If Applicable to Your Syllabus)

Depending on your specific IGCSE Maths syllabus, you may also need to know some basic calculus concepts.

• Differentiation: Finding the rate of change of a function.
• Integration: Finding the area under a curve.

Tips for Using Your IGCSE Maths Formula Sheet Effectively

• Understanding, not Memorization: Don't just memorize the formulas; understand what they represent and how they are derived. This will help you apply them correctly in different contexts.

• Practice Regularly: The best way to master these formulas is through consistent practice. Solve numerous problems, starting with simpler ones and gradually increasing the difficulty level.

• Identify Your Weak Areas: As you practice, identify the formulas and concepts you find most challenging. Focus your efforts on improving your understanding of these areas.

• Create Your Own Formula Sheet: Consider creating your own concise formula sheet from this extensive one. This process will reinforce your understanding and make it easier to access during the exam.

• Use Real-World Examples: Try to relate the formulas to real-world situations. This can make them more memorable and easier to apply.

• Seek Help When Needed: Don't hesitate to seek help from your teacher, tutor, or classmates if you encounter difficulties.

This comprehensive formula sheet and guide provide a solid foundation for your IGCSE Maths preparation. Remember that consistent effort and a deep understanding of the concepts are key to success. Good luck!