Mathematics (0580) Core Compact cheat sheet

    Mathematics (0580) · CAIE · Core

    Compact
    5 pages
    17 formulas, 25 concepts
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    All Mathematics (0580) Core cheat sheets

    Number

    • Percentage Change
      changeoriginal×100%\frac{\text{change}}{\text{original}} \times 100\%
      Use to find the percentage increase or decrease between two values.
    • Simple Interest
      I=PRT100I = \frac{PRT}{100}
      Calculate interest (I) given the Principal (P), Rate (R) and Time (T).

    Key concepts: **Prime Numbers**: A number greater than 1 with exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11)., **Rational vs Irrational**: Rational numbers can be written as a fraction (e.g., 0.5, 7, 1/3). Irrational numbers cannot (e.g., $\pi$, $\sqrt{2}$)., **Set Notation**: $\cap$ is intersection (elements in both sets), $\cup$ is union (elements in either set), and $\in$ means 'is an element of'.

    Exam tips

    • When using a calculator for multi-step problems, use the full answer from the previous step to avoid rounding errors.

    Algebra and graphs

    • Rules of Indices
      xa×xb=xa+bx^a \times x^b = x^{a+b}
      When multiplying powers with the same base, add the indices.
    • Nth Term of a Linear Sequence
      a+(n−1)da + (n-1)d
      Find any term in a sequence where 'a' is the first term and 'd' is the common difference.

    Key concepts: **Simplifying Expressions**: Collect 'like terms' (terms with the same variable and power) by adding or subtracting their coefficients., **Solving Linear Equations**: Isolate the unknown variable by performing the same inverse operation on both sides of the equation., **Factorising**: The reverse of expanding brackets; find the highest common factor (HCF) of all terms and place it outside the bracket.

    Exam tips

    • When solving simultaneous equations, always substitute your solutions back into the original equations to check your answer.

    Coordinate geometry

    • Gradient of a Line
      m=riserun=y2−y1x2−x1m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
      Calculates the steepness of a line passing through two points.
    • Equation of a Straight Line
      y=mx+cy = mx + c
      Represents a straight line where 'm' is the gradient and 'c' is the y-intercept.

    Key concepts: **Gradient (m)**: Measures the steepness. Positive gradient slopes up from left to right, negative slopes down., **Y-intercept (c)**: The point where the line crosses the y-axis. The coordinates are (0, c)., **Parallel Lines**: Lines that never intersect and have the exact same gradient (m).

    Exam tips

    • Be careful with negative signs when calculating the gradient. A common mistake is mixing up the order of coordinates.

    Geometry

    • Sum of Interior Angles
      (n−2)×180∘(n-2) \times 180^\circ
      Find the sum of all interior angles in any polygon with 'n' sides.

    Key concepts: **Angle Properties**: Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal., **Parallel Line Angles**: Corresponding angles are equal ('F' shape), alternate angles are equal ('Z' shape), and co-interior angles sum to 180° ('C' shape)., **Bearings**: Measured clockwise from North and always written as a 3-figure number (e.g., 045°).

    Exam tips

    • If a question asks you to 'give a reason' for your angle calculation, you must state the geometric rule you used.

    Mensuration

    • Area of a Trapezium
      A=12(a+b)hA = \frac{1}{2}(a+b)h
      Find the area of a trapezium where 'a' and 'b' are the parallel sides and 'h' is the height.
    • Circumference of a Circle
      C=2πrC = 2\pi r or C=πdC = \pi d
      Calculate the distance around the outside of a circle using radius 'r' or diameter 'd'.
    • Volume of a Cylinder
      V=πr2hV = \pi r^2 h
      Calculates the space inside a cylinder using the radius 'r' and height 'h'.

    Key concepts: **Perimeter**: The total distance around the outside of a 2D shape., **Area**: The amount of space inside a 2D shape, measured in square units (e.g., cm²)., **Volume**: The amount of space a 3D object occupies, measured in cubic units (e.g., m³).

    Exam tips

    • Always check if a question gives the radius or the diameter of a circle. Using the wrong one is a very common mistake.

    Trigonometry

    • Pythagoras' Theorem
      a2+b2=c2a^2 + b^2 = c^2
      Find a missing side in a right-angled triangle when two sides are known. 'c' is always the hypotenuse.
    • SOH CAH TOA
      sin⁡(x)=OH,cos⁡(x)=AH,tan⁡(x)=OA\sin(x) = \frac{O}{H}, \cos(x) = \frac{A}{H}, \tan(x) = \frac{O}{A}
      Use to find a missing side or angle in a right-angled triangle.

    Key concepts: **Right-Angled Triangle**: A triangle with one 90° angle. The longest side, opposite the right angle, is the hypotenuse., **Labelling Sides**: Identify the Hypotenuse (opposite right angle), Opposite (opposite the angle of interest), and Adjacent (next to the angle of interest).

    Exam tips

    • Before starting any trigonometry questions, make sure your calculator is in Degrees (DEG) mode.

    Transformations and vectors

    • Translation Vector
      (xy)\begin{pmatrix} x \\ y \end{pmatrix}
      Describes a translation. The top number is horizontal movement (right is +) and the bottom is vertical (up is +).

    Key concepts: **Reflection**: A mirror image of a shape across a given mirror line (e.g., y-axis, x=2)., **Rotation**: A turn described by a centre of rotation, an angle (e.g., 90°), and a direction (clockwise or anticlockwise)., **Enlargement**: Changes the size of a shape. Described by a centre of enlargement and a scale factor.

    Exam tips

    • For rotations, it can be helpful to use tracing paper to perform the rotation and then mark the new position on the grid.

    Probability

    • Probability of an Event
      P(A)=Number of successful outcomesTotal number of possible outcomesP(A) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}
      Calculates the likelihood of a single event occurring.
    • Complementary Events
      P(not A)=1−P(A)P(\text{not } A) = 1 - P(A)
      The probability of an event not happening is 1 minus the probability that it does happen.
    • Expected Frequency
      Probability×Number of trials\text{Probability} \times \text{Number of trials}
      Estimates how many times an event is expected to occur over a number of trials.

    Key concepts: **Probability Scale**: Probability is always a value between 0 (impossible) and 1 (certain)., **Relative Frequency**: An estimate of probability based on experimental results: $\frac{\text{number of times event occurs}}{\text{total number of trials}}$.

    Exam tips

    • Probability answers can be fractions, decimals, or percentages. Always simplify fractions unless the question says otherwise.

    Statistics

    • Mean
      Mean=Sum of all valuesNumber of values\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
      Calculates the mean average of a set of discrete data.

    Key concepts: **Averages**: Mean (sum ÷ count), Median (middle value of ordered data), and Mode (most frequent value)., **Range**: A measure of spread, calculated as the highest value minus the lowest value., **Correlation**: Describes the relationship between two variables on a scatter diagram (positive, negative, or no correlation).

    Exam tips

    • To find the median, you MUST arrange the data in order of size first. Forgetting this step is a very common error.