Mathematics (0580) Extended Compact cheat sheet

    Mathematics (0580) · CAIE · Extended

    Compact
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    27 formulas, 27 concepts
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    All Mathematics (0580) Extended cheat sheets

    Number

    • Exponential Growth/Decay
      A=P(1±r100)nA = P(1 \pm \frac{r}{100})^n
      Calculates the final amount after n periods of percentage increase/decrease.
    • Upper/Lower Bound Calculation
      Upper Bound of A/B=UB(A)LB(B)\text{Upper Bound of } A/B = \frac{\text{UB}(A)}{\text{LB}(B)}
      To find the maximum value of a division, use the upper bound of the numerator and lower bound of the denominator.
    • Standard Form
      A×10nA \times 10^n
      A compact way to write very large or small numbers, where 1 ≤ A < 10 and n is an integer.

    Key concepts: **Limits of Accuracy (Bounds)**: The upper and lower bounds of a number rounded to a certain degree of accuracy are found by adding/subtracting half of that accuracy. For a number 'x' rounded to the nearest 'a', the interval is [x - a/2, x + a/2)., **Set Notation**: Understand symbols like ∈ (is an element of), ⊂ (is a subset of), ∪ (union), ∩ (intersection), and A' (complement of A)., **Surds**: An irrational number left in square root form. Simplify using rules like √ab = √a × √b and rationalize the denominator by multiplying the numerator and denominator by the surd.

    Exam tips

    • For bounds questions, always determine the Upper Bound (UB) and Lower Bound (LB) of each individual measurement first before performing any calculations.

    Algebra and graphs

    • Quadratic Formula
      x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
      Solves any quadratic equation in the form ax² + bx + c = 0.
    • Gradient of a Curve (Differentiation)
      If y=axn, then dydx=anxn−1\text{If } y = ax^n, \text{ then } \frac{dy}{dx} = anx^{n-1}
      Finds the gradient function of a polynomial, which gives the gradient of the tangent at any point on the curve.
    • Nth Term of a Quadratic Sequence
      an2+bn+can^2 + bn + c
      The second difference (2a) is constant; use this to find 'a', then solve for 'b' and 'c'.

    Key concepts: **Algebraic Manipulation**: Includes expanding brackets, factorising expressions (common factor, difference of two squares, quadratics), and simplifying algebraic fractions by factorising then cancelling., **Functions**: Understand function notation f(x), composite functions fg(x) = f(g(x)), and inverse functions f⁻¹(x)., **Solving Inequalities**: Solve like equations, but remember to flip the inequality sign if you multiply or divide by a negative number. For quadratic inequalities, sketch the graph to find the correct region(s).

    Exam tips

    • When solving equations with algebraic fractions, multiply every term by the Lowest Common Multiple (LCM) of the denominators to eliminate them.

    Coordinate geometry

    • Gradient of a Line
      m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
      Calculates the steepness of a line passing through two points (x₁, y₁) and (x₂, y₂).
    • Equation of a Line
      y=mx+cy = mx + c
      Represents a straight line where 'm' is the gradient and 'c' is the y-intercept.
    • Perpendicular Gradients
      m1×m2=−1m_1 \times m_2 = -1
      The product of the gradients of two perpendicular lines is -1. One is the negative reciprocal of the other.

    Key concepts: **Midpoint of a Line Segment**: The midpoint is the average of the x-coordinates and the average of the y-coordinates: ( (x₁+x₂)/2, (y₁+y₂)/2 )., **Parallel Lines**: Parallel lines have the same gradient (m₁ = m₂)., **Finding the Equation of a Line**: To find the equation, you need the gradient (m) and one point (x, y). Substitute these into y = mx + c to find the y-intercept (c).

    Exam tips

    • To find the equation of a perpendicular bisector, you must: 1. Find the midpoint of the original line segment. 2. Find the gradient of the original line. 3. Calculate the negative reciprocal for the perpendicular gradient. 4. Use the midpoint and perpendicular gradient to find the new equation.

    Geometry

    • Sum of Interior Angles
      Sum=(n−2)×180∘\text{Sum} = (n-2) \times 180^{\circ}
      Calculates the sum of the interior angles of any polygon with 'n' sides.
    • Area Scale Factor
      Area Factor=(Length Factor)2\text{Area Factor} = (\text{Length Factor})^2
      The ratio of the areas of two similar shapes is the square of the ratio of their corresponding lengths.
    • Volume Scale Factor
      Volume Factor=(Length Factor)3\text{Volume Factor} = (\text{Length Factor})^3
      The ratio of the volumes of two similar solids is the cube of the ratio of their corresponding lengths.

    Key concepts: **Circle Theorems**: Key theorems include: angle at the centre is twice the angle at the circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; the angle in a semicircle is 90°., **Alternate Segment Theorem**: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment., **Similarity**: Two shapes are similar if their corresponding angles are equal and the ratio of their corresponding sides is constant.

    Exam tips

    • When answering geometry questions involving angles, always state the full circle theorem or geometric property you are using as a reason for each step.

    Mensuration

    • Volume of a Cone
      V=13πr2hV = \frac{1}{3}\pi r^2 h
      Calculates the volume of a cone with radius 'r' and perpendicular height 'h'.
    • Curved Surface Area of a Cone
      A=πrlA = \pi r l
      Calculates the area of the curved surface of a cone, where 'l' is the slant height.
    • Area of a Sector
      A=θ360∘×πr2A = \frac{\theta}{360^{\circ}} \times \pi r^2
      Calculates the area of a portion of a circle defined by angle 'θ' and radius 'r'.

    Key concepts: **Arc Length**: The length of part of the circumference of a circle. Calculated by (θ/360°) × 2πr., **Surface Area of 3D Solids**: The total area of all faces/surfaces of a 3D shape. For a cylinder, it's 2πr² + 2πrh. For a sphere, it's 4πr²., **Volume of a Prism**: The volume of any prism is the area of its cross-section multiplied by its length (V = A × l).

    Exam tips

    • For compound shapes, break the shape down into simpler, standard shapes. Calculate the area/volume for each part and then add or subtract them as required.

    Trigonometry

    • Sine Rule
      asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
      Use in non-right-angled triangles when you have a side and its opposite angle pair.
    • Cosine Rule
      a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A
      Use in non-right-angled triangles when you have two sides and the included angle (SAS) or all three sides (SSS).
    • Area of a Triangle
      Area=12absin⁡C\text{Area} = \frac{1}{2}ab \sin C
      Calculates the area of any triangle when given two sides and the angle between them.

    Key concepts: **SOH CAH TOA**: Trigonometric ratios for right-angled triangles: Sin(θ) = Opp/Hyp, Cos(θ) = Adj/Hyp, Tan(θ) = Opp/Adj., **Pythagoras' Theorem in 3D**: Apply Pythagoras' theorem twice. First on a 2D base to find a diagonal, then use that diagonal and the height to find the space diagonal., **Graphs of Trigonometric Functions**: Recognise the shapes, amplitudes, and periods of y = sin(x), y = cos(x), and y = tan(x).

    Exam tips

    • In 3D trigonometry problems, identify and sketch the 2D right-angled triangle that contains the length or angle you need to find. This simplifies the problem.

    Transformations and vectors

    • Magnitude of a Vector
      ∣v∣=x2+y2|\mathbf{v}| = \sqrt{x^2 + y^2}
      Calculates the length of a vector v = (x, y) using Pythagoras' theorem.
    • Vector Path
      AB⃗=OB⃗−OA⃗\vec{AB} = \vec{OB} - \vec{OA}
      Finds the vector from point A to point B using their position vectors relative to the origin O.
    • Enlargement Matrix
      (k00k)\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}
      Represents an enlargement with scale factor 'k' centred at the origin.

    Key concepts: **Describing Transformations**: Reflection requires a mirror line. Rotation requires a centre, angle, and direction. Translation requires a column vector. Enlargement requires a centre and a scale factor., **Vector Arithmetic**: Vectors can be added, subtracted, and multiplied by a scalar. Geometrically, adding vectors is done 'tip-to-tail'., **Parallel Vectors**: Two vectors are parallel if one is a scalar multiple of the other (e.g., vector **a** is parallel to 3**a**).

    Exam tips

    • For vector geometry proofs (e.g., show points A, B, C are collinear), show that the vector AB is a scalar multiple of vector BC, and that they share a common point (B).

    Probability

    • Independent Events (AND rule)
      P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)
      The probability of two independent events both occurring is the product of their individual probabilities.
    • Mutually Exclusive Events (OR rule)
      P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)
      The probability of one or the other of two mutually exclusive events occurring is the sum of their probabilities.
    • Conditional Probability
      P(B∣A)=P(A and B)P(A)P(B|A) = \frac{P(A \text{ and } B)}{P(A)}
      The probability of event B occurring, given that event A has already occurred.

    Key concepts: **Tree Diagrams**: Used for combined events. Multiply probabilities along the branches to find the probability of a specific outcome. Add probabilities down the list of outcomes for 'OR' scenarios., **Probability 'Without Replacement'**: When an item is not replaced, the total number of outcomes (the denominator) decreases for the next event, as does the number of that specific item (the numerator)., **Venn Diagrams**: Visually represent sets and their relationships. The intersection (A ∩ B) represents 'A and B', while the union (A ∪ B) represents 'A or B or both'.

    Exam tips

    • When a question asks for the probability of 'at least one' of something, it's often easier to calculate '1 - P(none)'.

    Statistics

    • Frequency Density (Histograms)
      Frequency Density=FrequencyClass Width\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}
      Used as the y-axis for histograms with unequal class widths. The area of the bar represents the frequency.
    • Estimate of the Mean (Grouped Data)
      xˉ=∑fx∑f\bar{x} = \frac{\sum fx}{\sum f}
      Calculates an estimate of the mean from a grouped frequency table, where 'x' is the midpoint of each class interval.
    • Interquartile Range (IQR)
      IQR=Q3−Q1IQR = Q_3 - Q_1
      Measures the spread of the middle 50% of the data. Q₃ is the upper quartile (75th percentile) and Q₁ is the lower quartile (25th percentile).

    Key concepts: **Cumulative Frequency**: A running total of the frequencies. A cumulative frequency graph (ogive) is used to estimate the median, quartiles, and percentiles., **Histograms**: Used for continuous data. Bars can have unequal widths, and their area (not height) is proportional to the frequency. The vertical axis is frequency density., **Scatter Diagrams**: Show the relationship between two variables. Describe the correlation as positive, negative, or no correlation. A line of best fit can be drawn to make predictions.

    Exam tips

    • When asked to compare two data sets, comment on both a measure of central tendency (e.g., median) and a measure of spread (e.g., interquartile range), using context from the question.