Mathematics (0580) Extended Comprehensive cheat sheet

    Mathematics (0580) · CAIE · Extended

    Comprehensive
    5 pages
    68 formulas, 60 concepts
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    All Mathematics (0580) Extended cheat sheets

    Number

    • Compound Interest
      A=P(1+r100)nA = P \left(1 + \frac{r}{100}\right)^n
      Calculates the final amount (A) after an initial principal (P) is invested for 'n' periods at an interest rate of 'r' percent per period.
    • Percentage Change
      Percentage Change=New Value−Original ValueOriginal Value×100\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100
      Used to find the percentage increase or decrease of a value. A positive result is an increase, a negative result is a decrease.
    • Speed, Distance, Time
      Speed=DistanceTime\text{Speed} = \frac{\text{Distance}}{\text{Time}}
      Relates speed, distance, and time. Can be rearranged to find any of the three variables (e.g., Distance = Speed × Time).
    • Density, Mass, Volume
      Density=MassVolume\text{Density} = \frac{\text{Mass}}{\text{Volume}}
      Relates density, mass, and volume. Useful in problems involving material properties and measurements.
    • Laws of Indices (Multiplication)
      am×an=am+na^m \times a^n = a^{m+n}
      When multiplying powers with the same base, add the indices.
    • Laws of Indices (Division)
      am÷an=am−na^m \div a^n = a^{m-n}
      When dividing powers with the same base, subtract the indices.
    • Laws of Indices (Power of a Power)
      (am)n=amn(a^m)^n = a^{mn}
      When raising a power to another power, multiply the indices.
    • Negative Indices
      a−n=1ana^{-n} = \frac{1}{a^n}
      A negative index indicates the reciprocal of the base raised to the positive index.
    • Fractional Indices
      amn=(an)m=amna^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}
      The denominator 'n' is the root, and the numerator 'm' is the power.

    Key concepts: **Rational vs Irrational Numbers**: A rational number can be written as a fraction $\frac{a}{b}$ where a and b are integers (e.g., 5, -0.75, $\frac{2}{3}$). An irrational number cannot be written as a simple fraction and has non-terminating, non-repeating decimals (e.g., $\pi$, $\sqrt{2}$)., **Prime Factorisation (for HCF/LCM)**: Expressing a number as a product of its prime factors (e.g., $12 = 2^2 \times 3$). This is used to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two or more numbers., **Upper and Lower Bounds**: For a number given to a certain degree of accuracy, the lower bound is the smallest possible value it could have been before rounding, and the upper bound is the smallest value it could not be., **Reverse Percentage**: Finding the original value of an item before a percentage increase or decrease was applied. This involves dividing by the percentage multiplier, not multiplying., **Standard Form**: A way of writing very large or very small numbers in the form $A \times 10^n$, where $1 \le A < 10$ and 'n' is an integer. It simplifies calculations and comparisons., **Set Notation**: Symbols used to describe sets and their relationships: $\in$ (is an element of), $\notin$ (is not an element of), $\cup$ (union), $\cap$ (intersection), $\subset$ (is a proper subset of), $\mathcal{E}$ (universal set), $A'$ (complement of A), $n(A)$ (number of elements in A)., **Surds**: A surd is an irrational number expressed as a root of a rational number, like $\sqrt{2}$ or $\sqrt{12}$. They can be simplified (e.g., $\sqrt{12} = 2\sqrt{3}$) and manipulated using specific rules.

    Exam tips

    • To find the maximum value of a division $\frac{A}{B}$, use the Upper Bound of A divided by the Lower Bound of B. For the minimum value, use the Lower Bound of A divided by the Upper Bound of B. It's a common mistake to use UB/UB.
    • In multi-step calculations, keep the full calculator display value for as long as possible. Only round your final answer to the required degree of accuracy (usually 3 significant figures unless specified otherwise).
    • To increase a value by 15%, multiply by 1.15. To decrease by 15%, multiply by 0.85. For reverse percentage problems, divide by these multipliers. This is faster and less error-prone than finding the percentage and adding/subtracting.
    • To find HCF, take the lowest power of each common prime factor and multiply them. To find LCM, take the highest power of every prime factor present in any of the numbers and multiply them.
    • When filling in a Venn diagram from a word problem, always start with the most central region (the intersection of all sets) and work your way outwards.

    Algebra and graphs

    • Quadratic Formula
      x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
      Used to find the solutions (roots) of a quadratic equation in the form $ax^2 + bx + c = 0$. This formula is given in the exam paper.
    • Gradient of a Straight Line
      m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
      Calculates the steepness (gradient) of a straight line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$.
    • 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 (the point where the line crosses the y-axis).
    • Power Rule of Differentiation
      If y=axn, then dydx=anxn−1\text{If } y = ax^n, \text{ then } \frac{dy}{dx} = anx^{n-1}
      Used to find the gradient function (derivative) of a polynomial term. This tells you the gradient of the curve at any point x.
    • Direct Proportion
      y=kxory∝xy = kx \quad \text{or} \quad y \propto x
      Describes a relationship where as one variable increases, the other increases at the same rate. 'k' is the constant of proportionality.
    • Inverse Proportion
      y=kxory∝1xy = \frac{k}{x} \quad \text{or} \quad y \propto \frac{1}{x}
      Describes a relationship where as one variable increases, the other decreases. 'k' is the constant of proportionality.
    • nth Term of a Linear Sequence
      an=dn+(a1−d)a_n = d n + (a_1 - d)
      Finds the general rule for a linear (arithmetic) sequence, where 'd' is the common difference and 'a_1' is the first term.
    • Completed Square Form
      a(x+p)2+qa(x+p)^2 + q
      An alternative form for a quadratic expression $ax^2+bx+c$. It easily reveals the vertex (turning point) of the parabola at $(-p, q)$.

    Key concepts: **Solving Simultaneous Equations**: Finding the set of values (e.g., x and y) that satisfy two or more equations at the same time. This corresponds to the point of intersection of their graphs., **Factorising**: The process of rewriting an algebraic expression as a product of its factors (e.g., writing $x^2 + 5x + 6$ as $(x+2)(x+3)$). It is the reverse of expanding., **Interpreting Kinematics Graphs**: For a speed-time graph, the gradient represents acceleration and the area under the graph represents distance travelled. For a distance-time graph, the gradient represents speed., **Stationary Points**: Points on a curve where the gradient is zero. They are found by setting the derivative $\frac{dy}{dx} = 0$ and can be classified as maximum, minimum, or points of inflexion., **Inverse Functions**: An inverse function, denoted $f^{-1}(x)$, reverses the action of the original function $f(x)$. To find it, set $y = f(x)$, swap $x$ and $y$, and then make $y$ the subject of the new equation., **Composite Functions**: A composite function, like $fg(x)$, is formed when one function is applied to the result of another. You evaluate the inner function first ($g(x)$) and then apply the outer function ($f$) to that result., **Quadratic Sequences (Second Difference)**: In a quadratic sequence of the form $an^2+bn+c$, the difference between consecutive terms is not constant, but the difference between those differences (the second difference) is constant and equal to $2a$.

    Exam tips

    • When solving a linear inequality, treat it like a normal equation. However, you MUST flip the inequality sign (e.g., < to >) if you multiply or divide both sides by a negative number.
    • Be extremely careful when expanding a bracket with a negative sign in front. Every term inside the bracket changes sign. For example, $5 - (2x - 3)$ becomes $5 - 2x + 3$, not $5 - 2x - 3$.
    • When adding or subtracting algebraic fractions, you must find a common denominator first by multiplying the denominators together (or finding the LCM). Don't just add the numerators and denominators separately.
    • A common mistake is to only divide the square root part by $2a$. The entire numerator, including the $-b$ term, must be divided by $2a$. Use brackets on your calculator: $(-b \pm \sqrt{...}) \div (2a)$.
    • After finding your values for x and y, substitute them back into BOTH original equations to ensure they work. This is a quick way to catch any calculation errors.

    Coordinate geometry

    • Gradient Formula
      m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
      Used to find the gradient (steepness) of a line segment connecting two points, $(x_1, y_1)$ and $(x_2, y_2)$.
    • Distance Formula
      d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
      Used to calculate the length of a line segment between two points, $(x_1, y_1)$ and $(x_2, y_2)$. Derived from Pythagoras' theorem.
    • Midpoint Formula
      M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
      Used to find the coordinates of the midpoint of a line segment connecting two points, $(x_1, y_1)$ and $(x_2, y_2)$.
    • Gradient-Intercept Form
      y=mx+cy = mx + c
      The standard equation of a straight line, where 'm' is the gradient and 'c' is the y-intercept (the y-coordinate where the line crosses the y-axis).
    • Parallel Lines Gradient Rule
      m1=m2m_1 = m_2
      Two non-vertical lines are parallel if and only if they have the same gradient.
    • Perpendicular Lines Gradient Rule
      m1×m2=−1orm2=−1m1m_1 \times m_2 = -1 \quad \text{or} \quad m_2 = -\frac{1}{m_1}
      Two non-vertical lines are perpendicular if and only if the product of their gradients is -1. Their gradients are negative reciprocals of each other.

    Key concepts: **Gradient (Slope)**: The gradient measures the steepness and direction of a line. A positive gradient means the line slopes upwards from left to right, while a negative gradient means it slopes downwards., **y-intercept**: The y-intercept is the point where a line crosses the vertical y-axis. In the equation y = mx + c, it is represented by 'c', and its coordinates are (0, c)., **Parallel Lines**: Parallel lines are lines in a plane that never intersect. They have the exact same gradient (slope) but different y-intercepts., **Perpendicular Lines**: Perpendicular lines are lines that intersect at a right angle (90°). The gradient of one line is the negative reciprocal of the other (e.g., if m=2, the perpendicular gradient is -1/2)., **Horizontal & Vertical Lines**: A horizontal line has the equation y = k and a gradient of 0. A vertical line has the equation x = k and an undefined gradient.

    Exam tips

    • A very common mistake is mishandling negative coordinates in the gradient and distance formulas. Always use parentheses, e.g., $y_2 - (-y_1)$ becomes $y_2 + y_1$.
    • To find a perpendicular gradient, you must do two things: flip the fraction (reciprocal) AND change the sign (negative). Forgetting to change the sign is a frequent error.
    • If a line's equation is not in the form $y = mx + c$ (e.g., $3x + 2y = 6$), you must first rearrange it by making 'y' the subject to correctly identify the gradient and y-intercept.
    • If you have a diagram, quickly check if your calculated gradient and y-intercept make sense. Does the line slope up (positive m)? Does it cross the y-axis where you expect (c)?

    Geometry

    • Sum of Interior Angles of a Polygon
      S=(n−2)×180∘S = (n-2) \times 180^\circ
      Calculates the sum (S) of all interior angles for a polygon with 'n' sides.
    • Length, Area, and Volume Scale Factors
      kA=kL2,kV=kL3k_A = k_L^2, \quad k_V = k_L^3
      If the length scale factor between two similar shapes is k_L, the area scale factor is k_A and the volume scale factor is k_V.
    • Exterior Angle of a Regular Polygon
      Exterior Angle=360∘n\text{Exterior Angle} = \frac{360^\circ}{n}
      Finds the size of one exterior angle for a regular polygon with 'n' sides. The sum of exterior angles is always 360°.
    • Interior Angle of a Regular Polygon
      Interior Angle=(n−2)×180∘n\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}
      Finds the size of one interior angle for a regular polygon with 'n' sides. Can also be found by 180° - Exterior Angle.

    Key concepts: **Angles in Parallel Lines**: When a transversal line crosses two parallel lines: alternate interior angles are equal (Z-angle), corresponding angles are equal (F-angle), and co-interior angles sum to 180° (C-angle)., **Circle Theorem: Angle at Centre**: The angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference., **Similarity**: Two shapes are similar if their corresponding angles are equal and the ratio of their corresponding side lengths is constant. This constant ratio is the length scale factor., **Circle Theorem: Angle in a Semicircle**: The angle in a semicircle is always a right angle (90°). This is a special case of the 'angle at the centre' theorem, as the angle at the centre is a straight line (180°)., **Circle Theorem: Cyclic Quadrilateral**: A quadrilateral with all four vertices on the circumference of a circle. The opposite angles of a cyclic quadrilateral sum to 180° (A+C=180°, B+D=180°)., **Circle Theorem: 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., **Loci**: A locus is a set of points that satisfy a certain rule. Key loci include points a fixed distance from a point (circle), a fixed distance from a line (parallel lines), and points equidistant from two points (perpendicular bisector)., **Bearings**: Bearings are used for navigation and are measured from the North line, in a clockwise direction, and are always written using three figures (e.g., 045°)., **Symmetry**: Line (or reflectional) symmetry is where a shape can be folded onto itself along a line. Rotational symmetry is when a shape looks the same after being rotated less than a full turn; the 'order' is the number of times it fits onto itself in a 360° turn.

    Exam tips

    • In any angle calculation problem, especially with circle theorems, you MUST write down the geometric reason for each step to get full marks. E.g., 'angle at centre = 2 x angle at circumference'.
    • A common mistake is using the length scale factor for area or volume questions. Remember to square it for area ($k_A = k_L^2$) and cube it for volume ($k_V = k_L^3$). If given the area factor, square root it to find the length factor.
    • For questions asking you to construct something (e.g., a perpendicular bisector or an angle bisector), do not erase your construction arcs. The examiner needs to see them to award marks.
    • After calculating an angle, look at the diagram. Does your answer make sense? If you calculated an angle to be 170° but it looks acute, you have likely made a mistake.

    Mensuration

    • Area of a Circle
      A=πr2A = \pi r^2
      Used to find the area of a circle given its radius (r).
    • Circumference of a Circle
      C=2πrC = 2\pi r
      Used to find the perimeter (circumference) of a circle given its radius (r).
    • Volume of a Cylinder
      V=πr2hV = \pi r^2 h
      Used to find the volume of a cylinder given its radius (r) and perpendicular height (h). This is a specific case of the Volume of a Prism (V=Al) where the area A is a circle.
    • Volume of a Cone
      V=13πr2hV = \frac{1}{3}\pi r^2 h
      Used to find the volume of a cone given its base radius (r) and perpendicular height (h).
    • Volume of a Sphere
      V=43πr3V = \frac{4}{3}\pi r^3
      Used to find the volume of a sphere given its radius (r).
    • Surface Area of a Sphere
      A=4πr2A = 4\pi r^2
      Used to find the total surface area of a sphere given its radius (r).
    • Area of a Triangle (Sine Rule)
      A=12absin⁡CA = \frac{1}{2}ab \sin C
      Used to find the area of any triangle given two sides (a, b) and the included angle (C). Essential when height is not known.
    • Area of a Trapezium
      A=12(a+b)hA = \frac{1}{2}(a+b)h
      Used to find the area of a trapezium given the lengths of the two parallel sides (a, b) and the perpendicular height (h) between them.
    • Arc Length
      L=θ360×2πrL = \frac{\theta}{360} \times 2\pi r
      Used to find the length of a part of a circle's circumference (an arc) defined by an angle (θ) at the center.
    • Area of a Sector
      A=θ360×πr2A = \frac{\theta}{360} \times \pi r^2
      Used to find the area of a 'slice' of a circle (a sector) defined by an angle (θ) at the center.
    • Curved Surface Area of a Cone
      A=πrlA = \pi r l
      Used to find the area of the slanted surface of a cone, given the base radius (r) and the slant height (l).
    • Volume of a Prism
      V=A×lV = A \times l
      Used to find the volume of any prism, where A is the area of the constant cross-section and l is the length (or height).
    • Volume of a Pyramid
      V=13AhV = \frac{1}{3} A h
      Used to find the volume of any pyramid, where A is the area of the base and h is the perpendicular height.
    • Total Surface Area of a Cone
      A=πrl+πr2A = \pi r l + \pi r^2
      The total surface area is the sum of the curved surface area (\pi rl) and the area of the circular base (\pi r^2).
    • Total Surface Area of a Cylinder
      A=2πrh+2πr2A = 2\pi r h + 2\pi r^2
      The total surface area is the sum of the curved surface area (2\pi rh) and the area of the two circular ends (2\pi r^2).

    Key concepts: **Compound Shapes Strategy**: To find the area or volume of a complex shape, break it down into simpler, standard shapes (e.g., rectangles, triangles, cylinders, cones). Calculate the measure for each part and then add or subtract them as required., **Surface Area vs. Volume**: Surface Area is the total area of all the faces of a 3D object (a 2D measure, units squared, e.g., cm²). Volume is the amount of space inside a 3D object (a 3D measure, units cubed, e.g., cm³)., **Slant Height (l) vs. Perpendicular Height (h)**: In cones and pyramids, 'h' is the perpendicular height from the apex to the center of the base. 'l' is the slant height, the distance along the slanted surface from the apex to the edge of the base. They form a right-angled triangle with the radius 'r'., **Arcs and Sectors**: An arc is a part of the circumference of a circle, and a sector is a part of the area of a circle (like a pizza slice). Both are defined by the angle at the center and are calculated as a fraction of the full circle's measure., **Unit Conversion (Area & Volume)**: When converting units for area, you must square the length conversion factor (e.g., 1 m² = 100² cm² = 10000 cm²). For volume, you must cube the length conversion factor (e.g., 1 m³ = 100³ cm³ = 1,000,000 cm³)., **Frustum**: A frustum is the shape remaining when the top of a cone or pyramid is cut off by a plane parallel to its base. Its volume is found by subtracting the volume of the small removed cone/pyramid from the volume of the original large one.

    Exam tips

    • Always double-check if the question gives the radius (r) or the diameter (d). All formulas use the radius. If given the diameter, remember to halve it first (r = d/2).
    • Ensure all lengths are in the same unit (e.g., all in cm or all in m) before you substitute them into a formula. If not, convert them first to avoid major errors.
    • When calculating surface area, especially for prisms or compound shapes, systematically list all the faces to ensure you don't miss any. For open shapes (like a bucket), remember to exclude the area of the open face.
    • In cone and pyramid problems, you often need to find the slant height (l) or perpendicular height (h). Use Pythagoras' theorem with the radius (r): $r^2 + h^2 = l^2$.
    • If you are given the volume or area and asked to find a length (like the radius or height), write down the formula, substitute the known values, and then carefully rearrange the equation to solve for the unknown.

    Trigonometry

    • Pythagoras' Theorem
      a2+b2=c2a^2 + b^2 = c^2
      Used in right-angled triangles to find the length of a third side when two sides are known. 'c' must be the hypotenuse (the side opposite the right angle).
    • SOHCAHTOA: Sine Ratio
      sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
      Used in right-angled triangles to relate an angle to the side opposite it and the hypotenuse.
    • SOHCAHTOA: Cosine Ratio
      cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
      Used in right-angled triangles to relate an angle to the side adjacent (next to) it and the hypotenuse.
    • SOHCAHTOA: Tangent Ratio
      tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}
      Used in right-angled triangles to relate an angle to the side opposite and the side adjacent to it.
    • Sine Rule
      asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
      Used in any triangle (non-right-angled) when you know a side and its opposite angle, plus one other side or angle.
    • Cosine Rule (to find a side)
      a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A
      Used in any triangle to find a third side when you know two sides and the angle between them (SAS).
    • Cosine Rule (to find an angle)
      cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}
      An arrangement of the Cosine Rule used to find an angle when you know all three sides (SSS).
    • Area of a Triangle
      Area=12absin⁡C\text{Area} = \frac{1}{2}ab \sin C
      Used to find the area of any triangle when you know two sides and the angle between them (SAS).
    • Exact Trigonometric Values
      θsin⁡θcos⁡θtan⁡θ0°01030°12321345°1212160°3212390°10undefined\begin{array}{|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta \\ \hline 0° & 0 & 1 & 0 \\ 30° & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{3}} \\ 45° & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 1 \\ 60° & \frac{\sqrt{3}}{2} & \frac{1}{2} & \sqrt{3} \\ 90° & 1 & 0 & \text{undefined} \\ \hline \end{array}
      Key values for angles 0°, 30°, 45°, 60°, and 90° that you should know for non-calculator questions.

    Key concepts: **Right-Angled Triangle Trigonometry**: Use Pythagoras' Theorem to find sides when two are known. Use SOHCAHTOA to find a missing side or angle when an angle and a side, or two sides, are known., **Non-Right-Angled Triangle Trigonometry**: For triangles without a 90° angle, use the Sine Rule or the Cosine Rule to find missing sides and angles. Use the formula Area = 1/2 ab sin(C) for the area., **3D Trigonometry Strategy**: Solve 3D problems by identifying and sketching relevant 2D triangles within the 3D shape. Often, you will need to use Pythagoras or SOHCAHTOA on one triangle to find a length needed for a second triangle., **Trigonometric Graphs & Functions**: The graphs of y=sin(x), y=cos(x), and y=tan(x) are periodic. You need to know their shapes, amplitudes (for sin/cos), and periods (360° for sin/cos, 180° for tan)., **Solving Trigonometric Equations**: To solve an equation like sin(x) = k, first find the principal value using your calculator (x = sin⁻¹(k)). Then, use the symmetry of the sine graph (or CAST diagram) to find all other solutions within the given range., **The Ambiguous Case (Sine Rule)**: When using the Sine Rule with two sides and a non-included angle (SSA), there might be two possible triangles. If finding an angle B and sin(B) = k, the two possible solutions are B and (180° - B).

    Exam tips

    • Before any calculation, ensure your calculator is in Degrees (DEG) mode, not Radians (RAD) or Gradians (GRA). This is the most common source of error in trigonometry questions.
    • For the Sine and Cosine Rules, always label your triangle so that side 'a' is opposite angle 'A', side 'b' is opposite angle 'B', etc. Incorrect labelling will lead to incorrect answers.
    • Is it a right-angled triangle? Use SOHCAHTOA/Pythagoras. Is it any other triangle? Use Sine Rule if you have a side-opposite-angle pair. Use Cosine Rule if you have two sides and the angle between them (SAS) or all three sides (SSS).
    • Remember that the cosine of an obtuse angle (between 90° and 180°) is negative. When using the Cosine Rule a² = b² + c² - 2bc cos(A), if A is obtuse, the '- 2bc cos(A)' term will become positive, making 'a' the longest side.

    Transformations and vectors

    • Magnitude of a Vector
      ∣v∣=∣(xy)∣=x2+y2|\mathbf{v}| = \left| \begin{pmatrix} x \\ y \end{pmatrix} \right| = \sqrt{x^2 + y^2}
      Calculates the length or magnitude of a column vector. It's an application of the Pythagorean theorem.
    • Vector Addition and Subtraction
      (ab)±(cd)=(a±cb±d)\begin{pmatrix} a \\ b \end{pmatrix} \pm \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a \pm c \\ b \pm d \end{pmatrix}
      Used to find the resultant vector from adding or subtracting two vectors. Add/subtract the corresponding components.
    • Scalar Multiplication
      k(xy)=(kxky)k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}
      Used to scale a vector by a factor 'k'. If k > 0, the direction is the same. If k < 0, the direction is reversed.
    • Vector between Two Points
      AB⃗=OB⃗−OA⃗\vec{AB} = \vec{OB} - \vec{OA}
      Finds the vector that goes from point A to point B using their position vectors (vectors from the origin O).
    • Transformation Matrix
      (abcd)(xy)=(ax+bycx+dy)\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix}
      Represents a transformation. Multiplying the matrix by the coordinate vector (x, y) gives the image coordinates.

    Key concepts: **Translation**: A transformation that slides every point of an object by the same amount in the same direction, described by a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$., **Rotation**: A transformation that turns an object about a fixed point (the center of rotation) through a given angle and direction (clockwise or anti-clockwise)., **Reflection**: A transformation that flips an object across a line (the mirror line or line of reflection), creating a mirror image., **Enlargement**: A transformation that changes the size of an object from a fixed point (the center of enlargement) by a given scale factor., **Negative Scale Factor**: In an enlargement, a negative scale factor means the image is on the opposite side of the center of enlargement and is inverted (upside down)., **Parallel Vectors**: Two vectors $\mathbf{a}$ and $\mathbf{b}$ are parallel if one is a scalar multiple of the other (i.e., $\mathbf{a} = k\mathbf{b}$ for some scalar $k$)., **Collinear Points**: Points A, B, and C are collinear (lie on the same straight line) if the vector $\vec{AB}$ is parallel to the vector $\vec{BC}$ and they share a common point (B)., **Position Vector**: A vector that starts from the origin (O) and ends at a specific point (P). It is denoted as $\vec{OP}$ and has the same components as the coordinates of P.

    Exam tips

    • To get full marks, you must give a complete description: Translation requires a vector. Reflection requires a line equation. Rotation requires a center, angle, and direction. Enlargement requires a center and scale factor.
    • Tracing paper is allowed in the exam and is extremely useful for performing and checking rotations, reflections, and combined transformations. Trace the object and the axes to help.
    • Remember that $\vec{AB} = -\vec{BA}$. Reversing the direction of a vector negates its components. This is a common source of error in vector geometry problems.
    • Applying transformation A then B is generally not the same as applying B then A. Pay close attention to the order specified in the question.

    Probability

    • Basic Probability
      P(A)=Number of favourable outcomesTotal number of possible outcomesP(A) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}
      Used to calculate the theoretical probability of a single event (A) occurring when all outcomes are equally likely.
    • Complementary Events
      P(A′)=1−P(A)P(A') = 1 - P(A)
      Used to find the probability of an event NOT happening (A'). This is very useful for 'at least one' type problems.
    • Independent Events (AND Rule)
      P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)
      Used to find the probability of two independent events (A and B) both occurring. Independent means the outcome of one does not affect the other.
    • Mutually Exclusive Events (OR Rule)
      P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)
      Used to find the probability of either of two mutually exclusive events (A or B) occurring. Mutually exclusive means they cannot happen at the same time.
    • Expected Frequency
      E(A)=n×P(A)E(A) = n \times P(A)
      Used to estimate how many times an event (A) is expected to occur in a certain number of trials (n).
    • General Addition Rule (Non-Mutually Exclusive)
      P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
      Used to find the probability of event A or event B (or both) occurring when they are not mutually exclusive. This is the formula for Venn diagrams.
    • Conditional Probability
      P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}
      Calculates the probability of event A happening, given that event B has already happened. The denominator is the probability of the 'given' event.

    Key concepts: **Probability Scale**: The probability of any event is a number between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain., **Independent vs Dependent Events**: Independent events do not affect each other (e.g., flipping a coin twice). Dependent events do affect each other (e.g., picking two cards from a deck without replacement)., **Tree Diagrams**: A diagram used to represent a sequence of events. Probabilities are written on the branches, and you multiply along the branches to find the probability of a specific sequence of outcomes., **Venn Diagrams**: A diagram that shows the relationship between sets of outcomes. The intersection (overlap) represents 'AND', while the total area of the circles represents 'OR'., **Conditional Probability**: The probability of an event occurring, given that another event has already occurred. This often involves restricting the 'total possible outcomes' to a specific row, column, or circle in a table or Venn diagram., **Relative Frequency**: An estimate of probability based on experimental data (e.g., how many times a biased coin lands on heads after 100 flips). It is calculated as Frequency of event / Total number of trials.

    Exam tips

    • For any event, the probability of it happening plus the probability of it not happening must equal 1. Also, the probabilities on branches coming from a single point in a tree diagram must sum to 1. Use this to check your work.
    • When an item is NOT replaced, the total number of items decreases for the next pick. Always remember to reduce both the numerator (if applicable) and the denominator for the second set of branches in a tree diagram.
    • If a question asks for the probability of 'at least one' of something, it is almost always easier to calculate 1 - P(none of them). For example, P(at least one 6 in two rolls) = 1 - P(no 6 on 1st AND no 6 on 2nd).
    • When filling in a Venn diagram with numbers, always start with the central intersection (the 'AND' part) and work your way outwards. Subtract this value from the totals for each circle to find the 'only' parts.

    Statistics

    • Estimate of the Mean (Grouped Data)
      xˉ≈∑fx∑f\bar{x} \approx \frac{\sum fx}{\sum f}
      Used to estimate the mean from a grouped frequency table, where 'x' is the midpoint of each class interval and 'f' is the frequency.
    • Frequency Density
      Frequency Density=FrequencyClass Width\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}
      Used to calculate the height of the bars in a histogram, especially when class widths are unequal. It ensures the area of the bar is proportional to the frequency.
    • Interquartile Range (IQR)
      IQR=Q3−Q1IQR = Q_3 - Q_1
      Calculates the range of the middle 50% of the data. Q3 is the upper quartile (75th percentile) and Q1 is the lower quartile (25th percentile).
    • Median Position (Discrete Data)
      Position=n+12\text{Position} = \frac{n+1}{2}
      Finds the position of the median in an ordered list of 'n' data points. If the result is a decimal (e.g., 4.5), it is halfway between the 4th and 5th values.
    • Range
      Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}
      Calculates the difference between the highest and lowest values in a dataset, indicating the total spread of the data.

    Key concepts: **Measures of Central Tendency**: These are single values that attempt to describe a set of data by identifying the central position. The main measures are the mean (the average), median (the middle value), and mode (the most frequent value)., **Histograms (Unequal Widths)**: A histogram uses bars to show the frequency of continuous data. For unequal class widths, the vertical axis must be Frequency Density, ensuring the area of each bar (not the height) is proportional to the frequency., **Cumulative Frequency**: This is the 'running total' of the frequencies. A cumulative frequency graph (ogive) plots this running total against the upper class boundaries, and is used to estimate the median, quartiles, and percentiles., **Measures of Spread**: These describe how spread out the data values are. The main measures are the range (total spread) and the interquartile range (IQR), which shows the spread of the middle 50% and is less affected by outliers., **Scatter Diagrams and Correlation**: A scatter diagram plots one variable against another to show the relationship (correlation) between them. Correlation can be positive (as one increases, the other increases), negative (as one increases, the other decreases), or no correlation., **Modal Class and Median Class**: For grouped data, the modal class is the class interval with the highest frequency. The median class is the first class interval where the cumulative frequency exceeds half the total frequency.

    Exam tips

    • Common Mistake: For histograms with unequal class widths, students often plot frequency on the y-axis. ALWAYS use Frequency Density. Remember: Area of bar = Frequency.
    • Exam Tip: When calculating the mean from a grouped frequency table, you must use the midpoint of each class interval as your 'x' value in the formula $\frac{\sum fx}{\sum f}$. For a class 10 < h ≤ 20, the midpoint is 15.
    • Common Mistake: When drawing a cumulative frequency graph, always plot the cumulative frequency against the UPPER boundary of each class interval. The first point should be at the lowest boundary with a cumulative frequency of 0.
    • Exam Tip: A line of best fit on a scatter diagram should pass through the mean point $(\bar{x}, \bar{y})$ and have roughly an equal number of points on either side of the line along its length.
    • Exam Tip: When asked to compare two datasets, always comment on both a measure of central tendency (e.g., 'Group A had a higher median score') and a measure of spread (e.g., 'Group B's scores were more consistent as their IQR was smaller').