BGCSE Scientific Mathematics 1237 cheat sheet

Place value: each position is ten times the value of the position to its right; in 4 706. 285 the digit 7 has value 700 and the digit 8 has value 0.08.

Order of operations (BODMAS): evaluate brackets, then orders (powers and roots), then multiplication and division left to right, then addition and subtraction left to right.

Rounding: to round to n decimal places, look at the (n+1)th digit; if it is 5 or more, increase the nth digit by 1. For significant figures, start counting at the first non-zero digit.

Fractions: a/b + c/d = (ad + bc)/(bd); a/b ÷ c/d = a/b × d/c. Convert mixed numbers to improper fractions before multiplying or dividing.

Percentages: percentage change = (change ÷ original) × 100. A 20% increase followed by a 20% decrease does not return to the original value; the net result is 0.96 of it.

Standard form: a × 10^n where 1 ≤ a < 10. To multiply, multiply the first factors and add the indices; to divide, divide the first factors and subtract the indices.

LCM and HCF from prime factors: take the highest power of each prime for the LCM and the lowest power of each shared prime for the HCF.

Negative numbers: multiplying or dividing two numbers with the same sign gives a positive result; different signs give a negative result. Subtracting a negative adds.

Expansion: a(b + c) = ab + ac and (x + a)(x + b) = x² + (a + b)x + ab. Squaring a binomial gives (a + b)² = a² + 2ab + b², not a² + b².

Factorising: always remove the highest common factor first; a quadratic x² + bx + c factorises when two numbers multiply to c and add to b.

Difference of two squares: a² − b² = (a + b)(a − b). This identity also allows rapid mental arithmetic, e.g. 53² − 47² = 100 × 6 = 600.

Solving linear equations: perform the same operation on both sides; collect variable terms on one side and constants on the other, then divide by the coefficient.

Simultaneous equations: elimination works by making the coefficients of one variable equal and subtracting; substitution works by expressing one variable in terms of the other.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac gives two, one, or no real roots as it is positive, zero, or negative.

Indices: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1 for a ≠ 0; a^(−n) = 1 ÷ a^n.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign; the solution set can be shown on a number line with open or closed circles.

Angle sums: angles in a triangle total 180°, in a quadrilateral 360°, and at a point 360°. Angles on a straight line total 180°.

Polygon angles: interior angle sum of an n-sided polygon = (n − 2) × 180°; each interior angle of a regular n-gon = (n − 2) × 180° ÷ n; exterior angles sum to 360°.

Circle theorems: the angle in a semicircle is 90°; the angle at the centre is twice the angle at the circumference on the same arc; angles in the same segment are equal.

Pythagoras' theorem: in a right-angled triangle a² + b² = c², where c is the hypotenuse. The converse (a² + b² = c²) proves a triangle is right-angled.

Congruence tests: SSS, SAS, ASA (with AAS), and RHS for right-angled triangles. Similar figures have equal corresponding angles and proportional corresponding sides.

Area formulas: triangle = ½ × base × perpendicular height; parallelogram = base × height; trapezium = ½(a + b)h; circle = πr².

Volume: prism = area of cross-section × length; cylinder = πr²h; cone = ⅓πr²h; sphere = 4/3 πr³.

Transformations: reflection reverses orientation, rotation preserves it, translation preserves both shape and orientation, and enlargement by scale factor k multiplies lengths by k and areas by k².

Metric conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm; 1 tonne = 1000 kg, 1 kg = 1000 g; 1 litre = 1000 ml and 1 cm³ = 1 ml.

Area conversions use squared factors (1 m² = 10 000 cm²) and volume conversions use cubed factors (1 m³ = 1 000 000 cm³).

Speed, distance, time: speed = distance ÷ time. Convert mixed units first, e.g. 72 km/h = 72 × 1000 ÷ 3600 = 20 m/s.

Density and rate problems: density = mass ÷ volume, so mass = density × volume. Keep units consistent before substituting.

Perimeter and circumference: circumference of a circle = 2πr = πd. Arc length = (θ ÷ 360) × 2πr for a sector with central angle θ.

Sector area = (θ ÷ 360) × πr². The area of a segment equals sector area minus triangle area.

Surface area: total surface area of a cuboid = 2(lw + lh + wh); of a cylinder = 2πr² + 2πrh; of a sphere = 4πr².

Time and timetables: elapsed time is found by counting forward in convenient steps; 1 hour = 60 minutes and 1 minute = 60 seconds, so time calculations use base 60, not base 10.

Averages from raw data: mean = sum of values ÷ number of values; median = middle value of ordered data (mean of the two middle values if n is even); mode = most frequent value.

Estimated mean from a grouped frequency table uses class midpoints: mean = Σ(f × x) ÷ Σf. The median lies in the class where the cumulative frequency reaches half the total.

Range = highest − lowest. Interquartile range = upper quartile − lower quartile and measures the spread of the middle half of the data.

Probability scale runs from 0 (impossible) to 1 (certain). P(not A) = 1 − P(A); for mutually exclusive events P(A or B) = P(A) + P(B).

Independent events: P(A and B) = P(A) × P(B). Without replacement, the denominator and numerator of later probabilities change.

Experimental probability = number of successful trials ÷ total trials; it approaches theoretical probability as the number of trials increases.

Scatter graphs: positive correlation means both variables increase together; the line of best fit passes through the mean point and is used for estimation within the data range.

Pie charts: angle for a category = (frequency ÷ total) × 360°. Bar charts compare categories; histograms with unequal class widths use frequency density = frequency ÷ class width.

A relation is any set of ordered pairs; a function is a relation in which each input has exactly one output. The vertical line test checks whether a graph represents a function.

Function notation: f(x) means the output for input x. To find f(3), substitute 3 everywhere x appears; the inverse f⁻¹ undoes f, so f(f⁻¹(x)) = x.

Composite functions: fg(x) means apply g first, then f. In general fg(x) ≠ gf(x), so order matters.

Straight line: y = mx + c, where m is the gradient and c the y-intercept. Gradient m = (y₂ − y₁) ÷ (x₂ − x₁); parallel lines have equal gradients.

Perpendicular lines have gradients whose product is −1, i.e. m₁ × m₂ = −1. A horizontal line has gradient 0; a vertical line's gradient is undefined.

Distance between two points = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).

Quadratic graphs y = ax² + bx + c are parabolas: a > 0 opens upward, a < 0 opens downward; the line of symmetry is x = −b ÷ 2a and the y-intercept is c.

Graph transformations: y = f(x) + k shifts up by k; y = f(x + k) shifts left by k; y = kf(x) stretches vertically by factor k; y = −f(x) reflects in the x-axis.