JCE Mathematics cheat sheet

Standard form: a × 10^n with 1 ≤ a < 10 and n an integer; multiply mantissas and add indices when multiplying, divide mantissas and subtract indices when dividing.

Rational numbers are those expressible as p/q with integers p, q and q ≠ 0; all terminating and recurring decimals are rational, while √2 and π are irrational.

Recurring decimals convert to fractions by placing 9s in the denominator for the repeating block, e.g. 0.4̄ = 4/9 and 0.12̄ = 11/90 after subtracting the non-repeating part.

Percentages: percentage change = (new − old)/old × 100; successive changes multiply factors (e.g. +20% then −10% gives 1.2 × 0.9 = 1.08, a net +8%).

Compound interest or growth: A = P(1 ± r/100)^n, where r is the rate per period and n the number of periods; simple interest is P × r/100 × n.

Rationalising denominators: multiply by the conjugate for √a ± √b forms, so 1/(√a + √b) = (√a − √b)/(a − b).

Rules of indices: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Upper and lower bounds: for a value rounded to a given unit, half the unit is the error; bounds of a sum add errors, bounds of a product use the extreme combinations of bounds.

HCF and LCM from prime factorisation: take the lowest power of each common prime for HCF and the highest power of every prime for LCM; HCF × LCM = product of the two numbers.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac > 0 gives two real roots, = 0 gives a repeated root, < 0 gives no real roots.

Factorising quadratics: for ax² + bx + c find two numbers with product ac and sum b, then split the middle term and factor by grouping.

Completing the square: x² + bx + c = (x + b/2)² + c − b²/4; the vertex of y = a(x − h)² + k is (h, k).

Simultaneous equations: eliminate one variable by matching coefficients and adding or subtracting; substitution is preferable when one equation is already solved for a variable.

Laws of logarithms: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^n) = n log a; solving a^x = b gives x = log b / log a.

Algebraic fractions: simplify by factorising numerator and denominator first, then cancelling common factors; add or subtract by finding a common denominator.

Inequalities: multiplying or dividing by a negative number reverses the inequality sign; represent solutions on a number line with open or closed circles.

Sequences: arithmetic nth term a + (n − 1)d; geometric nth term ar^(n−1); sum of a geometric series Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.

Rearranging formulae: perform the same operation on both sides and isolate the required variable; when the variable appears twice, collect it on one side and factorise it out.

Straight line y = mx + c has gradient m and y-intercept c; parallel lines share m, perpendicular gradients satisfy m₁m₂ = −1.

Gradient between two points = (y₂ − y₁)/(x₂ − x₁); the equation through a point is y − y₁ = m(x − x₁).

Transformations of y = f(x): f(x) + a shifts up by a, f(x + a) shifts left by a, af(x) stretches vertically by factor a, f(ax) stretches horizontally by factor 1/a.

Reflections: −f(x) reflects in the x-axis, f(−x) reflects in the y-axis; |f(x)| reflects any part below the x-axis above it.

Quadratic graph y = ax² + bx + c: opens upward if a > 0, downward if a < 0; axis of symmetry x = −b/(2a).

Exponential y = a^x passes through (0, 1); growth when a > 1, decay when 0 < a < 1; the graph never touches the x-axis.

Trigonometric graphs: y = sin x and y = cos x have period 360° and amplitude 1; y = tan x has period 180° and asymptotes where cos x = 0.

Inverse functions: swap x and y and solve; the graph of f⁻¹ is the reflection of f in the line y = x, and the domain of f⁻¹ is the range of f.

Composite functions: fg(x) means f(g(x)); apply the inner function first, and in general fg ≠ gf.

Circle equation: (x − a)² + (y − b)² = r² has centre (a, b) and radius r; expanding gives x² + y² + 2fx + 2gy + c = 0 with centre (−f, −g).

Pythagoras: a² + b² = c² for the hypotenuse c; the converse identifies right-angled triangles.

Trig ratios in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOHCAHTOA).

Sine rule: a/sin A = b/sin B = c/sin C; use when given a matching side–angle pair plus one more part.

Cosine rule: a² = b² + c² − 2bc cos A; use with two sides and the included angle, or all three sides.

Area of any triangle = ½ab sin C, where C is the angle between sides a and b.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3, sin 45° = cos 45° = √2/2, sin 60° = √3/2, cos 60° = ½, tan 45° = 1.

Circle theorems: angle at centre is twice the angle at circumference; angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°; tangent is perpendicular to the radius.

Volume and surface area: sphere V = 4/3 πr³ and S = 4πr²; cone V = 1/3 πr²h and curved surface πrl; cylinder V = πr²h.

Arc length = (θ/360) × 2πr and sector area = (θ/360) × πr², with θ in degrees; in radians use s = rθ and A = ½r²θ.

Similar shapes: lengths scale by k, areas by k², volumes by k³.

Mean = sum of values ÷ number of values; median is the middle value of ordered data; mode is the most frequent value; range = largest − smallest.

For grouped frequency data, estimate the mean using midpoints: Σ(fx)/Σf; the median lies in the class where cumulative frequency reaches half the total.

Cumulative frequency curves: plot upper class boundaries against cumulative frequency; quartiles are read at ¼, ½ and ¾ of the total, and IQR = UQ − LQ.

Probability scale runs from 0 to 1; P(not A) = 1 − P(A), and P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Tree diagrams: multiply along branches for combined outcomes and add between branches for alternative outcomes; without replacement, the second-branch denominator decreases.

Conditional probability: P(A|B) = P(A ∩ B)/P(B), valid when P(B) > 0.

Standard deviation measures spread about the mean; a small value indicates data clustered near the mean, a large value indicates wide spread.

Scatter graphs: correlation describes the direction and strength of a linear relationship, but correlation does not imply causation; the line of best fit passes through the mean point.

Modelling cycle: formulate a real situation as a mathematical model, solve it, interpret the result in context, then validate and refine assumptions.

Identify the variable and its units before writing an equation; check that the final answer's units and magnitude are plausible in the original context.

Use ratio and proportion to scale quantities: if a : b = c : d then ad = bc; unitary method finds the value of one unit first.

Rates: speed = distance/time, density = mass/volume, pressure = force/area; rearrange to solve for any of the three quantities.

Break multi-step problems into sub-goals; record intermediate results with units and check each against the constraints given.

Estimation and bounds: round inputs to one significant figure for a quick check, then compare with the exact answer to catch arithmetic errors.

Interpret graphs in context: gradient often represents a rate of change and area under the graph often represents a total accumulated quantity.

When a model's prediction conflicts with data, revise the assumptions or the function type rather than adjusting the data to fit.

Standard form: a × 10^n with 1 ≤ a < 10 and n an integer; multiply mantissas and add indices when multiplying, divide mantissas and subtract indices when dividing.

Rational numbers are those expressible as p/q with integers p, q and q ≠ 0; all terminating and recurring decimals are rational, while √2 and π are irrational.

Recurring decimals convert to fractions by placing 9s in the denominator for the repeating block, e.g. 0.4̄ = 4/9 and 0.12̄ = 11/90 after subtracting the non-repeating part.

Percentages: percentage change = (new − old)/old × 100; successive changes multiply factors (e.g. +20% then −10% gives 1.2 × 0.9 = 1.08, a net +8%).

Compound interest or growth: A = P(1 ± r/100)^n, where r is the rate per period and n the number of periods; simple interest is P × r/100 × n.

Rationalising denominators: multiply by the conjugate for √a ± √b forms, so 1/(√a + √b) = (√a − √b)/(a − b).

Rules of indices: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.

Upper and lower bounds: for a value rounded to a given unit, half the unit is the error; bounds of a sum add errors, bounds of a product use the extreme combinations of bounds.

HCF and LCM from prime factorisation: take the lowest power of each common prime for HCF and the highest power of every prime for LCM; HCF × LCM = product of the two numbers.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac > 0 gives two real roots, = 0 gives a repeated root, < 0 gives no real roots.

Factorising quadratics: for ax² + bx + c find two numbers with product ac and sum b, then split the middle term and factor by grouping.

Completing the square: x² + bx + c = (x + b/2)² + c − b²/4; the vertex of y = a(x − h)² + k is (h, k).

Simultaneous equations: eliminate one variable by matching coefficients and adding or subtracting; substitution is preferable when one equation is already solved for a variable.

Laws of logarithms: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^n) = n log a; solving a^x = b gives x = log b / log a.

Algebraic fractions: simplify by factorising numerator and denominator first, then cancelling common factors; add or subtract by finding a common denominator.

Inequalities: multiplying or dividing by a negative number reverses the inequality sign; represent solutions on a number line with open or closed circles.

Sequences: arithmetic nth term a + (n − 1)d; geometric nth term ar^(n−1); sum of a geometric series Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.

Rearranging formulae: perform the same operation on both sides and isolate the required variable; when the variable appears twice, collect it on one side and factorise it out.

Straight line y = mx + c has gradient m and y-intercept c; parallel lines share m, perpendicular gradients satisfy m₁m₂ = −1.

Gradient between two points = (y₂ − y₁)/(x₂ − x₁); the equation through a point is y − y₁ = m(x − x₁).

Transformations of y = f(x): f(x) + a shifts up by a, f(x + a) shifts left by a, af(x) stretches vertically by factor a, f(ax) stretches horizontally by factor 1/a.

Reflections: −f(x) reflects in the x-axis, f(−x) reflects in the y-axis; |f(x)| reflects any part below the x-axis above it.

Quadratic graph y = ax² + bx + c: opens upward if a > 0, downward if a < 0; axis of symmetry x = −b/(2a).

Exponential y = a^x passes through (0, 1); growth when a > 1, decay when 0 < a < 1; the graph never touches the x-axis.

Trigonometric graphs: y = sin x and y = cos x have period 360° and amplitude 1; y = tan x has period 180° and asymptotes where cos x = 0.

Inverse functions: swap x and y and solve; the graph of f⁻¹ is the reflection of f in the line y = x, and the domain of f⁻¹ is the range of f.

Composite functions: fg(x) means f(g(x)); apply the inner function first, and in general fg ≠ gf.

Circle equation: (x − a)² + (y − b)² = r² has centre (a, b) and radius r; expanding gives x² + y² + 2fx + 2gy + c = 0 with centre (−f, −g).

Pythagoras: a² + b² = c² for the hypotenuse c; the converse identifies right-angled triangles.

Trig ratios in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOHCAHTOA).

Sine rule: a/sin A = b/sin B = c/sin C; use when given a matching side–angle pair plus one more part.

Cosine rule: a² = b² + c² − 2bc cos A; use with two sides and the included angle, or all three sides.

Area of any triangle = ½ab sin C, where C is the angle between sides a and b.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3, sin 45° = cos 45° = √2/2, sin 60° = √3/2, cos 60° = ½, tan 45° = 1.

Circle theorems: angle at centre is twice the angle at circumference; angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°; tangent is perpendicular to the radius.

Volume and surface area: sphere V = 4/3 πr³ and S = 4πr²; cone V = 1/3 πr²h and curved surface πrl; cylinder V = πr²h.

Arc length = (θ/360) × 2πr and sector area = (θ/360) × πr², with θ in degrees; in radians use s = rθ and A = ½r²θ.

Similar shapes: lengths scale by k, areas by k², volumes by k³.

Mean = sum of values ÷ number of values; median is the middle value of ordered data; mode is the most frequent value; range = largest − smallest.

For grouped frequency data, estimate the mean using midpoints: Σ(fx)/Σf; the median lies in the class where cumulative frequency reaches half the total.

Cumulative frequency curves: plot upper class boundaries against cumulative frequency; quartiles are read at ¼, ½ and ¾ of the total, and IQR = UQ − LQ.

Probability scale runs from 0 to 1; P(not A) = 1 − P(A), and P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Tree diagrams: multiply along branches for combined outcomes and add between branches for alternative outcomes; without replacement, the second-branch denominator decreases.

Conditional probability: P(A|B) = P(A ∩ B)/P(B), valid when P(B) > 0.

Standard deviation measures spread about the mean; a small value indicates data clustered near the mean, a large value indicates wide spread.

Scatter graphs: correlation describes the direction and strength of a linear relationship, but correlation does not imply causation; the line of best fit passes through the mean point.

Modelling cycle: formulate a real situation as a mathematical model, solve it, interpret the result in context, then validate and refine assumptions.

Identify the variable and its units before writing an equation; check that the final answer's units and magnitude are plausible in the original context.

Use ratio and proportion to scale quantities: if a : b = c : d then ad = bc; unitary method finds the value of one unit first.

Rates: speed = distance/time, density = mass/volume, pressure = force/area; rearrange to solve for any of the three quantities.

Break multi-step problems into sub-goals; record intermediate results with units and check each against the constraints given.

Estimation and bounds: round inputs to one significant figure for a quick check, then compare with the exact answer to catch arithmetic errors.

Interpret graphs in context: gradient often represents a rate of change and area under the graph often represents a total accumulated quantity.

When a model's prediction conflicts with data, revise the assumptions or the function type rather than adjusting the data to fit.