LC Higher Maths cheat sheet

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / 2a; the discriminant b² − 4ac gives two real roots if positive, one repeated root if zero, and no real roots if negative.

Factor theorem: (x − a) is a factor of p(x) exactly when p(a) = 0; the remainder on dividing p(x) by (x − a) is p(a).

Cubic with integer coefficients: if p(x) has a rational root it is of the form ±(factor of constant term)/(factor of leading coefficient); test small integer divisors first.

Simultaneous equations, one linear and one quadratic: solve the linear equation for one variable, substitute into the quadratic, then solve the resulting quadratic in the other variable.

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a^(−n) = 1/aⁿ, a^(1/n) = ⁿ√a, a^(m/n) = ⁿ√(aᵐ).

Logs: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a, log_a a = 1, log_a 1 = 0, and log_a x = log_b x / log_b a for a change of base.

Solving aˣ = b: take logs of both sides to get x = log b / log a; for inequalities, reversing the inequality sign is required only when multiplying or dividing by a negative number.

Absolute value inequality |x − a| < d means a − d < x < a + d; |x − a| > d splits into x < a − d or x > a + d.

Proof by induction: establish the base case, assume the statement for n = k, then use that assumption to derive the statement for n = k + 1, and state the conclusion explicitly.

Domain restrictions: exclude values making a denominator zero and require the expression under an even root to be non-negative; state the domain explicitly.

Composition: (f ∘ g)(x) = f(g(x)) is defined only where x is in the domain of g and g(x) is in the domain of f.

Inverse function: f has an inverse only if it is one-to-one; find it by writing y = f(x), swapping x and y, and solving for y; the graph of f⁻¹ is the reflection of f in the line y = x.

Transformations of y = f(x): f(x) + k shifts up k, f(x + k) shifts left k, af(x) scales vertically by a, f(bx) scales horizontally by 1/b, and −f(x) reflects in the x-axis.

Exponential y = aˣ with a > 1 increases and passes through (0, 1); the natural exponential eˣ has slope equal to its value at every point.

Logarithmic function y = log_a x is the inverse of y = aˣ, has domain x > 0, and passes through (1, 0).

Quadratic in completed-square form y = a(x − h)² + k has vertex (h, k); the axis of symmetry is x = h and the sign of a fixes the direction of opening.

A function is even if f(−x) = f(x) for all x in its domain (symmetric about the y-axis) and odd if f(−x) = −f(x) (symmetric about the origin).

Sine rule: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius; use it when a side and its opposite angle are both known.

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

Area of a triangle = ½ab sin C, using two sides and the included angle.

Radian measure: π radians = 180°; arc length s = rθ and sector area A = ½r²θ, with θ in radians.

Identities: sin²θ + cos²θ = 1, tan θ = sin θ / cos θ, 1 + tan²θ = sec²θ, and the double-angle forms sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ.

Sine rule ambiguous case: when given two sides and a non-included angle, the sine rule may yield two possible angles (θ and 180° − θ); check whether both give valid triangles.

The equation of a circle with centre (h, k) and radius r is (x − h)² + (y − k)² = r²; expanding gives x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f) and radius √(g² + f² − c).

A line is tangent to a circle when the perpendicular distance from the centre to the line equals the radius; a chord's perpendicular bisector passes through the centre.

Theorem of Pythagoras and the converse: a triangle is right-angled exactly when the square of one side equals the sum of the squares of the other two.

Distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Line forms: y − y₁ = m(x − x₁) for point-slope, y = mx + c for slope-intercept, and x/a + y/b = 1 for intercepts a and b.

Perpendicular distance from (x₁, y₁) to the line ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²).

Two lines are concurrent with a third when the point of intersection of the first two satisfies the third equation; verify by substitution.

Divide a segment in the ratio m:n internally using the section formula ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

The angle θ between two lines satisfies tan θ = |(m₁ − m₂)/(1 + m₁m₂)|, the absolute value giving the acute angle.

A point lies on a locus defined by a geometric condition; translate the condition into an equation, e.g. equidistance from two fixed points gives the perpendicular bisector.

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0.

Multiplication rule: P(A ∩ B) = P(A) · P(B|A); for independent events P(A ∩ B) = P(A) · P(B).

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

Complement rule: P(A′) = 1 − P(A); use it whenever 'at least one' appears.

Counting: number of arrangements of n distinct objects is n!; number of ways to choose r from n is ⁿCᵣ = n! / (r!(n − r)!); permutations ⁿPᵣ = n! / (n − r)!.

Bernoulli trials: probability of exactly r successes in n independent trials with success probability p is ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ.

Expected value of a discrete random variable: E(X) = Σ x·P(X = x); variance Var(X) = E(X²) − (E(X))².

Odds in favour of an event with probability p are p : (1 − p); convert odds a : b to probability a/(a + b).

Mean of grouped data: x̄ = Σfx / Σf, where x is the mid-interval value of each class and f its frequency.

Standard deviation of a sample: s = √(Σ(x − x̄)² / (n − 1)); the population form divides by n. The standard deviation is never negative.

Median and quartiles come from cumulative frequency: the median is at n/2, the lower quartile at n/4, the upper quartile at 3n/4, with interpolation within the class.

Interquartile range = Q₃ − Q₁; outliers are commonly flagged as values below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR.

Correlation coefficient r lies between −1 and 1; r near 1 or −1 indicates strong linear association, r near 0 weak linear association. Correlation does not imply causation.

Line of best fit by least squares: slope b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² and intercept a = ȳ − b·x̄.

Adding a constant to every data value shifts the mean and median by that constant but leaves the standard deviation and range unchanged.

Empirical rule for approximately normal data: about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Percentage change = (new − old)/old × 100; a percentage increase of p% multiplies by (1 + p/100), a decrease by (1 − p/100).

Compound interest: A = P(1 + r)ⁿ with r as a decimal rate per period and n the number of periods; for depreciation use (1 − r)ⁿ.

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

Surface area: sphere 4πr², curved surface of a cylinder 2πrh, curved surface of a cone πrl where l is the slant height.

Similar figures: lengths scale by k, areas by k², volumes by k³; use this to relate ratios of corresponding quantities.

Scientific notation: a × 10ⁿ with 1 ≤ a < 10; multiply by adding indices and divide by subtracting them.

Units: 1 km = 1000 m, 1 m = 100 cm, 1 hectare = 10 000 m², 1 litre = 1000 cm³; convert all quantities to consistent units before calculating.

Absolute and percentage error: percentage error = (error / measured value) × 100; when adding or subtracting measurements, absolute errors add.

Differentiation from first principles: f′(x) = lim(h→0) [f(x + h) − f(x)] / h.

Power rule: d/dx (xⁿ) = n·xⁿ⁻¹ for any real n; also d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x.

Product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′)/v²; chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).

A stationary point occurs where f′(x) = 0; classify with the second derivative - f″ < 0 gives a local maximum, f″ > 0 a local minimum, and f″ = 0 requires a sign test.

The tangent at x = a has slope f′(a); the normal there has slope −1/f′(a). Write the line through (a, f(a)).

Integration: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫1/x dx = ln|x| + C; ∫cos x dx = sin x + C; ∫eˣ dx = eˣ + C.

Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a) where F is any antiderivative of f; this gives the signed area between the curve and the x-axis.

Area between two curves from x = a to x = b is ∫ₐᵇ (upper − lower) dx; find the limits from the points of intersection.

Rate of change: if s(t) is displacement, then velocity is s′(t) and acceleration is s″(t); the object is at rest when s′(t) = 0.

Show that questions require a derivation ending at the given result; do not assume the result, and finish by stating it explicitly.

When a result is given, use it in later parts even if an earlier part was not completed; marks in later parts are usually independent.

Verify a conjecture with several cases, then prove it generally; a single counterexample disproves a universal statement.

Translate word problems into equations by defining variables and units before solving; check the answer against the original context for reasonableness.

For optimisation, express the quantity as a function of one variable using the constraint, differentiate, solve for the stationary point, and justify that it is the required maximum or minimum.

Proof by contradiction: assume the negation of the statement and derive a contradiction with an established fact, such as the irrationality of √2.

Read the number of marks as a guide to the steps expected; a question worth 10 marks typically needs several distinct stages shown.

When asked to comment on a model, address assumptions, limitations, and whether the mathematical result fits the real-world data.