SPM Add Maths cheat sheet

A function maps each element of its domain to exactly one value; a one-to-one function also gives distinct outputs for distinct inputs.

Absolute value function: |x| = x for x ≥ 0 and |x| = −x for x < 0; its graph is V-shaped with vertex at the origin.

Composite function: (fg)(x) = f(g(x)); the range of g must lie inside the domain of f for fg to be defined.

fg generally ≠ gf; composition is associative but not commutative.

Inverse exists only for one-to-one functions; find it by writing y = f(x), swapping x and y, then solving for y.

Key inverse identities: f⁻¹(f(x)) = x for x in the domain, and f(f⁻¹(x)) = x for x in the range of f.

The graph of f⁻¹ is the reflection of the graph of f in the line y = x.

Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.

Roots of ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] / (2a).

Discriminant D = b² − 4ac: D > 0 two distinct real roots; D = 0 two equal real roots; D < 0 no real roots.

Sum of roots α + β = −b/a; product of roots αβ = c/a.

Form a quadratic from roots α, β: x² − (α + β)x + αβ = 0.

Completing the square: ax² + bx + c = a(x + b/2a)² + c − b²/4a.

Vertex of y = ax² + bx + c is at x = −b/(2a); y = a(x − h)² + k has vertex (h, k).

a > 0 opens upward with a minimum; a < 0 opens downward with a maximum.

Axis of symmetry x = −b/(2a) passes through the vertex; the curve is symmetric about it.

Solving inequalities: find roots, sketch the parabola, and read off where the graph lies above or below the x-axis.

Linear–linear in two unknowns: eliminate one variable by making coefficients equal and opposite, then add.

Linear–quadratic: solve the linear equation for one variable and substitute into the quadratic.

Substitution into a quadratic typically yields a quadratic; its discriminant tells how many intersection points exist.

D > 0 two solutions (two intersection points), D = 0 one solution (tangent), D < 0 no real solution.

For three unknowns, use three linear equations: eliminate one variable from two pairs to get two equations in two unknowns.

Graphical meaning: each solution pair (x, y) is a point where the two graphs intersect.

Always substitute candidate solutions back into both original equations to reject extraneous values.

A pair of linear equations has no solution when the lines are parallel (equal gradient, different intercept).

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ.

Fractional powers: a^(m/n) = ⁿ√(aᵐ); also (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ.

Definition: log_a N = x means aˣ = N, with a > 0, a ≠ 1, N > 0.

Log laws: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(xⁿ) = n log_a x.

log_a a = 1 and log_a 1 = 0 for every valid base a.

Change of base: log_a b = log_c b / log_c a; in particular log_a b = 1 / log_b a.

Solving aˣ = b: take logs to get x = log b / log a.

Solving log equations: combine into a single log, convert to exponential form, then check that all arguments stay positive.

Distance between (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²].

Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have m₁ = m₂.

Perpendicular lines satisfy m₁ × m₂ = −1.

Point–gradient form: y − y₁ = m(x − x₁); gradient–intercept form: y = mx + c.

Area of a polygon by coordinates: ½|x₁y₂ + x₂y₃ + … + xₙy₁ − y₁x₂ − y₂x₃ − … − yₙx₁|.

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

Locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.

Radian definition: 1 rad is the angle subtended at the centre by an arc equal in length to the radius.

Conversion: π rad = 180°; degrees to radians multiply by π/180; radians to degrees multiply by 180/π.

Arc length s = rθ, where θ is in radians.

Sector area A = ½r²θ (θ in radians), equivalently A = ½rs.

Area of segment = sector area − triangle area = ½r²θ − ½r² sin θ.

Triangle area from two sides and included angle: ½ab sin C.

Perimeter of a sector = 2r + rθ; perimeter of a segment = arc length + chord length.

Chord length for angle θ at the centre: 2r sin(θ/2).

Limit definition: f′(x) = lim(h→0) [f(x + h) − f(x)] / h.

Power rule: d/dx (axⁿ) = anxⁿ⁻¹; derivative of a constant is 0.

Product rule: d/dx (uv) = u dv/dx + v du/dx.

Quotient rule: d/dx (u/v) = (v du/dx − u dv/dx) / v².

Chain rule: d/dx [f(g(x))] = f′(g(x)) × g′(x).

Gradient of the tangent at x = a is f′(a); normal gradient is −1/f′(a).

Stationary points where f′(x) = 0; use f″(x) > 0 for a minimum, f″(x) < 0 for a maximum.

Equation of tangent at (a, f(a)): y − f(a) = f′(a)(x − a).

Rates of change: dy/dt = (dy/dx)(dx/dt); small changes δy ≈ f′(x) δx.

Indefinite integral: ∫axⁿ dx = axⁿ⁺¹/(n + 1) + c for n ≠ −1.

Definite integral ∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f.

Area between curve and x-axis: integrate; take the absolute value of parts below the axis separately.

Area between two curves: ∫ₐᵇ (upper − lower) dx over the intersection limits.

Volume of revolution about the x-axis: V = π∫ₐᵇ y² dx.

Reversing limits changes sign: ∫ₐᵇ f dx = −∫ᵦᵃ f dx.

Composite integrals of the form ∫k(ax + b)ⁿ dx = k(ax + b)ⁿ⁺¹ / [a(n + 1)] + c.

Integration is the reverse of differentiation; check an antiderivative by differentiating it back.

A vector has magnitude and direction; a scalar has magnitude only.

In component form, magnitude of xi + yj is √(x² + y²).

Unit vector in the direction of a is â = a/|a|.

Addition and subtraction are done component-wise; k a scales each component by k.

Position vector of a point relative to origin O is written OA with arrow; AB = OB − OA.

Parallel vectors satisfy a = k b for some scalar k; equal vectors have equal components.

Dot product a·b = |a||b| cos θ = x₁x₂ + y₁y₂; perpendicular vectors give a·b = 0.

Angle between vectors: cos θ = (a·b)/(|a||b|).

If P divides AB in ratio m : n, then OP = (n OA + m OB)/(m + n).

Sine rule: a/sin A = b/sin B = c/sin C (also true for the circumdiameter 2R).

Cosine rule: a² = b² + c² − 2bc cos A; rearrange to cos A = (b² + c² − a²)/(2bc).

Area of triangle = ½ab sin C.

Use sine rule when given two angles and a side, or two sides and a non-included angle (check the ambiguous case).

Use cosine rule when given three sides, or two sides and the included angle.

Ambiguous case: with two sides and a non-included angle, solutions may number 0, 1, or 2.

Angle sum: A + B + C = 180°; the largest side faces the largest angle.

Bearings are measured clockwise from north; convert to standard angles before applying the rules.

Index number I = (Q₁/Q₀) × 100, where Q₀ is the base-year value and Q₁ the current value.

Price index uses prices; quantity index uses quantities; both take base year = 100.

Composite index = Σ(W × I)/ΣW, where W is the weight and I each component index.

Weights are usually quantities, values, or percentages of expenditure.

Base-year value is always 100; an index of 125 means a 25% increase from base.

Changing the base year requires rebasing: new index = (old index / old index of new base) × 100.

Index numbers measure relative change, not absolute value, so they are unit-free.

Displacement s is position relative to a fixed point; velocity v = ds/dt; acceleration a = dv/dt = d²s/dt².

Velocity is zero at turning points; acceleration is zero when velocity is maximum or minimum.

Particle moves forward when v > 0 and backward when v < 0.

Total distance travelled requires splitting time intervals at each turning point and summing |displacement changes|.

Acceleration is positive when velocity increases (in the positive direction); negative acceleration means deceleration when moving forward.

Displacement from velocity: s = ∫v dt; velocity from acceleration: v = ∫a dt, using initial conditions to find constants.

Maximum or minimum velocity occurs where a = dv/dt = 0.

Constant acceleration formulas: v = u + at, s = ut + ½at², v² = u² + 2as.

A function maps each element of its domain to exactly one value; a one-to-one function also gives distinct outputs for distinct inputs.

Absolute value function: |x| = x for x ≥ 0 and |x| = −x for x < 0; its graph is V-shaped with vertex at the origin.

Composite function: (fg)(x) = f(g(x)); the range of g must lie inside the domain of f for fg to be defined.

fg generally ≠ gf; composition is associative but not commutative.

Inverse exists only for one-to-one functions; find it by writing y = f(x), swapping x and y, then solving for y.

Key inverse identities: f⁻¹(f(x)) = x for x in the domain, and f(f⁻¹(x)) = x for x in the range of f.

The graph of f⁻¹ is the reflection of the graph of f in the line y = x.

Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.

Roots of ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] / (2a).

Discriminant D = b² − 4ac: D > 0 two distinct real roots; D = 0 two equal real roots; D < 0 no real roots.

Sum of roots α + β = −b/a; product of roots αβ = c/a.

Form a quadratic from roots α, β: x² − (α + β)x + αβ = 0.

Completing the square: ax² + bx + c = a(x + b/2a)² + c − b²/4a.

Vertex of y = ax² + bx + c is at x = −b/(2a); y = a(x − h)² + k has vertex (h, k).

a > 0 opens upward with a minimum; a < 0 opens downward with a maximum.

Axis of symmetry x = −b/(2a) passes through the vertex; the curve is symmetric about it.

Solving inequalities: find roots, sketch the parabola, and read off where the graph lies above or below the x-axis.

Linear–linear in two unknowns: eliminate one variable by making coefficients equal and opposite, then add.

Linear–quadratic: solve the linear equation for one variable and substitute into the quadratic.

Substitution into a quadratic typically yields a quadratic; its discriminant tells how many intersection points exist.

D > 0 two solutions (two intersection points), D = 0 one solution (tangent), D < 0 no real solution.

For three unknowns, use three linear equations: eliminate one variable from two pairs to get two equations in two unknowns.

Graphical meaning: each solution pair (x, y) is a point where the two graphs intersect.

Always substitute candidate solutions back into both original equations to reject extraneous values.

A pair of linear equations has no solution when the lines are parallel (equal gradient, different intercept).

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ.

Fractional powers: a^(m/n) = ⁿ√(aᵐ); also (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ.

Definition: log_a N = x means aˣ = N, with a > 0, a ≠ 1, N > 0.

Log laws: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(xⁿ) = n log_a x.

log_a a = 1 and log_a 1 = 0 for every valid base a.

Change of base: log_a b = log_c b / log_c a; in particular log_a b = 1 / log_b a.

Solving aˣ = b: take logs to get x = log b / log a.

Solving log equations: combine into a single log, convert to exponential form, then check that all arguments stay positive.

Distance between (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²].

Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have m₁ = m₂.

Perpendicular lines satisfy m₁ × m₂ = −1.

Point–gradient form: y − y₁ = m(x − x₁); gradient–intercept form: y = mx + c.

Area of a polygon by coordinates: ½|x₁y₂ + x₂y₃ + … + xₙy₁ − y₁x₂ − y₂x₃ − … − yₙx₁|.

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

Locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.

Radian definition: 1 rad is the angle subtended at the centre by an arc equal in length to the radius.

Conversion: π rad = 180°; degrees to radians multiply by π/180; radians to degrees multiply by 180/π.

Arc length s = rθ, where θ is in radians.

Sector area A = ½r²θ (θ in radians), equivalently A = ½rs.

Area of segment = sector area − triangle area = ½r²θ − ½r² sin θ.

Triangle area from two sides and included angle: ½ab sin C.

Perimeter of a sector = 2r + rθ; perimeter of a segment = arc length + chord length.

Chord length for angle θ at the centre: 2r sin(θ/2).

Limit definition: f′(x) = lim(h→0) [f(x + h) − f(x)] / h.

Power rule: d/dx (axⁿ) = anxⁿ⁻¹; derivative of a constant is 0.

Product rule: d/dx (uv) = u dv/dx + v du/dx.

Quotient rule: d/dx (u/v) = (v du/dx − u dv/dx) / v².

Chain rule: d/dx [f(g(x))] = f′(g(x)) × g′(x).

Gradient of the tangent at x = a is f′(a); normal gradient is −1/f′(a).

Stationary points where f′(x) = 0; use f″(x) > 0 for a minimum, f″(x) < 0 for a maximum.

Equation of tangent at (a, f(a)): y − f(a) = f′(a)(x − a).

Rates of change: dy/dt = (dy/dx)(dx/dt); small changes δy ≈ f′(x) δx.

Indefinite integral: ∫axⁿ dx = axⁿ⁺¹/(n + 1) + c for n ≠ −1.

Definite integral ∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f.

Area between curve and x-axis: integrate; take the absolute value of parts below the axis separately.

Area between two curves: ∫ₐᵇ (upper − lower) dx over the intersection limits.

Volume of revolution about the x-axis: V = π∫ₐᵇ y² dx.

Reversing limits changes sign: ∫ₐᵇ f dx = −∫ᵦᵃ f dx.

Composite integrals of the form ∫k(ax + b)ⁿ dx = k(ax + b)ⁿ⁺¹ / [a(n + 1)] + c.

Integration is the reverse of differentiation; check an antiderivative by differentiating it back.

A vector has magnitude and direction; a scalar has magnitude only.

In component form, magnitude of xi + yj is √(x² + y²).

Unit vector in the direction of a is â = a/|a|.

Addition and subtraction are done component-wise; k a scales each component by k.

Position vector of a point relative to origin O is written OA with arrow; AB = OB − OA.

Parallel vectors satisfy a = k b for some scalar k; equal vectors have equal components.

Dot product a·b = |a||b| cos θ = x₁x₂ + y₁y₂; perpendicular vectors give a·b = 0.

Angle between vectors: cos θ = (a·b)/(|a||b|).

If P divides AB in ratio m : n, then OP = (n OA + m OB)/(m + n).

Sine rule: a/sin A = b/sin B = c/sin C (also true for the circumdiameter 2R).

Cosine rule: a² = b² + c² − 2bc cos A; rearrange to cos A = (b² + c² − a²)/(2bc).

Area of triangle = ½ab sin C.

Use sine rule when given two angles and a side, or two sides and a non-included angle (check the ambiguous case).

Use cosine rule when given three sides, or two sides and the included angle.

Ambiguous case: with two sides and a non-included angle, solutions may number 0, 1, or 2.

Angle sum: A + B + C = 180°; the largest side faces the largest angle.

Bearings are measured clockwise from north; convert to standard angles before applying the rules.

Index number I = (Q₁/Q₀) × 100, where Q₀ is the base-year value and Q₁ the current value.

Price index uses prices; quantity index uses quantities; both take base year = 100.

Composite index = Σ(W × I)/ΣW, where W is the weight and I each component index.

Weights are usually quantities, values, or percentages of expenditure.

Base-year value is always 100; an index of 125 means a 25% increase from base.

Changing the base year requires rebasing: new index = (old index / old index of new base) × 100.

Index numbers measure relative change, not absolute value, so they are unit-free.

Displacement s is position relative to a fixed point; velocity v = ds/dt; acceleration a = dv/dt = d²s/dt².

Velocity is zero at turning points; acceleration is zero when velocity is maximum or minimum.

Particle moves forward when v > 0 and backward when v < 0.

Total distance travelled requires splitting time intervals at each turning point and summing |displacement changes|.

Acceleration is positive when velocity increases (in the positive direction); negative acceleration means deceleration when moving forward.

Displacement from velocity: s = ∫v dt; velocity from acceleration: v = ∫a dt, using initial conditions to find constants.

Maximum or minimum velocity occurs where a = dv/dt = 0.

Constant acceleration formulas: v = u + at, s = ut + ½at², v² = u² + 2as.