ASSEB HSLC General Mathematics cheat sheet

Euclid's division lemma: for integers a and b with b > 0 there exist unique integers q and r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm for HCF: repeatedly replace (a, b) by (b, a mod b) until the remainder is 0; the last non-zero divisor is the HCF.

Fundamental Theorem of Arithmetic: every integer greater than 1 is a unique product of primes apart from the order of the factors.

HCF(a, b) × LCM(a, b) = a × b for positive integers a and b; use it to find the LCM once the HCF is known.

A rational number p/q in lowest terms has a terminating decimal exactly when q = 2^m·5^n; otherwise the decimal is non-terminating recurring.

Divisibility: by 3 or 9 if the digit sum is divisible by 3 or 9; by 4 if the last two digits form a multiple of 4; by 8 if the last three digits form a multiple of 8; by 11 if the alternating digit sum is divisible by 11.

Irrationality proof template: assume √p = a/b in lowest terms, square to get p·b² = a², so p divides a and then p divides b, contradicting lowest terms.

For any two positive integers, HCF divides LCM; also HCF ≤ min(a, b) and LCM ≥ max(a, b).

√(ab) = √a·√b and √(a/b) = √a/√b for a, b ≥ 0; these rules apply only to products and quotients, never to sums: √(a+b) ≠ √a + √b.

Rationalising a single surd: multiply numerator and denominator by √b to turn 1/√b into √b/b.

Rationalising a binomial surd: multiply by the conjugate; 1/(a + √b) becomes (a − √b)/(a² − b).

Comparing surds: raise to a common index, e.g. compare ∛2 and √3 by writing them as 6th roots: 2^(1/3) = ⁶√4 and 3^(1/2) = ⁶√27.

If a + √b = c + √d with a, c rational and √b, √d irrational, then a = c and b = d.

The sum or product of a rational and an irrational number is irrational; the product of two irrationals may be rational (√2·√2 = 2).

For real a, b ≥ 0: (√a + √b)² = a + b + 2√(ab); use this to simplify nested expressions such as √(7 + 4√3) = 2 + √3.

Laws of indices for real exponents: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(1/n) = ⁿ√a; these hold for a > 0.

Remainder Theorem: the remainder when p(x) is divided by (x − a) is p(a); if p(a) = 0 then (x − a) is a factor.

Factor Theorem is the Remainder Theorem with remainder zero; use it to test candidate roots that divide the constant term.

Division algorithm: p(x) = d(x)·q(x) + r(x) with deg r < deg d; the identity holds for all x.

For a quadratic ax² + bx + c with roots α and β: α + β = −b/a and αβ = c/a.

For a cubic ax³ + bx² + cx + d with roots α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

To build a quadratic from roots α and β: x² − (α + β)x + αβ = 0.

Symmetric expressions: α² + β² = (α + β)² − 2αβ; 1/α + 1/β = (α + β)/(αβ); α³ + β³ = (α + β)³ − 3αβ(α + β).

Graph shape: y = ax² + bx + c opens upward if a > 0 and downward if a < 0; the vertex is at x = −b/(2a).

Solving a pair by elimination: multiply the equations to make one coefficient equal in magnitude, then add or subtract to remove that variable.

A pair a₁x + b₁y = c₁, a₂x + b₂y = c₂ has a unique solution if a₁/a₂ ≠ b₁/b₂, no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂, and infinitely many if a₁/a₂ = b₁/b₂ = c₁/c₂.

Cross-multiplication form: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) for ax² + bx + c = 0 with a ≠ 0.

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

Roots are real only when D ≥ 0; the sum of roots is −b/a and the product is c/a.

Completing the square: ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)); the minimum or maximum occurs at x = −b/(2a).

Word problems with two unknowns need two independent conditions; state both equations before solving and check the answer against the original wording.

nth term of an AP: aₙ = a + (n − 1)d, where a is the first term and d the common difference.

Sum of the first n terms of an AP: Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l) where l is the last term.

The nth term from the end of an AP with last term l and difference d is l − (n − 1)d.

Three terms in AP are conveniently written a − d, a, a + d; four terms as a − 3d, a − d, a + d, a + 3d.

If a, b, c are in AP then 2b = a + c; b is the arithmetic mean of a and c.

aₙ = Sₙ − Sₙ₋₁ gives the nth term when only the sum formula is known.

Sum of the first n natural numbers: 1 + 2 + … + n = n(n + 1)/2; sum of first n odd numbers is n².

For a finite AP, the sum equals n × (average of first and last terms); use this when a and l are known but d is not.

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

Section formula: the point dividing the join of P and Q internally in ratio m:n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); the midpoint takes m = n = 1.

Slope through two points: m = (y₂ − y₁)/(x₂ − x₁); the slope is undefined for a vertical line.

Line forms: slope-intercept y = mx + c; point-slope y − y₁ = m(x − x₁); two-point (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁); intercept x/a + y/b = 1; general Ax + By + C = 0 with slope −A/B.

Parallel lines have equal slopes; perpendicular lines satisfy m₁·m₂ = −1.

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; zero area means the points are collinear.

Distance from point (x₁, y₁) to line Ax + By + C = 0 is |Ax₁ + By₁ + C|/√(A² + B²).

Angle θ between two lines satisfies tan θ = |(m₂ − m₁)/(1 + m₁m₂)|.

Standard values: sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1; cos runs the same list in reverse.

tan θ = sin θ/cos θ; cot θ = 1/tan θ; sec θ = 1/cos θ; cosec θ = 1/sin θ.

Pythagorean identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal.

In a right triangle, tan(angle of elevation) = height/horizontal distance; the line of sight is the hypotenuse.

Two-observer problems are solved by writing two tangent equations sharing the same height and eliminating it.

sin θ and cos θ each lie between −1 and 1; sec θ ≥ 1 or sec θ ≤ −1, so sec θ = 1/2 has no solution.

Angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal.

Parallel lines cut by a transversal: corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°.

In a triangle, the exterior angle equals the sum of the two interior opposite angles.

Angle sum of an n-sided polygon is (n − 2) × 180°; each interior angle of a regular n-gon is (n − 2)180°/n and each exterior angle is 360°/n.

Congruence criteria: SSS, SAS, ASA (with AAS), and RHS for right triangles.

Similar triangles have equal corresponding angles and proportional corresponding sides; areas are in the ratio of the squares of corresponding sides.

Basic Proportionality (Thales) Theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio.

The perpendicular from the centre of a circle to a chord bisects the chord.

Pythagoras: in a right triangle, hypotenuse² = sum of squares of the other two sides; converse also holds.

Tangent–radius theorem: a tangent is perpendicular to the radius at the point of contact.

Tangents drawn from an external point to a circle are equal in length.

Angle in a semicircle is 90°; the angle at the centre is twice the angle at the circumference standing on the same arc.

Angles in the same segment are equal; the opposite angles of a cyclic quadrilateral sum to 180°.

Alternate Segment Theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Tangent–secant (power of a point): for tangent PT and secant through P meeting the circle at A and B, PT² = PA·PB.

Two circles touch externally when the distance between centres equals r₁ + r₂ and internally when it equals |r₁ − r₂|.

Triangle area = ½ × base × height; Heron's formula: area = √[s(s − a)(s − b)(s − c)] with s = (a + b + c)/2.

Circle: circumference = 2πr, area = πr²; sector area = (θ/360)πr² and arc length = (θ/360)2πr.

Trapezium area = ½(a + b)h; parallelogram area = base × height; rhombus area = ½ × product of diagonals.

Cuboid: volume = lbh, total surface area = 2(lb + bh + hl); cube: volume = a³, surface area = 6a².

Cylinder: volume = πr²h, curved surface = 2πrh, total surface = 2πr(r + h).

Cone: volume = (1/3)πr²h, curved surface = πrl, total surface = πr(l + r), where slant height l = √(r² + h²).

Sphere: volume = (4/3)πr³, surface area = 4πr²; hemisphere total surface area = 3πr².

When a solid is melted and recast, volume is conserved: set the original volume equal to the new volume to find the unknown dimension.

Mean of raw data = (sum of observations)/n; for a frequency table, mean = Σfᵢxᵢ/Σfᵢ.

Median of n ordered values: the middle value when n is odd, the average of the two middle values when n is even.

Mode is the most frequent value; a data set may have no mode or more than one mode.

For grouped data, median = l + [(n/2 − cf)/f] × h and mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.

Empirical relation for moderately skewed data: mode = 3 × median − 2 × mean.

Class mark of a class is (lower limit + upper limit)/2 and is used as xᵢ in the grouped mean.

Probability of event E: P(E) = favourable outcomes/total equally likely outcomes, with 0 ≤ P(E) ≤ 1.

Complement rule: P(not E) = 1 − P(E); for mutually exclusive events P(A or B) = P(A) + P(B).

Number of subsets of a set with n elements is 2ⁿ; number of proper subsets is 2ⁿ − 1.

For finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract the three pairwise intersections and add back the triple intersection.

De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

A × B = {(a, b) : a ∈ A, b ∈ B}; n(A × B) = n(A) × n(B), and A × B ≠ B × A unless A = B.

A relation from A to B is any subset of A × B; the number of relations is 2^[n(A)·n(B)].

A function f: A → B assigns exactly one element of B to each element of A; every element of A must be used.

Domain is the set of allowed inputs, range is the set of actual outputs, and codomain is the declared target set; range ⊆ codomain.

One-one (injective) means distinct inputs give distinct outputs; onto (surjective) means range equals codomain; both together make the function bijective.

A statement is a sentence that is definitely true or definitely false; questions, commands, and exclamations are not statements.

Negation of 'all A are B' is 'some A are not B'; negation of 'some A are B' is 'no A is B'.

Converse of p → q is q → p; contrapositive is ¬q → ¬p; a conditional and its contrapositive are logically equivalent.

p → q is false only when p is true and q is false; in all other cases it is true.

'p and q' is true only when both are true; 'p or q' is true when at least one is true.

A statement is a tautology if true for every truth assignment and a contradiction if false for every assignment.

Proof by contradiction: assume the negation of the claim, derive a false consequence, and conclude the claim is true.

Counterexample method: one case where the hypothesis holds but the conclusion fails disproves a universal statement.