BSEH Secondary Mathematics - Basic 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; this underlies the Euclidean algorithm for HCF.

Fundamental Theorem of Arithmetic: every integer greater than 1 factors uniquely into primes up to order; HCF is the product of the smallest powers of common primes, LCM the product of the largest powers of all primes, so HCF × LCM = product of the two numbers.

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 repeating, with period determined by the multiplicative order of 10 modulo q after removing factors 2 and 5.

Irrationality proofs use contradiction: if √p = a/b in lowest terms then p·b² = a², forcing p | a and then p | b, contradicting coprimality; the same argument works for any prime p.

For positive reals a and b: √(ab) is the geometric mean, (a+b)/2 the arithmetic mean, and AM ≥ GM with equality iff a = b; between any two distinct rationals lies another rational, so rationals are dense.

Laws of exponents for real bases: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(m−n) = a^m/a^n, a^(1/n) = ⁿ√a, and a^(−n) = 1/a^n; rational exponents extend these laws consistently.

Decimal expansion of p/q is found by long division; the repeating block begins after at most n digits where q = 2^m·5^n·q′ and q′ is coprime to 10.

If a prime p divides a², then p divides a; this lemma is the key step in proving irrationality of √p and in unique factorization arguments.

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

For a quadratic ax² + bx + c with roots α and β: α + β = −b/a and αβ = c/a; the polynomial is k(x² − (α+β)x + αβ) for any nonzero k.

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.

Division algorithm for polynomials: p(x) = g(x)·q(x) + r(x) with deg r < deg g; the algorithm stops when the remainder has degree lower than the divisor.

A linear equation ax + by + c = 0 has infinitely many solutions; its graph is a straight line with slope −a/b and intercepts −c/a (x-axis) and −c/b (y-axis) when nonzero.

Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are consistent and unique-solving if a₁/a₂ ≠ b₁/b₂; parallel (no solution) if a₁/a₂ = b₁/b₂ ≠ c₁/c₂; coincident (infinitely many) if a₁/a₂ = b₁/b₂ = c₁/c₂.

Graphical solution method: plot both lines from two points each; the intersection coordinates satisfy both equations, and parallel lines signal an inconsistent pair.

Algebraic solution methods: substitution isolates one variable; elimination equalizes coefficients of one variable; cross-multiplication uses x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).

Quadratic formula: roots of ax² + bx + c = 0 (a ≠ 0) are x = (−b ± √(b² − 4ac))/(2a); the discriminant D = b² − 4ac decides two distinct real roots (D > 0), one repeated root (D = 0), or no real roots (D < 0).

Sum and product of roots: α + β = −b/a, αβ = c/a; a quadratic with given roots is x² − (α+β)x + αβ = 0 (up to a nonzero constant factor).

Completing the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a); this yields the vertex form and the formula in one step.

An arithmetic progression with first term a and common difference d has nth term aₙ = a + (n − 1)d and sum Sₙ = n/2[2a + (n − 1)d] = n/2(a + l) where l is the last term.

If three numbers are in AP, the middle is the arithmetic mean of the other two: 2b = a + c; inserting k arithmetic means between a and b gives d = (b − a)/(k + 1).

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

Sum of first n natural numbers is n(n+1)/2, an AP with a = 1, d = 1; the sum of first n odd numbers is n².

Word problems: quantities in AP are often written as a − d, a, a + d (three terms) or a − 3d, a − d, a + d, a + 3d (four terms) to simplify the sum equations.

Distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]; distance from origin is √(x² + y²).

Section formula: the point dividing the join of A(x₁, y₁) and B(x₂, y₂) internally in ratio m : n is ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)); midpoint uses m = n = 1.

The coordinates of the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) are ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).

Slope of the line through two points is m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes whose product is −1.

Equation forms: slope-intercept y = mx + c; point-slope y − y₁ = m(x − x₁); two-point (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁); intercept form x/a + y/b = 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.

Three points are collinear iff the slope between any two pairs is equal, or equivalently the triangle area formula gives zero.

To find a point on an axis equidistant from two given points, set the squared distances equal and use that the point has the form (x, 0) or (0, y).

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

Two triangles are similar if (AA) two angles are equal, (SSS) corresponding sides are proportional, or (SAS) two sides are proportional and the included angles are equal.

In similar triangles, corresponding sides, altitudes, medians, and perimeters are all in the same ratio k; areas are in the ratio k².

Pythagoras theorem: in a right triangle the square of the hypotenuse equals the sum of squares of the other two sides; the converse identifies right triangles.

Midpoint theorem: the segment joining midpoints of two sides of a triangle is parallel to the third side and half its length.

Angle bisector theorem: the bisector of an angle of a triangle divides the opposite side in the ratio of the adjacent sides.

Congruence criteria (SSS, SAS, ASA, AAS, RHS) give equal corresponding parts; similarity does not imply congruence unless the ratio is 1.

In an equilateral triangle of side a: altitude = (√3/2)a, area = (√3/4)a²; in a 30-60-90 triangle sides are in ratio 1 : √3 : 2, in a 45-45-90 triangle 1 : 1 : √2.

A tangent to a circle is perpendicular to the radius at the point of contact; from an external point the two tangents are equal in length.

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

Angle in a semicircle is a right angle; the angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining part of the circle.

Angles in the same segment are equal; the opposite angles of a cyclic quadrilateral sum to 180°, and its exterior angle equals the interior opposite angle.

The perpendicular from the centre to a chord bisects the chord; equal chords are equidistant from the centre.

Length of an arc subtending angle θ (in degrees) at the centre of a circle of radius r is (θ/360)·2πr; sector area is (θ/360)·πr².

Construction of a tangent from an external point: draw the circle with diameter equal to the segment joining the centre and the external point; its intersections with the given circle are the points of contact.

Construction of a triangle similar to a given triangle with a given scale factor uses a ray from one vertex divided into equal parts; the number of parts equals the larger of the ratio terms.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; cosec, sec, cot are their reciprocals.

Exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2, tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3; sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0.

Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ; complementary angles satisfy sin(90° − θ) = cos θ and tan(90° − θ) = cot θ.

Angle of elevation is measured upward from the horizontal to the line of sight; angle of depression is measured downward from the horizontal, and the two are equal when taken at opposite ends of the same horizontal level.

Standard heights-and-distances setup: draw a right triangle, label the known distance as adjacent or hypotenuse, then choose the ratio that links the known side to the unknown height.

If two angles of elevation from points on a line are α and β with the observer closer at β, the height h satisfies h = d·tan α·tan β/(tan α − tan β) where d is the distance between observation points.

tan θ = sin θ/cos θ and cot θ = cos θ/sin θ; multiplying an expression by (1 ± sin θ) or its conjugate often converts it into a standard identity.

Circle: circumference 2πr, area πr²; ring (annulus) area π(R² − r²); sector area (θ/360)πr² and arc length (θ/360)2πr.

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

Trapezium area = ½(a + b)h; rhombus area = ½ d₁d₂; parallelogram area = base × height; equilateral triangle area = (√3/4)a².

Cuboid: volume lbh, surface area 2(lb + bh + hl), diagonal √(l² + b² + h²); cube of side a: volume a³, surface area 6a², diagonal a√3.

Cylinder: volume πr²h, curved surface 2πrh, total surface 2πr(r + h); hollow cylinder volume πh(R² − r²).

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

Sphere: volume (4/3)πr³, surface area 4πr²; hemisphere: volume (2/3)πr³, curved surface 2πr², total surface 3πr².

Frustum of a cone: volume (1/3)πh(r₁² + r₁r₂ + r₂²), slant height l = √[h² + (r₁ − r₂)²], curved surface π(r₁ + r₂)l.

When a solid is melted and recast, volume is conserved; when a solid is immersed in a cylinder, the rise in water level equals volume of solid divided by base area of the container.

Mean of raw data: x̄ = (Σxᵢ)/n; for a frequency distribution x̄ = Σfᵢxᵢ / Σfᵢ.

Median of raw data: arrange in order; for odd n the median is the ((n+1)/2)th value, for even n the average of the (n/2)th and (n/2 + 1)th values.

Mode is the value with the highest frequency; a distribution may be bimodal or have no mode if all frequencies are equal.

For grouped data, median = l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class, cf the cumulative frequency before it, f its frequency, h the class size.

For grouped data, mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ is the modal class frequency, f₀ and f₂ the frequencies of the preceding and following classes.

Empirical relation for moderately skewed data: Mode = 3 Median − 2 Mean.

The mean of a grouped distribution computed by the assumed-mean or step-deviation method uses d = (xᵢ − a)/h and x̄ = a + h(Σfᵢdᵢ/Σfᵢ), which simplifies arithmetic.

Cumulative frequency curve (ogive): plotting upper limits against cumulative frequencies and joining by a smooth curve; the median is read at cumulative frequency n/2.

Classical probability: P(E) = number of favourable outcomes / total number of equally likely outcomes; 0 ≤ P(E) ≤ 1.

Complement rule: P(not E) = 1 − P(E); the probability of an impossible event is 0 and of a certain event is 1.

For a fair die: P(even) = 1/2, P(prime) = 1/2, P(number > 4) = 1/3; for two dice there are 36 equally likely ordered outcomes.

For a standard deck of 52 cards: 4 suits of 13 each, 26 red and 26 black, 12 face cards (J, Q, K), 4 aces; P(king) = 1/13, P(red face card) = 3/26.

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

Two coins give sample space {HH, HT, TH, TT}; P(exactly one head) = 1/2, P(at least one head) = 3/4.

A sure event has probability 1 and an impossible event 0; the sum of probabilities of all elementary outcomes in a sample space equals 1.

In problems with balls drawn from a bag, the total outcomes equal the total number of balls and favourable outcomes count balls of the required colour; drawing without replacement changes the denominator on the next draw.