UPMSP High School Mathematics cheat sheet

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

Fundamental Theorem of Arithmetic: every integer greater than 1 is a unique product of primes up to order.

HCF × LCM = product of the two numbers, for any two positive integers.

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.

Irrational numbers include √p for prime p, and sums or products of a nonzero rational with an irrational are irrational.

Between any two distinct real numbers there are infinitely many rationals and infinitely many irrationals.

For positive reals a, b: √(ab) is the geometric mean and (a+b)/2 is the arithmetic mean, with AM ≥ GM and equality only when a = b.

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, and a^0 = 1 for a ≠ 0.

Remainder Theorem: the remainder on dividing p(x) by (x − a) is p(a).

Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.

Division algorithm for polynomials: p(x) = g(x)·q(x) + r(x), where deg r < deg g or r = 0.

For a quadratic ax² + bx + c with roots α, 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.

A polynomial of degree n has at most n real roots, and its graph crosses the x-axis at each root of odd multiplicity.

To simplify a rational expression, factor numerator and denominator fully and cancel common factors, stating the excluded values where the denominator is zero.

Symmetric expressions such as α² + β² = (α + β)² − 2αβ are evaluated from the sum and product of roots without finding the roots.

Standard form: ax + by + c = 0 with a, b not both zero; its graph is a straight line.

A pair a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 is consistent and unique when a₁/a₂ ≠ b₁/b₂ (intersecting lines).

The pair has infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines).

The pair has no solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines).

Graphical method: draw both lines and read the intersection point; parallel lines signal an inconsistent pair.

Substitution method: express one variable from one equation and substitute into the other to obtain a single-variable equation.

Elimination method: multiply equations to equalise one coefficient, then add or subtract to eliminate that variable.

Cross-multiplication: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), valid when the last denominator is nonzero.

Standard form ax² + bx + c = 0 with a ≠ 0; the discriminant is D = b² − 4ac.

Roots by formula: x = (−b ± √D)/(2a), valid whenever D ≥ 0.

D > 0 gives two distinct real roots, D = 0 gives two equal real roots, D < 0 gives no real roots.

Factorisation: split the middle term into two parts whose product is ac and whose sum is b.

Sum of roots α + β = −b/a and product αβ = c/a; use them to build the equation x² − (α+β)x + αβ = 0.

Completing the square: ax² + bx + c = a(x + b/2a)² + (c − b²/4a), which locates the vertex of the parabola.

For a quadratic with rational coefficients, irrational roots occur in conjugate pairs p ± √q.

Word problems: form the equation from the stated relation, solve, then reject roots that violate the physical conditions of the problem.

An AP has a constant common difference d = aₙ − aₙ₋₁; the nth term is aₙ = a + (n − 1)d.

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

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

If a, b, c are in AP then 2b = a + c; the arithmetic mean of two numbers is their average.

In an AP, the sum of terms equidistant from the ends is constant: a₁ + aₙ = a₂ + aₙ₋₁, and generally aᵣ + a₍ₙ₊₁₋ᵣ₎ is constant.

Terms can be chosen symmetrically as a − d, a, a + d (three terms) or a − 3d, a − d, a + d, a + 3d (four terms) to simplify sums.

Given Sₙ, the nth term is aₙ = Sₙ − Sₙ₋₁ for n ≥ 2, with a₁ = S₁.

The sum of the first n natural numbers is n(n + 1)/2, the simplest AP with a = 1 and d = 1.

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

Distance of (x, y) from the origin is √(x² + y²).

Section formula: the point dividing the join of A and B in ratio m : n internally is ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).

Midpoint formula is the section formula with m = n = 1, giving ((x₁ + x₂)/2, (y₁ + y₂)/2).

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 collinear points.

Three points A, B, C are collinear exactly when AB + BC = AC or the triangle area is zero.

Centroid of a triangle is the average of its vertices: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).

A point on the x-axis has coordinates (x, 0) and a point on the y-axis has coordinates (0, y); use this to set up unknowns.

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

Converse of BPT: if a line divides two sides of a triangle in equal ratios, it is parallel to the third side.

Two triangles are similar by AA (or AAA), SSS ratio, or SAS ratio criteria.

Areas of similar triangles are in the ratio of the squares of corresponding sides.

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

Pythagoras theorem: in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Converse of Pythagoras: if a² + b² = c² in a triangle, the angle opposite side c is a right angle.

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

A tangent to a circle is perpendicular to the radius at the point of contact.

From an external point, the two tangents to a circle are equal in length.

The lengths of tangents from an external point P satisfy PA = PB, and OP bisects the angle between them.

Angle in a semicircle is a right angle; the angle in the major segment is half the central angle on the same arc.

Angles in the same segment of a circle are equal.

The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.

A cyclic quadrilateral has opposite angles supplementary, and its exterior angle equals the interior opposite angle.

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

To divide a segment AB in ratio m : n, draw a ray from A with m + n equal marks, join the last mark to B, and draw a parallel through the mth mark.

To construct a triangle similar to a given triangle with scale factor k < 1, use a ray with equal divisions and draw parallels to obtain the reduced sides.

To construct a tangent at a point on a circle, draw the radius to that point and erect a perpendicular at the point.

To construct tangents from an external point P, draw the circle with OP as diameter; its intersections with the given circle are the points of contact.

Angle bisector construction: arcs of equal radius from the two sides intersect on the bisector ray from the vertex.

Perpendicular bisector construction: equal arcs from both endpoints intersect at two points whose join is the perpendicular bisector.

Constructing an angle of 60° uses an equilateral triangle arc; 30° and 15° come from bisecting 60° and 30° respectively.

Justify each construction by naming the theorem used, such as BPT for division of a segment or the right-angle property for tangents.

Ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, with cosec, sec, cot as reciprocals.

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

Standard values: sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2; cos values are reversed; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

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 height-and-distance problems, draw a right triangle, mark the known angle at the observer, and use tan when both legs are involved.

When two angles of elevation from points on a line are given, set the common height h as the unknown and write two tangent equations.

The angle of elevation of a point equals the angle of depression of the observer from that point, since they are alternate angles.

Circle: area = πr², circumference = 2πr; 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 the semiperimeter.

Trapezium area = ½(a + b)h; rhombus area = ½ d₁d₂; parallelogram area = base × height.

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

Cone: slant height l = √(r² + h²), curved surface = πrl, total surface = πr(l + r), volume = ⅓πr²h.

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

Cube: total surface = 6a², volume = a³; cuboid: total surface = 2(lb + bh + hl), volume = lbh.

When a solid is melted and recast, volume is conserved; when it is immersed, the rise in water level times the base area equals the solid's volume.

Mean of raw data = Σxᵢ/n; for a frequency table, mean = Σfᵢxᵢ/Σfᵢ.

Median of grouped data = 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 width.

Mode of grouped data = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]·h, with f₁ the modal frequency and f₀, f₂ the neighbouring frequencies.

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

Cumulative frequency curves (ogives) meet at the median; the less-than and more-than ogives intersect at (median, n/2).

Class mark is the midpoint of a class interval and is used as xᵢ in the mean formula.

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

For complementary events P(E) + P(not E) = 1; for mutually exclusive events P(A or B) = P(A) + P(B).