Karnataka SSLC Mathematics (KSEAB) cheat sheet

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

Euclid's algorithm finds HCF by repeated division: HCF(a, b) = HCF(b, r), continuing until the remainder is 0.

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

For positive integers a and b: HCF(a, b) × LCM(a, b) = a × b; this identity fails for three or more numbers.

A rational number p/q (in lowest terms) has a terminating decimal exactly when q = 2^m × 5^n for non-negative integers m and n.

A real number is irrational if its decimal expansion is non-terminating and non-recurring; √2, √3, √5 and π are irrational.

The sum or difference of a rational and an irrational number is irrational; the product of a non-zero rational and an irrational is irrational.

If a prime p divides a², then p divides a; this is the key step in proving √p irrational for prime p.

An AP has a constant common difference d = aₙ₊₁ − aₙ; d may be positive, negative or zero.

nth term: aₙ = a + (n − 1)d, where a is the first term.

Sum of first n terms: Sₙ = (n/2)[2a + (n − 1)d].

Sum using first and last term: Sₙ = (n/2)(a + l), where l is the nth term.

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

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

A sequence is an AP if and only if aₙ₊₁ − aₙ is the same for every consecutive pair.

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

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare a₁/a₂, b₁/b₂, c₁/c₂ before solving.

Unique solution (intersecting lines): a₁/a₂ ≠ b₁/b₂.

Infinitely many solutions (coincident lines): a₁/a₂ = b₁/b₂ = c₁/c₂.

No solution (parallel lines): a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

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

Elimination method: multiply the equations to make one coefficient equal in magnitude, then add or subtract to eliminate that variable.

A pair of linear equations in two variables can have zero, one or infinitely many solutions - never exactly two.

Graphically, the solution is the point of intersection of the two lines; the lines may intersect, be parallel, or coincide.

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

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

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

A quadratic polynomial with given zeros α and β is k[x² − (α + β)x + αβ] for any non-zero constant k.

A polynomial of degree n has at most n real zeros; its graph meets the x-axis at no more than n points.

Zeros of a polynomial are the x-coordinates where its graph intersects the x-axis.

Standard form: ax² + bx + c = 0 with a ≠ 0.

Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a.

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

Sum of roots = −b/a; product of roots = c/a.

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

Completing the square converts ax² + bx + c = 0 into (x + b/2a)² = (b² − 4ac)/4a².

For a quadratic with equal roots, D = 0, so b² = 4ac.

Word problems: form the equation from the stated condition, reject roots that are invalid in context (such as negative lengths).

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 the same ratio, it is parallel to the third side.

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

Ratio of areas of two similar triangles equals the square of the ratio of any pair of corresponding sides.

Ratio of corresponding medians, altitudes, angle bisectors and perimeters of similar triangles equals the ratio of corresponding sides.

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.

In a right triangle, the altitude to the hypotenuse creates two triangles similar to each other and to the original.

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

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

Section formula for dividing AB internally in ratio m:n: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Midpoint of (x₁, y₁) and (x₂, y₂): ((x₁ + x₂)/2, (y₁ + y₂)/2).

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃): ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.

Three points are collinear exactly when the area of the triangle they form is zero.

A point on the x-axis has y = 0; a point on the y-axis has x = 0.

Centroid of a triangle is the average of the coordinates of its three vertices.

In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

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

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

sin 0° = 0, cos 0° = 1, tan 0° = 0; sin 90° = 1, cos 90° = 0, tan 90° is undefined.

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; angle of depression is measured downward from the horizontal.

In height-and-distance problems, drawing the right triangle and labelling the known side and angle is the first step before applying a ratio.

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

The lengths of the two tangents drawn from an external point to a circle are equal.

The tangent at any point of a circle is perpendicular to the radius through that point; the perpendicular distance from the centre to a tangent equals the radius.

Number of tangents from a point: 0 inside the circle, 1 on the circle, 2 from an external point.

A parallelogram circumscribing a circle is a rhombus.

The angle between two tangents from an external point is supplementary to the angle subtended by the line joining the centre to that point.

Construction of a tangent at a point on a circle: draw the radius to that point and construct a perpendicular at the point.

Construction of tangents from an external point: draw the circle with the given external point and the centre as diameter; its intersections with the original circle are the points of contact.

Circumference of a circle = 2πr; area of a circle = πr².

Length of an arc of sector angle θ: (θ/360°) × 2πr.

Area of a sector of angle θ: (θ/360°) × πr².

Area of a minor segment = area of sector − area of the triangle formed by the two radii and the chord.

Area of a ring between radii R and r is π(R² − r²).

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

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

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

When a solid is melted and recast, volume is conserved; when a hollow shape is filled, the volume transferred equals the capacity used.

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

Mean by assumed-mean method: x̄ = a + (Σfᵢdᵢ)/(Σfᵢ), where dᵢ = xᵢ − a.

Mean by step-deviation method: x̄ = a + h × (Σfᵢuᵢ)/(Σfᵢ), where uᵢ = (xᵢ − a)/h.

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

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

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

Cumulative frequency curve (ogive) gives the median as the x-coordinate where the cumulative frequency equals n/2.

Class mark of a class is the average of its upper and lower limits and is used as xᵢ in mean calculations.

For an event E, 0 ≤ P(E) ≤ 1; P(impossible event) = 0 and P(certain event) = 1.

The sum of probabilities of all elementary outcomes of an experiment is 1.

Complement rule: P(not E) = 1 − P(E).

Classical probability: P(E) = number of outcomes favourable to E divided by total number of equally likely outcomes.

For a fair die, each of the six outcomes has probability 1/6; for a fair coin, each face has probability ½.

A standard deck has 52 cards: 26 red, 26 black, 4 suits of 13 cards each, 4 aces and 12 face cards.

P(E) + P(not E) = 1, so the probability of at least one occurrence can be found through its complement.

In a pack of 52 cards, the probability of drawing a king is 4/52 = 1/13 and of drawing a red card is 26/52 = ½.