Karnataka SSLC Mathematics (KSEAB) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Euclid's division lemma

For positive integers a and b, there exist unique integers q and r such that a = bq + r with 0 ≤ r < b. It underlies the Euclidean algorithm for HCF.

HCF–LCM product rule

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. This identity does not extend to three or more numbers.

Terminating decimal condition

A rational number p/q in lowest terms has a terminating decimal expansion exactly when q is of the form 2^m × 5^n, where m and n are non-negative integers.

nth term of an AP

For an arithmetic progression with first term a and common difference d, the nth term is aₙ = a + (n − 1)d. This gives any term without listing earlier ones.

Sum of first n terms, AP

Sₙ = (n/2)[2a + (n − 1)d]. When the last term l is known, use Sₙ = (n/2)(a + l) instead.

Arithmetic mean of two terms

If a, b, c are in AP then b is the arithmetic mean of a and c, with b = (a + c)/2. The common difference is then (c − a)/2.

Unique solution condition

A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has a unique solution exactly when a₁/a₂ ≠ b₁/b₂, giving intersecting lines.

Coincident lines condition

The pair has infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂. The two equations then represent the same straight line.

Parallel lines condition

The pair has no solution when a₁/a₂ = b₁/b₂ but the ratios differ from c₁/c₂. The lines are parallel and never intersect.

Remainder theorem

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). If p(a) = 0, then (x − a) is a factor of p(x).

Sum and product of zeros

For the quadratic ax² + bx + c, the zeros α and β satisfy α + β = −b/a and αβ = c/a. These let you build or check a quadratic.

Degree and number of zeros

A polynomial of degree n has at most n real zeros. On a graph, these zeros are the x-coordinates where the curve crosses or touches the x-axis.

Discriminant of a quadratic

For ax² + bx + c = 0, D = b² − 4ac. D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots.

Quadratic formula

The roots of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)]/2a. The formula works even when factorisation is not obvious.

Sum and product of roots

For ax² + bx + c = 0 with roots α and β, α + β = −b/a and αβ = c/a. These relations help verify answers found by factorisation.

Basic Proportionality Theorem

A line drawn parallel to one side of a triangle intersects the other two sides in the same ratio. Its converse establishes that a line dividing two sides proportionally is parallel to the third.

Area ratio of similar triangles

The ratio of the areas of two similar triangles equals the square of the ratio of any pair of corresponding sides, medians or altitudes.

Pythagoras theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Its converse identifies a right angle from side lengths.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. Applied to the origin it reduces to √(x² + y²).

Section formula

The point dividing the segment joining (x₁, y₁) and (x₂, y₂) internally in the ratio m:n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

Collinearity by area

Three points are collinear exactly when the area of the triangle they form is zero. The area formula uses the coordinates of all three vertices.

Trigonometric ratios in right triangle

Relative to an acute angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. The reciprocals are cosec, sec and cot.

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Angle of elevation

The angle of elevation is the angle between the horizontal line of sight and the line of sight to an object above the observer's eye level.

Tangent perpendicular to radius

At the point of contact, a tangent to a circle is perpendicular to the radius drawn to that point. This gives a right angle for calculations.

Equal tangents from external point

The two tangent segments drawn from an external point to a circle are equal in length. Their lengths satisfy PT² = OP² − r².

Area of a sector

For a circle of radius r and a sector with central angle θ, the area is (θ/360°) × πr². The corresponding arc length is (θ/360°) × 2πr.

Cone slant height relation

For a right circular cone of radius r and height h, the slant height is l = √(r² + h²). Curved surface area is πrl and volume is ⅓πr²h.

Volume conservation on recasting

When a solid is melted and recast into another shape, the total volume remains unchanged. Equating the two volume formulas gives the unknown dimension.

Median of 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 and h the class size.

Empirical mean-median-mode relation

For moderately skewed distributions, mode = 3 median − 2 mean. It allows one measure to be estimated when the other two are known.

Classical probability

When outcomes are equally likely, P(E) equals the number of outcomes favourable to E divided by the total number of possible outcomes. Values always lie between 0 and 1.