Bihar BSEB Matric Mathematics flashcards

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Euclid's division lemma

For positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, with 0 ≤ r < b. It is the basis of Euclid's algorithm for finding HCF.

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order in which the prime factors are written.

Terminating decimal condition

A rational number p/q in lowest terms has a terminating decimal expansion exactly when the prime factorisation of q contains only 2s and 5s, that is q = 2^m × 5^n.

HCF–LCM relation

For any two positive integers a and b, the product of their HCF and LCM equals the product of the numbers: HCF(a, b) × LCM(a, b) = a × b.

Remainder theorem

If a polynomial p(x) is divided by the linear polynomial (x − a), then the remainder is p(a). The divisor need not be a factor for this to apply.

Sum and product of zeros

For the quadratic polynomial ax² + bx + c, the sum of the zeros is −b/a and the product of the zeros is c/a, where a is not zero.

Factor theorem

A polynomial p(x) has (x − a) as a factor if and only if p(a) = 0. It converts factor questions into simple substitution and evaluation.

Cubic zeros relations

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

Unique solution condition

A pair of linear equations has exactly one solution when a₁/a₂ ≠ b₁/b₂. Graphically the two straight lines intersect at a single point.

Parallel lines condition

A pair of linear equations has no solution when a₁/a₂ = b₁/b₂ but c₁/c₂ differs. The lines are parallel and never meet, so the pair is inconsistent.

Coincident lines condition

A pair of linear equations has infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂. The two equations represent the same line, so every point on it satisfies both.

Cross-multiplication method

For two linear equations, the solution is given by x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), valid when the last denominator is non-zero.

Discriminant of a quadratic

For ax² + bx + c = 0, the discriminant D = b² − 4ac decides the nature of roots: positive gives two distinct real roots, zero gives equal roots, negative gives no real roots.

Quadratic formula

The roots of ax² + bx + c = 0, with a not zero, are x = (−b ± √(b² − 4ac)) / 2a, obtained by completing the square on the general equation.

Equal roots condition

A quadratic equation ax² + bx + c = 0 has two equal real roots exactly when its discriminant is zero, that is when b² = 4ac.

nth term of an AP

In an arithmetic progression with first term a and common difference d, the nth term is aₙ = a + (n − 1)d. It grows linearly with n.

Sum of n terms of AP

The sum of the first n terms of an AP is Sₙ = (n/2)[2a + (n − 1)d], or equivalently Sₙ = (n/2)(a + l) when the last term l is known.

Arithmetic mean of three terms

Three numbers a, b, c are in arithmetic progression exactly when 2b = a + c, so the middle number b is the arithmetic mean of the outer two.

Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides those two sides in the same ratio.

Similar triangle area ratio

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

Pythagoras theorem

In a right-angled 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) in the plane is √[(x₂ − x₁)² + (y₂ − y₁)²], derived from the Pythagoras theorem.

Section formula

The point dividing the segment joining (x₁, y₁) and (x₂, y₂) internally in 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 formed by them is zero, which happens when the area formula gives ½ × 0.

Trigonometric ratio definitions

In a right triangle, sin θ = perpendicular/hypotenuse, cos θ = base/hypotenuse, and tan θ = perpendicular/base, with cosec, sec, and cot as their reciprocals.

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1. Dividing this identity by cos²θ gives 1 + tan²θ = sec²θ, and by sin²θ gives 1 + cot²θ = cosec²θ.

Angle of elevation

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

Tangent perpendicular to radius

A tangent to a circle is perpendicular to the radius drawn to the point of contact. This fact underlies most tangent-length and tangent-angle problems.

Equal tangents from external point

The two tangent segments drawn to a circle from an external point are equal in length, and they subtend equal angles at the centre of the circle.

Area of a sector

A sector of a circle of radius r with central angle θ has area (θ/360°) × πr², and its arc has length (θ/360°) × 2πr.

Volume of a sphere

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². A hemisphere has half the volume, (2/3)πr³, and total surface area 3πr².

Empirical mean-mode-median relation

For moderately skewed data, the three central measures satisfy 3 Median = Mode + 2 Mean. It allows one measure to be estimated when the other two are known.