Haryana BSEH Secondary Mathematics - Standard flashcards

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Euclid's division lemma

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

Fundamental Theorem of Arithmetic

Every integer greater than 1 can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors.

HCF–LCM product rule

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

Irrationality of √2

Suppose √2 = p/q in lowest terms; then p² = 2q², so p is even, say p = 2k. Then q is also even, contradicting lowest terms.

Remainder theorem

When a polynomial p(x) is divided by (x − a), the remainder equals p(a). Hence (x − a) is a factor exactly when p(a) = 0.

Sum and product of zeros

For ax² + bx + c with zeros α and β, the sum α + β equals −b/a and the product αβ equals c/a.

Polynomial division algorithm

For polynomials p(x) and non-zero g(x), there exist unique q(x) and r(x) with p(x) = g(x)q(x) + r(x), where r(x) is zero or has degree less than g(x).

Quadratic from given zeros

If α and β are the zeros, the polynomial is k[x² − (α + β)x + αβ] for any non-zero constant k, since the sum and product determine it.

Condition for unique solution

The pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has a unique solution exactly when a₁/a₂ ≠ b₁/b₂, meaning the lines intersect at one point.

Parallel lines condition

The pair has no solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂. The lines are parallel and the system is inconsistent.

Coincident lines condition

The pair has infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂. The equations represent the same line and the system is dependent.

Cross-multiplication method

For the standard pair, x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), provided a₁b₂ − a₂b₁ is not zero.

Quadratic formula

The roots of ax² + bx + c = 0, with a ≠ 0, are x = [−b ± √(b² − 4ac)]/(2a). The expression under the root is the discriminant.

Discriminant and nature of roots

For ax² + bx + c = 0, D = b² − 4ac. If D > 0 roots are real and distinct; D = 0 roots are equal; D < 0 no real roots exist.

Equal roots condition

A quadratic has equal real roots exactly when b² − 4ac = 0, and each root then equals −b/(2a).

nth term of an AP

The nth term of an arithmetic progression with first term a and common difference d is aₙ = a + (n − 1)d.

Sum of n terms of AP

Sₙ = (n/2)[2a + (n − 1)d], or equivalently Sₙ = (n/2)(a + l) when the last term l is known.

AP condition for three terms

Three numbers a, b, c are in arithmetic progression exactly when 2b = a + c, that is, b is the arithmetic mean of a and c.

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 parallelism from equal ratios.

Ratio of areas of similar triangles

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

Similarity criteria

Triangles are similar by AA (two equal angles), SSS (all three side ratios equal), or SAS (two sides in ratio with equal included angle).

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²], derived from the Pythagorean 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)).

Area of a triangle

Area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| for vertices (x₁,y₁), (x₂,y₂), (x₃,y₃). A zero result means the points are collinear.

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 upward line from the observer to an object above eye level.

Tangent–radius perpendicularity

A tangent to a circle is perpendicular to the radius drawn to the point of contact, so the radius and tangent form a right angle.

Equal tangents from external point

From a point outside a circle, the two tangent segments drawn to the circle are equal in length and equally inclined to the line joining the point to the centre.

Sector area and arc length

For a sector of angle θ in a circle of radius r, area = (θ/360°)πr² and arc length = (θ/360°) × 2πr.

Cone slant height

For a right circular cone with radius r and height h, the slant height is l = √(r² + h²), and curved surface area is πrl.

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.

Mode of grouped data

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ is the modal class frequency and f₀, f₂ are the frequencies of adjacent classes.

Classical probability

For equally likely outcomes, P(E) = number of favourable outcomes ÷ total number of outcomes, and 0 ≤ P(E) ≤ 1.

Complement rule

For any event E, P(E) + P(not E) = 1, so P(not E) = 1 − P(E). This simplifies 'at least one' probability questions.