Punjab PSEB Matric Mathematics 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 whole numbers q and r such that a = bq + r, where 0 ≤ r < b. It underpins the Euclidean algorithm for finding the HCF.

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this prime factorisation is unique apart from the order of the factors. It justifies using HCF and LCM through prime factors.

HCF–LCM product rule

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. So LCM = ab ÷ HCF. The rule does not extend directly to three numbers.

Terminating decimal condition

A rational number in lowest terms p/q has a terminating decimal expansion if and only if q has no prime factors other than 2 and 5, i.e. q = 2^m · 5^n.

Irrationality of √2

Assume √2 = a/b with a, b coprime. Then a² = 2b², so a is even; writing a = 2c gives b² = 2c², so b is even - contradicting coprimality.

Quadratic formula

The roots of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac)) / 2a. It applies to every quadratic, whether or not it factorises over the rationals.

Discriminant test

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

Sum and product of roots

If α and β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. These let you build a quadratic from its roots: x² − (α+β)x + αβ = 0.

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. The difference between consecutive terms is constant.

Sum of an AP

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

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). It follows from the Pythagoras theorem applied to the horizontal and vertical gaps.

Section formula

The point dividing the join of (x₁, y₁) and (x₂, y₂) internally in ratio m:n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). Setting m = n gives the midpoint.

Pythagorean identity

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

Standard trigonometric values

sin values for 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2. Cosines run in reverse order; tan θ = sin θ / cos θ.

Angle of elevation

The angle between the horizontal line of sight and the line of sight to an object above the observer. In problems, tan(elevation) = height ÷ horizontal distance.

Angle of depression

The angle between the horizontal and the line of sight to an object below the observer. It equals the angle of elevation measured from the object, since the horizontals are parallel.

Basic Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, it divides them in the same ratio. Its converse is used to prove that two lines are parallel.

Similar triangle criteria

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

Pythagoras theorem

In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides: c² = a² + b². The converse identifies right angles.

Tangent–radius theorem

The tangent at any point of a circle is perpendicular to the radius drawn to that point. This gives right triangles in tangent-length problems.

Equal tangents from a point

Tangents drawn to a circle from an external point are equal in length. The line from the centre to that point bisects the angle between the tangents.

Heron's formula

The area of a triangle with sides a, b, c is √(s(s−a)(s−b)(s−c)), where s = (a + b + c)/2 is the semi-perimeter. It needs no height.

Area of a sector

A sector of angle θ in a circle of radius r has area (θ/360) × πr² and arc length (θ/360) × 2πr. The full circle corresponds to θ = 360°.

Volume of a sphere

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². A hemisphere has half these values for volume, with total surface area 3πr².

Cone slant height

For a right circular cone of radius r and height h, slant height l = √(r² + h²). Curved surface area is πrl and volume is (1/3)πr²h.

Scaling rule for solids

If all linear dimensions of a solid are multiplied by k, surface areas multiply by k² and volumes by k³. This answers melting, recasting and ratio questions.

Mean of grouped data

Using the direct method, mean = Σfᵢxᵢ / Σfᵢ, where xᵢ are class marks. The assumed-mean and step-deviation methods give the same value with easier arithmetic.

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, f₀ the preceding and f₂ the following frequency, and h the class size.

Empirical relation of averages

For moderately skewed data, Mode ≈ 3 Median − 2 Mean. It allows one average to be estimated when the other two are known.

Classical probability

P(E) = number of outcomes favourable to E ÷ total number of equally likely outcomes. P(E) always lies between 0 and 1 inclusive, and P(not E) = 1 − P(E).

Probability of certain and impossible events

An impossible event has probability 0 and a sure event has probability 1. The probabilities of all elementary outcomes of an experiment sum to 1.