Punjab PSEB Matric Mathematics cheat sheet

Real numbers: Euclid's division lemma states a = bq + r with 0 ≤ r < b; the Fundamental Theorem of Arithmetic says every integer > 1 factors uniquely into primes up to order.

HCF × LCM = product of the two numbers, for any two positive integers a and b; use this to find the LCM when the HCF is known: LCM = ab / HCF.

A rational number p/q (in lowest terms) has a terminating decimal expansion exactly when q is of the form 2^m · 5^n; otherwise the expansion is non-terminating recurring.

Irrationality proof pattern: assume √p = a/b in lowest terms, square to get p·b² = a², so p divides a; then p divides b, contradicting coprimality.

Quadratic formula: for ax² + bx + c = 0 (a ≠ 0), x = (−b ± √(b² − 4ac)) / 2a; the discriminant D = b² − 4ac decides the nature of roots.

Discriminant rule: 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 = c/a.

Arithmetic progression: nth term aₙ = a + (n − 1)d; sum of first n terms Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + l) where l is the last term.

Coordinate geometry: distance = √((x₂ − x₁)² + (y₂ − y₁)²); section formula for point dividing AB in ratio m:n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); midpoint is the m = n case.

Trigonometric identities to quote: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ; standard values at 0°, 30°, 45°, 60°, 90° follow the pattern √0/2, √1/2, √2/2, √3/2, √4/2 for sine.

Similar triangles: if a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides them in the same ratio (Basic Proportionality Theorem); the converse also holds.

Pythagoras theorem and its converse: in a right triangle the square of the hypotenuse equals the sum of squares of the other two sides; conversely, if a² + b² = c² the triangle is right-angled.

Circle theorems: the tangent at any point is perpendicular to the radius through that point; tangents drawn from an external point are equal in length.

Area formulas: triangle = ½ × base × height (or Heron's formula √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2); circle area = πr², circumference = 2πr.

Surface areas and volumes: sphere = 4πr² and (4/3)πr³; cylinder = 2πr(r + h) and πr²h; cone = πr(r + l) and (1/3)πr²h with slant height l = √(r² + h²).

Statistics: mean of grouped data by direct method = Σfᵢxᵢ / Σfᵢ; median class is the class where cumulative frequency reaches n/2, and median = l + [(n/2 − cf)/f] × h.

Mode of grouped data = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where l is the lower limit of the modal class and h is its class size.

Probability: P(E) = number of favourable outcomes / total number of equally likely outcomes; 0 ≤ P(E) ≤ 1 and P(not E) = 1 − P(E).

Mensuration conversion rule: when every linear dimension is multiplied by k, areas scale by k² and volumes by k³; use this for combined-solid and frustum problems.