Odisha BSE Madhyama Mathematics cheat sheet

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

Polynomials: remainder theorem - if p(x) is divided by (x − a), the remainder is p(a); factor theorem - (x − a) is a factor of p(x) exactly when p(a) = 0.

Algebraic identities: (a ± b)² = a² ± 2ab + b²; a² − b² = (a − b)(a + b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Linear equations in two variables ax + by + c = 0 have infinitely many solutions; their graph is a straight line, and two such lines are parallel when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Coordinate geometry: distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2); point on x-axis is (x, 0).

Lines and angles: two lines cut by a transversal are parallel iff alternate interior angles are equal or co-interior angles sum to 180°; the exterior angle of a triangle equals the sum of the two remote interior angles.

Triangles: congruence criteria SSS, SAS, ASA, AAS and RHS; a point on the perpendicular bisector of a segment is equidistant from its endpoints, and angles opposite equal sides are equal.

Heron's formula: area of a triangle with sides a, b, c is √(s(s − a)(s − b)(s − c)) where s = (a + b + c)/2; area of an equilateral triangle of side a is (√3/4)a².

Statistics: mean of ungrouped data = Σx/n; median of n ordered observations is the ((n+1)/2)th value when n is odd, and the average of the (n/2)th and (n/2 + 1)th values when n is even.

Quadratic equations: for ax² + bx + c = 0 (a ≠ 0), roots are x = (−b ± √(b² − 4ac))/2a; the discriminant D = b² − 4ac gives D > 0 two distinct real roots, D = 0 equal roots, D < 0 no real roots.

Root relations: sum of roots = −b/a and product of roots = c/a; a quadratic with given roots α, β is x² − (α + β)x + αβ = 0.

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.

Similar triangles: if two triangles are similar, ratios of corresponding sides are equal and areas are in the ratio of the squares of corresponding sides (Theorem of Areas of Similar Triangles).

Pythagoras theorem: in a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides; its converse identifies a right angle from side lengths.

Circles: the tangent at any point of a circle is perpendicular to the radius through the point of contact; tangents drawn from an external point are equal in length.

Trigonometry: sin θ = perpendicular/hypotenuse, cos θ = base/hypotenuse, tan θ = perpendicular/base = sin θ/cos θ; sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

Standard trigonometric values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.

Mensuration: volume of a sphere = (4/3)πr³, curved surface area of a cylinder = 2πrh, total surface area of a cone = πr(l + r), and volume of a cone = (1/3)πr²h.

Statistics (grouped): mean by assumed-mean method x̄ = a + (Σfᵢdᵢ/Σfᵢ); median = l + ((n/2 − cf)/f)h; mode = l + ((f₁ − f₀)/(2f₁ − f₀ − f₂))h.