Mathematics Formula Sheet
All key formulas for BSE Telangana Class 10 Maths — organised by chapter. Bookmark this before your exam and revise the formulas you need most.
Ch. 1 — Real Numbers (includes Logarithms)
- ·HCF × LCM = Product of two numbers
- ·Euclid's Division Lemma: a = bq + r, where 0 ≤ r < b
- ·If p is prime and p | a², then p | a
- ·Terminating decimal ↔ denominator has only 2 and 5 as prime factors
- ·Logarithm definition: aˣ = N ⟺ logₐN = x (where a > 0, a ≠ 1, N > 0)
- ·Product law: logₐ(xy) = logₐx + logₐy
- ·Quotient law: logₐ(x/y) = logₐx − logₐy
- ·Power law: logₐ(xⁿ) = n·logₐx
- ·Key values: logₐa = 1; logₐ1 = 0; logₐ(aˣ) = x
- ·Change of base: logₐb = log b / log a (common log)
Ch. 2 — Sets
- ·n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
- ·n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
- ·A − B = A ∩ B' (elements in A but not in B)
- ·Number of subsets of a set with n elements = 2ⁿ
Ch. 3 — Polynomials
- ·Quadratic ax² + bx + c: Sum of zeros = −b/a, Product of zeros = c/a
- ·Cubic ax³+bx²+cx+d: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a
- ·Remainder Theorem: when p(x) ÷ (x−a), remainder = p(a)
- ·Factor Theorem: (x−a) is a factor of p(x) if and only if p(a) = 0
Ch. 4 — Pair of Linear Equations in Two Variables
- ·Graphical: intersecting lines → unique solution; parallel → no solution; coincident → infinite solutions
- ·Substitution / Elimination method for a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0
- ·Cross-multiplication: x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)
- ·Consistency: unique if a₁/a₂ ≠ b₁/b₂; inconsistent if a₁/a₂=b₁/b₂≠c₁/c₂; infinite if a₁/a₂=b₁/b₂=c₁/c₂
Ch. 5 — Quadratic Equations
- ·Quadratic Formula: x = [−b ± √(b² − 4ac)] / 2a
- ·Discriminant D = b² − 4ac: D > 0 → 2 distinct real roots; D = 0 → equal roots; D < 0 → no real roots
- ·If α, β are roots: α + β = −b/a and αβ = c/a
- ·Form equation from roots: x² − (α+β)x + αβ = 0
Ch. 6 — Progressions
- ·AP nth term: aₙ = a + (n−1)d
- ·AP sum of n terms: Sₙ = n/2 × [2a + (n−1)d] = n/2 × (a₁ + aₙ)
- ·GP nth term: aₙ = arⁿ⁻¹
- ·GP sum of n terms: Sₙ = a(rⁿ−1)/(r−1) for r≠1; Sₙ = na for r=1
- ·If Sₙ is given: aₙ = Sₙ − Sₙ₋₁
Ch. 7 — Coordinate Geometry
- ·Distance = √[(x₂−x₁)² + (y₂−y₁)²]
- ·Midpoint = [(x₁+x₂)/2, (y₁+y₂)/2]
- ·Section formula (m:n): x = (mx₂+nx₁)/(m+n), y = (my₂+ny₁)/(m+n)
- ·Area of triangle = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
- ·Slope of line through (x₁,y₁) and (x₂,y₂): m = (y₂−y₁)/(x₂−x₁)
Ch. 8 — Similar Triangles
- ·AA, SAS, SSS similarity criteria for triangles
- ·Basic Proportionality Theorem (Thales): DE ∥ BC ⟹ AD/DB = AE/EC
- ·If ΔABC ~ ΔPQR: AB/PQ = BC/QR = AC/PR
- ·Area ratio of similar triangles = (ratio of corresponding sides)²
- ·Pythagoras: In right-angled triangle, AC² = AB² + BC²
Ch. 9 — Tangents and Secants to a Circle
- ·Tangent from external point: PA = PB (equal tangent lengths)
- ·Tangent ⊥ radius at point of tangency: ∠OPA = 90°
- ·Length of tangent = √(d² − r²), where d = distance from external point to centre
- ·Angle between two tangents from external point + angle at centre = 180°
Ch. 10 — Mensuration
- ·Cylinder: V = πr²h, CSA = 2πrh, TSA = 2πr(r+h)
- ·Cone: V = ⅓πr²h, CSA = πrl, TSA = πr(r+l), slant l = √(r²+h²)
- ·Sphere: V = 4/3πr³, SA = 4πr²
- ·Hemisphere: V = ⅔πr³, CSA = 2πr², TSA = 3πr²
- ·Frustum of cone: V = πh/3(r₁²+r₂²+r₁r₂), CSA = π(r₁+r₂)l
Ch. 11 — Trigonometry
- ·sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
- ·sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
- ·sin 30°=½, cos 30°=√3/2, tan 30°=1/√3
- ·sin 45°=1/√2, cos 45°=1/√2, tan 45°=1
- ·sin 60°=√3/2, cos 60°=½, tan 60°=√3; sin 90°=1, cos 90°=0
Ch. 12 — Applications of Trigonometry
- ·Angle of elevation: looking up from horizontal to object
- ·Angle of depression: looking down from horizontal to object
- ·Height: h = d × tan(angle of elevation)
- ·If two observations: h = d × (tan α × tan β)/(tan α − tan β)
- ·Draw the right triangle and label sides before applying ratios
Ch. 13 — Probability
- ·P(E) = Favourable outcomes / Total equally likely outcomes
- ·0 ≤ P(E) ≤ 1
- ·P(E) + P(Ē) = 1
- ·P(impossible event) = 0; P(certain event) = 1
Ch. 14 — Statistics
- ·Mean (direct): x̄ = Σfᵢxᵢ / Σfᵢ
- ·Mean (assumed mean): x̄ = A + (Σfᵢdᵢ / Σfᵢ), dᵢ = xᵢ − A
- ·Mean (step deviation): x̄ = A + (Σfᵢuᵢ / Σfᵢ) × h, uᵢ = (xᵢ−A)/h
- ·Median = l + [(n/2 − cf) / f] × h
- ·Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h