Mathematics Formula Sheet
All key formulas for AP SSC Class 10 Mathematics — all 14 NCERT chapters organised by chapter. Bookmark this before your BSE AP board exam.
Ch. 1 — Real Numbers
- ·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 (used in irrationality proofs)
- ·Terminating decimal ↔ denominator has only 2 and 5 as prime factors
Ch. 2 — 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: p(x) ÷ (x−a) leaves remainder p(a)
- ·Factor Theorem: (x−a) is a factor of p(x) ↔ p(a) = 0
Ch. 3 — Pair of Linear Equations
- ·Graphical: intersecting → unique solution; parallel → no solution; coincident → infinite solutions
- ·Consistency: a₁/a₂ ≠ b₁/b₂ (unique); a₁/a₂=b₁/b₂≠c₁/c₂ (none); a₁/a₂=b₁/b₂=c₁/c₂ (infinite)
- ·Cross-multiplication: x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)
Ch. 4 — Quadratic Equations
- ·Quadratic Formula: x = [−b ± √(b²−4ac)] / 2a
- ·Discriminant D = b²−4ac: D > 0 (2 distinct real), D = 0 (equal), D < 0 (no real roots)
- ·Sum of roots α+β = −b/a; Product αβ = c/a
- ·Form equation from roots: x² − (α+β)x + αβ = 0
Ch. 5 — Arithmetic Progressions
- ·nth term: aₙ = a + (n−1)d
- ·Sum of n terms: Sₙ = n/2 × [2a + (n−1)d] = n/2 × (a₁ + aₙ)
- ·If Sₙ is given: aₙ = Sₙ − Sₙ₋₁
- ·Number of terms: n = (l − a)/d + 1, where l = last term
Ch. 6 — Triangles
- ·Basic Proportionality Theorem (Thales): DE ∥ BC ⟹ AD/DB = AE/EC
- ·Similarity criteria: AA, SAS, SSS
- ·Area ratio of similar triangles = (ratio of corresponding sides)²
- ·Pythagoras: AC² = AB² + BC² (right angle at B)
Ch. 7 — Coordinate Geometry
- ·Distance = √[(x₂−x₁)² + (y₂−y₁)²]
- ·Section formula (m:n): x = (mx₂+nx₁)/(m+n), y = (my₂+ny₁)/(m+n)
- ·Midpoint: [(x₁+x₂)/2, (y₁+y₂)/2]
- ·Area of triangle = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Ch. 8 — Introduction to Trigonometry
- ·sin²θ + cos²θ = 1 · 1 + tan²θ = sec²θ · 1 + cot²θ = cosec²θ
- ·sin 30°=½, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1
- ·cos 30°=√3/2, cos 45°=1/√2, cos 60°=½, cos 90°=0
- ·tan 30°=1/√3, tan 45°=1, tan 60°=√3; sin θ = cos(90°−θ)
Ch. 9 — Applications of Trigonometry
- ·Angle of elevation: looking up from horizontal to an object
- ·Angle of depression: looking down from horizontal to an object
- ·Height: h = d × tan(angle of elevation), where d = horizontal distance
- ·Two-point height: h = d × (tan α × tan β)/(tan α − tan β)
Ch. 10 — Circles
- ·Tangent from external point P: PA = PB (equal tangent lengths)
- ·Tangent ⊥ radius at point of tangency: ∠OAP = 90°
- ·Length of tangent = √(d²−r²), where d = distance from P to centre O
- ·Angle between two tangents from P + angle at centre = 180°
Ch. 11 — Areas Related to Circles
- ·Area of circle = πr²; Circumference = 2πr
- ·Area of sector (angle θ°) = (θ/360) × πr²
- ·Arc length (angle θ°) = (θ/360) × 2πr
- ·Area of segment = Area of sector − Area of triangle
Ch. 12 — Surface Areas and Volumes
- ·Cylinder: V = πr²h, CSA = 2πrh, TSA = 2πr(r+h)
- ·Cone: V = ⅓πr²h, CSA = πrl, TSA = πr(r+l), l = √(r²+h²)
- ·Sphere: V = 4/3πr³, SA = 4πr²
- ·Hemisphere: V = ⅔πr³, CSA = 2πr², TSA = 3πr²
- ·Frustum: V = πh/3(r₁²+r₂²+r₁r₂), CSA = π(r₁+r₂)l
Ch. 13 — 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
Ch. 14 — Probability
- ·P(E) = Number of favourable outcomes / Total equally likely outcomes
- ·0 ≤ P(E) ≤ 1
- ·P(E) + P(Ē) = 1
- ·P(impossible event) = 0; P(certain event) = 1