CBSE Class 10 Maths Formulas

01Real numbers and polynomials

• HCF and LCM HCF(a, b) × LCM(a, b) = a × b
• Irrational numbers √2, √3, √5 and √p for any prime p are irrational
• Quadratic polynomial ax² + bx + c sum of zeros α + β = −b/a,  product αβ = c/a
• Building a quadratic x² − (sum of zeros)x + (product of zeros)

02Pair of linear equations

• Unique solution (lines intersect) a₁/a₂ ≠ b₁/b₂
• No solution (parallel lines) a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Infinitely many solutions (same line) a₁/a₂ = b₁/b₂ = c₁/c₂

03Quadratic equations

• Quadratic formula x = [−b ± √(b² − 4ac)]/2a
• Discriminant D = b² − 4ac
• Nature of roots D > 0 two distinct real roots;  D = 0 two equal roots;  D < 0 no real roots

04Arithmetic progressions

• nth term a_n = a + (n − 1)d
• Sum of n terms S_n = (n/2)[2a + (n − 1)d] = (n/2)(a + l)
• Finding a term from sums a_n = S_n − S_n−1

05Triangles and coordinate geometry

• Basic proportionality theorem if DE ∥ BC in triangle ABC, then AD/DB = AE/EC
• Similarity criteria AAA (AA), SSS, SAS
• Pythagoras hypotenuse² = base² + perpendicular²
• Distance √[(x₂ − x₁)² + (y₂ − y₁)²]
• Section formula (ratio m:n) [(mx₂ + nx₁)/(m + n),  (my₂ + ny₁)/(m + n)]
• Midpoint [(x₁ + x₂)/2,  (y₁ + y₂)/2]

06Trigonometric values

07Trigonometry

• Ratios sinθ = P/H,  cosθ = B/H,  tanθ = P/B = sinθ/cosθ
• Reciprocals cosecθ = 1/sinθ,  secθ = 1/cosθ,  cotθ = 1/tanθ
• Identities sin²θ + cos²θ = 1;  1 + tan²θ = sec²θ;  1 + cot²θ = cosec²θ
• Heights and distances tan(angle of elevation) = height/horizontal distance

08Circles and areas

• Tangents a tangent is perpendicular to the radius at the point of contact;  tangents from an outside point are equal
• Circle circumference 2πr,  area πr²
• Arc of angle θ length (θ/360°) × 2πr
• Sector of angle θ area (θ/360°) × πr²
• Segment area of sector − area of the triangle

09Surface areas and volumes

• Cuboid TSA 2(lb + bh + hl),  volume lbh
• Cube TSA 6a²,  volume a³
• Cylinder CSA 2πrh,  TSA 2πr(r + h),  volume πr²h
• Cone slant height l = √(r² + h²),  CSA πrl,  TSA πr(l + r),  volume ⅓πr²h
• Sphere surface area 4πr²,  volume (4/3)πr³
• Hemisphere CSA 2πr²,  TSA 3πr²,  volume (2/3)πr³

10Statistics and probability

• Mean (direct) x̄ = Σf_ix_i/Σf_i
• Mean (assumed mean) x̄ = a + Σf_id_i/Σf_i,  d_i = x_i − a
• Median l + [(n/2 − cf)/f] × h
• Mode l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
• Empirical relation 3 Median = Mode + 2 Mean
• Probability P(E) = favourable outcomes/total outcomes;  0 ≤ P(E) ≤ 1;  P(not E) = 1 − P(E)
• Useful counts a deck has 52 cards, 26 red, 12 face cards;  two dice give 36 outcomes