CBSE Class 10 Maths All Formulas & Theorems
Complete chapter-wise revision sheet covering all 14 NCERT chapters. Master every identity, discriminant case, trigonometric table value, surface area, and statistical formula with 1-click chapter mock test drills.
Real Numbers
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Applicable strictly to any two positive integers a and b. (Does NOT hold directly for three numbers: HCF(a,b,c) × LCM(a,b,c) ≠ a·b·c).
For rational number p/q in simplest form (where HCF(p, q) = 1):
If prime factorisation of q contains any prime factor other than 2 or 5 (e.g., 3, 7, 11), the decimal expansion is non-terminating repeating (recurring).
Polynomials
For standard quadratic polynomial p(x) = ax² + bx + c (a ≠ 0) with zeroes α and β:
Where k is any non-zero real constant.
For p(x) = ax³ + bx² + cx + d with zeroes α, β, γ:
Pair of Linear Equations in Two Variables
For two linear equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
| Ratio Comparison | Graphical Representation | Algebraic Interpretation | System Consistency |
|---|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting Lines | Unique Solution (Exactly 1 point) | Consistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident Lines (Overlapping) | Infinitely Many Solutions | Consistent (Dependent) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel Lines | No Solution | Inconsistent |
Quadratic Equations
• If D > 0: Two distinct real roots.
• If D = 0: Two equal real roots (x = -b / 2a).
• If D < 0: No real roots (roots are imaginary / complex).
Arithmetic Progressions (AP)
Where a is first term, d is common difference (d = a₂ - a₁ = aₖ - aₖ₋₁), and n is term position.
Where l is the last term of the finite arithmetic progression.
Triangles
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio:
• AAA (or AA) Similarity: If two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar.
• SSS Similarity: If the corresponding sides of two triangles are proportional, their corresponding angles are equal, and hence the triangles are similar.
• SAS Similarity: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar.
Coordinate Geometry
Coordinates of point P(x, y) dividing line segment AB joining A(x₁, y₁) and B(x₂, y₂) internally in ratio m₁ : m₂:
Introduction to Trigonometry
| Ratio (θ) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Not defined |
| csc θ | Not defined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | Not defined |
| cot θ | Not defined | √3 | 1 | 1/√3 | 0 |
Some Applications of Trigonometry
• Angle of Elevation: Angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal plane.
• Angle of Depression: Angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal plane.
When moving a distance d towards a tower of height h changing angle from θ₁ to θ₂:
Circles
The tangent at any point of a circle is perpendicular to the radius through the point of contact:
The lengths of tangents drawn from an external point to a circle are equal:
Furthermore, tangents subtend equal angles at the center (∠AOP = ∠BOP) and are equally inclined to the line segment joining the center to that point (∠APO = ∠BPO).
Areas Related to Circles
Surface Areas and Volumes
| Solid Shape | Curved / Lateral Surface Area | Total Surface Area | Volume |
|---|---|---|---|
| Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
| Cube (side a) | 4a² | 6a² | a³ |
| Right Circular Cylinder (r, h) | 2πrh | 2πr(r + h) | πr²h |
| Right Circular Cone (r, h, l) | πrl [where l = √(r² + h²)] | πr(r + l) | (1/3) πr²h |
| Sphere (radius r) | 4πr² | 4πr² | (4/3) πr³ |
| Hemisphere (radius r) | 2πr² | 3πr² | (2/3) πr³ |
Statistics
Where l = lower limit of modal class, h = class size, f₁ = frequency of modal class, f₀ = frequency of preceding class, f₂ = frequency of succeeding class.
Where l = lower limit of median class, N = total frequency, cf = cumulative frequency of preceding class, f = frequency of median class, h = class width.
Probability
• Probability of an impossible event is 0.
• Probability of a sure (certain) event is 1.