Class 11 Maths Formulas (PDF) — All Important Formulas Free
Free class 11 maths formulas — all 53 key formulas on one page, downloadable as PDF for fast revision before CBSE board exams, JEE & NEET. No signup.
TL;DR: Free class 11 maths formulas — all 53 key formulas on one page, downloadable as PDF for fast revision before CBSE board exams, JEE & NEET. No signup.
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
Master Class 11 Maths formulas covering Trigonometry, Complex Numbers, Permutations & Combinations, Binomial Theorem, Sequences & Series, Straight Lines, Conic Sections, 3D Geometry, and Limits. Essential for JEE preparation and board exams.
Trigonometric Identities & Functions
| Formula | Expression |
|---|---|
| Addition formulas | sin(A + B) = sin A cos B + cos A sin B; sin(A - B) = sin A cos B - cos A sin B — Similarly for cos and tan |
| Cosine addition | cos(A + B) = cos A cos B - sin A sin B; cos(A - B) = cos A cos B + sin A sin B |
| Tangent addition | tan(A + B) = (tan A + tan B)/(1 - tan A tan B); tan(A - B) = (tan A - tan B)/(1 + tan A tan B) |
| Double angle formulas | sin 2A = 2 sin A cos A; cos 2A = cos² A - sin² A = 2cos² A - 1 = 1 - 2sin² A; tan 2A = 2tan A/(1 - tan² A) |
| Triple angle formulas | sin 3A = 3 sin A - 4 sin³ A; cos 3A = 4 cos³ A - 3 cos A; tan 3A = (3tan A - tan³ A)/(1 - 3tan² A) |
| Sum to product | sin C + sin D = 2 sin((C + D)/2) cos((C - D)/2); cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2) — Useful for simplifying trigonometric expressions |
| Product to sum | 2 sin A cos B = sin(A + B) + sin(A - B); 2 cos A cos B = cos(A + B) + cos(A - B) |
Complex Numbers
| Formula | Expression |
|---|---|
| Complex number definition | z = a + ib, where i² = -1 — a = real part, b = imaginary part |
| Modulus (absolute value) | |z| = √(a² + b²) — For z = a + ib |
| Conjugate | z* = a - ib — For z = a + ib; z × z* = |z|² |
| Argument (angle) | arg(z) = tan⁻¹(b/a) — Angle made with positive real axis; consider quadrant |
| Polar form | z = r(cos θ + i sin θ) = r e^(iθ) — where r = |z|, θ = arg(z) |
| De Moivre's Theorem | (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ — For integer n; also applies to zⁿ = rⁿ(cos nθ + i sin nθ) |
| nth roots of unity | 1^(1/n) = e^(2πik/n), k = 0, 1, 2, ..., n-1 — Sum of nth roots of unity = 0 |
Permutations & Combinations
| Formula | Expression |
|---|---|
| Factorial | n! = n × (n - 1) × (n - 2) × ... × 2 × 1; 0! = 1 — Product of all positive integers up to n |
| Permutation (nPr) | ⁿPᵣ = n!/(n - r)! — Number of ways to arrange r objects from n distinct objects |
| Combination (nCr) | ⁿCᵣ = n!/(r!(n - r)!) = ⁿPᵣ/r! — Number of ways to choose r objects from n distinct objects; order doesn't matter |
| Properties of combinations | ⁿC₀ = 1, ⁿCₙ = 1, ⁿCᵣ = ⁿCₙ₋ᵣ, ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ — Pascal's identity: ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ |
| Circular permutations | (n - 1)! — Arrangements of n distinct objects in a circle |
| Permutations with repetition | nʳ — r selections from n objects with replacement |
Binomial Theorem
| Formula | Expression |
|---|---|
| Binomial expansion | (x + y)ⁿ = Σ ⁿCᵣ xⁿ⁻ʳ yʳ (r = 0 to n) — Expansion of (x + y)ⁿ using binomial coefficients |
| General term | Tᵣ₊₁ = ⁿCᵣ xⁿ⁻ʳ yʳ — (r + 1)th term in binomial expansion |
| Middle term | If n is even: T₍ₙ/₂₎₊₁; If n is odd: T₍ₙ₊₁₎/₂ and T₍ₙ₊₃₎/₂ — There are two middle terms if n is odd |
| Sum of binomial coefficients | ⁿC₀ + ⁿC₁ + ⁿC₂ + ... + ⁿCₙ = 2ⁿ — Sum of all binomial coefficients |
| Binomial series | (1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + ... — For |x| < 1; used for fractional and negative exponents |
Sequences & Series (AP & GP)
| Formula | Expression |
|---|---|
| Arithmetic Progression (AP) | aₙ = a + (n - 1)d; Sₙ = n/2[2a + (n - 1)d] — a = first term, d = common difference |
| Geometric Progression (GP) | aₙ = ar^(n-1); Sₙ = a(1 - rⁿ)/(1 - r) if r ≠ 1 — a = first term, r = common ratio |
| Sum of infinite GP | S∞ = a/(1 - r) if |r| < 1 — Only converges if |r| < 1 |
| Arithmetic Mean (AM) | A = (a + b)/2 — Mean of two numbers |
| Geometric Mean (GM) | G = √(ab) — For two positive numbers a and b |
| Harmonic Mean (HM) | H = 2ab/(a + b) — Reciprocal of AM of reciprocals |
| AM-GM-HM inequality | A ≥ G ≥ H, with equality iff a = b — Useful inequality for two positive numbers |
Straight Lines
| Formula | Expression |
|---|---|
| Slope of line | m = tan θ = (y₂ - y₁)/(x₂ - x₁) — θ = angle with positive x-axis |
| Point-slope form | y - y₁ = m(x - x₁) — Line passing through (x₁, y₁) with slope m |
| Slope-intercept form | y = mx + c — m = slope, c = y-intercept |
| Two-point form | (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁) — Line through points (x₁, y₁) and (x₂, y₂) |
| Intercept form | x/a + y/b = 1 — a = x-intercept, b = y-intercept |
| Distance from point to line | d = |ax₀ + by₀ + c|/√(a² + b²) — Distance from (x₀, y₀) to line ax + by + c = 0 |
| Angle between two lines | tan θ = |m₁ - m₂|/(1 + m₁m₂) — Slopes m₁ and m₂; parallel if m₁ = m₂, perpendicular if m₁m₂ = -1 |
Conic Sections
| Formula | Expression |
|---|---|
| Circle | Standard: (x - h)² + (y - k)² = r²; General: x² + y² + 2gx + 2fy + c = 0 — Center: (h, k) or (-g, -f), Radius: r or √(g² + f² - c) |
| Parabola | Standard: y² = 4ax (focus at (a, 0)), x² = 4ay (focus at (0, a)) — Vertex at origin; directrix x = -a or y = -a |
| Ellipse | Standard: x²/a² + y²/b² = 1 (a > b); eccentricity e = √(1 - b²/a²) — Foci at (±c, 0) where c² = a² - b² |
| Hyperbola | Standard: x²/a² - y²/b² = 1; eccentricity e = √(1 + b²/a²) — Foci at (±c, 0) where c² = a² + b²; asymptotes: y = ±(b/a)x |
3D Geometry
| Formula | Expression |
|---|---|
| Distance formula in 3D | d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] — Distance between points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) |
| Section formula in 3D | P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)) — Point dividing line segment in ratio m:n |
| Direction cosines | If line makes angles α, β, γ with x, y, z axes, then l = cos α, m = cos β, n = cos γ; l² + m² + n² = 1 — Direction cosines satisfy l² + m² + n² = 1 |
| Equation of plane | Ax + By + Cz + D = 0 — General form; normal vector: (A, B, C) |
| Distance from point to plane | d = |Ax₀ + By₀ + Cz₀ + D|/√(A² + B² + C²) — Distance from point (x₀, y₀, z₀) to plane |
Limits & Continuity
| Formula | Expression |
|---|---|
| Limit definition | lim(x→a) f(x) = L means for every ε > 0, ∃ δ > 0 such that |f(x) - L| < ε whenever |x - a| < δ — Formal definition of limit |
| Limit laws | lim[f(x) + g(x)] = lim f(x) + lim g(x); lim[f(x) × g(x)] = lim f(x) × lim g(x); lim[f(x)/g(x)] = lim f(x)/lim g(x) if lim g(x) ≠ 0 — Assuming individual limits exist |
| Standard limits | lim(x→0) sin x/x = 1; lim(x→0) (1 - cos x)/x² = 1/2; lim(x→∞) (1 + 1/x)ˣ = e — Important limits to memorize |
| L'Hôpital's Rule | lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x) — If 0/0 or ∞/∞ form |
| Continuity at a point | f is continuous at x = a if lim(x→a) f(x) = f(a) — Left limit = Right limit = Function value |
FAQs
What's the difference between permutation and combination?
Permutation (ⁿPᵣ) counts arrangements where order matters. Combination (ⁿCᵣ) counts selections where order doesn't matter. For example, ABC and BAC are different permutations but the same combination. Formula: ⁿPᵣ = n!/(n-r)!, ⁿCᵣ = n!/(r!(n-r)!)
How do I use the binomial theorem?
The binomial theorem states (x + y)ⁿ = Σ ⁿCᵣ xⁿ⁻ʳ yʳ. Each term is ⁿCᵣ xⁿ⁻ʳ yʳ. To find a specific term, use Tᵣ₊₁ = ⁿCᵣ xⁿ⁻ʳ yʳ. The sum of all coefficients is 2ⁿ. This is useful for expanding (a+b)ⁿ and finding specific terms.
What are the trigonometric addition formulas and when do I use them?
sin(A±B) = sin A cos B ± cos A sin B and cos(A±B) = cos A cos B ∓ sin A sin B. Use them to find sin/cos of angles not in standard tables, to simplify expressions, or to prove identities. Double angle formulas (A = B) are: sin 2A = 2sin A cos A, cos 2A = cos²A - sin²A.
How do I find the distance from a point to a line?
For point (x₀, y₀) and line ax + by + c = 0, use: d = |ax₀ + by₀ + c|/√(a² + b²). The formula gives the perpendicular distance. Make sure the line equation is in the form ax + by + c = 0 before applying it.
What's the relationship between AM, GM, and HM?
For two positive numbers a and b: AM = (a+b)/2, GM = √(ab), HM = 2ab/(a+b). They satisfy AM ≥ GM ≥ HM, with equality only when a = b. This inequality is useful in optimization problems and proofs.