JEE Advanced Maths Formulas 2026: Free Formula Sheet PDF

CIRCLE

Area of a Circle:

In terms of radius: A = πr² In terms of diameter: A = (π/4) × d²

Surface Area of a Circle:

S = πr²

General Equation of a Circle:

(x − h)² + (y − k)² = r² Centre = (h, k), Radius = r

Standard Equation of a Circle:

(x − a)² + (y − b)² = r² Centre = (a, b), Radius = r

Diameter of a Circle:

D = 2 × radius

Circumference of a Circle:   

C = 2πr

General Form:

x² + y² + 2gx + 2fy + c = 0 Centre = (−g, −f), Radius = √(g² + f² − c)

Intercepts Made by Circle x² + y² + 2gx + 2fy + c = 0:

On x-axis: 2√(g² − c) On y-axis: 2√(f² − c)

Parametric Equations of a Circle:

x = h + r cosθ y = k + r sinθ

Tangent to Circle x² + y² = a²:

Slope form: y = mx ± a√(1 + m²) Point form: xx₁ + yy₁ = a² (or T = 0) Parametric form: x cosα + y sinα = a

Pair of Tangents from a Point:

SS₁ = T²

Length of a Tangent from External Point (x₁, y₁):

Length = √S₁ (where S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c)

Director Circle of x² + y² = a²:

x² + y² = 2a²

Chord of Contact:

T = 0

(i) Length of chord of contact = 2LR/√(R² + L²) (ii) Area of triangle formed by pair of tangents & chord of contact = RL³/√(R² + L²) (iii) Tangent of angle between pair of tangents from (x₁, y₁) = 2RL/(L² − R²) (iv) Equation of circle circumscribing triangle PT₁T₂: (x − x₁)(x + g) + (y − y₁)(y + f) = 0

Condition of Orthogonality of Two Circles:

2g₁g₂ + 2f₁f₂ = c₁ + c₂

Radical Axis:

S₁ − S₂ = 0, i.e., 2(g₁ − g₂)x + 2(f₁ − f₂)y + (c₁ − c₂) = 0

Family of Circles:

S₁ + KS₂ = 0 S + KL = 0

QUADRATIC EQUATIONS

General Form:

ax² + bx + c = 0; a, b, c are constants and a ≠ 0

Roots of Equation:

α = [−b + √(b² − 4ac)] / (2a) β = [−b − √(b² − 4ac)] / (2a)

Sum and Product of Roots:

If α and β are roots of ax² + bx + c = 0: Sum of roots: α + β = −b/a Product of roots: αβ = c/a

Discriminant:

D = b² − 4ac

Nature of Roots:

D = 0 → Roots are real and equal: α = β = −b/(2a) D > 0 → Roots are real and unequal D < 0 → Roots are imaginary and unequal (complex conjugates) D > 0 and D is a perfect square → Roots are rational and unequal D > 0 and D is not a perfect square → Roots are irrational and unequal

Formation of a Quadratic Equation with Given Roots:

(x − α)(x − β) = 0 x² − (α + β)x + αβ = 0 x² − (Sum of roots)x + (Product of roots) = 0

Common Roots:

If two equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 have:

Both roots common: a₁/a₂ = b₁/b₂ = c₁/c₂

Only one root α common: α = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁) = (b₁c₂ − b₂c₁)/(c₁a₂ − c₂a₁)

Range of Quadratic Expression f(x) = ax² + bx + c in Restricted Domain x ∈ [x₁, x₂]:

If −b/(2a) does not belong to [x₁, x₂]: f(x) ∈ [min{f(x₁), f(x₂)}, max{f(x₁), f(x₂)}]

If −b/(2a) ∈ [x₁, x₂]: f(x) ∈ [min{f(x₁), f(x₂), −D/(4a)}, max{f(x₁), f(x₂), −D/(4a)}]

Roots Under Special Cases (ax² + bx + c = 0):

c = 0 → One root is zero, other is −b/a b = 0 → Roots are equal in magnitude but opposite in sign b = c = 0 → Both roots are zero a = c → Roots are reciprocal to each other a + b + c = 0 → One root is 1, second root is c/a a = b = c = 0 → Equation becomes an identity (satisfied by every x)

Graph of a Quadratic Equation:

The graph is a parabola. a > 0 → Concave upwards (opens up) a < 0 → Concave downwards (opens down)

Maximum and Minimum Value of ax² + bx + c:

If a < 0 → Greatest (maximum) value at x = −b/(2a). Max value = −D/(4a) If a > 0 → Least (minimum) value at x = −b/(2a). Min value = −D/(4a)

Quadratic Expression in Two Variables:

ax² + 2hxy + by² + 2gx + 2fy + c

Condition for resolution into two linear rational factors:

|a h g| |h b f| = 0 |g f c|

i.e., abc + 2fgh − af² − bg² − ch² = 0 AND h² − ab > 0

BINOMIAL THEOREM

Binomial Theorem for Positive Integral Index:

(x + a)ⁿ = ⁿC₀xⁿa⁰ + ⁿC₁xⁿ⁻¹a + ⁿC₂xⁿ⁻²a² + ... + ⁿCᵣxⁿ⁻ʳaʳ + ... + ⁿCₙxaⁿ

General Term:

T_(r+1) = ⁿCᵣ xⁿ⁻ʳ aʳ

Deductions:

(1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + ⁿC₃x³ + ... + ⁿCᵣxʳ + ... + ⁿCₙxⁿ General term: T_(r+1) = ⁿCᵣ xʳ = [n(n−1)(n−2)...(n−r+1)/r!] × xʳ

(1 − x)ⁿ = ⁿC₀ − ⁿC₁x + ⁿC₂x² − ⁿC₃x³ + ... + (−1)ʳ ⁿCᵣxʳ + ... + (−1)ⁿ ⁿCₙxⁿ General term: T_(r+1) = (−1)ʳ ⁿCᵣ xʳ

Middle Term in (x + a)ⁿ:

Suppose n is even: Middle term = (n/2 + 1)th term. If n is odd: Middle terms are ((n+1)/2)th and ((n+3)/2)th terms. The binomial coefficient of the middle term is the greatest binomial coefficient.

To Determine a Particular Term:

In expansion of (xᵅ ± 1/xᵝ)ⁿ, if xᵐ occurs in T_(r+1): nα − r(α + β) = m → r = (nα − m)/(α + β)

Term independent of x: nα − r(α + β) = 0 → r = nα/(α + β)

To Find a Term from the End:

Tᵣ(E) = T_(n−r+2)(B) (rth term from end = (n − r + 2)th term from beginning)

Binomial Coefficients and Their Properties:

In (1 + x)ⁿ = C₀ + C₁x + C₂x² + ... + Cᵣxʳ + ... + Cₙxⁿ where C₀ = 1, C₁ = n, C₂ = n(n−1)/2!

(i) C₀ + C₁ + C₂ + ... + Cₙ = 2ⁿ (ii) C₀ − C₁ + C₂ − C₃ + ... = 0 (iii) C₀ + C₂ + C₄ + ... = C₁ + C₃ + C₅ + ... = 2ⁿ⁻¹ (iv) C₀² + C₁² + C₂² + ... + Cₙ² = (2n)!/(n!n!) (v) C₀ + C₁/2 + C₂/3 + ... + Cₙ/(n+1) = (2ⁿ⁺¹ − 1)/(n+1) (vi) C₀ − C₁/2 + C₂/3 − C₃/4 + ... + (−1)ⁿCₙ/(n+1) = 1/(n+1)

Greatest Term in (x + a)ⁿ: 

Greatest coefficient: If n is even: T_(n/2+1) If n is odd: T_((n+1)/2) and T_((n+3)/2)

Greatest term: If (n+1)a/(x+a) = p ∈ Z: T_p and T_(p+1) are both greatest If (n+1)a/(x+a) is not integer and q < (n+1)a/(x+a) < q+1: T_(q+1) is greatest

Multinomial Expansion (n ∈ N):

(x₁ + x₂ + x₃ + ... + xₖ)ⁿ = Σ [n!/(r₁!r₂!...rₖ!)] × x₁^r₁ × x₂^r₂ × ... × xₖ^rₖ

where sum is over all r₁ + r₂ + ... + rₖ = n

Binomial Theorem for Negative Integer or Fractional Indices:

(1 + x)ⁿ = 1 + nx + [n(n−1)/2!]x² + [n(n−1)(n−2)/3!]x³ + ... + [n(n−1)(n−2)...(n−r+1)/r!]xʳ + …

Valid for |x| < 1

T_(r+1) = [n(n−1)(n−2)...(n−r+1)/r!] × xʳ

PARABOLA

Estandar parábola y² = 4ax:

Focus: (a, 0), a > 0 Directrix: x = −a Vertex: (0, 0) Axis: y = 0 Length of latus rectum = 4a Ends of latus rectum: L(a, 2a) and L'(a, −2a)

Parametric Representation:

Point on parabola: (at², 2at)

Position of a Point (x₁, y₁) w.r.t. y² = 4ax:

y₁² − 4ax₁ > 0 → Outside y₁² − 4ax₁ = 0 → On the parabola y₁² − 4ax₁ < 0 → Inside

Length of Chord Intercepted by y² = 4ax on line y = mx + c:

Length = (4/m²)√[a(1 + m²)(a − mc)]

Tangent to y² = 4ax:

T = 0 y = mx + a/m, m ≠ 0 is tangent to y² = 4ax at point (a/m², 2a/m)

Normal to y² = 4ax:

At (x₁, y₁): y − y₁ = (−y₁/2a)(x − x₁)

Chord with Given Middle Point (x₁, y₁):

T = S₁ where S₁ = y₁² − 4ax₁

DEFINITE INTEGRATION

Definite Integral as Limit of Sum:

∫(a to b) f(x)dx = Σ(r=1 to n) h·f(a + rh) where h = (b − a)/n

Fundamental Theorem of Calculus:

∫(a to b) f(x)dx = F(b) − F(a), where F'(x) = f(x)

Properties of Definite Integrals:

1. ∫(a to b) f(x)dx = ∫(a to b) f(t)dt

2. ∫(a to b) f(x)dx = −∫(b to a) f(x)dx

3. ∫(a to b) cf(x)dx = c∫(a to b) f(x)dx

4. ∫(a to b) [f(x) ± g(x)]dx = ∫(a to b) f(x)dx ± ∫(a to b) g(x)dx

5. ∫(a to b) f(x)dx = ∫(a to c) f(x)dx + ∫(c to b) f(x)dx

6. ∫(a to b) f(x)dx = ∫(a to b) f(a + b − x)dx

7. ∫(0 to a) f(x)dx = ∫(0 to a) f(a − x)dx

8. ∫(0 to 2a) f(x)dx = 2∫(0 to a) f(x)dx if f(2a − x) = f(x)

9. ∫(0 to 2a) f(x)dx = 0 if f(2a − x) = −f(x)

10. ∫(−a to a) f(x)dx = 2∫(0 to a) f(x)dx if f(x) is even [f(−x) = f(x)]

11. ∫(−a to a) f(x)dx = 0 if f(x) is odd [f(−x) = −f(x)]

Definite Integrals Involving Rational/Irrational Expressions:

∫(0 to ∞) dx/(x² + a²) = π/(2a)

∫(0 to ∞) xᵐ dx/(xⁿ + aⁿ) = πa^(m−n+1) / [n sin((m+1)π/n)], 0 < m+1 < n

∫(0 to ∞) x^(p−1) dx/(1+x) = π/sin(pπ), 0 < p < 1

∫(−a to a) dx/√(a² − x²) = π/2

∫(−a to a) √(a² − x²) dx = πa²/4

Definite Integrals Involving Trigonometric Functions:

∫(0 to π) sin(mx)sin(nx)dx = 0 if m ≠ n; = π/2 if m = n (m, n positive integers)

∫(0 to π) cos(mx)cos(nx)dx = 0 if m ≠ n; = π/2 if m = n (m, n positive integers)

∫(0 to π) sin(mx)cos(nx)dx = 0 if m+n is even; = 2m/(m² − n²) if m+n is odd

∫(0 to π/2) sin²x dx = ∫(0 to π/2) cos²x dx = π/4

Walli's Formula:

∫(0 to π/2) sin²ᵐx dx = ∫(0 to π/2) cos²ᵐx dx = [1·3·5...(2m−1)]/[2·4·6...(2m)] × π/2, m = 1,2,...

∫(0 to π/2) sin²ᵐ⁺¹x dx = ∫(0 to π/2) cos²ᵐ⁺¹x dx = [2·4·6...(2m)]/[1·3·5...(2m+1)], m = 1,2,...

Periodic Function Properties:

∫(0 to nT) f(x)dx = n∫(0 to T) f(x)dx, n ∈ Z

∫(a to a+nT) f(x)dx = n∫(0 to T) f(x)dx, n ∈ Z

∫(mT to nT) f(x)dx = (n − m)∫(0 to T) f(x)dx, m, n ∈ Z

∫(nT to a+nT) f(x)dx = ∫(0 to a) f(x)dx

∫(a+nT to b+nT) f(x)dx = ∫(a to b) f(x)dx, n ∈ Z, a, b ∈ R

Leibniz Theorem:

If F(x) = ∫(g(x) to h(x)) f(t)dt, then: dF(x)/dx = h'(x)·f(h(x)) − g'(x)·f(g(x))

ELLIPSE

Standard Equation:

x²/a² + y²/b² = 1, where a > b and b² = a²(1 − e²)

Eccentricity: e = √(1 − b²/a²), 0 < e < 1 Directrices: x = ±a/e Foci: S = (±ae, 0) Vertices: A' = (−a, 0) and A = (a, 0) Length of major axis = 2a, minor axis = 2b Latus Rectum = 2b²/a = 2a(1 − e²)

Auxiliary Circle:

x² + y² = a²

Parametric Representation:

x = a cosθ, y = b sinθ

Position of Point (x₁, y₁) w.r.t. Ellipse:

x₁²/a² + y₁²/b² − 1 > 0 → Outside = 0 → On the ellipse < 0 → Inside

Line and Ellipse:

y = mx + c meets x²/a² + y²/b² = 1 in two points: Real if c² < a²m² + b² Coincident (tangent) if c² = a²m² + b² Imaginary if c² > a²m² + b²

Tangents to Ellipse:

Slope form: y = mx ± √(a²m² + b²) Point form: xx₁/a² + yy₁/b² = 1 Parametric form: (x cosθ)/a + (y sinθ)/b = 1

Normal to Ellipse:

At (x₁, y₁): a²x/x₁ − b²y/y₁ = a² − b² Parametric: ax secθ − by cosecθ = a² − b² Slope form: y = mx − (a² − b²)m/√(a² + b²m²)

Director Circle:

x² + y² = a² + b²

INVERSE TRIGONOMETRIC FUNCTIONS

Arcsine (sin⁻¹x):

sin⁻¹(−x) = −sin⁻¹(x), x ∈ [−1, 1] Domain: −1 ≤ x ≤ 1 Range: −π/2 ≤ y ≤ π/2 Derivative: d/dx[sin⁻¹(x)] = 1/√(1 − x²)

Arccosine (cos⁻¹x):

cos⁻¹(−x) = π − cos⁻¹(x), x ∈ [−1, 1] Domain: −1 ≤ x ≤ 1 Range: 0 ≤ y ≤ π Derivative: d/dx[cos⁻¹(x)] = −1/√(1 − x²)

Arctangent (tan⁻¹x):

tan⁻¹(−x) = −tan⁻¹(x), x ∈ R Domain: −∞ ≤ x ≤ ∞ Range: −π/2 ≤ y ≤ π/2 Derivative: d/dx[tan⁻¹(x)] = 1/(1 + x²)

Arc Cotangent (cot⁻¹x):

cot⁻¹(−x) = π − cot⁻¹(x), x ∈ R Domain: −∞ ≤ x ≤ ∞ Range: 0 ≤ y ≤ π Derivative: d/dx[cot⁻¹(x)] = −1/(1 + x²)

Arc Secant (sec⁻¹x):

sec⁻¹(−x) = π − sec⁻¹(x), |x| ≥ 1 Domain: −∞ ≤ x ≤ −1 or 1 ≤ x ≤ ∞ Range: 0 ≤ y ≤ π, y ≠ π/2 Derivative: d/dx[sec⁻¹(x)] = 1/(|x|√(x² − 1))

Arc Cosecant (cosec⁻¹x):

cosec⁻¹(−x) = −cosec⁻¹(x), |x| ≥ 1 Domain: −∞ ≤ x ≤ −1 or 1 ≤ x ≤ ∞ Range: −π/2 ≤ y ≤ π/2, y ≠ 0 Derivative: d/dx[cosec⁻¹(x)] = −1/(|x|√(x² − 1))

STRAIGHT LINES

Distance Formula:

d = √[(x₁ − x₂)² + (y₁ − y₂)²]

Section Formula:

Internal: x = (mx₂ + nx₁)/(m + n); y = (my₂ + ny₁)/(m + n) External: x = (mx₂ − nx₁)/(m − n); y = (my₂ − ny₁)/(m − n)

Centroid:

G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)

Incentre:

I = ((ax₁ + bx₂ + cx₃)/(a+b+c), (ay₁ + by₂ + cy₃)/(a+b+c))

Excentre (opposite to vertex A):

I₁ = ((−ax₁ + bx₂ + cx₃)/(−a+b+c), (−ay₁ + by₂ + cy₃)/(−a+b+c))

Area of Triangle:  

ΔABC = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| = ½ × determinant |x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1|

Slope Formula:

m = (y₁ − y₂)/(x₁ − x₂)

Condition of Collinearity of Three Points:

|x₁ y₁ 1| |x₂ y₂ 1| = 0 |x₃ y₃ 1|

Angle Between Two Straight Lines:

tanθ = |(m₁ − m₂)/(1 + m₁m₂)|

Bisector of Angles Between Two Lines:

(ax + by + c)/√(a² + b²) = ±(a'x + b'y + c')/√(a'² + b'²)

Condition of Concurrency (three lines a_ix + b_iy + c_i = 0, i = 1,2,3):

|a₁ b₁ c₁| |a₂ b₂ c₂| = 0 |a₃ b₃ c₃|

Pair of Straight Lines Through Origin:

ax² + 2hxy + by² = 0

Acute angle θ between them: tanθ = |2√(h² − ab)/(a + b)|

Two Lines ax + bx + c = 0 and a'x + b'y + c' = 0:

Parallel if: a/a' = b/b' ≠ c/c' Distance between parallel lines = |C₁ − C₂|/√(a² + b²) Perpendicular if: aa' + bb' = 0

Point and Line:   

Distance from point (x₁, y₁) to line ax + by + c = 0: = |ax₁ + by₁ + c|/√(a² + b²)

Reflection of point (x₁, y₁) about line ax + by + c = 0: (x − x₁)/a = (y − y₁)/b = −2(ax₁ + by₁ + c)/(a² + b²)

Foot of perpendicular from (x₁, y₁) on line ax + by + c = 0: (x − x₁)/a = (y − y₁)/b = −(ax₁ + by₁ + c)/(a² + b²)

INDEFINITE INTEGRATION

Basic Definition:

If g'(x) = f(x), then ∫f(x)dx = g(x) + c ⟺ d/dx{g(x) + c} = f(x)

Standard Formulas:

∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n+1)] + c, n ≠ −1 ∫dx/(ax + b) = (1/a) ln|ax + b| + c ∫e^(ax+b) dx = (1/a)e^(ax+b) + c ∫a^(px+q) dx = (1/p) × a^(px+q)/ln(a) + c, a > 0 ∫sin(ax + b) dx = −(1/a)cos(ax + b) + c ∫cos(ax + b) dx = (1/a)sin(ax + b) + c ∫tan(ax + b) dx = (1/a) ln|sec(ax + b)| + c ∫cot(ax + b) dx = (1/a) ln|sin(ax + b)| + c ∫sec²(ax + b) dx = (1/a)tan(ax + b) + c ∫cosec²(ax + b) dx = −(1/a)cot(ax + b) + c ∫sec(x) dx = ln|sec x + tan x| + c = ln|tan(π/4 + x/2)| + c ∫cosec(x) dx = ln|cosec x − cot x| + c = ln|tan(x/2)| + c

Inverse Trigonometric Integrals:

∫dx/√(a² − x²) = sin⁻¹(x/a) + c ∫dx/(a² + x²) = (1/a)tan⁻¹(x/a) + c ∫dx/(|x|√(x² − a²)) = (1/a)sec⁻¹(x/a) + c

Integrals with Square Roots:

∫dx/√(x² + a²) = ln|x + √(x² + a²)| + c ∫dx/√(x² − a²) = ln|x + √(x² − a²)| + c ∫dx/(a² − x²) = (1/2a) ln|(a + x)/(a − x)| + c ∫dx/(x² − a²) = (1/2a) ln|(x − a)/(x + a)| + c

∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2)sin⁻¹(x/a) + c ∫√(x² + a²) dx = (x/2)√(x² + a²) + (a²/2) ln|(x + √(x² + a²))/a| + c ∫√(x² − a²) dx = (x/2)√(x² − a²) − (a²/2) ln|(x + √(x² − a²))/a| + c

Integration by Substitution:

If f(x) = t, then f'(x)dx = dt

Integration by Parts:

∫f(x)g(x)dx = f(x)∫g(x)dx − ∫[d/dx(f(x)) × ∫g(x)dx]dx

Integration of Type ∫dx/(ax² + bx + c), ∫dx/√(ax² + bx + c), ∫√(ax² + bx + c)dx:

Substitute x + b/(2a) = t

Trigonometric Integrals:

For ∫dx/(a + b cos²x) or ∫dx/(a + b sin²x) or ∫dx/(a sin²x + b sinx cosx + c cos²x): Put tan x = t

For ∫dx/(a + b sinx) or ∫dx/(a + b cosx) or ∫dx/(a + b sinx + c cosx): Put tan(x/2) = t

Integration of Type ∫(x² + 1)/(x⁴ + Kx² + 1) dx:

Divide numerator and denominator by x², then put x ∓ 1/x = t

APPLICATION OF DERIVATIVES

Equation of Tangent at (x₁, y₁):

y − y₁ = f'(x₁)(x − x₁) (f'(x₁) must be real)

Equation of Normal at (x₁, y₁):  

y − y₁ = −[1/f'(x₁)](x − x₁) (f'(x₁) must be non-zero and real)

Tangent from External Point (a, b) not on curve y = f(x):

Find point of contact (h, f(h)) by solving: f'(h) = [f(h) − b]/(h − a) Then tangent: y − b = [(f(h) − b)/(h − a)](x − a)

Lengths of Tangent, Normal, Subtangent, Subnormal:   

(where k = y₁, m = dy/dx at the point)

Length of tangent PT = |k|√(1 + 1/m²) Length of normal PN = |k|√(1 + m²) Length of subtangent TM = |k/m| Length of subnormal MN = |km|

Angle Between Two Curves:   

tanθ = |(m₁ − m₂)/(1 + m₁m₂)| (m₁, m₂ are slopes of tangents at point of intersection)

Rolle's Theorem:

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c ∈ (a, b) such that f'(c) = 0.

Lagrange's Mean Value Theorem (LMVT):

If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c ∈ (a, b) such that: [f(b) − f(a)]/(b − a) = f'(c)

Formulae of Mensuration:

Volume of cuboid = lbh Surface area of cuboid = 2(lb + bh + hl) Volume of cube = a³ Surface area of cube = 6a² Volume of cone = (1/3)πr²h Curved surface area of cone = πrl (l = slant height) Curved surface area of cylinder = 2πrh Total surface area of cylinder = 2πrh + 2πr² Volume of sphere = (4/3)πr³ Surface area of sphere = 4πr² Area of circular sector = (1/2)r²θ (θ in radians) Volume of prism = (area of base) × height Lateral surface area of prism = (perimeter of base) × height Total surface area of prism = (lateral surface area) + 2(area of base) Volume of pyramid = (1/3)(area of base) × height Curved surface area of pyramid = (1/2)(perimeter of base) × (slant height)

SEQUENCE & SERIES

Arithmetic Progression (A.P.):

a, a+d, a+2d, ..., a+(n−1)d

nth term: tₙ = a + (n − 1)d

Sum of first n terms: Sₙ = (n/2)[2a + (n − 1)d] = (n/2)[a + l] (l = last term)

rth term when sum is given: tᵣ = Sᵣ − Sᵣ₋₁  

Properties of A.P.:

If a, b, c are in A.P.: 2b = a + c. If a, b, c, and d are in A.P.: a + d = b + c. Sum of terms equidistant from beginning and end = sum of first and last term

Arithmetic Mean:

If a, b, and c are in A.P., then b is the A.M. of a and c.

n Arithmetic Means between a and b (a, A₁, A₂, ..., Aₙ, b in A.P.): A₁ = a + (b−a)/(n+1) A₂ = a + 2(b−a)/(n+1) ... Aₙ = a + n(b−a)/(n+1)

Σ(r=1 to n) Aᵣ = nA, where A is the single A.M. between a and b

Geometric Progression (G.P.):

a, ar, ar², ar³, ar⁴, …

nth term = arⁿ⁻¹

Sum of first n terms: Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1 Sₙ = na, r = 1

Sum to infinity (|r| < 1): S∞ = a/(1 − r)

Harmonic Mean:

If a, b, c are in H.P.: b = 2ac/(a + c)

H.M. of a₁, a₂, ..., aₙ: 1/H = (1/n)[1/a₁ + 1/a₂ + ... + 1/aₙ]

Relation Between Means:

G² = AH A.M. ≥ G.M. ≥ H.M. A.M. = G.M. = H.M. if a₁ = a₂ = a₃ = ... = aₙ

Important Summation Results:

Σ(r=1 to n)(aᵣ ± bᵣ) = Σaᵣ ± Σbᵣ Σ(r=1 to n) k·aᵣ = k·Σaᵣ Σ(r=1 to n) k = nk (k = constant) Σ(r=1 to n) r = n(n+1)/2 Σ(r=1 to n) r² = n(n+1)(2n+1)/6 Σ(r=1 to n) r³ = [n(n+1)/2]² = n²(n+1)²/4

HYPERBOLA

Standard Equation:

x²/a² − y²/b² = 1, where b² = a²(e² − 1)

Foci: S ≡ (±ae, 0) Directrices: x = ±a/e Vertices: A ≡ (±a, 0) Eccentricity: e > 1 Latus Rectum: l = 2b²/a = 2a(e² − 1)

Conjugate Hyperbola:

x²/a² − y²/b² = 1 and −x²/a² + y²/b² = 1 are conjugate hyperbolas of each other.

Auxiliary Circle:

x² + y² = a²

Parametric Representation:

x = a secθ, y = b tanθ

Position of Point (x₁, y₁) w.r.t. Hyperbola:

S₁ = x₁²/a² − y₁²/b² − 1 S₁ > 0 → Inside S₁ = 0 → On the hyperbola S₁ < 0 → Outside

Tangents:

Slope form: y = mx ± √(a²m² − b²) Point form: xx₁/a² − yy₁/b² = 1 Parametric form: (x secθ)/a − (y tanθ)/b = 1

Normal:

At (x₁, y₁): a²x/x₁ + b²y/y₁ = a² + b² = a²e². At (a secθ, b tanθ): ax/secθ + by/tanθ = a² + b² = a²e². Slope form: y = mx ± (a² + b²)m/√(a² − b²m²)

Asymptotes:

x/a + y/b = 0 and x/a − y/b = 0 Pair of asymptotes: x²/a² − y²/b² = 0

Rectangular (Equilateral) Hyperbola xy = c²:

Eccentricity = √2 Vertices: (±c, ±c) Foci: (±√2 c, ±√2 c) Directrices: x + y = ±√2 c Latus Rectum: l = 2√2 c = T.A. = C.A.

Parametric equation: x = ct, y = c/t, t ∈ R − {0}

Tangent at (x₁, y₁): x/x₁ + y/y₁ = 2 Tangent at P(t): x/t + ty = 2c Normal at P(t): xt³ − yt = c(t⁴ − 1) Chord with midpoint (h, k): kx + hy = 2hk

Maths is the highest-scoring subject in JEE Advanced if you know your formulas cold. Students who revise formulas daily score 20-30 marks more than those who rely on memory alone. This formula sheet is designed to make that daily revision fast and easy.

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