JEE Maths Formula Sheet: Every JEE Main Unit on One Printable Page

This JEE maths formula sheet puts the key formulas for all 14 units of the official JEE Main 2026 Mathematics syllabus on one printable page. It follows the order of the NTA syllabus, and each unit ends with a short note on how to use the formulas and the traps that cost students marks. Print it (Ctrl + P or "Print" on your phone) and keep it next to your practice papers, not in a drawer.

A formula sheet helps only if you already understand where each result comes from. If whole units feel unfamiliar, work with a JEE Maths tutor in Gurgaon first and use this page for revision. When you want a plan for turning formulas into marks, read our guide on how to score 90+ in JEE Main Maths.

By the Ajay Vatsyayan Classes Home Tutors Team. Reviewed by Ajay Vatsyayan (B.Tech; Maths and Physics, Class 11–12 and JEE). Last reviewed: 28 September 2026.

How to use this JEE maths formula sheet

Use this sheet to recall formulas, not to learn them for the first time. The units follow the official JEE Main 2026 syllabus published by NTA, so nothing here is outside what the syllabus names, except for a few standard shortcuts that we label as such. NTA can revise the syllabus for each session, so check the current one on the official JEE Main website before your exam.

Read the notation. ⁿC_r means "n choose r", |z| is the modulus of z, and a bar over z (z̄) is its conjugate. Vectors are shown in bold (a, b).
Cover and recall. Cover the right-hand column of a table and write each formula from memory. Anything you miss goes into your error log.
Read the trap notes. Most lost marks come from using a formula outside its conditions, not from forgetting it.

JEE Main asks many questions that test whether you can apply a result quickly. Our calculus for JEE and coordinate geometry for JEE guides show worked problems for the two largest blocks on this sheet.

1. Sets, relations and functions

Result	Formula
Union of two sets	n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Union of three sets	n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)
De Morgan's laws	(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′
Power set of a set with n elements	n(P(A)) = 2ⁿ
Relations from A (m elements) to B (n elements)	2^mn
Reflexive relations on a set of n elements	2ⁿ⁽ⁿ⁻¹⁾
Symmetric relations on a set of n elements	2^n(n+1)/2
Functions from A (m elements) to B (n elements)	n^m
One-one functions (m ≤ n)	ⁿP_m = n! / (n − m)!
Onto functions (m ≥ n)	Σ_k=0ⁿ (−1)^k ⁿC_k (n − k)^m
Composition and inverse	(g ∘ f)(x) = g(f(x)); (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹

How to use / common traps: An equivalence relation must be reflexive, symmetric and transitive; students often check two and assume the third. For g ∘ f to exist, the range of f must sit inside the domain of g. A function has an inverse only if it is both one-one and onto. For "onto" counts with small numbers (for example 4 elements onto 2), direct counting (2⁴ − 2 = 14) is faster than the general formula.

2. Complex numbers and quadratic equations

Result	Formula
Powers of i	i² = −1, i³ = −i, i⁴ = 1, so i^4k = 1
Modulus of z = a + ib	\|z\| = √(a² + b²); z z̄ = \|z\|²
Polar (Euler) form	z = r(cos θ + i sin θ) = re^iθ, principal argument θ ∈ (−π, π]
Product and quotient	\|z₁z₂\| = \|z₁\|\|z₂\|; arg(z₁z₂) = arg z₁ + arg z₂ (up to a multiple of 2π); \|z₁/z₂\| = \|z₁\|/\|z₂\|
Triangle inequality	\|z₁ + z₂\| ≤ \|z₁\| + \|z₂\|; \|\|z₁\| − \|z₂\|\| ≤ \|z₁ − z₂\|
Geometry in the Argand plane	\|z₁ − z₂\| = distance between the points; \|z − z₀\| = r is a circle with centre z₀ and radius r
De Moivre's theorem (standard shortcut)	(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
Cube roots of unity	1, ω, ω², with ω = (−1 + i√3)/2, ω³ = 1, 1 + ω + ω² = 0
Roots of ax² + bx + c = 0 (a ≠ 0)	x = (−b ± √D) / 2a, where D = b² − 4ac
Nature of roots (real coefficients)	D > 0: real and distinct; D = 0: real and equal; D < 0: complex conjugate pair
Sum and product of roots	α + β = −b/a; αβ = c/a
Useful identities	α² + β² = (α + β)² − 2αβ; \|α − β\| = √D / \|a\|
Equation with given roots	x² − (α + β)x + αβ = 0
Common root of a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0	(c₁a₂ − c₂a₁)² = (b₁c₂ − b₂c₁)(a₁b₂ − a₂b₁)
Both roots greater than k	D ≥ 0, a·f(k) > 0 and −b/2a > k

How to use / common traps: Complex roots come in conjugate pairs only when the coefficients are real. The rule arg(z₁z₂) = arg z₁ + arg z₂ can take you outside the principal range, so adjust by 2π at the end. Many "find the value of an expression in α and β" questions are really sum-and-product questions, so rewrite the expression before you try to find the roots.

3. Matrices and determinants

Result	Formula (A is a square matrix of order n)
Transpose and inverse of a product	(AB)^T = B^TA^T; (AB)⁻¹ = B⁻¹A⁻¹
Adjoint identity	A (adj A) = (adj A) A = \|A\| I
Inverse	A⁻¹ = (adj A) / \|A\|, only when \|A\| ≠ 0
Determinant rules	\|A^T\| = \|A\|; \|AB\| = \|A\|\|B\|; \|kA\| = kⁿ\|A\|; \|A⁻¹\| = 1/\|A\|
Adjoint results	\|adj A\| = \|A\|ⁿ⁻¹; adj(adj A) = \|A\|ⁿ⁻² A
Symmetric and skew-symmetric	A^T = A and A^T = −A; any A = ½(A + A^T) + ½(A − A^T); a skew-symmetric matrix of odd order has determinant 0
Orthogonal matrix	AA^T = I, so \|A\| = ±1
Trace	tr(AB) = tr(BA); tr(A + B) = tr A + tr B
Area of a triangle	½ \|det of the 3 × 3 matrix with rows (x₁, y₁, 1), (x₂, y₂, 1), (x₃, y₃, 1)\|
Cramer's rule, Δ ≠ 0	Unique solution: x = Δ₁/Δ, y = Δ₂/Δ, z = Δ₃/Δ
Cramer's rule, Δ = 0	If any Δ_i ≠ 0: no solution. If all Δ_i = 0: infinitely many solutions or none (check further)
Homogeneous system AX = O	Only the trivial solution if \|A\| ≠ 0; non-trivial solutions exist if \|A\| = 0

How to use / common traps: |kA| = kⁿ|A|, not k|A|. This single slip is one of the most common errors we see. When Δ = 0 and all Δ_i = 0, do not write "infinitely many solutions" automatically for a 3 × 3 system; two of the planes can still be parallel. Matrix multiplication is not commutative, so (A + B)² = A² + AB + BA + B².

From our tutors: in matrices questions, students often lose time multiplying out a 3 × 3 product that the question never needed. Before you calculate, ask whether a determinant property (|AB| = |A||B|, |adj A| = |A|ⁿ⁻¹) gives the answer directly. It usually does.

4. Permutations and combinations

Result	Formula
Permutations (arrangements) of r from n distinct objects	ⁿP_r = n! / (n − r)!
Combinations (selections)	ⁿC_r = n! / [r! (n − r)!]
Key identities	ⁿC_r = ⁿC_n−r; ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r; ⁿP_r = r! · ⁿC_r
Arrangements with repeated objects	n! / (p! q! r!), where p, q, r objects are alike
Arrangements with repetition allowed	n^r
Circular arrangements of n distinct objects	(n − 1)!; if clockwise and anticlockwise are the same (necklace, garland): (n − 1)!/2
Selections from n distinct objects	Any number: 2ⁿ; at least one: 2ⁿ − 1
n identical objects into r groups (empty allowed)	^n+r−1C_r−1
n identical objects into r groups (none empty)	ⁿ⁻¹C_r−1
Diagonals of an n-sided polygon	ⁿC₂ − n = n(n − 3)/2
Derangements (standard shortcut)	D_n = n! [1 − 1/1! + 1/2! − … + (−1)ⁿ/n!]; D₃ = 2, D₄ = 9

How to use / common traps: Decide first whether order matters (arrangement) or not (selection), and whether objects are distinct or identical. Most wrong answers in this unit come from that first decision, not from arithmetic. For small cases, list the possibilities to check your formula; a two-minute check is worth four marks.

5. Binomial theorem

Result	Formula (n a positive integer)
Expansion	(x + a)ⁿ = Σ_r=0ⁿ ⁿC_r x^n−r a^r
General term	T_r+1 = ⁿC_r x^n−r a^r
Number of terms	n + 1
Middle term	n even: T_{n/2 + 1}; n odd: T_(n+1)/2 and T_(n+3)/2
Sum of coefficients	ⁿC₀ + ⁿC₁ + … + ⁿC_n = 2ⁿ (put x = a = 1)
Even and odd coefficients	ⁿC₀ + ⁿC₂ + … = ⁿC₁ + ⁿC₃ + … = 2ⁿ⁻¹
Weighted sum	Σ r · ⁿC_r = n · 2ⁿ⁻¹
Sum of squares	(ⁿC₀)² + (ⁿC₁)² + … + (ⁿC_n)² = ²ⁿC_n
Term independent of x	Write the power of x in T_r+1 and set it equal to 0

Worked example: find the term independent of x in (x² + 1/x)⁶. T_r+1 = ⁶C_r (x²)^6−r (1/x)^r = ⁶C_r x^{12 − 3r}. Setting 12 − 3r = 0 gives r = 4, so the term is ⁶C₄ = 15.

How to use / common traps: The general term is T_r+1, not T_r. "Coefficient of the term" and "the term" are different answers when the term contains x. Keep negative signs inside the bracket: in (x − a)ⁿ, the general term carries (−a)^r.

6. Sequences and series

Result	Formula
AP: nth term	a_n = a + (n − 1)d
AP: sum of n terms	S_n = (n/2)[2a + (n − 1)d] = (n/2)(a + l), where l is the last term
GP: nth term	a_n = arⁿ⁻¹
GP: sum of n terms	S_n = a(rⁿ − 1)/(r − 1), r ≠ 1
GP: sum to infinity	S_∞ = a/(1 − r), only when \|r\| < 1
Means of two positive numbers a, b	AM = (a + b)/2; GM = √(ab); HM = 2ab/(a + b)
Inequality of means	AM ≥ GM ≥ HM (equality only when a = b); for two numbers, GM² = AM × HM
Inserting n means between a and b	n AMs: common difference d = (b − a)/(n + 1); n GMs: common ratio r = (b/a)^1/(n+1)
nth term from the sum	a_n = S_n − S_n−1 (n ≥ 2)
Standard sums (useful tools)	Σk = n(n + 1)/2; Σk² = n(n + 1)(2n + 1)/6; Σk³ = [n(n + 1)/2]²

How to use / common traps: AM ≥ GM needs positive numbers; using it with negative values gives wrong minimum values. S_∞ exists only for |r| < 1. In questions such as "three numbers in AP", take them as a − d, a, a + d so the sum gives a immediately; for three numbers in GP, take a/r, a, ar.

Know the formulas but still lose marks on application? A one-to-one JEE Maths tutor at home in Gurgaon or online can find the exact gaps in a free demo class.

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7. Limits, continuity and differentiability

Standard limits

Limit (x → 0 unless stated)	Value
(sin x)/x, (tan x)/x	1 (x in radians)
(1 − cos x)/x²	1/2
(e^x − 1)/x; ln(1 + x)/x	1
(a^x − 1)/x, a > 0	ln a
(1 + x)^1/x	e
(xⁿ − aⁿ)/(x − a), as x → a	n aⁿ⁻¹
1^∞ form: f(x) → 1 and g(x) → ∞	lim f^g = e^{lim g(f − 1)}

Derivatives

f(x)	f′(x)	f(x)	f′(x)
xⁿ	n xⁿ⁻¹	sin x	cos x
e^x	e^x	cos x	−sin x
a^x	a^x ln a	tan x	sec² x
ln x	1/x	cot x	−cosec² x
sin⁻¹ x	1/√(1 − x²)	sec x	sec x tan x
cos⁻¹ x	−1/√(1 − x²)	cosec x	−cosec x cot x
tan⁻¹ x	1/(1 + x²)	cot⁻¹ x	−1/(1 + x²)

Rule	Formula
Product and quotient	(uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²
Chain rule	d/dx f(g(x)) = f′(g(x)) · g′(x)
Parametric form	dy/dx = (dy/dt)/(dx/dt)
Continuity at x = a	LHL = RHL = f(a)
Differentiability at x = a	Left derivative = right derivative; differentiable ⇒ continuous (not the reverse)
Monotonicity	f′(x) > 0 on an interval ⇒ increasing; f′(x) < 0 ⇒ decreasing
Local maximum/minimum (second-derivative test)	f′(c) = 0 and f″(c) < 0 ⇒ maximum; f″(c) > 0 ⇒ minimum

How to use / common traps: The standard limits hold only in radians. |x| is continuous at 0 but not differentiable there, a favourite JEE example. If f″(c) = 0, the second-derivative test tells you nothing; check the sign of f′ on both sides. On a closed interval, compare critical values with the values at the end points before naming the absolute maximum.

8. Integral calculus

Standard integrals (add + C to each)

Integral	Result
∫ xⁿ dx, n ≠ −1	xⁿ⁺¹/(n + 1)
∫ (1/x) dx	ln \|x\|
∫ e^x dx; ∫ a^x dx	e^x; a^x/ln a
∫ sin x dx; ∫ cos x dx; ∫ sec² x dx	−cos x; sin x; tan x
∫ tan x dx; ∫ cot x dx	ln \|sec x\|; ln \|sin x\|
∫ sec x dx; ∫ cosec x dx	ln \|sec x + tan x\|; ln \|cosec x − cot x\|
∫ dx/(x² + a²)	(1/a) tan⁻¹(x/a)
∫ dx/(x² − a²)	(1/2a) ln \|(x − a)/(x + a)\|
∫ dx/(a² − x²)	(1/2a) ln \|(a + x)/(a − x)\|
∫ dx/√(a² − x²)	sin⁻¹(x/a)
∫ dx/√(x² + a²)	ln \|x + √(x² + a²)\|
∫ dx/√(x² − a²)	ln \|x + √(x² − a²)\|
∫ √(a² − x²) dx	(x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a)
∫ √(x² + a²) dx	(x/2)√(x² + a²) + (a²/2) ln \|x + √(x² + a²)\|
∫ √(x² − a²) dx	(x/2)√(x² − a²) − (a²/2) ln \|x + √(x² − a²)\|
∫ e^x[f(x) + f′(x)] dx	e^x f(x)

Methods and definite-integral properties

Result	Formula
Integration by parts	∫ u v dx = u ∫ v dx − ∫ [u′ ∫ v dx] dx; choose u in the order ILATE (inverse trig, log, algebraic, trig, exponential)
Fundamental theorem of calculus	∫_a^b f(x) dx = F(b) − F(a), where F′ = f; d/dx ∫_a^x f(t) dt = f(x)
King's property	∫_a^b f(x) dx = ∫_a^b f(a + b − x) dx; ∫₀^a f(x) dx = ∫₀^a f(a − x) dx
Even and odd functions	∫_−a^a f(x) dx = 2∫₀^a f(x) dx if f is even; 0 if f is odd
Doubling property	∫₀^2a f(x) dx = 2∫₀^a f(x) dx if f(2a − x) = f(x); 0 if f(2a − x) = −f(x)
Periodic function (period T)	∫₀^nT f(x) dx = n ∫₀^T f(x) dx
Area between curves	Area = ∫_a^b [upper curve − lower curve] dx

Worked example: I = ∫₀^π/2 sin x / (sin x + cos x) dx. Using f(π/2 − x), I = ∫₀^π/2 cos x / (cos x + sin x) dx. Adding the two forms gives 2I = ∫₀^π/2 1 dx = π/2, so I = π/4.

How to use / common traps: The modulus in ln |x| matters when x can be negative. In area questions, split the interval wherever the curves cross; integrating "upper minus lower" blindly can cancel positive and negative areas. Always sketch before you integrate an area.

From our tutors: when we check students' definite-integral work, the most common lost mark is not the method but the limits after a substitution. We teach one habit: the moment you write t = g(x), change the limits on the same line. Students who do this stop "solving correctly" and getting the wrong number.

9. Differential equations

Type	Method
Order and degree	Order = highest derivative; degree = power of that derivative once the equation is a polynomial in derivatives (not defined otherwise, e.g. with sin(dy/dx))
Variables separable: dy/dx = f(x) g(y)	∫ dy/g(y) = ∫ f(x) dx + C
Homogeneous: dy/dx = F(y/x)	Put y = vx, so dy/dx = v + x dv/dx, then separate variables
Linear: dy/dx + P(x) y = Q(x)	Integrating factor IF = e^{∫P dx}; solution y · IF = ∫ Q · IF dx + C
Linear in x: dx/dy + P(y) x = Q(y)	IF = e^{∫P dy}; x · IF = ∫ Q · IF dy + C

Worked example: solve dy/dx + y = e^x. Here P = 1, so IF = e^x. Then y e^x = ∫ e^2x dx = e^2x/2 + C, giving y = e^x/2 + C e^−x. Check: y′ + y = (e^x/2 − Ce^−x) + (e^x/2 + Ce^−x) = e^x.

How to use / common traps: Put the equation in standard linear form (coefficient of dy/dx equal to 1) before finding P. Do not forget the constant C; many JEE questions give a starting condition such as y(0) = 1 to fix it, and the answer choices are built around students who skip that step.

10. Coordinate geometry

Points and straight lines

Result	Formula
Distance	√[(x₂ − x₁)² + (y₂ − y₁)²]
Section formula, ratio m : n	Internal: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)); external: replace + with − in each numerator and denominator
Centroid	((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
Centroid, orthocentre, circumcentre	They lie on one line; the centroid divides the orthocentre–circumcentre segment in the ratio 2 : 1 (from the orthocentre)
Slope	m = (y₂ − y₁)/(x₂ − x₁) = tan θ; parallel: m₁ = m₂; perpendicular: m₁m₂ = −1
Forms of a line	y = mx + c; y − y₁ = m(x − x₁); x/a + y/b = 1; x cos α + y sin α = p
Angle between two lines	tan θ = \|(m₁ − m₂)/(1 + m₁m₂)\|
Distance from (x₁, y₁) to ax + by + c = 0	\|ax₁ + by₁ + c\| / √(a² + b²)
Distance between parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0	\|c₁ − c₂\| / √(a² + b²)
Three lines a_ix + b_iy + c_i = 0 concurrent	The 3 × 3 determinant of their coefficients is 0 (and no two lines are parallel)

Circle and conic sections (standard forms)

Curve	Key formulas
Circle	(x − h)² + (y − k)² = r²; general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c); diameter form (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0
Line meeting x² + y² = a²	y = mx + c touches the circle when c² = a²(1 + m²); cuts it in two points when c² < a²(1 + m²)
Parabola y² = 4ax	Focus (a, 0); directrix x = −a; latus rectum 4a; parametric point (at², 2at); y = mx + a/m is a tangent
Ellipse x²/a² + y²/b² = 1 (a > b)	b² = a²(1 − e²); foci (±ae, 0); directrices x = ±a/e; latus rectum 2b²/a; SP + S′P = 2a; y = mx + c is a tangent when c² = a²m² + b²
Hyperbola x²/a² − y²/b² = 1	b² = a²(e² − 1); foci (±ae, 0); latus rectum 2b²/a; \|SP − S′P\| = 2a; asymptotes y = ±(b/a)x; y = mx + c is a tangent when c² = a²m² − b²

How to use / common traps: The general circle equation is a real circle only if g² + f² − c > 0, and the coefficients of x² and y² must first be made equal to 1. For an ellipse with b > a, the major axis is along the y-axis, so the foci move to (0, ±be) and the formulas swap. The syllabus names conic sections "in standard forms"; tangent conditions are included here as time-saving shortcuts.

11. Three-dimensional geometry

Result	Formula
Distance between two points	√[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
Section formula, ratio m : n (internal)	((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n))
Direction cosines from direction ratios a, b, c	l = a/√(a² + b² + c²), and similarly m, n; l² + m² + n² = 1
Angle between two lines	cos θ = \|a₁a₂ + b₁b₂ + c₁c₂\| / [√(a₁² + b₁² + c₁²) √(a₂² + b₂² + c₂²)]
Perpendicular / parallel lines	a₁a₂ + b₁b₂ + c₁c₂ = 0 / a₁/a₂ = b₁/b₂ = c₁/c₂
Equation of a line	Vector: r = a + λb; Cartesian: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c
Shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂	d = \|(a₂ − a₁) · (b₁ × b₂)\| / \|b₁ × b₂\|
Distance between parallel lines (common direction b)	d = \|b × (a₂ − a₁)\| / \|b\|

How to use / common traps: Two non-parallel lines intersect exactly when the shortest distance is 0. The skew-line formula fails for parallel lines because b₁ × b₂ = 0, so use the parallel-line formula instead. The JEE Main 2026 syllabus text for this unit lists points, direction cosines, lines and skew lines; it does not name the plane. Check the current syllabus before spending revision time on plane formulas for JEE Main (they remain useful for school exams and JEE Advanced).

12. Vector algebra

Result	Formula
Magnitude and unit vector	\|a\| = √(a₁² + a₂² + a₃²); unit vector = a/\|a\|
Scalar (dot) product	a · b = \|a\|\|b\| cos θ = a₁b₁ + a₂b₂ + a₃b₃
Vector (cross) product	\|a × b\| = \|a\|\|b\| sin θ; a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
Perpendicular and parallel	a · b = 0 (perpendicular); a × b = 0 (parallel)
Projection of a on b	(a · b)/\|b\|
Areas	Parallelogram with sides a, b: \|a × b\|; triangle: ½\|a × b\|; parallelogram with diagonals d₁, d₂: ½\|d₁ × d₂\|
Magnitude of a sum	\|a + b\|² = \|a\|² + \|b\|² + 2 a · b
Lagrange's identity	\|a × b\|² + (a · b)² = \|a\|²\|b\|²
Scalar triple product (used in the skew-line formula)	[a b c] = a · (b × c) = determinant of the components; 0 means the vectors are coplanar

How to use / common traps: The cross product is not commutative: b × a = −(a × b). The middle (j) component of the determinant carries a minus sign, which is where most arithmetic errors happen. Projection is a scalar; the projection vector is [(a · b)/|b|²] b.

13. Statistics and probability

Statistics

Measure	Formula
Mean	x̄ = Σx_i/n; grouped: x̄ = Σf_ix_i/Σf_i
Median (grouped)	l + [(N/2 − C)/f] × h, where C is the cumulative frequency before the median class
Mode (grouped)	l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
Mean deviation about a (mean or median)	Σ\|x_i − a\|/n; grouped: Σf_i\|x_i − a\|/Σf_i
Variance and standard deviation	σ² = Σ(x_i − x̄)²/n = Σx_i²/n − x̄²; σ = √(variance)
Change of data	Adding a constant to every value: variance unchanged. Multiplying every value by k: variance × k², SD × \|k\|

Probability

Result	Formula
Addition theorem	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Conditional probability	P(A \| B) = P(A ∩ B)/P(B), P(B) ≠ 0
Multiplication theorem	P(A ∩ B) = P(B) P(A \| B); for independent events P(A ∩ B) = P(A) P(B)
Total probability (E₁, …, E_n a partition)	P(A) = Σ P(E_i) P(A \| E_i)
Bayes' theorem	P(E_i \| A) = P(E_i) P(A \| E_i) / Σ_j P(E_j) P(A \| E_j)
Random variable: mean and variance	E(X) = Σ x_i p_i; Var(X) = E(X²) − [E(X)]²
Binomial distribution (standard shortcut)	P(X = r) = ⁿC_r p^r q^n−r, q = 1 − p; mean np; variance npq

How to use / common traps: Mutually exclusive and independent are different ideas; two events with non-zero probability cannot be both. In a probability distribution, the probabilities must add to 1, which often gives you the unknown constant in the first line. In Bayes' questions, draw a two-level tree first; it prevents mixing up P(A | E) and P(E | A).

14. Trigonometry

Result	Formula
Pythagorean identities	sin² x + cos² x = 1; 1 + tan² x = sec² x; 1 + cot² x = cosec² x
Compound angles	sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Double angle	sin 2A = 2 sin A cos A = 2 tan A/(1 + tan² A); cos 2A = cos² A − sin² A = 1 − 2 sin² A = 2 cos² A − 1 = (1 − tan² A)/(1 + tan² A); tan 2A = 2 tan A/(1 − tan² A)
Triple angle	sin 3A = 3 sin A − 4 sin³ A; cos 3A = 4 cos³ A − 3 cos A; tan 3A = (3 tan A − tan³ A)/(1 − 3 tan² A)
Sum to product	sin C + sin D = 2 sin[(C + D)/2] cos[(C − D)/2]; sin C − sin D = 2 cos[(C + D)/2] sin[(C − D)/2]; cos C + cos D = 2 cos[(C + D)/2] cos[(C − D)/2]; cos C − cos D = −2 sin[(C + D)/2] sin[(C − D)/2]
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); 2 sin A sin B = cos(A − B) − cos(A + B)
Range of a sin x + b cos x	From −√(a² + b²) to +√(a² + b²)
Useful values	sin 15° = (√6 − √2)/4; cos 15° = (√6 + √2)/4; tan 15° = 2 − √3; sin 18° = (√5 − 1)/4; cos 36° = (√5 + 1)/4

Inverse trigonometric functions

Function	Domain	Principal range
sin⁻¹ x	[−1, 1]	[−π/2, π/2]
cos⁻¹ x	[−1, 1]	[0, π]
tan⁻¹ x	all real x	(−π/2, π/2)
cot⁻¹ x	all real x	(0, π)
sec⁻¹ x	\|x\| ≥ 1	[0, π] except π/2
cosec⁻¹ x	\|x\| ≥ 1	[−π/2, π/2] except 0

sin⁻¹ x + cos⁻¹ x = π/2 (|x| ≤ 1); tan⁻¹ x + cot⁻¹ x = π/2
sin⁻¹(−x) = −sin⁻¹ x; tan⁻¹(−x) = −tan⁻¹ x; cos⁻¹(−x) = π − cos⁻¹ x
tan⁻¹ x + tan⁻¹ y = tan⁻¹[(x + y)/(1 − xy)] when xy < 1

How to use / common traps: sin⁻¹(sin x) = x only when x is in [−π/2, π/2]; outside that interval, bring the angle back into the principal range first. The tan⁻¹ addition formula changes when xy > 1 (add or subtract π depending on the signs), which is exactly the case JEE likes to test.

A 10-minute daily revision routine with this sheet

A JEE maths formula chart on the wall does little by itself. Formulas stay in memory when you recall them under light pressure and then use them in problems. This is the routine our tutors give students across Gurgaon and Gurugram:

Minutes 1–4: blind recall. Pick one unit. Cover the formula column and write every result you remember on rough paper.
Minutes 5–6: mark and log. Compare with the sheet. Copy each missed or wrong formula into your error log with the date.
Minutes 7–10: use it once. Solve one previous-year question from that unit that needs the formula you missed. Our guide to the JEE Main maths study plan explains how this fits into weekly revision.

Rotate through the 14 units so that each one comes round roughly every two weeks. Units you score poorly on in mock tests should come round more often.

From our tutors: students who only re-read a formula sheet feel confident but freeze in the exam. The students who improve fastest write formulas from memory first and check afterwards. The act of getting a formula wrong, then correcting it, is what makes it stick.

Printing tips

Use your browser's print option. The table of contents and booking boxes are hidden when you print, so the sheet prints as formulas and notes only.
Print on both sides and keep the pages in the same file as your error log.
Add your own shortcuts in the margins. A sheet with your handwriting on it is revised far more often than a clean one.

Want a tutor to test you on these formulas and fix the units that cost you marks? Ajay Vatsyayan Classes offers one-to-one JEE Maths tuition at home across Gurgaon and online. Start with a free demo class.

Book a Free Demo Class +91 92204 75088

Frequently asked questions

Is this JEE maths formula sheet enough for JEE Main?

It covers the key formulas for every unit in the JEE Main 2026 Mathematics syllabus, but formulas alone are not enough. JEE Main tests whether you can choose and apply the right result quickly, so pair this sheet with previous-year questions and timed practice.

Can I download this as a PDF?

We do not offer a separate download. The page is printable: use your browser's "Print" option and choose "Save as PDF" if you want a copy on your device. The table of contents and booking boxes are hidden when printed.

Does JEE Main give a formula sheet in the exam?

No formula sheet is described for the JEE Main Mathematics paper, so you need to know the formulas yourself. Always read the current NTA information bulletin for exam-day instructions.

Which JEE mains maths formula sheet units should I learn first?

Start with the units that other chapters depend on: trigonometry, quadratic equations, and functions. Calculus and coordinate geometry use them constantly. After that, follow your school or coaching order, and give extra revision time to the units where your mock-test accuracy is lowest.

How many formulas are there in JEE Maths?

There is no official count, and counting is not useful. What matters is that you can recall and apply the core results in each of the 14 syllabus units. This sheet keeps to results that regularly help in JEE Main questions.

Are plane formulas needed for JEE Main 2026?

The JEE Main 2026 syllabus text for three-dimensional geometry names points, direction cosines, lines and skew lines, and does not mention the plane. Syllabi can change, so check the current NTA syllabus. Plane formulas are still needed for Class 12 boards and JEE Advanced.

How often should I revise the formula chart?

A short daily session of about 10 minutes, rotating through the units, works better than one long session a week. Increase the frequency for units that cost you marks in mocks.

Can a home tutor help with memorising formulas?

A tutor helps most by showing where each formula comes from and by testing recall in problems, which makes memorising far easier. Ajay Vatsyayan Classes tutors teach JEE Maths one-to-one at home in Gurgaon and online.

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JEE Maths Guide

Part of our JEE Maths guide