All JEE Main Maths Formulas

💡 Always non-negative. |z| = 0 iff z = 0.

💡 Check the quadrant! tan⁻¹ alone gives values in (-π/2, π/2).

💡 r = |z|, θ = arg(z). Euler's form is faster for multiplication.

💡 z·z̄ = |z|² is used everywhere - division, modulus proofs, locus.

💡 Arguments add for product, subtract for quotient.

💡 Works for all integers n. Key for finding nth roots.

💡 They form a regular n-gon on the unit circle. Sum = 0, Product = (-1)^{n+1}.

💡 ω = (-1+i√3)/2. Used heavily in factoring and symmetric expressions.

💡 Equality in upper bound when arg(z₁) = arg(z₂). Lower bound when arg differ by π.

💡 |z − a| = |z − b| is the perpendicular bisector of segment ab.

💡 arg((z-a)/(z-b)) = θ gives an arc of a circle through a and b.

💡 Complex roots always come in conjugate pairs for real polynomials.

💡 a = first term, d = common difference. Works for any integer n.

💡 l = last term. Second form is faster when you know the last term.

💡 a = first term, r = common ratio. Valid for all positive integers n.

💡 Use (1-r^n)/(1-r) when |r|<1 to avoid sign confusion.

💡 Only converges when |r|<1. This is the most tested GP formula in JEE.

💡 Building block for all summation formulas.

💡 Frequently appears in method-of-differences and series summation.

💡 Sum of cubes = (sum of first n numbers) squared. Elegant identity.

💡 Multiply S by r, subtract from S to reduce the AP part. Works every time.

💡 If first differences form an AP or GP, use this to find the general term.

💡 Order matters. Number of ways to arrange r items from n distinct items.

💡 Order does not matter. Selection of r items from n distinct items.

💡 n total items with p_1 identical of type 1, p_2 of type 2, etc.

💡 Fix one object and arrange the rest. For necklace/bracelet, divide by 2.

💡 n identical items into r distinct groups (each group can be empty).

💡 Basis of Pascal's triangle. Useful for recursive counting arguments.

💡 Number of permutations where no element is in its original position.

💡 Divisibility by 3/9: digit sum divisible. By 4: last 2 digits. By 8: last 3 digits.

💡 n must be a non-negative integer. Total (n+1) terms.

💡 The (r+1)th term. r starts from 0. For finding a specific coefficient, set the power of x equal to the required value and solve for r.

💡 Even n: one middle term. Odd n: two middle terms. The middle term often has the greatest binomial coefficient.

💡 Put x=1 in (1+x)^n. Sum of all coefficients of (1+x)^n is 2^n.

💡 Put x=-1 in (1+x)^n. Even-indexed and odd-indexed coefficients have equal sum.

💡 To find sum of all coefficients in any expansion, substitute x=1. Works for any polynomial expression.

💡 For (a^{1/p} + b^{1/q})^n, the term is rational only when both exponents are integers.

💡 Write the base as (multiple of divisor ± small number). All terms except the last are divisible.

💡 Product of main diagonal minus product of off-diagonal.

💡 Expand along the row/column with most zeros to simplify calculation.

💡 Works for square matrices of same order. Also |A^n| = |A|^n.

💡 adj(A) = transpose of cofactor matrix. |adj(A)| = |A|^{n-1} for n×n matrix.

💡 Each of the n rows gets multiplied by k, so the determinant picks up k^n.

💡 D ≠ 0: unique solution. D = 0 and all Dᵢ = 0: infinite or no solutions. D = 0 and some Dᵢ ≠ 0: no solution.

💡 tr(A) = sum of diagonal elements = sum of eigenvalues. |A| = product of eigenvalues.

💡 For 2×2: A² - tr(A)·A + |A|·I = O. Use this to express A⁻¹ in terms of A and I.

💡 Rotation matrices are orthogonal. If det = 1, it's a proper rotation.

💡 For adj applied k times: det = |A|^{(n-1)^k}. Very common in JEE numerical problems.

💡 Read as 'probability of A given B'. Reduces the sample space to B.

💡 Use when you know the result and want to find which cause produced it. 'Reverse' conditional probability.

💡 E₁, E₂, ..., Eₙ must be a partition of the sample space (mutually exclusive, exhaustive).

💡 n independent trials, each with success probability p. Mean = np, Variance = npq.

💡 For binomial, Var(X) < E(X) since q < 1. If Var > Mean, it's NOT binomial.

💡 Independence ≠ mutually exclusive. If A and B are mutually exclusive and both have non-zero probability, they are NOT independent.

💡 Var(aX+b) = a²Var(X). E(aX+b) = aE(X)+b. Variance is always non-negative.

💡 For mutually exclusive events: P(A ∪ B) = P(A) + P(B). Extend for 3 events using inclusion-exclusion.

💡 m = slope, c = y-intercept. Slope = tan(θ) where θ is angle with positive x-axis.

💡 Slope is undefined for vertical lines (x₁ = x₂).

💡 Line: ax + by + c = 0. Don't forget the absolute value. Distance between parallel lines: |c₁-c₂|/√(a²+b²).

💡 For perpendicular lines: m₁m₂ = -1. For parallel lines: m₁ = m₂.

💡 Passes through the intersection of L₁ = 0 and L₂ = 0 for all values of λ. Choose λ to satisfy additional conditions.

💡 Area = 0 means the points are collinear. Can also use determinant form.

💡 Internal division: both +. External division: replace n with -n. Midpoint: m = n = 1.

💡 Line: ax + by + c = 0. The foot of perpendicular uses the same formula but with -1 instead of -2.

💡 For n = -1: ∫(1/x)dx = ln|x| + C. Always add the constant of integration for indefinite integrals.

💡 Choose substitution to simplify the integrand. Don't forget to convert dx to dt and change limits for definite integrals.

💡 ILATE rule for choosing u: Inverse trig > Log > Algebraic > Trig > Exponential.

💡 Most powerful property for definite integrals. Use when f(x) + f(a+b-x) simplifies nicely.

💡 Check parity first for symmetric limits. Saves huge computation for odd functions (integral = 0).

💡 For sin²x, cos²x: period = π. For |sinx|, |cosx|: period = π.

💡 k = π/2 if n is even, k = 1 if n is odd. Double factorial: n!! = n(n-2)(n-4)...

💡 For repeated roots: A/(x-a) + B/(x-a)². For irreducible quadratic: (Ax+B)/(x²+px+q).

💡 Replace r/n → x, 1/n → dx. Limits: r=1 gives x=0, r=n gives x=1.

💡 Useful for integrals of the form ∫₀¹ xᵃ(1-x)ᵇ dx. Γ(n) = (n-1)! for positive integers.

💡 For cos, the sign flips: cos(A+B) has minus, cos(A-B) has plus.

💡 Undefined when denominator is zero, i.e., tan A tan B = 1 for A+B case.

💡 cos 2A has three equivalent forms. Pick the one that matches the unknowns in the problem.

💡 The sign of the square root depends on the quadrant of A/2, not A. Derived from cos 2A = 1 - 2sin^2A (rearrange for sin A/2) and cos 2A = 2cos^2A - 1 (rearrange for cos A/2).

💡 Useful for integrating products of trig functions and simplifying series. Derived by adding/subtracting compound angle formulas.

💡 For sin C - sin D, it becomes 2 cos((C+D)/2) sin((C-D)/2). For cos C - cos D, it becomes -2 sin((C+D)/2) sin((C-D)/2).

💡 Always express the general solution. For specific intervals, substitute integer values of n.

💡 R is the circumradius. Use when you know an angle and its opposite side, or need to find the circumradius.

💡 Use when you know all three sides (SSS), or two sides and included angle (SAS).

💡 Also equals abc/(4R) using the sine rule, and rs where r is the inradius and s is the semi-perimeter.

💡 Works for radians only. Also: lim(tan x / x) = 1 and lim(sin(kx) / (kx)) = 1 as x approaches 0.

💡 For the general form: lim [1 + f(x)]^(1/f(x)) = e when f(x) approaches 0. Use this to handle 1^infinity forms.

💡 Useful for converting exponential limits into simpler forms. Substitute t = e^x - 1 when needed.

💡 Here log means log base e (natural logarithm). This follows from writing a^x = e^(x log a) and using the (e^t - 1)/t limit.

💡 Valid for all real n. This is essentially the derivative of x^n at x = a from first principles.

💡 Always verify the 0/0 or infinity/infinity indeterminate form before applying. May need to be applied repeatedly.

💡 All three must exist and be equal: left-hand limit, right-hand limit, and the function value at the point.

💡 Also called the limit definition of derivative. Both left-hand derivative (h approaches 0 from negative side) and right-hand derivative must be equal for differentiability.

💡 All three conditions must hold: continuity on [a,b], differentiability on (a,b), and f(a) = f(b). If any fails, the theorem cannot be applied.

💡 LMVT is a generalization of Rolle's theorem (Rolle's is the special case when f(a) = f(b)). Geometrically, the tangent at c is parallel to the secant joining (a, f(a)) and (b, f(b)).

💡 dy/dx gives the instantaneous rate of change of y with respect to x. For related rates, use chain rule: dy/dt = (dy/dx)(dx/dt).

💡 The slope of the tangent at (x1, y1) is f'(x1). If f'(x1) = 0, the tangent is horizontal: y = y1.

💡 Normal is perpendicular to the tangent. Its slope = -1/f'(x1). If f'(x1) = 0, the normal is vertical: x = x1.

💡 Length of tangent = |y1|sqrt(1 + 1/[f'(x1)]^2). Length of normal = |y1|sqrt(1 + [f'(x1)]^2). These are measured from the point to the x-axis intercept along tangent/normal.

💡 f'(x) >= 0 (with equality only at isolated points) also gives strictly increasing. Check open interval for derivative sign.

💡 Similar to increasing: f'(x) <= 0 (with equality only at isolated points) also gives strictly decreasing.

💡 Find critical points where f'(x) = 0 or f'(x) does not exist. Check sign of f'(x) on either side of each critical point.

💡 If f''(c) = 0, the test is inconclusive. Fall back to the first derivative test in that case.

💡 Evaluate f at all critical points inside (a, b) AND at both endpoints a and b. The largest value is the global max, smallest is the global min.

💡 Used to approximate values like sqrt(4.01), (3.98)^(1/2), etc. Choose x as the nearest value where f is easy to compute.

💡 Order is always defined. Degree is defined only when the DE is a polynomial in its derivatives. If sin(y') or e^(y'') appears, degree is not defined.

💡 Separate all y terms (with dy) to one side and all x terms (with dx) to the other, then integrate both sides.

💡 After solving for v, substitute back v = y/x to get the solution in terms of x and y.

💡 First rewrite the DE in standard linear form. The integrating factor (IF) multiplies both sides. Remember: the formula also works as dx/dy + P(y)x = Q(y).

💡 After substituting v = y^(1-n), the equation reduces to a linear DE in v. Solve that linear DE, then convert back to y.

💡 Solution: integrate M w.r.t. x (treating y as constant), then add terms from N that are not already present. The solution is F(x,y) = C.

💡 The order of the resulting DE equals the number of arbitrary constants in the family of curves.

💡 Orthogonal trajectories cut the given family at right angles. After replacement, solve the new DE to get the trajectory equation.

💡 k > 0 for growth, k < 0 for decay. Half-life: t_{1/2} = log(2)/|k|. Use log, not ln, per JEE convention.

💡 T_s is the surrounding temperature, T_0 is the initial temperature of the body, and k > 0. The body cools exponentially toward T_s.

💡 Vertex at origin. Axis along x-axis. For y² = -4ax, the parabola opens leftward. Latus rectum (LR) passes through the focus, perpendicular to the axis.

💡 Here a > b. If b > a, the major axis is along the y-axis and eccentricity uses a²/b². For an ellipse, 0 < e < 1 always.

💡 For a hyperbola, e > 1 always. The conjugate hyperbola is x²/a² - y²/b² = -1. Relationship: b² = a²(e² - 1).

💡 This is the tangent to y² = 4ax with slope m. The point of contact is (a/m², 2a/m). For m = 0, the tangent is at infinity.

💡 The condition for y = mx + c to be tangent to the ellipse is c² = a²m² + b². Two tangents exist for each slope (one on each side).

💡 For parabola y² = 4ax: c = a/m. Note the sign difference between ellipse (+b²) and hyperbola (-b²).

💡 For parabola y² = 4ax, the semi-latus rectum l = 2a. If PQ is a focal chord with parameters t₁ and t₂, then t₁t₂ = -1.

💡 The director circle is the locus of the point from which two perpendicular tangents are drawn. For hyperbola, it exists only when a > b.

💡 T = 0 gives chord of contact from external point (x₁, y₁). Same equation works for tangent at a point on the curve. For pair of tangents: SS₁ = T².

💡 Parametric form simplifies tangent and normal equations. For parabola, slope of tangent at parameter t is 1/t.

💡 Also called the modulus or length. Always non-negative. |a| = 0 only for the zero vector.

💡 Result is a scalar. theta is the angle between the two vectors. Dot product is commutative: a.b = b.a.

💡 Result is a vector perpendicular to both a and b (right-hand rule). Cross product is NOT commutative: a x b = -(b x a).

💡 Result is a scalar. Cyclic permutation does not change the value: [a b c] = [b c a] = [c a b]. Swapping two vectors changes the sign.

💡 This gives a scalar (the component of a along b). For the vector projection, multiply by the unit vector of b.

💡 Here a and b are vectors representing two sides of the triangle from a common vertex.

💡 Parallelogram area is exactly twice the triangle area formed by the same two vectors.

💡 Take the absolute value of the scalar triple product. Volume is always non-negative.

💡 Three vectors are coplanar if and only if their scalar triple product is zero. Equivalently, the 3x3 determinant of their components is zero.

💡 Divides the line segment from A (position vector a) to B (position vector b) in the ratio m:n internally. For external division, use m:(-n).

💡 Direction cosines (l, m, n) are the cosines of the angles a line makes with the positive x, y, z axes. They always satisfy l^2 + m^2 + n^2 = 1.

💡 Here (x1, y1, z1) is a point on the line and (a, b, c) are direction ratios of the line. If any DR is zero, the corresponding numerator must also be zero.

💡 Here (a, b, c) are the direction ratios of the normal to the plane. The normal vector is n = a i + b j + c k.

💡 Use direction ratios (a1, b1, c1) and (a2, b2, c2) of the two lines. Take the absolute value of the numerator to get the acute angle.

💡 theta is measured between the line and the plane (not the normal). The formula uses sin, not cos. Here (l, m, n) are DRs of the line and (a, b, c) are DRs of the normal to the plane.

💡 Substitute the point (x1, y1, z1) directly into ax + by + cz + d. The absolute value ensures a non-negative distance.

💡 Lines: r = a1 + t*b1 and r = a2 + s*b2. Compute b1 x b2 first, then dot it with (a2 - a1). Take the absolute value and divide by |b1 x b2|.

💡 Two lines are coplanar (intersect or are parallel) if and only if the shortest distance between them is zero. This is equivalent to the scalar triple product being zero.

💡 The foot lies on the line through (x1, y1, z1) with DRs (a, b, c) (the normal direction). Substitute the parametric point into the plane equation to find the parameter value.

💡 The image is obtained by going twice the distance from the point to the plane along the normal. The parameter value is exactly double that of the foot of perpendicular.

💡 Center is (-g, -f) and radius is sqrt(g^2 + f^2 - c). The radius is real only when g^2 + f^2 - c > 0.

💡 Center is (h, k) and radius is r. This is the most direct form when center and radius are known.

💡 For the circle x^2 + y^2 = a^2, the tangent y = mx + c requires c^2 = a^2(1 + m^2). From an external point, there are exactly two tangents.

💡 S1 is obtained by substituting the external point (x1, y1) into the circle equation. This works only when the point is outside the circle (S1 > 0).

💡 The chord of contact from an external point (x1, y1) to the circle is obtained by replacing x^2 with xx1, y^2 with yy1, x with (x+x1)/2, and y with (y+y1)/2.

💡 Power is positive if the point is outside, zero if on the circle, and negative if inside. It equals (distance from center)^2 - r^2.

💡 The radical axis is the locus of points having equal power with respect to both circles. It is always perpendicular to the line joining the centers.

💡 Two circles are orthogonal when their tangents at the intersection points are perpendicular. This condition comes from the Pythagorean theorem applied to the triangle formed by the two centers and a point of intersection.

💡 Any point on the circle (x-h)^2 + (y-k)^2 = r^2 can be written as (h + r cos(theta), k + r sin(theta)). Useful for finding points on the circle satisfying additional conditions.

💡 The director circle of x^2 + y^2 = a^2 is the locus of the point from which two perpendicular tangents can be drawn to the circle. Its radius is a*sqrt(2).

💡 Domain of sin inverse and cos inverse is [-1, 1]. Domain of tan inverse is all real numbers. The principal value is the unique value in the specified range. cosec inverse, sec inverse, and cot inverse follow from these.

💡 Valid for all x in the respective domains. These are the most frequently used identities in JEE. They allow you to convert between inverse trig functions quickly.

💡 The condition xy < 1 vs xy > 1 determines whether the sum stays in (-pi/2, pi/2) or shifts by pi. This is the most tested formula in inverse trig.

💡 These identities connect tan inverse to sin inverse and cos inverse via double-angle substitution. The domain restrictions are critical and are the main source of errors.

💡 This comes from substituting x = sin(theta). The split at 1/sqrt(2) corresponds to theta = pi/4. JEE often tests this with x = sqrt(3)/2 or x = 1/sqrt(2).

💡 Draw a right triangle with the known side to find the other trig ratio. If cos inverse(x) = theta, then cos(theta) = x, so sin(theta) = sqrt(1 - x^2).

💡 cosec inverse and sec inverse have domains |x| >= 1 (excludes the interval (-1, 1)). cot inverse has domain all real numbers. These domain checks are often the first step in solving problems.

💡 The derivative of cos inverse is the negative of sin inverse's derivative. Similarly, cot inverse's derivative is the negative of tan inverse's. These are essential for integration as well.

💡 This is the difference version of the sum formula. The condition xy > -1 ensures the result is in (-pi/2, pi/2). Used heavily in telescoping series problems.

💡 The key trick: 1/(1 + r + r^2) = ((r+1) - r)/(1 + r(r+1)). This makes each term a difference of two tan inverse values, leading to telescoping cancellation.

💡 For three sets: n(A union B union C) = n(A) + n(B) + n(C) - n(A cap B) - n(B cap C) - n(A cap C) + n(A cap B cap C). Always subtract the pairwise overlaps and add back the triple overlap.

💡 Complement of union is intersection of complements. Complement of intersection is union of complements. These hold for any number of sets.