Mathematics Formulas

Calculus552 Qs

Functions & Relations

Domain, range, injective (one-one), surjective (onto), composite functions, inverse functions, and functional equations.

Calculus590 Qs

Differentiation & Methods

Product rule, quotient rule, chain rule, implicit differentiation, logarithmic differentiation, and higher derivatives.

Calculus969 Qs

Application of Derivatives (AOD)

Tangents, normals, monotonicity, maxima & minima, Rolle's theorem, and Lagrange Mean Value Theorem (LMVT).

Calculus765 Qs

Indefinite Integration

Integration by substitution, integration by parts (ILATE), partial fractions, and standard trigonometric integrals.

9 Formulas

Calculus838 Qs

Definite Integration & Properties

Definite integrals, symmetry properties, King's rule, periodic functions, and differentiation under the integral sign.

Calculus828 Qs

Differential Equations

First-order linear differential equations, integrating factor (IF), homogeneous forms, and variable separable methods.

Calculus346 Qs

Area Under Curves

Area bounded by curves, lines, parabolas, and standard shortcut formulas.

Coordinate Geometry773 Qs

Straight Lines & Pair of Lines

Slope-intercept, point-slope, distance from a point, angle between lines, foot of perpendicular, and homogeneous pair of lines.

Coordinate Geometry666 Qs

Circle & System of Circles

Standard circle, general equation, tangent conditions, chord of contact, director circle, and orthogonality.

9 Formulas Diagram

Coordinate Geometry358 Qs

Parabola & Conics

Standard parabola y² = 4ax, parametric form, tangent, normal in slope form, focal chords, and reflection property.

Coordinate Geometry350 Qs

Ellipse

Standard ellipse x²/a² + y²/b² = 1, eccentricity, foci, auxiliary circle, director circle, and tangent conditions.

7 Formulas Diagram

Coordinate Geometry320 Qs

Hyperbola & Rectangular Hyperbola

Standard hyperbola x²/a² - y²/b² = 1, asymptotes, conjugate hyperbola, rectangular hyperbola xy = c².

7 Formulas Diagram

Algebra1,015 Qs

Matrices & Determinants

Matrix algebra, determinant expansion, adjoint properties, inverse, Cramer's rule, and systems of linear equations.

9 Formulas

Algebra1,001 Qs

Probability & Distributions

Conditional probability, Bayes theorem, independent events, total probability, and binomial distribution.

Algebra658 Qs

Complex Numbers

Argand plane, polar & Euler forms, triangle inequalities, cube roots of unity, De Moivre's theorem, and geometric loci.

Algebra599 Qs

Permutations & Combinations (P&C)

Fundamental principle of counting, nPr, nCr, circular permutations, distribution into groups, and de-arrangement.

Algebra551 Qs

Sequences & Series

Arithmetic, geometric, arithmetico-geometric (AGP), harmonic progressions, AM-GM-HM inequality, and telescoping series.

Algebra443 Qs

Binomial Theorem

General term T_{r+1}, middle term, greatest term, binomial coefficients identities, and remainder theorem.

Algebra315 Qs

Quadratic Equations

Roots of quadratic, discriminant nature, Vieta relations, condition for common roots, and location of roots.

Algebra332 Qs

Statistics

Mean, variance, standard deviation, mean deviation, and effect of linear change of origin and scale.

Vectors & 3D Geometry1,174 Qs

Three Dimensional Geometry (3D)

Direction cosines & ratios, equations of lines in 3D, shortest distance between skew and parallel lines.

9 Formulas Diagram

Vectors & 3D Geometry1,073 Qs

Vector Algebra

Dot product, cross product, scalar triple product, vector triple product, and Lagrange's identity.

7 Formulas Diagram

Trigonometry426 Qs

Trigonometric Ratios & Identities

Compound angles, transformation formulas, multiple angles, conditional identities, and general trigonometric equations.

10 Formulas Diagram

Trigonometry470 Qs

Inverse Trigonometric Functions (ITF)

Principal branches, domain & range, complementary angle relations, and sum of inverse tangents.

Trigonometry351 Qs

Properties of Triangles (SOT)

Sine rule, cosine rule, projection formula, radii of circumcircle, incircle, and excircles.

Formulas Reference (All)

26 Chapters match

CalculusVery High Yield (3-4 Qs in JEE)868 PYQs

Limits, Continuity & Differentiability

Fundamental algebraic, exponential, logarithmic, and trigonometric limits, continuity conditions, and derivative theorems.

Standard Algebraic & Trig Limits

\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan x}{x} = 1, \quad \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}

Fundamental trigonometric and polynomial limits for resolving 0/0 indeterminate forms.

Angle x MUST be measured in radians.

Exponential & Logarithmic Limits

\lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a, \quad \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1

Standard limits for natural exponential and logarithmic expressions.

Useful substitution: lim_{x->0} (ln(1+ax))/x = a.

Exponential 1^∞ Indeterminate Form

\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} g(x)[f(x) - 1]} \quad (\text{when } f(x) \to 1, g(x) \to \infty)

Converts 1^∞ indeterminate power limits into standard exponential products directly.

Very frequent in JEE Main: lim_{x->0} (1 + ax)^(b/x) = e^(ab).

Maclaurin Series Expansions

\sin x = x - \frac{x^3}{6} + \dots, \quad e^x = 1 + x + \frac{x^2}{2} + \dots, \quad \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots

Taylor-Maclaurin series for resolving complex indeterminate forms without repetitive L'Hôpital differentiation.

Keep terms up to the lowest non-vanishing power of x in the denominator.

Continuity Condition at a Point

\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c) \iff \text{LHL} = \text{RHL} = f(c)

A function f(x) is continuous at x = c if and only if both one-sided limits exist and equal the functional value.

Greatest integer [x], fractional part {x}, and signum sgn(x) are classic discontinuity test cases.

Differentiability from First Principles

f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}, \quad \text{LHD} = \text{RHD} \implies \text{Differentiable}

Differentiability implies continuity, but continuity does NOT imply differentiability.

Sharp vertices, cusps, and vertical tangents signify non-differentiable points.

L'Hôpital's Rule for Indeterminate Forms

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \left(\text{Valid strictly for } \frac{0}{0} \text{ and } \frac{\infty}{\infty} \text{ forms}\right)

Differentiate numerator and denominator independently until the indeterminate form is resolved.

Do NOT apply the quotient rule when applying L'Hôpital's rule. Check that the indeterminate form persists before differentiating repeatedly.

Sandwich (Squeeze) Theorem & Limits

g(x) \le f(x) \le h(x) \quad \text{and} \quad \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \implies \lim_{x \to a} f(x) = L

Bounding technique for evaluating limits of functions involving oscillatory factors like [x], {x}, or sin(1/x).

Classic example: lim_{x->0} x² sin(1/x) = 0 because -x² ≤ x² sin(1/x) ≤ x².

Intermediate Value Theorem (IVT)

f(x) \in C[a, b] \text{ and } f(a) \cdot f(b) < 0 \implies \exists c \in (a, b) \text{ such that } f(c) = 0

Fundamental root existence theorem for continuous real-valued functions across opposite-sign endpoints.

Every odd degree real polynomial has at least one real root.

CalculusHigh Yield (2 Qs in JEE)552 PYQs

Functions & Relations

Domain, range, injective (one-one), surjective (onto), composite functions, inverse functions, and functional equations.

Domain & Range Rules

\sqrt{f(x)} \implies f(x) \ge 0, \quad \ln(f(x)) \implies f(x) > 0, \quad \frac{1}{f(x)} \implies f(x) \ne 0

Fundamental algebraic domain restrictions for real-valued functions.

Domain of sin⁻¹(f(x)) and cos⁻¹(f(x)) requires -1 ≤ f(x) ≤ 1.

One-One (Injective) Test

f(x_1) = f(x_2) \implies x_1 = x_2 \quad \text{or} \quad f'(x) > 0 \; \forall x \text{ (Strictly Monotonic)}

Horizontal line test: every horizontal line intersects the graph at most once.

Strictly increasing or strictly decreasing continuous functions on an interval are always one-one.

Onto (Surjective) Condition

\text{Range}(f) = \text{Codomain}(f)

Every element in the codomain has at least one pre-image in the domain.

Odd degree polynomial functions with codomain R are always onto R.

Composite & Inverse Functions

(f \circ g)(x) = f(g(x)), \quad f(f^{-1}(x)) = x, \quad (f \circ g)^{-1} = g^{-1} \circ f^{-1}

Function composition and reversal law for inverse functions.

Graph of f⁻¹(x) is the mirror reflection of f(x) across the line y = x.

Standard Functional Equations

f(xy) = f(x) + f(y) \implies f(x) = k\ln x; \quad f(x+y) = f(x)f(y) \implies f(x) = a^x; \quad f(x) + f(1/x) = f(x)f(1/x) \implies f(x) = 1 \pm x^n

Classic functional identities frequently tested in JEE Advanced.

Substitute x = 0, 1, or -1 to find base constants.

Logarithm Fundamental Identities & Base Change

\log_a(xy) = \log_a x + \log_a y, \quad \log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y, \quad \log_b a = \frac{\log_c a}{\log_c b}, \quad a^{\log_b c} = c^{\log_b a}, \quad \log_{b^\beta}(a^\alpha) = \frac{\alpha}{\beta}\log_b a

Exhaustive logarithmic laws and base-change properties for solving exponential and log expressions.

Always ensure base a > 0, a ≠ 1 and argument x > 0 before manipulating equations.

Logarithmic Inequalities & Characteristic

\text{For } \log_a x > \log_a y: \begin{cases} x > y > 0 & \text{if } a > 1 \\ 0 < x < y & \text{if } 0 < a < 1 \end{cases}; \quad x = [x] + \{x\}, \quad \log_{10} N = \text{Char} + \text{Mantissa}

Inequality reversal depending on base a, and number of digits in N is equal to (Characteristic + 1).

Number of leading zeroes before first significant digit after decimal in N (0 < N < 1) is |Characteristic| - 1.

Even, Odd & Periodic Functions

f(-x) = f(x) \iff \text{Even (Symmetric about y-axis)}; \quad f(-x) = -f(x) \iff \text{Odd (Symmetric about origin)}; \quad f(x+T) = f(x) \iff \text{Periodic with Period } T

Symmetry and periodicity properties. Derivative of an even function is odd, and vice-versa.

Period of |sin x|, |cos x|, |tan x| is π. Period of sin(ax + b) is 2π/|a|.

CalculusHigh Yield (2 Qs in JEE)590 PYQs

Differentiation & Methods

Product rule, quotient rule, chain rule, implicit differentiation, logarithmic differentiation, and higher derivatives.

Product & Quotient Rules

\frac{d}{dx}(uv) = u'v + uv', \quad \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}

Fundamental derivative rules for algebraic combinations.

Numerator of quotient rule starts with derivative of numerator: u'v.

Chain Rule & Parametric Derivatives

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}, \quad \frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

Derivative of composite functions and second derivative of parametric functions.

Do NOT just divide d²y/dt² by d²x/dt² for second derivative.

Logarithmic Differentiation

y = [u(x)]^{v(x)} \implies \frac{dy}{dx} = [u(x)]^{v(x)} \left[ v'(x)\ln(u(x)) + \frac{v(x)u'(x)}{u(x)} \right]

Differentiating variable-base to variable-exponent functions.

Standard result: d/dx (x^x) = x^x(1 + ln x).

Standard Inverse Trig Derivatives

\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}, \quad \frac{d}{dx}(\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}}

Derivatives of inverse trigonometric functions.

Derivative of co-functions (cos⁻¹x, cot⁻¹x, csc⁻¹x) has identical form with a minus sign.

Leibniz's Rule for n-th Derivative

\frac{d^n}{dx^n}(uv) = \sum_{r=0}^n {}^nC_r \frac{d^{n-r}u}{dx^{n-r}} \frac{d^r v}{dx^r}

Binomial-type expansion for calculating the n-th derivative of a product of two functions.

Set v as the polynomial term whose higher derivatives vanish after a few steps.

Differentiation of Determinants

\frac{d}{dx} \begin{vmatrix} f & g \\ u & v \end{vmatrix} = \begin{vmatrix} f' & g' \\ u & v \end{vmatrix} + \begin{vmatrix} f & g \\ u' & v' \end{vmatrix}

Differentiate one row (or column) at a time, keeping all other rows unchanged, and sum the resulting determinants.

For 3×3 determinants, the result is the sum of 3 determinants corresponding to differentiating row 1, row 2, and row 3.

Implicit Function Differentiation Shortcut

F(x, y) = 0 \implies \frac{dy}{dx} = -\frac{\partial F / \partial x}{\partial F / \partial y}

Fast evaluation using partial derivatives treating y as constant in numerator and x as constant in denominator.

Massive time saver in JEE for finding slopes of tangents to implicit algebraic curves.

CalculusVery High Yield (3-4 Qs in JEE)969 PYQs

Application of Derivatives (AOD)

Tangents, normals, monotonicity, maxima & minima, Rolle's theorem, and Lagrange Mean Value Theorem (LMVT).

Tangent and Normal Equations

y - y_1 = m(x - x_1) \quad \left(m = \left.\frac{dy}{dx}\right|_{(x_1, y_1)}\right), \quad \text{Normal: } y - y_1 = -\frac{1}{m}(x - x_1)

Point-slope equations for tangent and perpendicular normal lines to curve y = f(x).

If m = 0, tangent is horizontal (y = y_1). If m -> ∞, tangent is vertical (x = x_1).

Length of Subtangent & Subnormal

\text{Subtangent} = \left|\frac{y}{m}\right|, \quad \text{Subnormal} = |y \cdot m|, \quad \text{Tangent length} = \left|\frac{y}{m}\right|\sqrt{1 + m^2}

Geometric lengths of projections on the x-axis for curve tangents and normals.

Subnormal is proportional to y for parabola y² = 4ax (constant subnormal = 2a).

Monotonicity Criteria

f'(x) \ge 0 \; \forall x \in (a, b) \implies f(x) \text{ Increasing}; \quad f'(x) \le 0 \implies f(x) \text{ Decreasing}

Derivative test for strictly increasing and decreasing functions on open intervals.

f'(x) = 0 at discrete isolated points does NOT destroy strict monotonicity.

Maxima & Minima (First & Second Derivative)

f'(c) = 0: \quad f''(c) < 0 \implies \text{Local Maxima}, \quad f''(c) > 0 \implies \text{Local Minima}

Stationary points and concavity curvature tests for extreme values.

If f''(c) = 0, test higher derivatives: odd order = inflection point, even order = extremum.

Lagrange's Mean Value Theorem (LMVT)

\frac{f(b) - f(a)}{b - a} = f'(c) \quad \text{for at least one } c \in (a, b)

Secant slope across [a, b] equals the tangent derivative slope at some interior point c.

Rolle's Theorem is the special case when f(a) = f(b) (giving f'(c) = 0).

Angle of Intersection of Two Curves & Orthogonality

\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|, \quad \text{Orthogonal Condition: } m_1 m_2 = -1

Acute angle between tangents to curves y = f(x) and y = g(x) at their point of intersection.

Two intersecting confocal conics (e.g. ellipse and hyperbola with same foci) always intersect orthogonally.

Concavity, Convexity & Point of Inflection

f''(x) > 0 \implies \text{Concave Upward } (\cup), \quad f''(x) < 0 \implies \text{Concave Downward } (\cap), \quad f''(c) = 0 \; (\& \text{ changes sign}) \implies \text{Inflection}

Curvature direction and inflection point where tangent crosses the curve.

At a point of inflection, the tangent crosses the curve (e.g. y = x³ at the origin).

Cauchy's Mean Value Theorem (CMVT) & Approximations

\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)} \quad (c \in (a,b)), \quad \Delta y \approx f'(x)\,\Delta x, \quad \frac{\Delta y}{y} \approx \left[\frac{x f'(x)}{f(x)}\right]\frac{\Delta x}{x}

Generalized mean value theorem for two functions, and differential approximation for calculating percentage errors.

Percentage error in y is (% error in x) multiplied by the power index n when y = k x^n.

CalculusVery High Yield (2-3 Qs in JEE)765 PYQs

Indefinite Integration

Integration by substitution, integration by parts (ILATE), partial fractions, and standard trigonometric integrals.

Integration by Parts (ILATE Rule)

\int u v \, dx = u \int v \, dx - \int \left( u' \int v \, dx \right) dx

Integration of product of two functions with priority order: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential.

For ∫ ln x dx, treat as ∫ (ln x) · 1 dx (u = ln x, v = 1).

Special Exponential Cancellation Integral

\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C, \quad \int [f(x) + x f'(x)] \, dx = x f(x) + C

Classic JEE pattern where derivative cancels out directly.

Always try to express awkward integrals in the form e^x [f(x) + f'(x)].

Standard Algebraic Quadratic Integrals

\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C, \quad \int \frac{dx}{\sqrt{x^2 \pm a^2}} = \ln|x + \sqrt{x^2 \pm a^2}| + C

Standard formulas for quadratic denominators.

∫ dx / (a² + x²) = (1/a) tan⁻¹(x/a) + C.

Trigonometric Substitution Guidelines

\sqrt{a^2 - x^2} \implies x = a\sin\theta; \quad \sqrt{a^2 + x^2} \implies x = a\tan\theta; \quad \sqrt{x^2 - a^2} \implies x = a\sec\theta

Standard substitutions that eliminate radicals via trigonometric identities.

For √( (a - x)/(a + x) ), substitute x = a cos 2θ.

Three Fundamental Radical Integrals

\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C, \quad \int \sqrt{x^2 \pm a^2} dx = \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\ln|x + \sqrt{x^2 \pm a^2}| + C

Indispensable formulas for area calculations and conic integrals in JEE Advanced.

Memory trick: All three start with (x/2)·[original integrand], followed by (a²/2)·[integral of 1/[original integrand]].

Symmetric Biquadratic Rational Integrals

\int \frac{x^2 \pm 1}{x^4 + kx^2 + 1} dx = \int \frac{1 \pm 1/x^2}{(x \mp 1/x)^2 + (k \pm 2)} dx

Divide numerator and denominator by x² and substitute t = x ∓ 1/x.

If numerator is 1 or x², express as 1/2 · [(x² + 1) ± (x² - 1)].

Linear over Quadratic & Radical Integrals

\int \frac{px + q}{ax^2 + bx + c} dx = \frac{p}{2a} \int \frac{2ax + b}{ax^2 + bx + c} dx + \left(q - \frac{pb}{2a}\right) \int \frac{dx}{ax^2 + bx + c}

Split numerator into derivative of quadratic denominator plus a constant.

The first part integrates directly to (p/2a) ln|ax² + bx + c|, while the second is resolved by completing the square.

Walli's Formula (Definite Integral Reduction)

\int_0^{\pi/2} \sin^n x \, dx = \int_0^{\pi/2} \cos^n x \, dx = \begin{cases} \dfrac{(n-1)!!}{n!!} \cdot \dfrac{\pi}{2} & n \text{ even} \\[6pt] \dfrac{(n-1)!!}{n!!} & n \text{ odd} \end{cases}

Walli's (Wallis) reduction for ∫₀^{π/2} sinⁿx dx. The double-factorial n!! = n·(n-2)·(n-4)···. E.g., ∫₀^{π/2} sin⁴x dx = (3·1)/(4·2) · π/2 = 3π/16.

Quick check: ∫₀^{π/2} sin²x dx = π/4, ∫₀^{π/2} sin³x dx = 2/3, ∫₀^{π/2} sin⁵x dx = 8/15. Walli's eliminates repeated IBP in JEE Advanced.

Reduction Formulas for sin^n x and cos^n x

\int \sin^n x \, dx = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2}x \, dx, \quad \int \cos^n x \, dx = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n} \int \cos^{n-2}x \, dx

Recursive IBP-based formulas to reduce power n by 2 each step until reaching ∫sin x dx or ∫dx. Terminates at base cases n = 0 (→ x) or n = 1 (→ -cos x / sin x).

Also: ∫ sec³x dx = (1/2)[sec x tan x + ln|sec x + tan x|] + C — needed in arc-length and surface-area integrals.

CalculusVery High Yield (3-4 Qs in JEE)838 PYQs

Definite Integration & Properties

Definite integrals, symmetry properties, King's rule, periodic functions, and differentiation under the integral sign.

King's Definite Integral Property

\int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx

Transforming x into (a + b - x) leaves the integral value invariant.

Add the new integral to the original integral I to cancel awkward denominators (2I technique).

Symmetric Bounds (Even & Odd Functions)

\int_{-a}^a f(x) \, dx = \begin{cases} 2 \int_0^a f(x) \, dx & \text{if } f(-x) = f(x) \text{ (even)} \\ 0 & \text{if } f(-x) = -f(x) \text{ (odd)} \end{cases}

Symmetry test across origin drastically reduces definite integral calculations.

Check odd symmetry immediately for symmetric limits [-a, a].

Periodic Function Integral Rule

\int_0^{nT} f(x) \, dx = n \int_0^T f(x) \, dx, \quad \int_a^{a + T} f(x) \, dx = \int_0^T f(x) \, dx

If f(x + T) = f(x), integrals over multiple periods pull factor n outside.

The integral over any full period T is completely independent of the starting limit a.

Leibniz Differentiation Rule

\frac{d}{dx} \int_{\alpha(x)}^{\beta(x)} f(t) \, dt = f(\beta(x)) \beta'(x) - f(\alpha(x)) \alpha'(x)

Differentiating a definite integral with variable boundary limits with respect to x.

Crucial for applying L'Hôpital's Rule on definite integral indeterminate forms.

Definite Integral as Limit of a Sum

\lim_{n \to \infty} \sum_{r=1}^n \frac{1}{n} f\left(\frac{r}{n}\right) = \int_0^1 f(x) \, dx

Converts infinite Riemann series summations into standard definite integrals (r/n -> x, 1/n -> dx).

Lower bound = lim (r_min/n), Upper bound = lim (r_max/n).

Walli's Reduction Formula

\int_0^{\pi/2} \sin^n x \, dx = \frac{(n-1)(n-3)\dots}{(n)(n-2)\dots} \times K \quad \left(K = \frac{\pi}{2} \text{ if } n \text{ even, } 1 \text{ if odd}\right)

Instant evaluation of definite powers of sine and cosine over [0, π/2].

Saves up to 5 minutes compared to standard reduction integration.

Half-Integral Invariance Property

\int_0^{2a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f(2a - x) = f(x) \\ 0 & \text{if } f(2a - x) = -f(x) \end{cases}

Splits interval [0, 2a] based on reflection symmetry across the midpoint x = a.

Essential for integrating ∫₀^π x f(sin x) dx: first remove x using King's property, then apply half-integral rule.

Definite Integral Estimation & Cauchy-Schwarz Inequality

m(b - a) \le \int_a^b f(x)\,dx \le M(b - a), \quad \left(\int_a^b f(x)g(x)\,dx\right)^2 \le \left(\int_a^b [f(x)]^2\,dx\right)\left(\int_a^b [g(x)]^2\,dx\right)

Bounding bounds for definite integrals where m and M are infimum and supremum of f(x) on [a, b].

Modulus inequality: |∫ f(x) dx| ≤ ∫ |f(x)| dx.

CalculusHigh Yield (2-3 Qs in JEE)828 PYQs

Differential Equations

First-order linear differential equations, integrating factor (IF), homogeneous forms, and variable separable methods.

Linear Differential Equation (First Order)

\frac{dy}{dx} + P(x) y = Q(x) \implies y \cdot e^{\int P dx} = \int \left( Q \cdot e^{\int P dx} \right) dx + C

Multiplying by Integrating Factor IF = e^(∫ P dx) makes LHS a total derivative.

If dy/dx looks complicated, check if dx/dy + P(y)x = Q(y) is simpler.

Bernoulli's Differential Equation

\frac{dy}{dx} + P(x) y = Q(x) y^n \implies \frac{1}{1-n} \frac{dz}{dx} + P(x) z = Q(x) \quad (z = y^{1-n})

Reduces non-linear Bernoulli equations into standard first-order linear form.

Divide entire equation by y^n before substituting z = y^(1-n).

Homogeneous Differential Equation

\frac{dy}{dx} = f\left(\frac{y}{x}\right) \implies y = vx, \quad \frac{dy}{dx} = v + x\frac{dv}{dx}

Substitute y = vx to convert homogeneous equations into variable separable form.

Always substitute back v = y/x into the final expression.

Exact Differential Form Substitutions

x \, dy + y \, dx = d(xy), \quad \frac{x \, dy - y \, dx}{x^2} = d\left(\frac{y}{x}\right), \quad \frac{x \, dx + y \, dy}{x^2 + y^2} = d\left(\frac{1}{2}\ln(x^2 + y^2)\right)

Recognizing exact differentials allows one-line integration without lengthy substitutions.

Group x dy and y dx terms together first.

Orthogonal Trajectories

\text{Replace } \frac{dy}{dx} \longrightarrow -\frac{dx}{dy} \text{ in the differential equation}

Finds the family of curves that intersect the given family at right angles everywhere.

Eliminate any arbitrary constants before replacing the derivative.

Clairaut's Equation & Singular Solution

y = px + f(p) \; \left(p = \frac{dy}{dx}\right) \implies \text{General: } y = cx + f(c), \quad \text{Singular: Eliminate } p \text{ via } x + f'(p) = 0

Standard non-linear differential equation whose general solution is a family of straight lines and singular solution is the envelope.

Substitute dy/dx = c directly into the differential equation to get the general solution in 2 seconds.

Linear Fractional Form Reductions

\frac{dy}{dx} = \frac{a_1 x + b_1 y + c_1}{a_2 x + b_2 y + c_2} \implies \begin{cases} \text{If } a_2 = -b_1: \text{ Cross-multiply and integrate directly} \\ \text{If } \frac{a_1}{a_2} = \frac{b_1}{b_2}: \text{ Substitute } t = a_1 x + b_1 y \end{cases}

Shortcuts for non-homogeneous rational fractional linear differential equations.

When a₂ = -b₁, (b₁ y dx + a₂ x dy) becomes exact: d(b₁ x y).

Order, Degree & Newton's Law of Cooling

\text{Degree is undefined if derivatives are within transcendental functions}; \quad \frac{dT}{dt} = -k(T - T_0) \implies T(t) = T_0 + \Delta T_0 \, e^{-kt}

Criteria for polynomial degree in derivatives, and standard differential model for thermal cooling.

dy/dx = sin(y) has degree 1, but sin(dy/dx) = y has undefined degree.

CalculusHigh Yield (1-2 Qs in JEE)346 PYQs

Area Under Curves

Area bounded by curves, lines, parabolas, and standard shortcut formulas.

Area Bounded Between Two Curves

\text{Area} = \int_a^b [y_{\text{upper}}(x) - y_{\text{lower}}(x)] \, dx = \int_c^d [x_{\text{right}}(y) - x_{\text{left}}(y)] \, dy

Area enclosed between two intersecting curves.

Find points of intersection first to determine integration limits a and b.

Area Between Parabola and Line (Shortcut)

\text{Area bounded by } y^2 = 4ax \text{ and } y = mx \implies \text{Area} = \frac{8a^2}{3m^3}

Direct closed-form formula saving 5 minutes of definite integration.

Area between y² = 4ax and x² = 4by is (16/3)ab.

Standard Ellipse Area

\text{Area of } \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \text{Area} = \pi a b

Total area enclosed by ellipse with semi-axes a and b.

Area of circle of radius r is special case: π r² (a = b = r).

Area Between Two Intersecting Parabolas

\text{Area bounded by } y^2 = 4ax \text{ and } x^2 = 4by \implies \text{Area} = \frac{16}{3} a b

Legendary JEE shortcut for enclosed area between standard horizontal and vertical parabolas.

For y² = 4x and x² = 4y (a = 1, b = 1), the area is exactly 16/3 sq units.

Area Between Parabola & Latus Rectum

\text{Area bounded by } y^2 = 4ax \text{ and } x = a \implies \text{Area} = \frac{8}{3} a^2

Area bounded by the parabola and the focal chord perpendicular to its principal axis.

Area bounded by y² = 4ax and double ordinate x = x₁ is (4/3) x₁ y₁.

Loop Area of Algebraic Curves

\text{Loop of } a y^2 = x^2 (a - x) \implies \text{Area} = \frac{8}{15} a^2, \quad \text{Arch of Cycloid } = 3\pi a^2

Closed-form definite integration results for parametric and looped algebraic geometries.

Area of leaf of Folium of Descartes x³ + y³ = 3axy is (3/2) a².

Area Enclosed by Sine & Cosine Waves

\int_0^\pi \sin x\,dx = 2, \quad \text{Area between } \sin x \text{ and } \cos x \; \left[\frac{\pi}{4}, \frac{5\pi}{4}\right] = 2\sqrt{2}

Fundamental cycle areas for periodic trigonometric functions.

Area of a single full lobe between consecutive zero crossings of y = sin(kx) is 2/k.

Average & Root Mean Square (RMS) Value of Function

f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx, \quad f_{\text{rms}} = \sqrt{\frac{1}{b - a}\int_a^b [f(x)]^2\,dx}

Continuous integral definitions of the mean value and effective quadratic average over interval [a, b].

For alternating sinusoidal function f(t) = I₀ sin(ωt): f_avg over half-cycle is 2I₀/π ≈ 0.637 I₀; f_rms is I₀/√2 ≈ 0.707 I₀.

Coordinate GeometryHigh Yield (2-3 Qs in JEE)773 PYQs

Straight Lines & Pair of Lines

Slope-intercept, point-slope, distance from a point, angle between lines, foot of perpendicular, and homogeneous pair of lines.

Distance from Point to Line

d = \frac{|a x_1 + b y_1 + c|}{\sqrt{a^2 + b^2}}

Perpendicular distance from point P(x₁, y₁) to line ax + by + c = 0.

Distance between parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is |c₁ - c₂| / √(a² + b²).

Angle Between Two Lines

\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|

Acute angle between lines with slopes m₁ and m₂.

Lines are perpendicular if m₁ m₂ = -1; parallel if m₁ = m₂.

Foot of Perpendicular & Reflection Image

\frac{x - x_1}{a} = \frac{y - y_1}{b} = -k \frac{a x_1 + b y_1 + c}{a^2 + b^2} \quad (k = 1 \text{ for Foot}, \; k = 2 \text{ for Image})

Coordinates of the projection foot and mirror reflection of point (x₁, y₁) across ax + by + c = 0.

Memory trick: k = 1 for foot (1 step), k = 2 for image (2 steps).

Angle Between Pair of Lines through Origin

a x^2 + 2hxy + b y^2 = 0 \implies \tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|}

Joint homogeneous equation of two lines passing through the origin.

Perpendicular lines condition: a + b = 0 (coefficient of x² + coefficient of y² = 0).

General Second Degree Condition for Lines

\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0, \quad \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0

Determinant condition for general equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 to represent two straight lines.

If Δ = 0 and h² ≥ ab, equation factors into two straight lines.

Angle Bisectors of Two Lines

\frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}} \quad (c_1, c_2 > 0)

Equations of internal and external angle bisectors between two intersecting lines.

With c₁, c₂ > 0: positive sign gives origin-containing bisector. If a₁a₂ + b₁b₂ > 0, it is the obtuse bisector; if < 0, it is acute.

Family of Lines & Concurrency

L_1 + \lambda L_2 = 0, \quad \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0

Equation of any straight line passing through intersection of L₁ = 0 and L₂ = 0, and concurrency condition for 3 lines.

Area of parallelogram bounded by y = m₁x + c₁, y = m₁x + c₂, y = m₂x + d₁, y = m₂x + d₂ is |(c₁ - c₂)(d₁ - d₂)| / |m₁ - m₂|.

Coordinate GeometryHigh Yield (2-3 Qs in JEE)666 PYQs

Circle & System of Circles

Standard circle, general equation, tangent conditions, chord of contact, director circle, and orthogonality.

General Circle Equation & Radius

x^2 + y^2 + 2gx + 2fy + c = 0 \implies \text{Center: } (-g, -f), \quad r = \sqrt{g^2 + f^2 - c}

Standard parameters of general circle.

Always ensure the coefficients of x² and y² are 1 before identifying g, f, and c.

Tangent in Slope Form

y = mx \pm r\sqrt{1 + m^2}

Equation of tangent with slope m to circle x² + y² = r².

Condition of tangency: c² = r²(1 + m²).

Tangent at a Point (T = 0)

x x_1 + y y_1 + g(x + x_1) + f(y + y_1) + c = 0

Tangent equation at point P(x₁, y₁) lying on the circle.

Replace x² -> xx₁, y² -> yy₁, 2x -> (x + x₁), 2y -> (y + y₁).

Chord of Contact & Length of Tangent

\text{Chord of Contact: } T = 0, \quad \text{Length of Tangent: } L = \sqrt{S_1} = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}

Line joining contact points of tangents drawn from external point P(x₁, y₁).

Area of quadrilateral formed by tangents and radii = rL.

Condition of Orthogonal Intersection

2 g_1 g_2 + 2 f_1 f_2 = c_1 + c_2

Condition for two circles to cut each other at right angles (tangents are perpendicular).

Both circles must have coefficients of x² and y² equal to 1.

Director Circle of a Circle

x^2 + y^2 = 2 r^2

Locus of point of intersection of two perpendicular tangents to x² + y² = r².

Concentrical circle with radius √2 times the original circle radius.

Family of Circles & Radical Axis

S_1 + \lambda S_2 = 0 \quad (\lambda \ne -1), \quad S + \lambda L = 0, \quad \text{Radical Axis: } S_1 - S_2 = 0

Locus of points from which tangents drawn to two circles are of equal length. Perpendicular to line of centers.

Radical center is the point of concurrence of radical axes of three circles taken in pairs.

Lengths of Common Tangents

L_{\text{DCT}} = \sqrt{d^2 - (r_1 - r_2)^2}, \quad L_{\text{TCT}} = \sqrt{d^2 - (r_1 + r_2)^2} \quad (d = \text{center distance})

Direct and Transverse common tangents between two separated circles.

If d > r₁ + r₂: 4 common tangents; if d = r₁ + r₂: 3; if |r₁ - r₂| < d < r₁ + r₂: 2; if d = |r₁ - r₂|: 1; if d < |r₁ - r₂|: 0.

Chord with Given Midpoint & Length of Intercept

T = S_1, \quad \text{Chord Length } = 2\sqrt{r^2 - p^2}

Equation of chord of circle S = 0 bisected at point (x₁, y₁), where p is perpendicular distance from center to chord.

T = x x₁ + y y₁ + g(x+x₁) + f(y+y₁) + c, and S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c.

Coordinate GeometryHigh Yield (2-3 Qs in JEE)358 PYQs

Parabola & Conics

Standard parabola y² = 4ax, parametric form, tangent, normal in slope form, focal chords, and reflection property.

Standard Parabola Parameters

y^2 = 4ax \implies \text{Focus: } (a, 0), \quad \text{Directrix: } x = -a, \quad \text{Latus Rectum: } 4a

Standard horizontal parabola opened to the right with vertex at origin.

Parametric coordinates are (at², 2at) with parameter t.

Tangent in Slope Form

y = mx + \frac{a}{m} \quad (m \ne 0)

Equation of tangent to y² = 4ax with slope m.

Point of contact is (a/m², 2a/m).

Normal in Slope Form

y = mx - 2am - am^3

Equation of normal with slope m to y² = 4ax.

Three normals can be drawn from a point (h, k) to a parabola. Sum of slopes m₁ + m₂ + m₃ = 0.

Focal Chord Property

t_1 t_2 = -1, \quad \text{Length of Focal Chord} = a\left(t + \frac{1}{t}\right)^2 = 4a\csc^2\theta

Parametric relation connecting endpoints of a chord passing through the focus (a, 0).

Minimum length of any focal chord is the latus rectum (4a) at θ = 90°.

Tangent at Parameter t

ty = x + at^2

Parametric tangent to parabola at point P(at², 2at).

Intersection of tangents at t₁ and t₂ is (a t₁ t₂, a(t₁ + t₂)).

Normal at Parameter t & Re-intersection

y + tx = 2at + at^3, \quad t_2 = -t_1 - \frac{2}{t_1}

Parametric normal at (at₁², 2at₁) intersects the parabola again at parameter t₂.

Intersection of normals at t₁ and t₂ is (2a + a(t₁² + t₂² + t₁t₂), -at₁t₂(t₁ + t₂)).

Co-Normal Points & Centroid Properties

m_1 + m_2 + m_3 = 0, \quad y_1 + y_2 + y_3 = 0, \quad x_1 + x_2 + x_3 = 2(h - 2a)

Properties of the 3 points whose normals concur at (h, k).

Centroid of the triangle formed by 3 co-normal points has y-coordinate 0 (always lies on the axis of parabola).

Director Circle of Parabola

x = -a \quad (\text{The Directrix})

Locus of point of intersection of two mutually perpendicular tangents to parabola y² = 4ax.

Tangents drawn from any point on the directrix always meet at right angles (90°).

Coordinate GeometryHigh Yield (2-3 Qs in JEE)350 PYQs

Ellipse

Standard ellipse x²/a² + y²/b² = 1, eccentricity, foci, auxiliary circle, director circle, and tangent conditions.

Eccentricity & Foci Parameters

e = \sqrt{1 - \frac{b^2}{a^2}}, \quad \text{Foci: } (\pm ae, 0), \quad \text{Directrices: } x = \pm \frac{a}{e} \quad (a > b)

Fundamental eccentricity relation for horizontal ellipse.

Latus rectum length = 2b²/a.

Focal Distance Property (SP + S'P)

SP + S'P = 2a

The sum of distances from any point on the ellipse to both foci is constant and equals the major axis length 2a.

Commonly used to identify ellipses from locus equations |z - z₁| + |z - z₂| = 2a.

Tangent in Slope Form

y = mx \pm \sqrt{a^2 m^2 + b^2}

Condition of tangency: c² = a² m² + b².

Point of contact is (∓ a²m / √(a²m² + b²), ± b² / √(a²m² + b²)).

Director Circle of Ellipse

x^2 + y^2 = a^2 + b^2

Locus of point of intersection of two mutually perpendicular tangents to the ellipse.

Radius of director circle is √(a² + b²).

Parametric Coordinates & Tangent

P(a\cos\theta, b\sin\theta), \quad \text{Tangent: } \frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1

Parametric eccentric angle θ representation of points and tangent line.

Eccentric angle θ is the angle made by corresponding point on the auxiliary circle with the major axis.

Normal in Slope & Parametric Forms

y = mx \pm \frac{m(a^2 - b^2)}{\sqrt{a^2 + b^2 m^2}}, \quad ax\sec\theta - by\csc\theta = a^2 - b^2

Equation of normal with slope m, and parametric normal at eccentric angle θ.

Maximum 4 normals can be drawn from an arbitrary point (h, k) to an ellipse.

Auxiliary Circle & Product of Focal Perpendiculars

x^2 + y^2 = a^2, \quad p_1 \cdot p_2 = b^2

The product of lengths of perpendiculars drawn from the two foci (±ae, 0) to any tangent is constant and equals b².

Feet of perpendiculars drawn from the foci to any tangent always lie on the auxiliary circle x² + y² = a².

Coordinate GeometryHigh Yield (2 Qs in JEE)320 PYQs

Hyperbola & Rectangular Hyperbola

Standard hyperbola x²/a² - y²/b² = 1, asymptotes, conjugate hyperbola, rectangular hyperbola xy = c².

Standard Hyperbola Parameters

e = \sqrt{1 + \frac{b^2}{a^2}}, \quad \text{Foci: } (\pm ae, 0), \quad \text{Directrices: } x = \pm \frac{a}{e}

Standard horizontal hyperbola opening along x-axis.

Eccentricity e is strictly greater than 1.

Focal Distance Property (|SP - S'P|)

|SP - S'P| = 2a

Difference of distances from any point on hyperbola to the two foci is constant 2a.

Locus definition: |z - z₁| - |z - z₂| = 2a represents hyperbola.

Tangent in Slope Form

y = mx \pm \sqrt{a^2 m^2 - b^2}

Condition of tangency: c² = a² m² - b².

Real tangents exist only when m² > b²/a².

Asymptotes of Hyperbola

y = \pm \frac{b}{a} x \iff \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0

Straight lines passing through the origin asymptotic to the hyperbola branches.

Angle between asymptotes is 2 tan⁻¹(b/a) = 2 sec⁻¹(e).

Rectangular Hyperbola (xy = c²)

xy = c^2, \quad \text{Parametric: } \left(ct, \frac{c}{t}\right), \quad e = \sqrt{2}

Asymptotes are mutually perpendicular coordinate axes.

Tangent at parameter t is x + y t² = 2ct; normal is y - c/t = t²(x - ct) or x t³ - y t - ct⁴ + c = 0.

Conjugate Hyperbola & Eccentricity Relation

-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \frac{1}{e_1^2} + \frac{1}{e_2^2} = 1

Hyperbola whose transverse and conjugate axes are interchanged. Foci lie on y-axis at (0, ±be₂).

The hyperbola, its asymptotes, and conjugate hyperbola differ only by constant terms in their joint equations.

Normal in Slope & Parametric Forms

y = mx \mp \frac{m(a^2 + b^2)}{\sqrt{a^2 - b^2 m^2}}, \quad ax\cos\theta + by\cot\theta = a^2 + b^2

Normal to hyperbola with slope m and at parametric point (a sec θ, b tan θ).

Director Circle of Hyperbola is x² + y² = a² - b². Real director circle exists only if a > b.

AlgebraTop Priority (3-4 Qs in JEE)1,015 PYQs

Matrices & Determinants

Matrix algebra, determinant expansion, adjoint properties, inverse, Cramer's rule, and systems of linear equations.

Determinant Properties

|A^T| = |A|, \quad |AB| = |A||B|, \quad |kA| = k^n |A| \quad (n \times n \text{ matrix})

Scalar multiplication pulls out factor k raised to the dimension power n.

|A⁻¹| = 1 / |A|.

Adjoint Properties

A \cdot \text{adj}(A) = |A| I_n, \quad |\text{adj}(A)| = |A|^{n-1}, \quad |\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}

Crucial adjoint determinant power formulas tested almost every year in JEE Main.

For 3x3 matrix (n = 3): |adj(A)| = |A|² and |adj(adj(A))| = |A|⁴.

Matrix Inverse Formula

A^{-1} = \frac{1}{|A|} \text{adj}(A) \quad (|A| \ne 0)

Inverse exists if and only if matrix A is non-singular (|A| ≠ 0).

(AB)⁻¹ = B⁻¹ A⁻¹ (reversal law).

Cramer's Rule for Linear Systems

x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}

Determinant method for solving system of linear equations.

Δ ≠ 0: Unique solution. Δ = 0 with any Δ_i ≠ 0: Inconsistent (No solution).

Symmetric & Skew-Symmetric Matrices

A = \frac{A + A^T}{2} + \frac{A - A^T}{2}, \quad \text{Skew-Symmetric: } A^T = -A \implies a_{ii} = 0

Every square matrix is uniquely decomposable into symmetric and skew-symmetric parts.

Determinant of odd-order skew-symmetric matrix is ALWAYS zero.

Orthogonal Matrix Property

A \cdot A^T = I \implies |A| = \pm 1, \quad A^{-1} = A^T

Orthogonal matrices preserve lengths and angles.

Rows and columns form orthonormal vectors.

Cayley-Hamilton Theorem & Matrix Powers

\det(A - \lambda I) = 0 \implies P(A) = O, \quad A^2 - \text{tr}(A)A + |A|I = O \; (2 \times 2)

Every square matrix satisfies its own characteristic polynomial equation. Enables computing high powers and inverses.

To find A⁻¹, multiply the characteristic equation throughout by A⁻¹.

Special Matrix Types (Idempotent, Involutory, Nilpotent)

A^2 = A \; (\text{Idempotent}), \quad A^2 = I \; (\text{Involutory}), \quad A^k = O \; (\text{Nilpotent}), \quad \text{tr}(AB) = \text{tr}(BA)

Key algebraic matrix classes frequently tested in eigenvalue and matrix polynomial questions.

Eigenvalues of idempotent matrices can only be 0 or 1. Determinant of involutory matrix is ±1.

Homogeneous System of Linear Equations (AX = 0)

|A| \ne 0 \implies X = O \; (\text{Trivial Unique Solution}), \quad |A| = 0 \implies \text{Infinitely Many Non-Trivial Solutions}

Criterion for existence of non-zero solutions to homogeneous linear equations.

If a homogeneous system has a non-trivial solution, its determinant MUST equal 0.

AlgebraHigh Yield (2-3 Qs in JEE)1,001 PYQs

Probability & Distributions

Conditional probability, Bayes theorem, independent events, total probability, and binomial distribution.

Conditional Probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given that B has already occurred.

If events are independent, P(A|B) = P(A).

Bayes' Theorem

P(E_i|A) = \frac{P(E_i) P(A|E_i)}{\sum_{j=1}^n P(E_j) P(A|E_j)}

Calculates reverse (posterior) probability using total probability denominator.

Write partition events E₁, E₂... clearly before evaluating branch likelihoods.

Binomial Distribution (Mean & Variance)

P(X = r) = {}^nC_r p^r q^{n-r}, \quad \text{Mean } \mu = np, \quad \text{Variance } \sigma^2 = npq

Probability distribution for r successes in n independent Bernoulli trials (p + q = 1).

Variance is ALWAYS strictly less than Mean for Binomial distributions (npq < np since q < 1).

Total Probability Theorem

P(A) = \sum_{i=1}^n P(E_i) \cdot P(A|E_i)

Law of total probability over a partition of sample space.

Forms the denominator of Bayes theorem.

Addition Theorem & 3-Event Inclusion-Exclusion

P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(AB) - P(BC) - P(CA) + P(ABC)

Probability of occurrence of at least one of three arbitrary events.

Probability of exactly two events occurring = P(AB) + P(BC) + P(CA) - 3 P(ABC).

Odds in Favor & Odds Against

\text{Odds in favor of } A = \frac{P(A)}{P(A')} = \frac{m}{n} \implies P(A) = \frac{m}{m + n}, \quad \text{Odds against } = \frac{n}{m}

Ratio of favorable outcomes to unfavorable outcomes.

If odds in favor of an event are 3:5, its probability is 3/(3 + 5) = 3/8.

Expectation & Variance of Discrete Random Variable

E(X) = \sum x_i p_i, \quad \text{Var}(X) = E(X^2) - [E(X)]^2 = \sum x_i^2 p_i - \mu^2, \quad \text{Var}(aX + b) = a^2 \text{Var}(X)

Mathematical expectation (weighted mean) and spread of a discrete probability distribution.

E(aX + b) = a E(X) + b. For independent random variables X and Y: Var(X ± Y) = Var(X) + Var(Y).

AlgebraVery High Yield (2-3 Qs in JEE)658 PYQs

Complex Numbers

Argand plane, polar & Euler forms, triangle inequalities, cube roots of unity, De Moivre's theorem, and geometric loci.

Polar & Euler Form

z = x + iy = r(\cos\theta + i\sin\theta) = r e^{i\theta} \quad (r = |z| = \sqrt{x^2 + y^2}, \; \theta = \arg(z))

Exponential Euler representation for complex numbers.

e^(iπ) + 1 = 0 (Euler's identity).

Triangle Inequalities

||z_1| - |z_2|| \le |z_1 \pm z_2| \le |z_1| + |z_2|

Bounds for modulus of sum and difference of complex numbers.

Equality holds when origin, z₁, and z₂ are collinear.

Cube Roots of Unity (1, ω, ω²)

1 + \omega + \omega^2 = 0, \quad \omega^3 = 1, \quad \omega = \frac{-1 + i\sqrt{3}}{2} = e^{i 2\pi/3}

Roots of equation z³ - 1 = 0.

1 + ω^n + ω^(2n) = 3 if n is multiple of 3, else 0.

De Moivre's Theorem

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) = e^{in\theta}

Powers and roots of trigonometric complex forms for integer and rational n.

Always convert to polar form before taking high powers like z²⁰²⁴.

Geometric Locus in Complex Plane

|z - z_0| = r \; (\text{Circle}), \quad |z - z_1| = |z - z_2| \; (\text{Perpendicular Bisector}), \quad \arg\left(\frac{z - z_1}{z - z_2}\right) = \theta \; (\text{Circular Arc})

Translating complex equations into 2D Cartesian geometric figures.

If θ = ±π/2, the locus is a circle with z₁ and z₂ as diametric endpoints.

Rotation Theorem (Coni's Theorem)

\frac{z_3 - z_1}{z_2 - z_1} = \frac{|z_3 - z_1|}{|z_2 - z_1|} e^{i\alpha}

Rotates directed line segment (z₂ - z₁) by an angle α counter-clockwise around pivot point z₁ to obtain direction of (z₃ - z₁).

For an equilateral triangle with vertices z₁, z₂, z₃: z₁² + z₂² + z₃² = z₁z₂ + z₂z₃ + z₃z₁.

Apollonius Circle & Conic Loci in Argand Plane

\frac{|z - z_1|}{|z - z_2|} = k \; (k \ne 1 \implies \text{Circle}), \quad |z - z_1| + |z - z_2| = 2a \; (2a > |z_1 - z_2| \implies \text{Ellipse})

Modulus ratio represents an Apollonius circle, while sum and difference of distances represent ellipses and hyperbolas.

If |z - z₁| + |z - z₂| = |z₁ - z₂|, locus is the line segment joining z₁ and z₂.

n-th Roots of Unity & Factorization

z^n = 1 \implies z_k = e^{i \frac{2k\pi}{n}} \quad (k = 0, 1, \dots, n-1), \quad \sum_{k=0}^{n-1} z_k = 0, \quad \prod_{k=0}^{n-1} z_k = (-1)^{n-1}

The n-th roots of unity form vertices of a regular n-gon inscribed in unit circle centered at origin.

Factorization identity: x^n - 1 = (x - 1)(x - α)(x - α²) ... (x - α^(n-1)).

AlgebraHigh Yield (2 Qs in JEE)599 PYQs

Permutations & Combinations (P&C)

Fundamental principle of counting, nPr, nCr, circular permutations, distribution into groups, and de-arrangement.

Basic Permutation & Combination Formulas

{}^nP_r = \frac{n!}{(n-r)!}, \quad {}^nC_r = \frac{n!}{r!(n-r)!}, \quad {}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r

Fundamental counting formulas (Pascal's rule).

^nC_r = ^nC_{n-r}.

Number of Non-Negative Integral Solutions

x_1 + x_2 + \dots + x_r = n \implies \text{Solutions} = {}^{n + r - 1}C_{r - 1}

Stars and bars method: distributing n identical items into r distinct boxes.

For positive integral solutions (x_i ≥ 1), solutions = ^{n - 1}C_{r - 1}.

De-arrangement Formula (D_n)

D_n = n! \left[ 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(-1)^n}{n!} \right]

Number of permutations such that none of the n objects appear in their original natural position.

D₁ = 0, D₂ = 1, D₃ = 2, D₄ = 9, D₅ = 44.

Division into Groups

\text{Equal Groups: } \frac{(mn)!}{(m!)^n n!}, \quad \text{Unequal Groups: } \frac{(m+n+p)!}{m! n! p!}

Formulas for partition into distinct or identical sets.

Divide by k! whenever k groups have the same size and are undistinguished.

Circular Permutations & Necklaces

P_{\text{circle}} = (n - 1)!, \quad P_{\text{necklace}} = \frac{(n - 1)!}{2}

Arrangements around a round table versus a necklace/garland where flip-over (clockwise = anticlockwise) makes halves identical.

If arranging n men and n women alternatively around a round table: (n - 1)! × n!.

Grid / Lattice Path Counting

\text{Paths from } (0, 0) \to (m, n) = \frac{(m + n)!}{m! n!} = {}^{m+n}C_m

Number of shortest monotonic paths on an m × n grid moving only rightward (R) and upward (U).

If passing through an intermediate point (p, q): Total Paths = {}^{p+q}C_p × {}^{(m-p)+(n-q)}C_{m-p}.

Distribution of Distinct Objects into Boxes

r^n \; (\text{Distinct items, Distinct boxes}), \quad \sum_{k=0}^r (-1)^{r-k} {}^rC_k k^n \; (\text{No empty box})

Fundamental distribution theorem with restrictions on empty containers using inclusion-exclusion.

Total number of functions from set of n elements to set of r elements is r^n; total onto functions is r! · S(n, r).

AlgebraHigh Yield (2-3 Qs in JEE)551 PYQs

Sequences & Series

Arithmetic, geometric, arithmetico-geometric (AGP), harmonic progressions, AM-GM-HM inequality, and telescoping series.

Arithmetic Progression (AP)

T_n = a + (n - 1)d, \quad S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)

Standard formulas for linear progressions.

Three terms in AP are best assumed as (a - d, a, a + d).

Geometric Progression (GP)

T_n = a r^{n-1}, \quad S_n = \frac{a(1 - r^n)}{1 - r}, \quad S_\infty = \frac{a}{1 - r} \quad (|r| < 1)

Exponential series with common ratio r.

Three terms in GP are best assumed as (a/r, a, ar).

AM-GM-HM Inequality

\frac{a + b}{2} \ge \sqrt{ab} \ge \frac{2ab}{a + b} \iff \text{AM} \ge \text{GM} \ge \text{HM}

Fundamental algebraic inequality for positive real numbers. Equality holds iff all terms are equal.

Premier tool for finding minimum values of expressions like x + 1/x ≥ 2.

Sum of First n Powers of Natural Numbers

\sum n = \frac{n(n+1)}{2}, \quad \sum n^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum n^3 = \left[\frac{n(n+1)}{2}\right]^2

Standard power sum formulas for polynomial series evaluation.

Notice that ∑ n³ = (∑ n)².

Arithmetico-Geometric Progression (AGP) Sums

S_\infty = \frac{a}{1 - r} + \frac{d \cdot r}{(1 - r)^2}, \quad S_n = \frac{a}{1 - r} + \frac{d r(1 - r^{n-1})}{(1 - r)^2} - \frac{[a + (n-1)d]r^n}{1 - r}

Sum of finite and infinite series whose terms are products of an AP and a GP.

To evaluate manually: multiply by r and subtract (1 - r)S = a + d·r + d·r² + ... - [a + (n-1)d]r^n.

Method of Differences & Telescoping Series

S_n = \sum_{r=1}^n [f(r) - f(r-1)] = f(n) - f(0), \quad \frac{1}{r(r+1)(r+2)} = \frac{1}{2}\left[\frac{1}{r(r+1)} - \frac{1}{(r+1)(r+2)}\right]

Splitting general term T_r into difference of consecutive functions cancels all intermediate terms.

For products of factors in AP in the denominator: multiply and divide by (last factor - first factor).

Cauchy-Schwarz & Weighted AM-GM Inequalities

(\sum a_i^2)(\sum b_i^2) \ge (\sum a_i b_i)^2, \quad \frac{\sum m_i x_i}{\sum m_i} \ge \left(\prod x_i^{m_i}\right)^{\frac{1}{\sum m_i}}

Powerful analytical inequalities for finding minimum values of multivariable constrained expressions in JEE Advanced.

Equality holds in Cauchy-Schwarz iff a₁/b₁ = a₂/b₂ = ... = a_n/b_n.

Harmonic Progression (HP) & Relation Between Means

H = \frac{2ab}{a+b}, \quad G^2 = A \cdot H, \quad A \ge G \ge H \quad (a, b > 0)

Sequence whose reciprocals form an AP. Geometric mean is the geometric mean between AM and HM.

There is no general summation formula for HP series; solve HP problems by converting terms to AP reciprocals.

AlgebraHigh Yield (2 Qs in JEE)443 PYQs

Binomial Theorem

General term T_{r+1}, middle term, greatest term, binomial coefficients identities, and remainder theorem.

General Term in Binomial Expansion

(x + y)^n = \sum_{r=0}^n {}^nC_r x^{n-r} y^r, \quad T_{r+1} = {}^nC_r x^{n-r} y^r

Expansion has (n + 1) terms.

Independent of x term occurs when power of x in T_{r+1} equals 0.

Binomial Coefficient Sum Identities

\sum_{r=0}^n {}^nC_r = 2^n, \quad \sum_{r=0}^n (-1)^r {}^nC_r = 0, \quad \sum_{r=1}^n r \cdot {}^nC_r = n \cdot 2^{n-1}

Classic combinatorial summation identities.

Differentiate (1 + x)^n to find sums with r in the coefficient.

Middle Term in Expansion

n \text{ even} \implies T_{\frac{n}{2} + 1}; \quad n \text{ odd} \implies T_{\frac{n+1}{2}} \text{ and } T_{\frac{n+3}{2}}

Has the greatest binomial coefficient ^nC_{max}.

For even n, single middle term has maximum coefficient ^nC_{n/2}.

Fractional / Negative Index Binomial Series

(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots \quad (|x| < 1), \quad T_{r+1} = \frac{n(n-1)\cdots(n-r+1)}{r!}x^r

Infinite convergent series for any rational n (negative or fractional). General term uses falling factorial in numerator.

Common cases: (1+x)^{-1} = 1 - x + x² - x³ + ⋯, (1-x)^{-1} = 1 + x + x² + x³ + ⋯, (1+x)^{-2} = 1 - 2x + 3x² - 4x³ + ⋯. All valid for |x| < 1.

Weighted Binomial Coefficient Sums

\sum_{r=1}^n r \cdot {}^nC_r = n \cdot 2^{n-1}, \quad \sum_{r=1}^n r^2 \cdot {}^nC_r = n(n+1) \cdot 2^{n-2}, \quad \sum_{r=0}^n \frac{1}{r+1}{}^nC_r = \frac{2^{n+1} - 1}{n + 1}

Weighted sums obtained by differentiating or integrating (1+x)^n and substituting x = 1.

Differentiate (1+x)^n once → Σ r·C(n,r)·x^(r-1); differentiate again → Σ r(r-1)·C(n,r)·x^(r-2). Add to get r²·C(n,r) sum.

Multinomial Theorem & Number of Terms

(x_1 + x_2 + \dots + x_k)^n = \sum \frac{n!}{r_1! r_2! \dots r_k!} x_1^{r_1} x_2^{r_2} \dots x_k^{r_k}, \quad \text{Total Terms} = {}^{n+k-1}C_{k-1}

Expansion of sum of k variables raised to integer power n where r₁ + r₂ + ... + r_k = n.

Coefficient of x₁^{r₁} x₂^{r₂}...x_k^{r_k} is the multinomial coefficient n! / (r₁! r₂!...r_k!).

Numerically Greatest Term (NGT)

\left| \frac{T_{r+1}}{T_r} \right| = \frac{n - r + 1}{r} \left| \frac{y}{x} \right| \ge 1 \implies r \le \frac{n + 1}{1 + |x/y|}

Determines the term having the largest magnitude in the expansion of (x + y)^n.

If m = (n+1)/(1 + |x/y|) is an integer, both T_m and T_{m+1} are equal and numerically greatest.

Divisibility & Remainder Finding

(1 + k)^n = 1 + nk + {^nC_2} k^2 + \dots \equiv 1 + nk \pmod{k^2}

Express base as (multiple of divisor ± remainder) to find remainders of large powers.

For finding remainder of 7¹⁰³ divided by 25: rewrite as 7 · (49)⁵¹ = 7 · (50 - 1)⁵¹.

AlgebraHigh Yield (1-2 Qs in JEE)315 PYQs

Quadratic Equations

Roots of quadratic, discriminant nature, Vieta relations, condition for common roots, and location of roots.

Quadratic Formula & Vieta's Relations

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}

Roots and coefficients of ax² + bx + c = 0.

Difference of roots: |α - β| = √(D) / |a|.

Nature of Roots & Discriminant D

D = b^2 - 4ac: \quad D > 0 \; (\text{Real, Distinct}), \quad D = 0 \; (\text{Real, Equal}), \quad D < 0 \; (\text{Complex Conjugates})

Discriminant determines the geometric intersection with the x-axis.

If a, b, c are rational and D is not a perfect square, roots occur in irrational conjugate pairs (p ± √q).

Location of Roots (Both Roots > k)

D \ge 0, \quad -\frac{b}{2a} > k, \quad a \cdot f(k) > 0

Simultaneous necessary and sufficient conditions for both roots to lie to the right of value k.

If roots lie on opposite sides of k, the ONLY condition needed is a * f(k) < 0.

Condition for Both Roots Common

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

Condition for two quadratic equations to have identical solutions.

If one equation has imaginary roots (D < 0), having ONE common root implies BOTH roots are common.

Condition for Exactly One Common Root

(c_1 a_2 - c_2 a_1)^2 = (a_1 b_2 - a_2 b_1)(b_1 c_2 - b_2 c_1), \quad x = \frac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1}

Cross-multiplication determinant condition for a single shared root between two quadratics.

The common root can also be found directly by subtracting equations to eliminate the x² term.

Newton's Sum Theorem for Polynomial Roots

S_n = \alpha^n + \beta^n \implies a S_n + b S_{n-1} + c S_{n-2} = 0

Powerful recurrence relation connecting powers of roots for ax² + bx + c = 0.

Very frequent in JEE Main: e.g. finding (a₁₀ - 2a₈) / 2a₉ without solving for roots explicitly.

Transformation of Quadratic Equations

\text{Roots } k\alpha, k\beta \implies a(x/k)^2 + b(x/k) + c = 0; \quad \text{Roots } 1/\alpha, 1/\beta \implies c x^2 + b x + a = 0; \quad \text{Roots } \alpha \pm h \implies a(x \mp h)^2 + b(x \mp h) + c = 0

Symmetric transformations of equations without recalculating individual roots.

For roots α² and β², substitute x = √y and square to eliminate radicals.

AlgebraHigh Yield (1-2 Qs in JEE)332 PYQs

Statistics

Mean, variance, standard deviation, mean deviation, and effect of linear change of origin and scale.

Mean & Variance Formulas

\bar{x} = \frac{\sum x_i}{n}, \quad \sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2 = \frac{\sum (x_i - \bar{x})^2}{n}

Measures of central tendency and dispersion.

Variance = (Mean of squares) - (Square of mean).

Change of Origin and Scale

y_i = a x_i + b \implies \bar{y} = a\bar{x} + b, \quad \sigma_y^2 = a^2 \sigma_x^2, \quad \sigma_y = |a| \sigma_x

Shift of origin (b) does not affect variance or standard deviation.

Adding constant to every observation leaves variance completely unchanged.

Combined Mean & Variance

\bar{x}_{12} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}, \quad \sigma_{12}^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2} \quad (d_i = \bar{x}_i - \bar{x}_{12})

Pooled variance of two combined statistical groups.

Standard deviation cannot be simply averaged.

Mean Deviation about Mean & Median

MD(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n}, \quad MD(M) = \frac{\sum |x_i - M|}{n}

Mean absolute dispersion of observations from central tendency. Mean deviation is strictly MINIMUM when measured about the Median.

Coefficient of MD = (Mean Deviation) / (Mean or Median).

Coefficient of Variation (C.V.) & Consistency

\text{C.V.} = \frac{\sigma}{\bar{x}} \times 100\%

Relative measure of dispersion used to compare consistency between two datasets.

The series or team with smaller C.V. is considered more consistent and uniform.

Mean & Variance of First n Natural Numbers

\bar{x} = \frac{n + 1}{2}, \quad \sigma^2 = \frac{n^2 - 1}{12}, \quad \sigma = \sqrt{\frac{n^2 - 1}{12}}

Direct closed-form results for arithmetic mean and variance of first n positive integers {1, 2, ..., n}.

Frequently asked direct question in JEE Main (e.g. variance of first 50 natural numbers is (2500 - 1)/12 = 208.25).

Empirical Relation of Central Tendencies

\text{Mode} = 3\,\text{Median} - 2\,\text{Mean} \iff \text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})

Karl Pearson's empirical formula connecting mean, median, and mode for moderately skewed unimodal frequency distributions.

For perfectly symmetrical distribution, Mean = Median = Mode.

Variance of First n Even & Odd Numbers

\text{Even } \{2, 4, \dots, 2n\}: \bar{x} = n + 1, \; \sigma^2 = \frac{n^2 - 1}{3}; \quad \text{Odd } \{1, 3, \dots, 2n-1\}: \bar{x} = n, \; \sigma^2 = \frac{n^2 - 1}{3}

Variance of first n consecutive even and odd positive integers.

Notice that variance of first n even numbers and first n odd numbers is identical: (n² - 1)/3.

Vectors & 3D GeometryTop Priority (3-4 Qs in JEE)1,174 PYQs

Three Dimensional Geometry (3D)

Direction cosines & ratios, equations of lines in 3D, shortest distance between skew and parallel lines.

Direction Cosines & Direction Ratios

l = \cos\alpha, \; m = \cos\beta, \; n = \cos\gamma \implies l^2 + m^2 + n^2 = 1, \quad l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}

Direction cosines represent components of the unit direction vector in 3D.

cos²α + cos²β + cos²γ = 1 implies sin²α + sin²β + sin²γ = 2.

Equation of a Line in 3D

\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = \lambda \iff \vec{r} = \vec{a} + \lambda \vec{b}

Symmetric and vector forms of line passing through (x₁, y₁, z₁) with direction ratios <a, b, c>.

Any general point on the line is (x₁ + aλ, y₁ + bλ, z₁ + cλ).

Shortest Distance Between Skew Lines

d = \left| \frac{(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|} \right|

Minimum distance between non-parallel, non-intersecting skew lines in 3D.

Lines intersect if and only if (a₂ - a₁) · (b₁ × b₂) = 0.

Shortest Distance Between Parallel Lines

d = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}

Distance between parallel lines r = a₁ + λb and r = a₂ + μb.

Uses cross product magnitude divided by |b|.

Angle Between Two Lines in 3D

\cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}} = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|}

Acute angle between lines with direction ratios.

Lines are perpendicular if a₁a₂ + b₁b₂ + c₁c₂ = 0.

Condition for Coplanarity of Two Lines

\begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \iff (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0

Criterion for two straight lines in 3D space to lie in the same unique plane.

If coplanar, either the lines intersect at a point or they are parallel.

Distance of a Point from a Line in 3D

d = \frac{|(\vec{p} - \vec{a}) \times \vec{b}|}{|\vec{b}|} = \sqrt{|\vec{p} - \vec{a}|^2 - \left(\frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{|\vec{b}|}\right)^2}

Perpendicular distance from point P(p) to line r = a + λb.

Foot of perpendicular F: let F = a + λb, then (p - F) · b = 0 determines λ directly.

Equation of Sphere & Its Properties

x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 \implies \text{Centre} = (-u,-v,-w), \; r = \sqrt{u^2 + v^2 + w^2 - d}

General second-degree equation of a sphere. Standard form with centre (a,b,c): (x-a)² + (y-b)² + (z-c)² = r².

Sphere through origin: d = 0. Diameter-form sphere with endpoints (x₁,y₁,z₁) and (x₂,y₂,z₂): (x-x₁)(x-x₂) + (y-y₁)(y-y₂) + (z-z₁)(z-z₂) = 0. For a point to lie on sphere: substitute and check if the equation is satisfied.

Image of a Point in a Plane & Foot of Perpendicular

\frac{x' - x_1}{a} = \frac{y' - y_1}{b} = \frac{z' - z_1}{c} = -2\cdot\frac{ax_1 + by_1 + cz_1 + d}{a^2 + b^2 + c^2}

Coordinates of image (x', y', z') of point (x₁, y₁, z₁) in plane ax + by + cz + d = 0. Foot of perpendicular F has parameter k = −(ax₁+by₁+cz₁+d)/(a²+b²+c²).

F (foot of ⊥) = (x₁ + ak, y₁ + bk, z₁ + ck). Image point = (x₁ + 2ak, y₁ + 2bk, z₁ + 2ck). Distance from point to plane: |ax₁+by₁+cz₁+d| / √(a²+b²+c²).

Vectors & 3D GeometryTop Priority (3-4 Qs in JEE)1,073 PYQs

Vector Algebra

Dot product, cross product, scalar triple product, vector triple product, and Lagrange's identity.

Scalar (Dot) Product & Projection

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta, \quad \text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Dot product yields scalar; zero if vectors are orthogonal.

Projection vector is ((a · b) / |b|²) b.

Vector (Cross) Product

\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \, \hat{n}, \quad \text{Area of Parallelogram} = |\vec{a} \times \vec{b}|

Perpendicular vector directed by right-hand thumb rule.

Area of triangle with adjacent vectors a and b is (1/2)|a × b|.

Scalar Triple Product (Box Product)

[\vec{a} \; \vec{b} \; \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

Represents the volume of parallelepiped formed by coterminous vectors a, b, c.

Coplanarity condition: [a b c] = 0.

Vector Triple Product (BAC-CAB Rule)

\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}

Expands double cross product into a linear combination of vectors b and c.

Memory trick: BAC minus CAB.

Lagrange's Vector Identity

|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2

Connects cross product and dot product magnitudes directly.

Derived from sin²θ + cos²θ = 1.

Scalar & Vector Product of Four Vectors

(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \begin{vmatrix} \vec{a}\cdot\vec{c} & \vec{a}\cdot\vec{d} \\ \vec{b}\cdot\vec{c} & \vec{b}\cdot\vec{d} \end{vmatrix}, \quad (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = [\vec{a}\;\vec{b}\;\vec{d}]\vec{c} - [\vec{a}\;\vec{b}\;\vec{c}]\vec{d}

Determinant expansion for dot product of two cross products, and linear resolution of cross product of two cross products.

Crucial identity for finding line of intersection of two planes defined by normal vectors.

Reciprocal System of Vectors & Tetrahedron Volume

\vec{a}' = \frac{\vec{b} \times \vec{c}}{[\vec{a}\,\vec{b}\,\vec{c}]}, \quad \vec{a} \cdot \vec{a}' = 1, \; \vec{a} \cdot \vec{b}' = 0, \quad V_{\text{tetrahedron}} = \frac{1}{6}|[\vec{a}\,\vec{b}\,\vec{c}]|

Reciprocal triad of vectors satisfying orthogonal dot relations, and volume of tetrahedron with coterminous edges a, b, c.

The scalar triple product of reciprocal system is [a' b' c'] = 1 / [a b c].

TrigonometryHigh Yield (2-3 Qs in JEE)426 PYQs

Trigonometric Ratios & Identities

Compound angles, transformation formulas, multiple angles, conditional identities, and general trigonometric equations.

Compound Angle Formulas

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B, \quad \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

Fundamental addition and subtraction identities.

Notice the minus-plus inversion in cos(A ± B).

Transformation (Product to Sum & Sum to Product)

2\sin A\cos B = \sin(A+B) + \sin(A-B), \quad \cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}

Converts trigonometric products into sums and vice versa for easy calculus integration.

cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2) = 2 sin((C+D)/2) sin((D-C)/2).

Multiple Angle Formulas (Double & Half)

\sin 2A = \frac{2\tan A}{1 + \tan^2 A}, \quad \cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A} = 2\cos^2 A - 1 = 1 - 2\sin^2 A

Expresses double angle functions entirely in terms of tan A.

1 - cos 2A = 2 sin²A, and 1 + cos 2A = 2 cos²A.

Maximum and Minimum of a sin θ + b cos θ

-\sqrt{a^2 + b^2} \le a\sin\theta + b\cos\theta \le \sqrt{a^2 + b^2}

Extremum values of linear harmonic combination of sine and cosine.

Useful for range questions in both Mathematics and Physics SHM.

General Solutions of Trigonometric Equations

\sin\theta = \sin\alpha \implies \theta = n\pi + (-1)^n \alpha, \quad \cos\theta = \cos\alpha \implies \theta = 2n\pi \pm \alpha, \quad \tan\theta = \tan\alpha \implies \theta = n\pi + \alpha

Fundamental general solutions for periodic trigonometric equations (where n ∈ ℤ).

For a·cos θ + b·sin θ = c to have real solutions, condition is |c| ≤ √(a² + b²).

Quadratic & Squared Trigonometric Equations

\sin^2\theta = \sin^2\alpha \iff \cos^2\theta = \cos^2\alpha \iff \tan^2\theta = \tan^2\alpha \implies \theta = n\pi \pm \alpha

Universal general solution for all squared trigonometric equation forms.

Always check for extraneous roots when squaring both sides of a trigonometric equation.

Triple Angle Identities

\sin 3A = 3\sin A - 4\sin^3 A, \quad \cos 3A = 4\cos^3 A - 3\cos A, \quad \tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}

Expansion of triple angles, frequently applied in cubic root and polynomial evaluations.

tan A · tan(60° - A) · tan(60° + A) = tan 3A, and sin A · sin(60° - A) · sin(60° + A) = (1/4) sin 3A.

Summation of Sines & Cosines in A.P.

\sum_{k=0}^{n-1} \sin(\alpha + k\beta) = \frac{\sin(n\beta/2)}{\sin(\beta/2)} \sin\left(\alpha + \frac{(n-1)\beta}{2}\right), \quad \sum_{k=0}^{n-1} \cos(\alpha + k\beta) = \frac{\sin(n\beta/2)}{\sin(\beta/2)} \cos\left(\alpha + \frac{(n-1)\beta}{2}\right)

Closed-form summation for harmonic series whose angles progress in an arithmetic sequence.

Crucial for interference of n waves in Physics Wave Optics as well as JEE Advanced Maths.

Conditional Identities in Triangles (A + B + C = π)

\tan A + \tan B + \tan C = \tan A \tan B \tan C, \quad \sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C, \quad \cos^2 A + \cos^2 B + \cos^2 C = 1 - 2\cos A\cos B\cos C

Standard cyclic identities holding inside any Euclidean triangle.

If A + B + C = π/2, then tan A tan B + tan B tan C + tan C tan A = 1.

Heights & Distances Shortcut Formulas

h = \frac{d}{\cot\alpha - \cot\beta} = \frac{d\sin\alpha\sin\beta}{\sin(\beta - \alpha)}, \quad H_{\text{cloud}} = h\cdot\frac{\tan\beta + \tan\alpha}{\tan\beta - \tan\alpha}

Height of tower h from two collinear points distance d apart with elevation angles α and β (β > α). Cloud height H above lake from observer at height h above water with elevation α and reflection depression β.

Tower between two observation points: h = d / (cot α + cot β). Flagstaff of height p on top of tower: tower height h = p tan α / (tan β - tan α).

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Inverse Trigonometric Functions (ITF)

Principal branches, domain & range, complementary angle relations, and sum of inverse tangents.

Principal Value Domains & Ranges

\sin^{-1}x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right], \quad \cos^{-1}x \in [0, \pi], \quad \tan^{-1}x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

Standard principal branches defined to make trigonometric functions bijective.

Always reduce angles outside principal branch into the fundamental domain first.

Complementary Angles Property

\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}, \quad \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \quad (x \in \text{Domain})

Sum of co-functions equals π/2 everywhere in their common domain.

sec⁻¹x + csc⁻¹x = π/2 for |x| ≥ 1.

Sum of Inverse Tangents

\tan^{-1}x + \tan^{-1}y = \begin{cases} \tan^{-1}\left(\frac{x+y}{1-xy}\right) & xy < 1 \\ \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right) & x,y > 0, \; xy > 1 \end{cases}

Combines two inverse tangents into one.

Verify the sign of xy before applying the formula without π adjustment.

Multiple Angle Inverse Formula

2\tan^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \tan^{-1}\left(\frac{2x}{1-x^2}\right)

Interconversion between 2 tan⁻¹x and other inverse functions.

Check domain conditions: cos⁻¹((1-x²)/(1+x²)) = 2 tan⁻¹x holds for x ≥ 0.

Negative Argument Relations

\sin^{-1}(-x) = -\sin^{-1}x, \quad \cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x

Odd vs supplementary behavior for negative arguments.

sec⁻¹(-x) = π - sec⁻¹x and cot⁻¹(-x) = π - cot⁻¹x.

Telescoping Series of Inverse Tangents

\sum_{r=1}^n \tan^{-1}\left(\frac{x_r - x_{r-1}}{1 + x_r x_{r-1}}\right) = \tan^{-1}x_n - \tan^{-1}x_0, \quad \tan^{-1}\left(\frac{1}{1 + r + r^2}\right) = \tan^{-1}(r+1) - \tan^{-1}r

Decomposition of rational algebraic arguments into differences of consecutive inverse tangents for infinite summation.

For ∑_{r=1}^∞ tan⁻¹(1/(r² + r + 1)): Sum = lim_{n→∞} (tan⁻¹(n+1) - tan⁻¹ 1) = π/2 - π/4 = π/4.

Addition Formulas for Inverse Sine & Cosine

\sin^{-1}x \pm \sin^{-1}y = \sin^{-1}\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2}\right), \quad \cos^{-1}x \pm \cos^{-1}y = \cos^{-1}\left(xy \mp \sqrt{1-x^2}\sqrt{1-y^2}\right)

Standard addition and subtraction formulas for inverse sines and cosines within primary domains.

For sin⁻¹x + sin⁻¹y: if x² + y² > 1 and x,y > 0, the sum exceeds π/2 and equals π - sin⁻¹(x√(1-y²) + y√(1-x²)).

Extreme Values of Inverse Trigonometric Sums

(\sin^{-1}x)^2 + (\cos^{-1}x)^2 \in \left[\frac{\pi^2}{8}, \frac{5\pi^2}{4}\right], \quad (\sin^{-1}x)^3 + (\cos^{-1}x)^3 \in \left[\frac{\pi^3}{32}, \frac{7\pi^3}{8}\right]

Ranges of quadratic and cubic symmetric combinations of inverse sine and cosine.

Minimum occurs when sin⁻¹x = cos⁻¹x = π/4 (i.e. x = 1/√2). Maximum occurs at x = -1.

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Properties of Triangles (SOT)

Sine rule, cosine rule, projection formula, radii of circumcircle, incircle, and excircles.