JEE Mathematics — Complete Formula Sheet

For domain/range questions with composite functions, always find the domain of the INNER function first, then check where the outer function is defined for those output values — working outside-in causes most domain errors.

For cube roots of unity problems, always remember 1+ω+ω²=0 and ω³=1 — these two identities alone solve almost every JEE problem involving ω without needing to expand complex expressions manually.

For 'location of roots' problems (like 'both roots greater than k'), always combine THREE conditions together — the sign of f(k), the sign of 'a', and the vertex position — using only one condition alone is the most common incomplete-answer mistake.

For AM-GM-HM inequality questions, remember AM ≥ GM ≥ HM always holds for positive real numbers, with equality only when all numbers are equal — this single inequality solves many 'prove that' JEE Advanced problems instantly.

For 'circular arrangement with restrictions' problems, fix one person/object's position first to break the circular symmetry, then arrange the rest linearly — this single trick converts a confusing circular problem into a standard linear permutation.

Use standard determinant row/column operations (like R1→R1+R2) to create zeros BEFORE expanding a 3x3 determinant — expanding directly without simplification is far more error-prone and time-consuming.

To check if a line is tangent to, secant to, or doesn't touch a circle, just compare the perpendicular distance from the centre to the line against the radius — distance=radius (tangent), distance<radius (secant), distance>radius (no intersection).

Eccentricity instantly tells you which conic you're dealing with: e=0 is a circle, 0<e<1 is an ellipse, e=1 is a parabola, e>1 is a hyperbola — memorise this single number scale to identify any conic immediately.

When a limit gives an indeterminate 0/0 or ∞/∞ form directly on substitution, try L'Hôpital's rule or standard limit formulas immediately rather than attempting long algebraic manipulation first — it's usually the fastest path in JEE Main time constraints.

For optimisation word problems, always express the quantity to be maximised/minimised as a function of a SINGLE variable first (using given constraints to eliminate other variables) before differentiating — trying to differentiate a two-variable expression directly is a common setup mistake.

Use the ILATE rule (Inverse, Log, Algebraic, Trig, Exponential) to decide which function to treat as 'u' in integration by parts — picking the wrong function as u makes the resulting integral harder, not easier.

Before brute-force solving a definite integral, always check if it fits one of the standard properties (like the a-x substitution property or even/odd function symmetry) — these tricks often reduce a hard integral to a one-line answer.

Always rearrange the given differential equation into the standard linear form (coefficient of dy/dx equal to 1) BEFORE identifying P and Q — attempting to find the integrating factor from a non-standardised equation is the most common setup error.

Remember 'Dot gives a number (angle-related), Cross gives a vector (perpendicular direction)' — this single distinction resolves most confusion about which product to use in a given 3D geometry problem.

Memorise the three general solution templates (for sin, cos, tan equal to a standard value) as fixed formulas — nearly every trigonometric equation question in JEE Main reduces to applying one of these three templates correctly.

Bayes' theorem questions always describe an 'effect' (like a positive test result) and ask for the probability of a 'cause' — spotting this reversed direction immediately tells you it's a Bayes' theorem problem, not a simple conditional probability one.

For 'rate of change' word problems involving two related quantities (like a ladder sliding down a wall), always write BOTH quantities as functions of time and differentiate implicitly with respect to t — this is the standard related-rates technique tested repeatedly.

For combined mean/variance of two data sets, use the direct combination formulas rather than recomputing from raw data — this saves significant calculation time in JEE Main's strict per-question time limit.