JEE Mathematics Formula Sheet

$A'$ and $B'$ represent the complements of sets $A$ and $B$, while $\cup$ and $\cap$ represent union and intersection.

$n(S)$ represents the number of elements in set $S$; $\cup$ and $\cap$ represent union and intersection respectively.

$R$ is relation, $A$ is domain set, $B$ is co-domain set, and $(a,b)$ represents ordered pairs in $R$.

$n(S)$ represents the number of elements in set $S$, and $n$ is the cardinality of set $A$.

$g(x)$ is a real-valued sub-function, and $\implies$ represents constraints for mathematical definedness.

$f(x)$ is a real-valued function, and $T > 0$ is the smallest positive constant satisfying the relation.

$a = n(A)$ is the size of the domain, $b = n(B)$ is the size of the co-domain, and $P(b,a)$ represents permutations.

$g \circ f$ is the composite function of $f$ and $g$, and $f^{-1}$ is the inverse function of $f$.

$z_1, z_2$ are complex numbers, $\overline{z}$ represents the complex conjugate of $z$, and $|z|$ represents the modulus.

$r$ is the modulus, $\theta$ is the argument (amplitude), and $i = \sqrt{-1}$ is the imaginary unit.

$|z| = \sqrt{a^2 + b^2}$, and $\operatorname{sgn}(b)$ is $1$ if $b \ge 0$ and $-1$ if $b < 0$.

$z_1$ and $z_2$ are complex numbers, and $|z|$ represents the modulus of $z$.

$a, b, c$ are coefficients, $D$ is the discriminant, and $\alpha, \beta$ are the roots.

$x$ is the variable, $S$ is the sum of the roots, $P$ is the product of the roots, and $\alpha, \beta$ are the roots.

$a_1, b_1, c_1$ and $a_2, b_2, c_2$ are the coefficients of the two quadratic equations respectively.

$A^T$ is the transpose of square matrix $A$, $I$ is the identity matrix, and $-A$ is the negative matrix of $A$.

$A^{-1}$ is the inverse of matrix $A$, $|A|$ is the determinant of $A$, and $\operatorname{adj}(A)$ is the adjoint matrix of $A$.

$a,b,c,d$ are elements of a $2\times 2$ matrix; $a_i, b_i, c_i$ are elements of a $3\times 3$ matrix expanding along the first row.

$A^T$ is the transpose, $k$ is a scalar constant, $n$ is the dimension of the square matrices, and $|A|$ is the determinant of $A$.

$\Delta$ is the coefficient determinant, and $\Delta_x, \Delta_y, \Delta_z$ are determinants obtained by replacing columns with the constant terms.

$m$ is the number of ways task 1 can occur, and $n$ is the number of ways task 2 can occur.

$n$ is the total number of items, and $r$ is the number of items to arrange or select.

$n$ is the number of vertices/points, and $p$ is the number of collinear points among them.

$n$ is a positive integer, $T_{r+1}$ represents the $(r+1)$-th term in the expansion, and $C(n,r)$ is the binomial coefficient.

$C_r$ represents the binomial coefficient $C(n,r)$ for a positive integral index $n$.

$n$ is the positive power index, $k$ is the number of variables inside the brackets, and $r_i$ are non-negative integer powers.

$a$ is the first term, $d$ is the common difference, $r$ is the common ratio, and $S_n$ represents the sum of $n$ terms.

$a, b$ are positive real numbers.

$a$ is first term of AP/AGP, $d$ is common difference, $r$ is common ratio of GP part, and $S_{\infty}$ is sum to infinity.

$x$ is a real variable approaching $0$, and $e$ is Euler's number.

$f(x)$ is the function, $c$ is the point of interest, and the left-hand and right-hand limits must match the value $f(c)$.

$f'(x)$ is the derivative, and $h$ is an infinitesimally small change in the input variable $x$.

$f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and $c$ is a point inside the interval.

$u, v, y$ are differentiable functions of a real variable $x$, and $u$ is a function of $x$.

$x$ is a real variable within the valid domain of each corresponding function.

$y$ is a function of $x$, and both are differentiable functions of time $t$.

$f(x)$ is a differentiable function, and $m_{tangent}$ is the slope of the tangent at $x_0$.

$f(x)$ is a twice-differentiable function, and $c$ is a critical point inside its domain.

$(x_1, y_1)$ is the point on the curve, and $f'(x_1)$ is the slope of the tangent.

$n$ is a constant exponent, $x$ is a real variable, and $C$ is the integration constant.

$u, v, f(x), g(x)$ are differentiable functions of $x$.

$p,q,a,b$ are constants, and $A,B$ are solved numerical coefficients.

$f$ is continuous, $F$ is antiderivative, and $\phi(x), \psi(x)$ are differentiable boundary functions.

$f(x)$ is an integrable function on the specified intervals, and $a, b$ are real limits.

$A$ is the area of the region, and $x = a$ to $x = b$ are boundary lines.

$A$ is the bounded area, and $f(x), g(x)$ are curves intersecting or bounded at $x=a$ and $x=b$.

$x$ is independent variable, $y$ is dependent variable, and $y^{(n)}$ is the $n$-th derivative of $y$.

$v$ is a temporary helper variable, $x, y$ are coordinate variables, and $f, g$ are functions.

$P$ and $Q$ are functions of $x$ only, I.F. is the integrating factor, and $C$ is the integration constant.

$m$ is the slope, $c$ is the y-intercept, $a, b$ are axes intercepts, and $(x_i, y_i)$ are points on the line.

$d$ is the perpendicular distance, $(x_1, y_1)$ is the coordinates of the point, and $a, b, c$ are the coefficients of the line.

$m_1, m_2$ are slopes of the intersecting lines, $d_{\text{parallel}}$ is distance between lines $ax+by+c_1=0$ and $ax+by+c_2=0$.

$(x_1, y_1)$ is the given point, $(x_p, y_p)$ is the foot of the perpendicular, and $(x_i, y_i)$ is the mirror image point.

$L_1, L_2$ are lines, $\lambda$ is a parameter, and $a_i, b_i, c_i$ are line coefficients.

$(h,k)$ is the center, $r$ is the radius, and $g, f, c$ are general equation parameters.

$m$ is the slope of the tangent, and $a$ is the radius of the circle.

$(x_1, y_1)$ is an external point, and $a$ is the radius of the circle.

$a, b$ are semi-axes lengths, and $e$ represents the eccentricity of the ellipse or hyperbola.

$m$ is the slope of the tangent line, and $a, b$ are conic parameters.

$l, m, n$ are direction cosines; $a, b, c$ are direction ratios.

$\vec{a}$ (or $(x_1, y_1, z_1)$) is a point on the line, $\vec{b}$ (or $b_x, b_y, b_z$) is the direction vector, and $\lambda$ is a scalar parameter.

$d_{shortest}$ is the shortest distance; the lines are defined by $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$.

$a,b,c$ are normal coefficients, and $\theta$ is the angle between planes with coefficients $a_i, b_i, c_i$.

$d$ is the perpendicular distance, and $a, b, c, d$ are coefficients of the plane equation.

$\vec{b}$ is the direction vector of the line, $\vec{n}$ is the normal vector of the plane, and $\theta$ is the angle between them.