JEE Mathematics Formulas

Cheatsheet Content

### Quadratic Equations - **Standard Form:** $ax^2 + bx + c = 0$, where $a \neq 0$. - **Discriminant:** $\Delta = b^2 - 4ac$. - If $\Delta > 0$, roots are real and distinct: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$. - If $\Delta = 0$, roots are real and equal: $x = -\frac{b}{2a}$. - If $\Delta ### Sequences and Series #### Arithmetic Progression (AP) - **$n^{th}$ term:** $a_n = a + (n-1)d$. - **Sum of $n$ terms:** $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + a_n)$. - **Arithmetic Mean (AM):** If $a, b, c$ are in AP, $b = \frac{a+c}{2}$. #### Geometric Progression (GP) - **$n^{th}$ term:** $a_n = ar^{n-1}$. - **Sum of $n$ terms:** $S_n = \frac{a(r^n - 1)}{r-1}$ if $r \neq 1$. $S_n = na$ if $r=1$. - **Sum to infinity:** $S_\infty = \frac{a}{1-r}$ if $|r| ### Complex Numbers - **Standard form:** $z = x + iy$, where $i = \sqrt{-1}$. - **Conjugate:** $\bar{z} = x - iy$. - **Modulus:** $|z| = \sqrt{x^2 + y^2}$. - **Argument/Amplitude:** $\arg(z) = \theta$, where $x = |z|\cos\theta$, $y = |z|\sin\theta$. - $\theta = \tan^{-1}\left(\frac{y}{x}\right)$, considering quadrant. - **Polar form:** $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ (Euler's form). - **De Moivre's Theorem:** $( \cos\theta + i\sin\theta )^n = \cos(n\theta) + i\sin(n\theta)$. - **Cube roots of unity:** $1, \omega, \omega^2$. - $\omega = e^{i2\pi/3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$. - $\omega^2 = e^{i4\pi/3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$. - $1 + \omega + \omega^2 = 0$. - $\omega^3 = 1$. - **Properties:** - $|z_1z_2| = |z_1||z_2|$ - $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$ - $\arg(z_1z_2) = \arg(z_1) + \arg(z_2)$ - $\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$ - $z\bar{z} = |z|^2$ - **Locus of a point:** - $|z-z_0| = r$: Circle with center $z_0$ and radius $r$. - $|z-z_1| = |z-z_2|$: Perpendicular bisector of the line segment joining $z_1$ and $z_2$. - $\arg(\frac{z-z_1}{z-z_2}) = \alpha$: Arc of a circle. ### Permutations and Combinations - **Factorial:** $n! = n \times (n-1) \times ... \times 2 \times 1$. $0! = 1$. - **Permutations ($P$):** Arrangement of $n$ distinct items taken $r$ at a time. - $P(n, r) = {}^nP_r = \frac{n!}{(n-r)!}$. - Number of permutations of $n$ items where $p_1$ are of one type, $p_2$ of another, etc.: $\frac{n!}{p_1! p_2! ... p_k!}$. - **Combinations ($C$):** Selection of $n$ distinct items taken $r$ at a time. - $C(n, r) = {}^nC_r = \frac{n!}{r!(n-r)!}$. - **Properties of Combinations:** - ${}^nC_r = {}^nC_{n-r}$. - ${}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r$. - ${}^nC_0 = 1, {}^nC_n = 1$. ### Binomial Theorem - **For positive integer $n$:** $(x+y)^n = {}^nC_0 x^n y^0 + {}^nC_1 x^{n-1} y^1 + ... + {}^nC_r x^{n-r} y^r + ... + {}^nC_n x^0 y^n$. - **General term:** $T_{r+1} = {}^nC_r x^{n-r} y^r$. - **Middle term:** - If $n$ is even, $(n/2 + 1)^{th}$ term: $T_{n/2+1} = {}^{n}C_{n/2} x^{n/2} y^{n/2}$. - If $n$ is odd, two middle terms: $(\frac{n+1}{2})^{th}$ and $(\frac{n+3}{2})^{th}$ terms. - **Binomial Theorem for any index (negative or fractional):** $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + ...$ - Valid for $|x| ### Matrices and Determinants #### Matrices - **Order:** $m \times n$ (rows $\times$ columns). - **Types:** Row, Column, Square, Diagonal, Scalar, Identity, Null. - **Operations:** - **Addition/Subtraction:** Element-wise (same order). - **Multiplication:** $AB$ defined if columns of $A$ = rows of $B$. $(AB)_{ij} = \sum_k A_{ik}B_{kj}$. - Not commutative ($AB \neq BA$). - Associative $(A(BC) = (AB)C)$. - Distributive ($A(B+C) = AB+AC$). - **Transpose ($A^T$):** $(A^T)_{ij} = A_{ji}$. - $(A^T)^T = A$. - $(A+B)^T = A^T + B^T$. - $(kA)^T = kA^T$. - $(AB)^T = B^T A^T$. - **Symmetric Matrix:** $A^T = A$. - **Skew-Symmetric Matrix:** $A^T = -A$. (Diagonal elements are zero). - **Inverse ($A^{-1}$):** $AA^{-1} = A^{-1}A = I$. - $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$. - $(AB)^{-1} = B^{-1}A^{-1}$. - $(A^T)^{-1} = (A^{-1})^T$. - **Orthogonal Matrix:** $A A^T = A^T A = I$. (So $A^{-1} = A^T$). - **Idempotent Matrix:** $A^2 = A$. - **Nilpotent Matrix:** $A^k = 0$ for some positive integer $k$. - **Involutory Matrix:** $A^2 = I$. #### Determinants - **For a $2 \times 2$ matrix:** $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad-bc$. - **For a $3 \times 3$ matrix:** $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$. $\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$. - **Properties:** - $\det(A^T) = \det(A)$. - If two rows/columns are identical or proportional, $\det(A) = 0$. - If a row/column is all zeros, $\det(A) = 0$. - Interchanging two rows/columns changes sign of determinant. - Multiplying a row/column by $k$ multiplies determinant by $k$. - $\det(AB) = \det(A)\det(B)$. - $\det(A^n) = (\det(A))^n$. - $\det(kA) = k^n \det(A)$ for $n \times n$ matrix $A$. - $\text{adj}(A) = (C_{ij})^T$, where $C_{ij}$ is cofactor of $a_{ij}$. - $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A) \cdot I$. - $\det(\text{adj}(A)) = (\det(A))^{n-1}$. - **Cramer's Rule (for $AX=B$):** - $x = \frac{\Delta_1}{\Delta}$, $y = \frac{\Delta_2}{\Delta}$, $z = \frac{\Delta_3}{\Delta}$. - $\Delta = \det(A)$. $\Delta_i$ is determinant of matrix $A$ with $i^{th}$ column replaced by $B$. - If $\Delta \neq 0$, unique solution. - If $\Delta = 0$ and at least one $\Delta_i \neq 0$, no solution (inconsistent). - If $\Delta = 0$ and all $\Delta_i = 0$, infinitely many solutions (consistent). ### Logarithms - **Definition:** If $a^x = N$, then $x = \log_a N$. ($a > 0, a \neq 1, N > 0$). - **Properties:** - $\log_a 1 = 0$. - $\log_a a = 1$. - $\log_a (MN) = \log_a M + \log_a N$. - $\log_a (\frac{M}{N}) = \log_a M - \log_a N$. - $\log_a (M^k) = k \log_a M$. - **Change of Base:** $\log_a N = \frac{\log_b N}{\log_b a}$. - $\log_a b \cdot \log_b a = 1$. - $a^{\log_a N} = N$. - $\log_{a^k} N = \frac{1}{k} \log_a N$. - $\log_{a^k} N^m = \frac{m}{k} \log_a N$. ### Trigonometric Ratios and Identities #### Basic Identities - $\sin^2\theta + \cos^2\theta = 1$. - $1 + \tan^2\theta = \sec^2\theta$. - $1 + \cot^2\theta = \csc^2\theta$. #### Sum and Difference Formulas - $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$. - $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$. - $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$. - $\cot(A \pm B) = \frac{\cot A \cot B \mp 1}{\cot B \pm \cot A}$. #### Double Angle Formulas - $\sin 2A = 2\sin A \cos A = \frac{2\tan A}{1+\tan^2 A}$. - $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1-\tan^2 A}{1+\tan^2 A}$. - $\tan 2A = \frac{2\tan A}{1-\tan^2 A}$. #### Triple Angle Formulas - $\sin 3A = 3\sin A - 4\sin^3 A$. - $\cos 3A = 4\cos^3 A - 3\cos A$. - $\tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}$. #### Half Angle Formulas (in terms of $t = \tan(A/2)$) - $\sin A = \frac{2t}{1+t^2}$. - $\cos A = \frac{1-t^2}{1+t^2}$. - $\tan A = \frac{2t}{1-t^2}$. #### Product-to-Sum and Sum-to-Product Formulas - $2\sin A \cos B = \sin(A+B) + \sin(A-B)$. - $2\cos A \sin B = \sin(A+B) - \sin(A-B)$. - $2\cos A \cos B = \cos(A+B) + \cos(A-B)$. - $2\sin A \sin B = \cos(A-B) - \cos(A+B)$. - $\sin C + \sin D = 2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$. - $\sin C - \sin D = 2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)$. - $\cos C + \cos D = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$. - $\cos C - \cos D = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)$. #### General Solutions of Trigonometric Equations - $\sin x = \sin \alpha \implies x = n\pi + (-1)^n \alpha$, $n \in \mathbb{Z}$. - $\cos x = \cos \alpha \implies x = 2n\pi \pm \alpha$, $n \in \mathbb{Z}$. - $\tan x = \tan \alpha \implies x = n\pi + \alpha$, $n \in \mathbb{Z}$. - $\sin^2 x = \sin^2 \alpha \implies x = n\pi \pm \alpha$, $n \in \mathbb{Z}$. - $\cos^2 x = \cos^2 \alpha \implies x = n\pi \pm \alpha$, $n \in \mathbb{Z}$. - $\tan^2 x = \tan^2 \alpha \implies x = n\pi \pm \alpha$, $n \in \mathbb{Z}$. ### Inverse Trigonometric Functions #### Principal Value Branch - $\sin^{-1} x$: $[-\pi/2, \pi/2]$, $x \in [-1,1]$. - $\cos^{-1} x$: $[0, \pi]$, $x \in [-1,1]$. - $\tan^{-1} x$: $(-\pi/2, \pi/2)$, $x \in \mathbb{R}$. - $\csc^{-1} x$: $[-\pi/2, \pi/2] - \{0\}$, $x \in (-\infty, -1] \cup [1, \infty)$. - $\sec^{-1} x$: $[0, \pi] - \{\pi/2\}$, $x \in (-\infty, -1] \cup [1, \infty)$. - $\cot^{-1} x$: $(0, \pi)$, $x \in \mathbb{R}$. #### Important Identities - $\sin^{-1} x + \cos^{-1} x = \pi/2$, for $x \in [-1, 1]$. - $\tan^{-1} x + \cot^{-1} x = \pi/2$, for $x \in \mathbb{R}$. - $\csc^{-1} x + \sec^{-1} x = \pi/2$, for $|x| \ge 1$. - $\tan^{-1} x \pm \tan^{-1} y = \tan^{-1}\left(\frac{x \pm y}{1 \mp xy}\right)$. - $2\tan^{-1} x = \tan^{-1}\left(\frac{2x}{1-x^2}\right) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$. - $\sin^{-1}(-x) = -\sin^{-1} x$. - $\cos^{-1}(-x) = \pi - \cos^{-1} x$. - $\tan^{-1}(-x) = -\tan^{-1} x$. ### Properties of Triangle - **Sine Rule:** $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$, where $R$ is circumradius. - **Cosine Rule:** - $a^2 = b^2 + c^2 - 2bc \cos A$. - $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$. - **Projection Rule:** - $a = b \cos C + c \cos B$. - $b = c \cos A + a \cos C$. - $c = a \cos B + b \cos A$. - **Area of Triangle ($\Delta$):** - $\Delta = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B = \frac{1}{2}ab \sin C$. - $\Delta = \sqrt{s(s-a)(s-b)(s-c)}$ (Heron's formula), where $s = \frac{a+b+c}{2}$. - $\Delta = \frac{abc}{4R}$. - $\Delta = rs$, where $r$ is inradius. - **Inradius ($r$):** - $r = \frac{\Delta}{s} = 4R \sin(A/2)\sin(B/2)\sin(C/2) = (s-a)\tan(A/2)$. - **Exradii ($r_a, r_b, r_c$):** - $r_a = \frac{\Delta}{s-a} = 4R \sin(A/2)\cos(B/2)\cos(C/2)$. - **Napier's Analogy (Tangent Rule):** - $\tan\left(\frac{B-C}{2}\right) = \frac{b-c}{b+c}\cot\left(\frac{A}{2}\right)$. - **Half Angle Formulas:** - $\sin(A/2) = \sqrt{\frac{(s-b)(s-c)}{bc}}$. - $\cos(A/2) = \sqrt{\frac{s(s-a)}{bc}}$. - $\tan(A/2) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} = \frac{r}{s-a}$. ### Straight Lines - **Distance Formula:** $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. - **Section Formula:** - **Internal Division:** $(x,y) = \left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$. - **External Division:** $(x,y) = \left(\frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}\right)$. - **Midpoint:** $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. - **Area of Triangle:** $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$. - **Slope of a Line ($m$):** $m = \tan\theta = \frac{y_2-y_1}{x_2-x_1}$. - **Equation of a Line:** - **Slope-intercept form:** $y = mx + c$. - **Point-slope form:** $y - y_1 = m(x - x_1)$. - **Two-point form:** $y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x - x_1)$. - **Intercept form:** $\frac{x}{a} + \frac{y}{b} = 1$. - **Normal form:** $x\cos\alpha + y\sin\alpha = p$. - **General form:** $Ax + By + C = 0$. - **Angle between two lines:** $\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|$. - Parallel lines: $m_1 = m_2$. - Perpendicular lines: $m_1m_2 = -1$. - **Distance of a point from a line:** $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$. - **Distance between parallel lines:** $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$, for $Ax+By+C_1=0$ and $Ax+By+C_2=0$. - **Equation of angle bisectors:** $\frac{Ax+By+C}{\sqrt{A^2+B^2}} = \pm \frac{A'x+B'y+C'}{\sqrt{A'^2+B'^2}}$. ### Circles - **Standard Equation:** $(x-h)^2 + (y-k)^2 = r^2$, center $(h,k)$, radius $r$. - **General Equation:** $x^2 + y^2 + 2gx + 2fy + c = 0$. - Center: $(-g, -f)$. - Radius: $\sqrt{g^2 + f^2 - c}$. - **Equation of Tangent:** - At $(x_1, y_1)$ to $x^2+y^2=r^2$: $xx_1 + yy_1 = r^2$. - At $(x_1, y_1)$ to $x^2+y^2+2gx+2fy+c=0$: $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$. - With slope $m$ to $x^2+y^2=r^2$: $y = mx \pm r\sqrt{1+m^2}$. - **Length of Tangent from $(x_1, y_1)$ to $x^2+y^2+2gx+2fy+c=0$:** $\sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$. - **Power of a point:** $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$. - **Radical Axis of two circles $S_1=0$ and $S_2=0$:** $S_1 - S_2 = 0$. - **Condition for orthogonality of two circles:** $2g_1g_2 + 2f_1f_2 = c_1 + c_2$. ### Conic Sections #### Parabola - **Standard Equation:** $y^2 = 4ax$. - Vertex: $(0,0)$. Focus: $(a,0)$. Directrix: $x = -a$. Axis: $y=0$. - **Parametric Form:** $(at^2, 2at)$. - **Equation of Tangent:** - At $(x_1, y_1)$: $yy_1 = 2a(x+x_1)$. - With slope $m$: $y = mx + \frac{a}{m}$. - At parameter $t$: $yt = x + at^2$. #### Ellipse - **Standard Equation:** $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a>b$). - Center: $(0,0)$. Vertices: $(\pm a, 0)$. Co-vertices: $(0, \pm b)$. - Foci: $(\pm ae, 0)$, where $e$ is eccentricity. - Directrices: $x = \pm \frac{a}{e}$. - **Eccentricity ($e$):** $b^2 = a^2(1-e^2)$. $e 1$. - Length of Latus Rectum: $\frac{2b^2}{a}$. - Asymptotes: $y = \pm \frac{b}{a}x$. - **Parametric Form:** $(a\sec\theta, b\tan\theta)$. - **Equation of Tangent:** - At $(x_1, y_1)$: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$. - With slope $m$: $y = mx \pm \sqrt{a^2m^2 - b^2}$. - At parameter $\theta$: $\frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1$. - **Conjugate Hyperbola:** $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$. - **Rectangular Hyperbola:** $xy = c^2$. Asymptotes are coordinate axes. ### Vectors and 3D Geometry #### Vectors - **Scalar Product (Dot Product):** $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_xb_x + a_yb_y + a_zb_z$. - $\vec{a} \cdot \vec{b} = 0 \implies \vec{a} \perp \vec{b}$ (if $\vec{a}, \vec{b} \neq \vec{0}$). - **Vector Product (Cross Product):** $\vec{a} \times \vec{b} = (|\vec{a}||\vec{b}|\sin\theta)\hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$. - $|\vec{a} \times \vec{b}|$ = Area of parallelogram. - $\vec{a} \times \vec{b} = \vec{0} \implies \vec{a} || \vec{b}$ (if $\vec{a}, \vec{b} \neq \vec{0}$). - **Scalar Triple Product:** $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}$. - Volume of parallelepiped. - If $[\vec{a} \vec{b} \vec{c}] = 0$, vectors are coplanar. - **Vector Triple Product:** $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$. #### 3D Geometry - **Distance between two points:** $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. - **Direction Cosines (l, m, n):** $\cos\alpha, \cos\beta, \cos\gamma$. $l^2+m^2+n^2=1$. - **Direction Ratios (a, b, c):** Proportional to direction cosines. $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$. - **Equation of a Line:** - **Vector Form:** $\vec{r} = \vec{a} + \lambda\vec{b}$. - **Cartesian Form:** $\frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n}$. - **Angle between two lines:** $\cos\theta = |l_1l_2 + m_1m_2 + n_1n_2|$. - **Shortest distance between two skew lines:** - $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$. - $d = \left|\frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|}\right|$. - **Equation of a Plane:** - **Normal Form:** $\vec{r} \cdot \hat{n} = d$. - **Cartesian Form:** $Ax+By+Cz+D=0$. - **Through three non-collinear points:** $\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$. - **Distance of a point from a plane:** $d = \frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$. - **Angle between two planes:** $\cos\theta = \left|\frac{A_1A_2+B_1B_2+C_1C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right|$. - **Angle between a line and a plane:** $\sin\theta = \left|\frac{Al+Bm+Cn}{\sqrt{A^2+B^2+C^2}\sqrt{l^2+m^2+n^2}}\right|$. ### Functions - **Domain:** Set of all possible input values. - **Range:** Set of all possible output values. - **Types of Functions:** - **One-one (Injective):** $f(x_1) = f(x_2) \implies x_1 = x_2$. - **Onto (Surjective):** Range = Codomain. - **Bijective:** Both one-one and onto. - **Composite Function:** $(f \circ g)(x) = f(g(x))$. - **Inverse Function:** $f^{-1}(y) = x \iff f(x) = y$. - $(f^{-1} \circ f)(x) = x$ and $(f \circ f^{-1})(y) = y$. - The graph of $f^{-1}(x)$ is the reflection of $f(x)$ across $y=x$. - **Even Function:** $f(-x) = f(x)$. Graph symmetric about y-axis. - **Odd Function:** $f(-x) = -f(x)$. Graph symmetric about origin. - **Periodic Function:** $f(x+T) = f(x)$ for some $T>0$. Smallest such $T$ is period. ### Limits, Continuity, and Differentiability #### Limits - **Definition:** $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that $0 ### Differentiation - **Basic Derivatives:** - $\frac{d}{dx}(c) = 0$. - $\frac{d}{dx}(x^n) = nx^{n-1}$. - $\frac{d}{dx}(\sin x) = \cos x$. - $\frac{d}{dx}(\cos x) = -\sin x$. - $\frac{d}{dx}(\tan x) = \sec^2 x$. - $\frac{d}{dx}(\cot x) = -\csc^2 x$. - $\frac{d}{dx}(\sec x) = \sec x \tan x$. - $\frac{d}{dx}(\csc x) = -\csc x \cot x$. - $\frac{d}{dx}(e^x) = e^x$. - $\frac{d}{dx}(a^x) = a^x \ln a$. - $\frac{d}{dx}(\ln x) = \frac{1}{x}$. - $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$. - $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$. - $\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$. - $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$. - **Rules of Differentiation:** - **Sum/Difference Rule:** $\frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}$. - **Product Rule:** $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$. - **Quotient Rule:** $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$. - **Chain Rule:** $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$. - **Implicit Differentiation:** Differentiate both sides with respect to $x$, treating $y$ as a function of $x$. - **Parametric Differentiation:** If $x=f(t), y=g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. ### Applications of Derivatives - **Rate of Change:** $\frac{dy}{dx}$ is the rate of change of $y$ with respect to $x$. - **Tangents and Normals:** - Slope of tangent at $(x_1, y_1)$ is $\left(\frac{dy}{dx}\right)_{(x_1, y_1)}$. - Equation of tangent: $y - y_1 = \left(\frac{dy}{dx}\right)_{(x_1, y_1)}(x - x_1)$. - Slope of normal: $-\frac{1}{(dy/dx)_{(x_1, y_1)}}$. - Equation of normal: $y - y_1 = -\frac{1}{(dy/dx)_{(x_1, y_1)}}(x - x_1)$. - **Monotonicity:** - $f'(x) > 0$: strictly increasing. - $f'(x) 0 \implies$ local minimum at $x=c$. - $f''(c) ### Integration #### Indefinite Integrals - **Basic Formulas:** - $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, ($n \neq -1$). - $\int \frac{1}{x} dx = \ln|x| + C$. - $\int \sin x dx = -\cos x + C$. - $\int \cos x dx = \sin x + C$. - $\int \sec^2 x dx = \tan x + C$. - $\int \csc^2 x dx = -\cot x + C$. - $\int \sec x \tan x dx = \sec x + C$. - $\int \csc x \cot x dx = -\csc x + C$. - $\int e^x dx = e^x + C$. - $\int a^x dx = \frac{a^x}{\ln a} + C$. - $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$. - $\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$. - $\int \frac{1}{x\sqrt{x^2-a^2}} dx = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C$. - $\int \frac{1}{x^2-a^2} dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$. - $\int \frac{1}{a^2-x^2} dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$. - $\int \frac{1}{\sqrt{x^2 \pm a^2}} dx = \ln|x + \sqrt{x^2 \pm a^2}| + C$. - $\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$. - $\int \sqrt{x^2 \pm a^2} dx = \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\ln|x + \sqrt{x^2 \pm a^2}| + C$. - **Integration by Substitution:** $\int f(g(x))g'(x) dx = \int f(t) dt$ (where $t=g(x)$). - **Integration by Parts:** $\int u dv = uv - \int v du$. (Choose $u$ using LIATE rule). #### Definite Integrals - **Fundamental Theorem of Calculus:** $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$. - **Properties of Definite Integrals:** - $\int_a^b f(x) dx = -\int_b^a f(x) dx$. - $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$. - $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. - $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$. - $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f(x)$ is even. - $\int_{-a}^a f(x) dx = 0$ if $f(x)$ is odd. - $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ if $f(2a-x) = f(x)$. - $\int_0^{2a} f(x) dx = 0$ if $f(2a-x) = -f(x)$. ### Differential Equations - **Order:** Order of the highest derivative. - **Degree:** Power of the highest order derivative (after making it polynomial in derivatives). - **Formation of DE:** Eliminate arbitrary constants. - **Solution Methods:** - **Variable Separable:** $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x) dx$. - **Homogeneous DE:** $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$. Substitute $y=vx \implies v+x\frac{dv}{dx} = f(v)$. - **Linear DE:** $\frac{dy}{dx} + P(x)y = Q(x)$. - **Integrating Factor (IF):** $e^{\int P(x) dx}$. - **Solution:** $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) dx + C$. - **Exact DE:** $M(x,y)dx + N(x,y)dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. - Solution: $\int M dx$ (treating $y$ as constant) $+ \int N' dy$ (terms of $N$ not containing $x$) $= C$. ### Area Under Curve - **Area between curve $y=f(x)$, x-axis, and lines $x=a, x=b$:** $A = \int_a^b |f(x)| dx$. - **Area between curve $x=g(y)$, y-axis, and lines $y=c, y=d$:** $A = \int_c^d |g(y)| dy$. - **Area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$:** $A = \int_a^b |f(x)-g(x)| dx$. ### Statistics - **Mean:** $\bar{x} = \frac{\sum x_i}{n}$ (for ungrouped data). $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ (for grouped data). - **Median:** Middle value when data is arranged in order. - **Mode:** Most frequent value. - **Variance ($\sigma^2$):** $\frac{\sum (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - \bar{x}^2$. - **Standard Deviation ($\sigma$):** $\sqrt{\text{Variance}}$. - **Mean Deviation:** - About Mean: $\frac{\sum |x_i - \bar{x}|}{n}$. - About Median: $\frac{\sum |x_i - \text{Median}|}{n}$. - **Coefficient of Variation (CV):** $\frac{\sigma}{\bar{x}} \times 100$. ### Probability - **Classical Probability:** $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. - **Addition Theorem:** $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. - For mutually exclusive events: $P(A \cup B) = P(A) + P(B)$. - **Conditional Probability:** $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(B) \neq 0$. - **Multiplication Theorem:** $P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)$. - For independent events: $P(A \cap B) = P(A)P(B)$. - **Bayes' Theorem:** $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{j=1}^n P(A|E_j)P(E_j)}$. - **Binomial Distribution:** $P(X=r) = {}^nC_r p^r q^{n-r}$, where $p+q=1$. - Mean $= np$. Variance $= npq$. ### Mathematical Reasoning - **Statements:** Declarative sentences that are either true or false. - **Connectives:** - **And ($\land$):** Conjunction. True only if both statements are true. - **Or ($\lor$):** Disjunction. False only if both statements are false. - **Not ($\neg$):** Negation. - **If...then... ($\implies$):** Implication. False only if antecedent is true AND consequent is false. - **If and only if ($\iff$):** Biconditional. True if both statements have same truth value. - **Truth Tables:** Used to determine truth values of compound statements. - **Tautology:** A statement that is always true. - **Contradiction:** A statement that is always false. - **Contingency:** A statement that is neither a tautology nor a contradiction. - **Quantifiers:** - **Universal Quantifier ($\forall$):** "For all", "For every". - **Existential Quantifier ($\exists$):** "There exists", "For some". - **Contrapositive:** For $p \implies q$, the contrapositive is $\neg q \implies \neg p$. (Logically equivalent). - **Converse:** For $p \implies q$, the converse is $q \implies p$. - **Inverse:** For $p \implies q$, the inverse is $\neg p \implies \neg q$.