JEE Mains Maths Formulae

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### Quadratic Equations For a quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$: - **Roots:** $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - **Discriminant ($\Delta$ or $D$):** $\Delta = b^2 - 4ac$ - If $\Delta > 0$, roots are real and distinct. - If $\Delta = 0$, roots are real and equal. - If $\Delta ### Complex Numbers - **Form:** $z = x + iy$, where $i = \sqrt{-1}$ - **Conjugate:** $\bar{z} = x - iy$ - **Modulus:** $|z| = \sqrt{x^2 + y^2}$ - **Argument (Amplitude):** $\theta = \arg(z)$, where $\cos\theta = \frac{x}{|z|}$ and $\sin\theta = \frac{y}{|z|}$ - **Polar Form:** $z = r(\cos\theta + i\sin\theta)$ - **Euler's Form:** $z = re^{i\theta}$ - **De Moivre's Theorem:** $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ - **Cube Roots of Unity:** $1, \omega, \omega^2$ - $1 + \omega + \omega^2 = 0$ - $\omega^3 = 1$ ### Sequences & Series #### Arithmetic Progression (AP) - **General term:** $a_n = a + (n-1)d$ - **Sum of n terms:** $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + a_n)$ #### Geometric Progression (GP) - **General term:** $a_n = ar^{n-1}$ - **Sum of n terms:** $S_n = \frac{a(r^n - 1)}{r-1}$ if $r \neq 1$ - **Sum of infinite GP:** $S_\infty = \frac{a}{1-r}$ if $|r| ### Permutations & Combinations - **Factorial:** $n! = n \times (n-1) \times ... \times 2 \times 1$ - **Permutations ($P(n,r)$ or $^nP_r$):** Number of ways to arrange $r$ items from $n$ distinct items. $^nP_r = \frac{n!}{(n-r)!}$ - **Combinations ($C(n,r)$ or $^nC_r$):** Number of ways to select $r$ items from $n$ distinct items. $^nC_r = \frac{n!}{r!(n-r)!}$ - **Properties:** - $^nC_r = ^nC_{n-r}$ - $^nC_r + ^nC_{r-1} = ^{n+1}C_r$ ### Binomial Theorem - **For positive integer n:** $(x+y)^n = \sum_{r=0}^{n} ^nC_r x^{n-r} y^r$ - **General term ($T_{r+1}$):** $^nC_r x^{n-r} y^r$ - **Middle term(s):** - If $n$ is even, middle term is $(\frac{n}{2} + 1)^{th}$ term. - If $n$ is odd, middle terms are $(\frac{n+1}{2})^{th}$ and $(\frac{n+3}{2})^{th}$ terms. - **Binomial Theorem for any index:** $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ...$ (valid for $|x| ### Matrices & Determinants #### Matrices - **Addition/Subtraction:** Element-wise - **Multiplication:** $(AB)_{ij} = \sum_k A_{ik}B_{kj}$ - **Transpose ($A^T$):** $(A^T)_{ij} = A_{ji}$ - **Symmetric Matrix:** $A^T = A$ - **Skew-Symmetric Matrix:** $A^T = -A$ - **Idempotent Matrix:** $A^2 = A$ - **Involutory Matrix:** $A^2 = I$ - **Nilpotent Matrix:** $A^k = 0$ for some positive integer $k$ - **Orthogonal Matrix:** $AA^T = A^TA = I$ #### Determinants - **Determinant of a $2 \times 2$ matrix:** If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then $\det(A) = ad - bc$. - **Properties:** - $\det(AB) = \det(A)\det(B)$ - $\det(A^T) = \det(A)$ - $\det(kA) = k^n \det(A)$ for an $n \times n$ matrix $A$ - **Adjoint of a matrix:** $\text{adj}(A) = (C_{ij})^T$, where $C_{ij}$ is the cofactor of $a_{ij}$. - **Inverse of a matrix:** $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$ (if $\det(A) \neq 0$) - **Cramer's Rule** (for system $Ax=B$): $x_i = \frac{\det(A_i)}{\det(A)}$, where $A_i$ is matrix $A$ with $i^{th}$ column replaced by $B$. ### Vector Algebra - **Position Vector:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - **Magnitude:** $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$ - **Dot Product:** $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_xb_x + a_yb_y + a_zb_z$ - If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a} \perp \vec{b}$ - **Cross Product:** $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$ - Area of parallelogram = $|\vec{a} \times \vec{b}|$ - Area of triangle = $\frac{1}{2}|\vec{a} \times \vec{b}|$ - **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 = $|[\vec{a} \vec{b} \vec{c}]|$ - 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 #### Direction Cosines & Ratios - **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. - $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$, etc. #### Straight Line - **Equation of line through $(x_1, y_1, z_1)$ with direction ratios $(a,b,c)$:** $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} = r$ - **Angle between two lines:** $\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ - **Shortest distance between two skew lines:** $SD = \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$ (lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$) #### Plane - **Equation of 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 $(x_1, y_1, z_1)$ from plane $Ax+By+Cz+D=0$:** $d = \frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$ - **Angle between two planes:** $\cos\theta = \frac{A_1A_2+B_1B_2+C_1C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}$ - **Angle between a line and a plane:** $\sin\theta = \frac{Aa+Bb+Cc}{\sqrt{A^2+B^2+C^2}\sqrt{a^2+b^2+c^2}}$ ### Differentiation - **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:** $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ #### Standard Derivatives - $\frac{d}{dx}(x^n) = nx^{n-1}$ - $\frac{d}{dx}(\sin x) = \cos x$ - $\frac{d}{dx}(\cos x) = -\sin x$ - $\frac{d}{dx}(\tan x) = \sec^2 x$ - $\frac{d}{dx}(\cot x) = -\csc^2 x$ - $\frac{d}{dx}(\sec x) = \sec x \tan x$ - $\frac{d}{dx}(\csc x) = -\csc x \cot x$ - $\frac{d}{dx}(e^x) = e^x$ - $\frac{d}{dx}(a^x) = a^x \ln a$ - $\frac{d}{dx}(\ln x) = \frac{1}{x}$ - $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$ - $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$ - $\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$ - $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$ ### Applications of Derivatives - **Rate of Change:** $\frac{dy}{dx}$ - **Tangents and Normals:** - **Slope of tangent:** $m_T = \frac{dy}{dx}|_{(x_1, y_1)}$ - **Equation of tangent:** $y - y_1 = m_T(x - x_1)$ - **Slope of normal:** $m_N = -\frac{1}{m_T}$ - **Equation of normal:** $y - y_1 = m_N(x - x_1)$ - **Increasing/Decreasing Functions:** - $f'(x) > 0 \implies f(x)$ is increasing - $f'(x) 0$, local minimum. - If $f'(c) = 0$ and $f''(c) = 0$, test fails, use first derivative test. - **Rolle's Theorem:** If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c \in (a,b)$ such that $f'(c)=0$. - **Lagrange's Mean Value Theorem:** If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$. ### Integration #### Indefinite Integrals - $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$) - $\int \frac{1}{x} dx = \ln|x| + C$ - $\int \sin x dx = -\cos x + C$ - $\int \cos x dx = \sin x + C$ - $\int \sec^2 x dx = \tan x + C$ - $\int \csc^2 x dx = -\cot x + C$ - $\int \sec x \tan x dx = \sec x + C$ - $\int \csc x \cot x dx = -\csc x + C$ - $\int e^x dx = e^x + C$ - $\int a^x dx = \frac{a^x}{\ln a} + C$ - $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{1}{x\sqrt{x^2-a^2}} dx = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{1}{x^2-a^2} dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$ - $\int \frac{1}{a^2-x^2} dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$ - $\int \frac{1}{\sqrt{x^2 \pm a^2}} dx = \ln|x + \sqrt{x^2 \pm a^2}| + C$ - **Integration by Parts:** $\int u dv = uv - \int v du$ #### Definite Integrals - **Newton-Leibniz Formula:** $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$ - **Properties:** - $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ - $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ - $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f(x)$ is even - $\int_{-a}^a f(x) dx = 0$ if $f(x)$ is odd ### Differential Equations - **Order:** Highest order derivative. - **Degree:** Power of highest order derivative (after making it polynomial in derivatives). #### Methods of Solving - **Variable Separable:** $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x) dx$ - **Homogeneous:** $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$, substitute $y=vx \implies \frac{dy}{dx} = v + x\frac{dv}{dx}$ - **Linear Differential Equation:** $\frac{dy}{dx} + Py = Q$, where $P, Q$ are functions of $x$. - **Integrating Factor (IF):** $e^{\int P dx}$ - **Solution:** $y \cdot (IF) = \int Q \cdot (IF) dx + C$ - **Exact Differential Equation:** $M dx + N dy = 0$ if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ ### Coordinate Geometry #### Straight Line - **Distance between $(x_1, y_1)$ and $(x_2, y_2)$:** $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ - **Section Formula:** - **Internal division:** $\left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)$ - **External division:** $\left(\frac{m x_2 - n x_1}{m-n}, \frac{m y_2 - n y_1}{m-n}\right)$ - **Slope of a line:** $m = \tan\theta = \frac{y_2-y_1}{x_2-x_1}$ - **Equation of a line:** - **Point-slope form:** $y - y_1 = m(x - x_1)$ - **Slope-intercept form:** $y = mx + c$ - **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:** $x\cos\alpha + y\sin\alpha = p$ - **Angle between two lines:** $\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|$ - **Distance of a point $(x_1, y_1)$ from line $Ax+By+C=0$:** $\frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$ #### Circle - **Equation of circle with center $(h,k)$ and radius $r$:** $(x-h)^2 + (y-k)^2 = r^2$ - **General equation:** $x^2 + y^2 + 2gx + 2fy + c = 0$ - Center: $(-g, -f)$, Radius: $\sqrt{g^2+f^2-c}$ - **Condition for tangency of line $y=mx+c$ to circle $x^2+y^2=a^2$:** $c^2 = a^2(1+m^2)$ #### Parabola - **Standard equation:** $y^2 = 4ax$ - Vertex: $(0,0)$, Focus: $(a,0)$, Directrix: $x=-a$ - **Parametric form:** $(at^2, 2at)$ #### Ellipse - **Standard equation:** $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a>b$) - Center: $(0,0)$, Foci: $(\pm ae, 0)$, Vertices: $(\pm a, 0)$ - Eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$ - Latus Rectum: $\frac{2b^2}{a}$ #### Hyperbola - **Standard equation:** $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ - Center: $(0,0)$, Foci: $(\pm ae, 0)$, Vertices: $(\pm a, 0)$ - Eccentricity: $e = \sqrt{1 + \frac{b^2}{a^2}}$ - Latus Rectum: $\frac{2b^2}{a}$ ### Trigonometry #### Basic Identities - $\sin^2\theta + \cos^2\theta = 1$ - $\sec^2\theta - \tan^2\theta = 1$ - $\csc^2\theta - \cot^2\theta = 1$ #### Sum/Difference Formulas - $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ - $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ - $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ #### Double Angle Formulas - $\sin 2A = 2 \sin A \cos A = \frac{2\tan A}{1+\tan^2 A}$ - $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1-\tan^2 A}{1+\tan^2 A}$ - $\tan 2A = \frac{2\tan A}{1-\tan^2 A}$ #### Half Angle Formulas - $\sin A = 2 \sin(A/2) \cos(A/2)$ - $1 - \cos A = 2\sin^2(A/2)$ - $1 + \cos A = 2\cos^2(A/2)$ - $\tan(A/2) = \sqrt{\frac{1-\cos A}{1+\cos A}}$ #### Product to Sum Formulas - $2\sin A \cos B = \sin(A+B) + \sin(A-B)$ - $2\cos A \sin B = \sin(A+B) - \sin(A-B)$ - $2\cos A \cos B = \cos(A+B) + \cos(A-B)$ - $2\sin A \sin B = \cos(A-B) - \cos(A+B)$ #### Inverse Trigonometric Functions - $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$ - $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$ - $\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2}$ - $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$ (if $xy -1$) ### Statistics - **Mean (for ungrouped data):** $\bar{x} = \frac{\sum x_i}{n}$ - **Mean (for grouped data):** $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ - **Variance ($\sigma^2$):** $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - (\bar{x})^2$ - **Standard Deviation ($\sigma$):** $\sigma = \sqrt{\text{Variance}}$ - **Coefficient of Variation (CV):** $\frac{\sigma}{\bar{x}} \times 100\%$ ### Probability - **Probability of an event E:** $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$ - **Addition Rule:** $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ - **Conditional Probability:** $P(A|B) = \frac{P(A \cap B)}{P(B)}$ - **Multiplication Rule:** $P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)$ - **Independent Events:** $P(A \cap B) = P(A)P(B)$ - **Bayes' Theorem:** $P(B_i|A) = \frac{P(A|B_i)P(B_i)}{\sum_{j=1}^n P(A|B_j)P(B_j)}$ - **Binomial Distribution:** $P(X=k) = ^nC_k p^k q^{n-k}$, where $p+q=1$ - Mean: $np$, Variance: $npq$ ### Mathematical Reasoning - **Connectives:** - **AND ($\land$):** True if both statements are true. - **OR ($\lor$):** True if at least one statement is true. - **NOT ($\neg$):** Reverses truth value. - **Implication ($p \to q$):** "If $p$ then $q$". False only if $p$ is true and $q$ is false. - **Biconditional ($p \leftrightarrow q$):** "p if and only if q". True if $p$ and $q$ have same truth value. - **Contrapositive:** The contrapositive of $p \to q$ is $\neg q \to \neg p$. (Logically equivalent) - **Converse:** The converse of $p \to q$ is $q \to p$. - **Inverse:** The inverse of $p \to q$ is $\neg p \to \neg q$. - **Tautology:** A statement always true. - **Contradiction:** A statement always false. ### Limits, Continuity & Differentiability #### Limits - **Limit definition:** $\lim_{x \to a} f(x) = L$ - **L'Hopital's Rule:** If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ #### Standard Limits - $\lim_{x \to 0} \frac{\sin x}{x} = 1$ - $\lim_{x \to 0} \frac{\tan x}{x} = 1$ - $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ - $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$ - $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$ - $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$ - $\lim_{x \to 0} (1+x)^{1/x} = e$ - $\lim_{x \to a} \frac{x^n - a^n}{x-a} = na^{n-1}$ #### Continuity - A function $f(x)$ is continuous at $x=a$ if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$. #### Differentiability - A function $f(x)$ is differentiable at $x=a$ if $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists. - Differentiability implies continuity, but not vice-versa.