JEE Maths Formulas (Class 11-1

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### Sets, Relations & Functions - **Sets:** - Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$ - Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$ - Difference: $A - B = \{x : x \in A \text{ and } x \notin B\}$ - Complement: $A' = U - A$ - De Morgan's Laws: $(A \cup B)' = A' \cap B'$, $(A \cap B)' = A' \cup B'$ - **Relations:** - Reflexive: $(a,a) \in R$ - Symmetric: $(a,b) \in R \implies (b,a) \in R$ - Transitive: $(a,b) \in R \text{ and } (b,c) \in R \implies (a,c) \in R$ - Equivalence Relation: Reflexive, Symmetric, Transitive - **Functions:** - Domain, Codomain, Range - One-to-one (Injective), Onto (Surjective), Bijective - Composite Function: $(g \circ f)(x) = g(f(x))$ - Inverse Function: $f^{-1}(y) = x \iff f(x) = y$ ### Trigonometric Functions - **Basic Identities:** - $\sin^2\theta + \cos^2\theta = 1$ - $1 + \tan^2\theta = \sec^2\theta$ - $1 + \cot^2\theta = \csc^2\theta$ - **Compound Angles:** - $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ - $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ - $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ - **Double/Half Angle Formulas:** - $\sin 2A = 2 \sin A \cos A = \frac{2 \tan A}{1 + \tan^2 A}$ - $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1 - \tan^2 A}{1 + \tan^2 A}$ - $\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$ - **Transformation Formulas:** - $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$ - $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$ - $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$ - $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$ - $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$ - $\cos C + \cos D = 2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}$ - **General Solutions:** - $\sin x = \sin y \implies x = n\pi + (-1)^n y$ - $\cos x = \cos y \implies x = 2n\pi \pm y$ - $\tan x = \tan y \implies x = n\pi + y$ ### Complex Numbers - **Form:** $z = x + iy$, where $i = \sqrt{-1}$ - **Conjugate:** $\bar{z} = x - iy$ - **Modulus:** $|z| = \sqrt{x^2 + y^2}$ - **Polar Form:** $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$, $\theta = \arg(z)$ - **Euler's Form:** $z = re^{i\theta}$ - **De Moivre's Theorem:** $( \cos\theta + i\sin\theta )^n = \cos(n\theta) + i\sin(n\theta)$ - **Roots of Unity:** $x^n = 1 \implies x = e^{i \frac{2k\pi}{n}}$, where $k = 0, 1, ..., n-1$ ### Quadratic Equations - **Standard Form:** $ax^2 + bx + c = 0$ - **Discriminant:** $\Delta = b^2 - 4ac$ - **Roots:** $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ - **Nature of Roots:** - $\Delta > 0$: Real & distinct - $\Delta = 0$: Real & equal - $\Delta ### Sequences & Series - **Arithmetic Progression (AP):** - $a_n = a + (n-1)d$ - $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + l)$ - **Geometric Progression (GP):** - $a_n = ar^{n-1}$ - $S_n = \frac{a(r^n - 1)}{r-1}$ (for $r \ne 1$) - $S_\infty = \frac{a}{1-r}$ (for $|r| ### Permutations & Combinations - **Factorial:** $n! = n \times (n-1) \times ... \times 1$ - **Permutations:** $P(n,r) = \frac{n!}{(n-r)!}$ (arrangement) - **Combinations:** $C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ (selection) - **Properties:** - $C(n,r) = C(n, n-r)$ - $C(n,r) + C(n, r-1) = C(n+1, r)$ ### Binomial Theorem - **Expansion:** $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$ - **General Term:** $T_{r+1} = \binom{n}{r} x^{n-r} y^r$ - **Middle Term(s):** - If n is even, $(n/2 + 1)^{th}$ term - If n is odd, $((n+1)/2)^{th}$ and $((n+3)/2)^{th}$ terms ### Straight Lines - **Distance Formula:** $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ - **Section Formula:** $(x,y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)$ - **Slope:** $m = \tan\theta = \frac{y_2-y_1}{x_2-x_1}$ - **Equation of Line:** - Slope-intercept: $y = mx + c$ - Point-slope: $y - y_1 = m(x - x_1)$ - Two-point: $y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x - x_1)$ - Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ - Normal form: $x \cos\alpha + y \sin\alpha = p$ - **Angle between two lines:** $\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$ - **Parallel lines:** $m_1 = m_2$ - **Perpendicular lines:** $m_1 m_2 = -1$ - **Distance of a point from a line:** $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ - **Distance between parallel lines:** $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$ ### Conic Sections #### Circle - **Equation:** $(x-h)^2 + (y-k)^2 = r^2$ - **General Equation:** $x^2 + y^2 + 2gx + 2fy + c = 0$ #### Parabola - **Standard Equation:** $y^2 = 4ax$ - **Focus:** $(a,0)$ - **Directrix:** $x = -a$ - **Latus Rectum:** $4a$ #### Ellipse - **Standard Equation:** $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (for $a>b$) - **Foci:** $(\pm ae, 0)$ - **Directrices:** $x = \pm a/e$ - **Eccentricity:** $e = \sqrt{1 - b^2/a^2}$ - **Latus Rectum:** $2b^2/a$ #### Hyperbola - **Standard Equation:** $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ - **Foci:** $(\pm ae, 0)$ - **Directrices:** $x = \pm a/e$ - **Eccentricity:** $e = \sqrt{1 + b^2/a^2}$ - **Latus Rectum:** $2b^2/a$ ### 3D Geometry - **Distance Formula:** $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ - **Direction Cosines:** $l = \cos\alpha, m = \cos\beta, n = \cos\gamma$; $l^2+m^2+n^2=1$ - **Direction Ratios:** $a, b, c$ (proportional to $l, m, n$) - **Equation of Line:** - Vector form: $\vec{r} = \vec{a} + \lambda\vec{b}$ - Cartesian form: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ - **Equation of Plane:** - Normal form: $\vec{r} \cdot \hat{n} = d$ - Cartesian form: $Ax+By+Cz+D=0$ - Intercept form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ - **Angle between two lines:** $\cos\theta = |\frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}|$ - **Angle between two planes:** $\cos\theta = |\frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}||\vec{n_2}|}|$ - **Angle between line and plane:** $\sin\theta = |\frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}|$ ### Limits, Continuity & Differentiability - **Limits:** - $\lim_{x \to a} f(x) = L$ - L'Hopital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of 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: - $\lim_{x \to 0} \frac{\sin x}{x} = 1$ - $\lim_{x \to 0} \frac{\tan x}{x} = 1$ - $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ - $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$ - $\lim_{x \to 0} (1+x)^{1/x} = e$ - $\lim_{x \to \infty} (1+1/x)^x = e$ - **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. ### Differentiation - **Basic Rules:** - $\frac{d}{dx}(c) = 0$ - $\frac{d}{dx}(x^n) = nx^{n-1}$ - $\frac{d}{dx}(e^x) = e^x$ - $\frac{d}{dx}(\ln x) = 1/x$ - $\frac{d}{dx}(\sin x) = \cos x$ - $\frac{d}{dx}(\cos x) = -\sin x$ - $\frac{d}{dx}(\tan x) = \sec^2 x$ - **Product Rule:** $(uv)' = u'v + uv'$ - **Quotient Rule:** $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$ - **Chain Rule:** $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ ### Applications of Derivatives - **Rate of Change:** $\frac{dy}{dx}$ - **Increasing/Decreasing Functions:** - $f'(x) > 0$: Increasing - $f'(x) 0$ (Minima), $f''(c) ### Integrals - **Indefinite Integrals:** - $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \ne -1$) - $\int \frac{1}{x} dx = \ln|x| + C$ - $\int e^x dx = e^x + C$ - $\int \sin x dx = -\cos x + C$ - $\int \cos x dx = \sin x + C$ - $\int \sec^2 x dx = \tan x + C$ - $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}(\frac{x}{a}) + C$ - $\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}(\frac{x}{a}) + C$ - **Integration by Parts:** $\int u dv = uv - \int v du$ - **Definite Integrals:** $\int_a^b f(x) dx = F(b) - F(a)$ - **Properties of Definite Integrals:** - $\int_a^b f(x) dx = \int_a^b f(t) dt$ - $\int_a^b f(x) dx = -\int_b^a f(x) dx$ - $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ - $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ - $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ - $\int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a-x) dx$ - $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f(x)$ is even - $\int_{-a}^a f(x) dx = 0$ if $f(x)$ is odd ### Differential Equations - **Order:** Highest order derivative present - **Degree:** Power of highest order derivative (after removing radicals) - **Methods of Solving:** - Variable Separable: $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x) dx$ - Homogeneous: $\frac{dy}{dx} = f(\frac{y}{x})$, substitute $y=vx$ - Linear: $\frac{dy}{dx} + Py = Q$, where P, Q are functions of $x$. - Integrating Factor (IF): $e^{\int P dx}$ - Solution: $y \cdot (IF) = \int Q \cdot (IF) dx + C$ ### Vectors - **Representation:** $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ - **Magnitude:** $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$ - **Dot Product:** $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$ - **Cross Product:** $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$ - **Scalar Triple Product:** $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$ (volume of parallelepiped) - **Vector Triple Product:** $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$ ### Probability - **Basic Formula:** $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$ - **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:** $P(A \cap B) = P(A)P(B)$ - **Total Probability Theorem:** $P(A) = \sum_{i=1}^n P(A|E_i)P(E_i)$ - **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:** $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ - Mean: $np$ - Variance: $np(1-p)$ ### Matrices & Determinants - **Matrix Operations:** Addition, Subtraction, Multiplication - **Determinant:** $|A|$ - **Properties of Determinants:** - $|A^T| = |A|$ - $|AB| = |A||B|$ - If two rows/columns are identical, $|A|=0$ - If a row/column is all zeros, $|A|=0$ - **Adjoint of a Matrix:** $adj(A) = (C_{ij})^T$, where $C_{ij}$ is cofactor - **Inverse of a Matrix:** $A^{-1} = \frac{1}{|A|} adj(A)$ (if $|A| \ne 0$) - **System of Linear Equations (Cramer's Rule):** - $x = D_x/D, y = D_y/D, z = D_z/D$ - $D = \det(A)$ - Consistent if $D \ne 0$ (unique solution) - If $D=0$ and $D_x=D_y=D_z=0$: Infinitely many solutions - If $D=0$ and at least one of $D_x, D_y, D_z$ is non-zero: No solution