Class 12 Maths Formula Sheet

Cheatsheet Content

### Relations and Functions - **Reflexive:** $(a,a) \in R$ for all $a \in A$ - **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 - **One-One (Injective):** $f(x_1) = f(x_2) \implies x_1 = x_2$ - **Onto (Surjective):** For every $y \in B$, there exists $x \in A$ s.t. $f(x) = y$ - **Bijective:** Both one-one and onto - **Composition of Functions:** $(g \circ f)(x) = g(f(x))$ - **Inverse Function:** $f^{-1}(y) = x \iff f(x) = y$. $(f^{-1} \circ f)(x) = x$, $(f \circ f^{-1})(y) = y$ ### Inverse Trigonometric Functions - $\sin^{-1}x + \cos^{-1}x = \pi/2$, $x \in [-1,1]$ - $\tan^{-1}x + \cot^{-1}x = \pi/2$, $x \in R$ - $\sec^{-1}x + \csc^{-1}x = \pi/2$, $|x| \ge 1$ - $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$, $xy -1$ - $2\tan^{-1}x = \tan^{-1}\left(\frac{2x}{1-x^2}\right) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ - $\sin^{-1}(-x) = -\sin^{-1}x$ - $\cos^{-1}(-x) = \pi - \cos^{-1}x$ - $\tan^{-1}(-x) = -\tan^{-1}x$ - $\cot^{-1}(-x) = \pi - \cot^{-1}x$ - $\sec^{-1}(-x) = \pi - \sec^{-1}x$ - $\csc^{-1}(-x) = -\csc^{-1}x$ ### Matrices - **Addition:** $(A+B)_{ij} = A_{ij} + B_{ij}$ - **Scalar Multiplication:** $(kA)_{ij} = k A_{ij}$ - **Matrix Multiplication:** $(AB)_{ij} = \sum_{k} A_{ik} B_{kj}$ (Rows by Columns) - **Transpose:** $(A^T)_{ij} = A_{ji}$ - $(A^T)^T = A$ - $(A+B)^T = A^T + B^T$ - $(kA)^T = kA^T$ - $(AB)^T = B^T A^T$ - **Symmetric Matrix:** $A^T = A$ - **Skew-Symmetric Matrix:** $A^T = -A$ (Diagonal elements are 0) - **Every square matrix $A$ can be expressed as:** $A = \frac{1}{2}(A+A^T) + \frac{1}{2}(A-A^T)$ - **Inverse of a Matrix:** $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$ (if $\det(A) \ne 0$) - $(AB)^{-1} = B^{-1}A^{-1}$ - $(A^T)^{-1} = (A^{-1})^T$ - **Adjoint:** $\text{adj}(A) = (C_{ij})^T$, where $C_{ij}$ is cofactor of $a_{ij}$ ### Determinants - **For a $2 \times 2$ matrix** $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad-bc$ - **Properties:** - $\det(A^T) = \det(A)$ - If two rows/columns are identical or proportional, $\det(A) = 0$ - If a row/column is all zeros, $\det(A) = 0$ - $\det(kA) = k^n \det(A)$ for an $n \times n$ matrix $A$ - $\det(AB) = \det(A)\det(B)$ - $\det(A^{-1}) = 1/\det(A)$ - $\text{adj}(A) = \det(A) A^{-1}$ - $\det(\text{adj}(A)) = (\det(A))^{n-1}$ - $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A) I$ - **Area of a triangle:** $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| = \frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|$ - **Consistent system (unique solution):** $\det(A) \ne 0$, $X = A^{-1}B$ - **Inconsistent system (no solution):** $\det(A) = 0$ and $\text{adj}(A)B \ne O$ - **Consistent system (infinitely many solutions):** $\det(A) = 0$ and $\text{adj}(A)B = O$ ### Continuity and Differentiability - **Continuity at $x=a$:** $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$ - **Differentiability at $x=a$:** - L.H.D. = $\lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$ - R.H.D. = $\lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$ - L.H.D. = R.H.D. - **Every differentiable function is continuous, but not vice-versa.** - **Derivative of $y=f(x)$ w.r.t. $x$:** $\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ - **Chain Rule:** $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ - **Implicit Differentiation:** Differentiate terms w.r.t. $x$ (treat $y$ as $f(x)$) - **Parametric Differentiation:** $x=f(t), y=g(t) \implies \frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ ### Common Derivatives - $\frac{d}{dx}(c) = 0$ - $\frac{d}{dx}(x^n) = nx^{n-1}$ - $\frac{d}{dx}(e^x) = e^x$ - $\frac{d}{dx}(a^x) = a^x \log_e a$ - $\frac{d}{dx}(\log_e x) = \frac{1}{x}$ - $\frac{d}{dx}(\log_a x) = \frac{1}{x \log_e a}$ - $\frac{d}{dx}(\sin x) = \cos x$ - $\frac{d}{dx}(\cos x) = -\sin x$ - $\frac{d}{dx}(\tan x) = \sec^2 x$ - $\frac{d}{dx}(\cot x) = -\csc^2 x$ - $\frac{d}{dx}(\sec x) = \sec x \tan x$ - $\frac{d}{dx}(\csc x) = -\csc x \cot x$ - $\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}$ - $\frac{d}{dx}(\cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}$ - $\frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}$ - $\frac{d}{dx}(\cot^{-1}x) = -\frac{1}{1+x^2}$ - $\frac{d}{dx}(\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}}$ - $\frac{d}{dx}(\csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2-1}}$ - **Product Rule:** $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ - **Quotient Rule:** $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ ### Applications of Derivatives - **Rate of Change:** $\frac{dy}{dx}$ - **Equation of Tangent:** $y - y_0 = m(x - x_0)$, where $m = \left(\frac{dy}{dx}\right)_{(x_0,y_0)}$ - **Equation of Normal:** $y - y_0 = -\frac{1}{m}(x - x_0)$ - **Increasing Function:** $f'(x) > 0$ - **Decreasing Function:** $f'(x) 0$ - **Point of Inflection:** $f''(c)=0$ and $f''(x)$ changes sign around $c$. - **Approximation:** $f(x+\Delta x) \approx f(x) + f'(x)\Delta x$ ### Integrals - **Fundamental Theorem of Calculus:** $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$ - **Properties of Definite Integrals:** - $\int_a^b f(x) dx = -\int_b^a f(x) dx$ - $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ - $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ - $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ - $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f(x)$ is even ($f(-x)=f(x)$) - $\int_{-a}^a f(x) dx = 0$ if $f(x)$ is odd ($f(-x)=-f(x)$) - $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ if $f(2a-x)=f(x)$ - $\int_0^{2a} f(x) dx = 0$ if $f(2a-x)=-f(x)$ - **Integration by Parts:** $\int u dv = uv - \int v du$ (LIATE rule for choosing $u$) ### Common Integrals - $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $n \ne -1$ - $\int \frac{1}{x} dx = \log|x| + C$ - $\int e^x dx = e^x + C$ - $\int a^x dx = \frac{a^x}{\log a} + C$ - $\int \sin x dx = -\cos x + C$ - $\int \cos x dx = \sin x + C$ - $\int \sec^2 x dx = \tan x + C$ - $\int \csc^2 x dx = -\cot x + C$ - $\int \sec x \tan x dx = \sec x + C$ - $\int \csc x \cot x dx = -\csc x + C$ - $\int \tan x dx = \log|\sec x| + C = -\log|\cos x| + C$ - $\int \cot x dx = \log|\sin x| + C$ - $\int \sec x dx = \log|\sec x + \tan x| + C = \log\left|\tan\left(\frac{\pi}{4} + \frac{x}{2}\right)\right| + C$ - $\int \csc x dx = \log|\csc x - \cot x| + C = \log\left|\tan\left(\frac{x}{2}\right)\right| + C$ - $\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{dx}{a^2+x^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{dx}{x\sqrt{x^2-a^2}} = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right| + C$ - $\int \frac{dx}{a^2-x^2} = \frac{1}{2a}\log\left|\frac{a+x}{a-x}\right| + C$ - $\int \frac{dx}{\sqrt{x^2-a^2}} = \log|x+\sqrt{x^2-a^2}| + C$ - $\int \frac{dx}{\sqrt{x^2+a^2}} = \log|x+\sqrt{x^2+a^2}| + C$ - $\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$ - $\int \sqrt{x^2-a^2} dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\log|x+\sqrt{x^2-a^2}| + C$ - $\int \sqrt{x^2+a^2} dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| + C$ - $\int e^{ax}\sin(bx) dx = \frac{e^{ax}}{a^2+b^2}(a\sin(bx) - b\cos(bx)) + C$ - $\int e^{ax}\cos(bx) dx = \frac{e^{ax}}{a^2+b^2}(a\cos(bx) + b\sin(bx)) + C$ ### Applications of Integrals - **Area under simple curves:** $\int_a^b y dx$ or $\int_c^d x dy$ - **Area between two curves:** $\int_a^b (y_2 - y_1) dx$ or $\int_c^d (x_2 - x_1) dy$ ### Differential Equations - **Order:** Highest order derivative present - **Degree:** Power of highest order derivative (after making it free from radicals/fractions) - **General Solution:** Contains arbitrary constants equal to the order - **Particular Solution:** Obtained by applying boundary conditions - **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 $y=vx \implies \frac{dy}{dx} = v + x\frac{dv}{dx}$ - **Linear DE:** $\frac{dy}{dx} + Py = Q$, where $P, Q$ are functions of $x$ or constants. - **Integrating Factor (IF):** $e^{\int P dx}$ - **Solution:** $y \cdot (\text{IF}) = \int Q \cdot (\text{IF}) dx + C$ - **Linear DE (type 2):** $\frac{dx}{dy} + P'x = Q'$, where $P', Q'$ are functions of $y$ or constants. - **Integrating Factor (IF):** $e^{\int P' dy}$ - **Solution:** $x \cdot (\text{IF}) = \int Q' \cdot (\text{IF}) dy + C$ ### Vector Algebra - **Position Vector:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - **Magnitude:** $|\vec{r}| = \sqrt{x^2+y^2+z^2}$ - **Unit Vector:** $\hat{r} = \frac{\vec{r}}{|\vec{r}|}$ - **Vector joining two points:** $\vec{PQ} = \vec{OQ} - \vec{OP} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$ - **Section Formula:** Position vector of $R$ dividing $PQ$ in ratio $m:n$: - **Internally:** $\frac{n\vec{a} + m\vec{b}}{m+n}$ - **Externally:** $\frac{n\vec{a} - m\vec{b}}{n-m}$ - **Scalar (Dot) Product:** $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$ - $\vec{a} \cdot \vec{a} = |\vec{a}|^2$ - $\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$ (for non-zero vectors) - **Projection of $\vec{a}$ on $\vec{b}$:** $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ - **Vector (Cross) Product:** $\vec{a} \times \vec{b} = (|\vec{a}||\vec{b}|\sin\theta)\hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$ - $|\vec{a} \times \vec{b}|$ = Area of parallelogram with adjacent sides $\vec{a}, \vec{b}$ - $\frac{1}{2}|\vec{a} \times \vec{b}|$ = Area of triangle with adjacent sides $\vec{a}, \vec{b}$ - $\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \parallel \vec{b}$ (for non-zero vectors) - **Scalar Triple Product:** $[\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$ - Volume of parallelepiped = $|[\vec{a}, \vec{b}, \vec{c}]|$ - Vectors are coplanar if $[\vec{a}, \vec{b}, \vec{c}] = 0$ ### Three Dimensional Geometry - **Direction Cosines (DC's):** $l = \cos\alpha, m=\cos\beta, n=\cos\gamma$. $l^2+m^2+n^2=1$ - **Direction Ratios (DR's):** $a, b, c$. $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$, etc. - **Line Equation:** - **Vector form:** $\vec{r} = \vec{a} + \lambda\vec{b}$ ($\vec{a}$ is position vector of a point, $\vec{b}$ is direction vector) - **Cartesian form:** $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ - **Angle between two lines:** $\cos\theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}$ or $\cos\theta = |l_1l_2+m_1m_2+n_1n_2|$ - **Shortest Distance between two skew lines:** - $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ - $d = \left|\frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|}\right|$ - **Shortest Distance between parallel lines:** - $\vec{r} = \vec{a_1} + \lambda\vec{b}$ and $\vec{r} = \vec{a_2} + \mu\vec{b}$ - $d = \left|\frac{\vec{b} \times (\vec{a_2} - \vec{a_1})}{|\vec{b}|}\right|$ - **Plane Equation:** - **Normal form:** $\vec{r} \cdot \hat{n} = d$ - **Perpendicular from origin:** $lx+my+nz=p$ - **Through a point $(\vec{a})$ and perpendicular to $\vec{N}$:** $(\vec{r} - \vec{a}) \cdot \vec{N} = 0$ - **Cartesian form for a point $(x_1, y_1, z_1)$ and normal DR's $(A,B,C)$:** $A(x-x_1) + B(y-y_1) + C(z-z_1) = 0$ - **Intercept form:** $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ - **Through three non-collinear points:** $\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$ - **Angle between two planes:** $\cos\theta = \left|\frac{\vec{N_1} \cdot \vec{N_2}}{|\vec{N_1}||\vec{N_2}|}\right|$ - **Angle between a line and a plane:** $\sin\theta = \left|\frac{\vec{b} \cdot \vec{N}}{|\vec{b}||\vec{N}|}\right|$ - **Distance of a point from a plane:** - Point $(x_1, y_1, z_1)$ to plane $Ax+By+Cz+D=0$: $d = \left|\frac{Ax_1+By_1+Cz_1+D}{\sqrt{A^2+B^2+C^2}}\right|$ ### Linear Programming - **Objective Function:** $Z = ax+by$ (to be maximized/minimized) - **Constraints:** Linear inequalities - **Feasible Region:** Area satisfying all constraints (bounded or unbounded) - **Optimal Solution:** Occurs at a corner point of the feasible region. ### Probability - **Conditional Probability:** $P(E|F) = \frac{P(E \cap F)}{P(F)}$, $P(F) \ne 0$ - **Multiplication Theorem:** $P(E \cap F) = P(F)P(E|F) = P(E)P(F|E)$ - **Independent Events:** $P(E \cap F) = P(E)P(F)$. Also $P(E|F) = P(E)$ - **Total Probability Theorem:** $P(E) = P(A_1)P(E|A_1) + P(A_2)P(E|A_2) + \dots + P(A_n)P(E|A_n)$ for a partition $A_1, \dots, A_n$ of sample space $S$. - **Bayes' Theorem:** $P(A_i|E) = \frac{P(A_i)P(E|A_i)}{\sum_{j=1}^n P(A_j)P(E|A_j)}$ - **Random Variable:** A real-valued function whose domain is the sample space of a random experiment. - **Probability Distribution:** A table/function assigning probabilities to values of a random variable. - **Mean (Expectation) of a random variable $X$:** $E(X) = \sum x_i P(X=x_i)$ - **Variance of $X$:** $Var(X) = E(X^2) - (E(X))^2 = \sum x_i^2 P(X=x_i) - (E(X))^2$ - **Binomial Distribution:** $P(X=r) = {}^n C_r p^r q^{n-r}$, where $q=1-p$ - Mean $= np$ - Variance $= npq$ ### Trigonometric Identities (for reference) - $\sin^2\theta + \cos^2\theta = 1$ - $\sec^2\theta - \tan^2\theta = 1$ - $\csc^2\theta - \cot^2\theta = 1$ - $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ - $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ - $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ - $\sin 2A = 2\sin A \cos A = \frac{2\tan A}{1+\tan^2 A}$ - $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1-\tan^2 A}{1+\tan^2 A}$ - $\tan 2A = \frac{2\tan A}{1-\tan^2 A}$ - $1+\cos 2A = 2\cos^2 A$ - $1-\cos 2A = 2\sin^2 A$ - $\sin 3A = 3\sin A - 4\sin^3 A$ - $\cos 3A = 4\cos^3 A - 3\cos A$ ### Standard Limits (for continuity/differentiability) - $\lim_{x \to 0} \frac{\sin x}{x} = 1$ - $\lim_{x \to 0} \frac{\tan x}{x} = 1$ - $\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}$ - $\lim_{x \to 0} \frac{e^x-1}{x} = 1$ - $\lim_{x \to 0} \frac{\log(1+x)}{x} = 1$ - $\lim_{x \to a} \frac{x^n - a^n}{x-a} = na^{n-1}$ - $\lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x = e$ - $\lim_{x \to 0} (1+x)^{1/x} = e$ ### Most Forgotten but Important Formulas - **Area of parallelogram (vector):** $|\vec{a} \times \vec{b}|$ - **Area of triangle (vector):** $\frac{1}{2}|\vec{a} \times \vec{b}|$ - **Volume of parallelepiped (vector):** $|(\vec{a} \times \vec{b}) \cdot \vec{c}|$ - **Coplanarity of vectors:** $(\vec{a} \times \vec{b}) \cdot \vec{c} = 0$ - **Distance of a point from a plane:** $d = \left|\frac{Ax_1+By_1+Cz_1+D}{\sqrt{A^2+B^2+C^2}}\right|$ - **Shortest distance between skew lines:** $d = \left|\frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|}\right|$ - **Derivative of $\log_a x$:** $\frac{1}{x \log_e a}$ - **Derivative of $a^x$:** $a^x \log_e a$ - **Integral of $\sec x$ and $\csc x$:** - $\int \sec x dx = \log|\sec x + \tan x| + C$ - $\int \csc x dx = \log|\csc x - \cot x| + C$ - **General solution of Linear DE:** $y \cdot (\text{IF}) = \int Q \cdot (\text{IF}) dx + C$ - **Inverse of a matrix:** $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$ - **$\det(\text{adj}(A)) = (\det(A))^{n-1}$** - **$A \cdot \text{adj}(A) = \det(A) I$** - **$2\tan^{-1}x$ formulas:** $\tan^{-1}\left(\frac{2x}{1-x^2}\right) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ - **Properties of Definite Integrals (especially $\int_0^a f(x) dx = \int_0^a f(a-x) dx$)**