JEE Main & Advanced Maths

Cheatsheet Content

### Quadratic Equations - **Standard Form:** $ax^2 + bx + c = 0$, where $a \neq 0$. - **Discriminant ($\Delta$):** $\Delta = b^2 - 4ac$. - $\Delta > 0$: Two distinct real roots. - $\Delta = 0$: Two equal real roots. - $\Delta ### Complex Numbers - **Form:** $z = x + iy$, where $x, y \in \mathbb{R}$ and $i = \sqrt{-1}$. - **Conjugate:** $\bar{z} = x - iy$. - **Modulus:** $|z| = \sqrt{x^2 + y^2}$. - **Polar Form:** $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$, where $r = |z|$ and $\theta = \arg(z)$. - **De Moivre's Theorem:** $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$. - **Cube Roots of Unity:** $1, \omega, \omega^2$, where $\omega = e^{i2\pi/3}$. - $1 + \omega + \omega^2 = 0$. - $\omega^3 = 1$. ### Sequences & Series - **Arithmetic Progression (AP):** $a, a+d, a+2d, ...$ - $n^{th}$ term: $a_n = a + (n-1)d$. - Sum of $n$ terms: $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + a_n)$. - **Geometric Progression (GP):** $a, ar, ar^2, ...$ - $n^{th}$ term: $a_n = ar^{n-1}$. - Sum of $n$ terms: $S_n = \frac{a(r^n - 1)}{r-1}$ for $r \neq 1$. - Sum to infinity (if $|r| ### Permutations & Combinations - **Permutations ($P(n,r)$ or $_nP_r$):** Arrangement of $r$ items from $n$ distinct items. - $P(n,r) = \frac{n!}{(n-r)!}$. - **Combinations ($C(n,r)$ or $_nC_r$):** Selection of $r$ items from $n$ distinct items. - $C(n,r) = \frac{n!}{r!(n-r)!}$. - **Binomial Theorem:** $(x+y)^n = \sum_{k=0}^n C(n,k) x^{n-k} y^k$. - General term: $T_{k+1} = C(n,k) x^{n-k} y^k$. ### Binomial Theorem - **Expansion:** $(x+a)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} a^r$. - **General Term:** $T_{r+1} = \binom{n}{r} x^{n-r} a^r$. - **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. - **Properties of Binomial Coefficients:** - $\binom{n}{r} = \binom{n}{n-r}$. - $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$. - Sum of coefficients: $\sum_{r=0}^{n} \binom{n}{r} = 2^n$. ### Matrices & Determinants - **Determinant of $2 \times 2$ matrix:** $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$. - **Determinant of $3 \times 3$ matrix:** $a(ei-fh) - b(di-fg) + c(dh-eg)$. - **Adjoint of a Matrix:** $adj(A) = (C_{ij})^T$, where $C_{ij}$ is cofactor of $a_{ij}$. - **Inverse of a Matrix:** $A^{-1} = \frac{1}{|A|} adj(A)$ (if $|A| \neq 0$). - **System of Linear Equations (Cramer's Rule):** - $a_1x + b_1y + c_1z = d_1$ - $a_2x + b_2y + c_2z = d_2$ - $a_3x + b_3y + c_3z = d_3$ - $D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$, $D_x = \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{vmatrix}$, etc. - If $D \neq 0$, $x = D_x/D$, $y = D_y/D$, $z = D_z/D$ (unique solution). - If $D = 0$: - If $D_x=D_y=D_z=0$: Infinite solutions. - If at least one of $D_x, D_y, D_z$ is non-zero: No solution. ### Trigonometry - **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 Angle Formulas:** - $\sin 2\theta = 2\sin\theta\cos\theta = \frac{2\tan\theta}{1+\tan^2\theta}$. - $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta = \frac{1-\tan^2\theta}{1+\tan^2\theta}$. - $\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}$. - **Half Angle Formulas:** - $\sin\theta = 2\sin(\theta/2)\cos(\theta/2)$. - $\cos\theta = \cos^2(\theta/2) - \sin^2(\theta/2)$. - **Sum-to-Product & Product-to-Sum Formulas.** - **General Solutions:** - $\sin\theta = \sin\alpha \implies \theta = n\pi + (-1)^n\alpha$. - $\cos\theta = \cos\alpha \implies \theta = 2n\pi \pm \alpha$. - $\tan\theta = \tan\alpha \implies \theta = n\pi + \alpha$. ### Inverse Trigonometric Functions - **Principal Value Branches:** - $\sin^{-1}x: [-\pi/2, \pi/2]$. - $\cos^{-1}x: [0, \pi]$. - $\tan^{-1}x: (-\pi/2, \pi/2)$. - **Properties:** - $\sin^{-1}x + \cos^{-1}x = \pi/2$. - $\tan^{-1}x + \cot^{-1}x = \pi/2$. - $\sec^{-1}x + \csc^{-1}x = \pi/2$. - $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$, if $xy ### Straight Lines - **Distance Formula:** $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. - **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 ($m$):** $\frac{y_2-y_1}{x_2-x_1} = \tan\theta$. - **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:** $\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$. - **Parallel Lines:** $m_1 = m_2$. - **Perpendicular Lines:** $m_1 m_2 = -1$. - **Distance from a Point to a Line:** $\frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$. ### Conic Sections - **Circle:** $(x-h)^2 + (y-k)^2 = r^2$. - **Parabola:** $y^2 = 4ax$ (or $x^2 = 4ay$). - Focus: $(a,0)$. Directrix: $x = -a$. - **Ellipse:** $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (for $a>b$). - Foci: $(\pm ae, 0)$. Eccentricity: $e = \sqrt{1 - b^2/a^2}$. - **Hyperbola:** $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. - Foci: $(\pm ae, 0)$. Eccentricity: $e = \sqrt{1 + b^2/a^2}$. ### Three Dimensional Geometry - **Distance Formula:** $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. - **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. - **Equation of a 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 a Plane:** - **Normal Form:** $\vec{r} \cdot \hat{n} = d$. - **Cartesian Form:** $Ax+By+Cz=D$. - **Distance from a Point to a Plane:** $\frac{|Ax_1 + By_1 + Cz_1 - D|}{\sqrt{A^2+B^2+C^2}}$. ### Limits, Continuity & Differentiability - **Limit:** $\lim_{x \to a} f(x) = L$. - **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 $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$. - **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 Derivatives:** - $\frac{d}{dx}(x^n) = nx^{n-1}$. - $\frac{d}{dx}(\sin x) = \cos x$. - $\frac{d}{dx}(\cos x) = -\sin x$. - $\frac{d}{dx}(e^x) = e^x$. - $\frac{d}{dx}(\ln x) = \frac{1}{x}$. - **Product Rule:** $(uv)' = u'v + uv'$. - **Quotient Rule:** $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$. - **Chain Rule:** $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. ### Applications of Derivatives - **Rate of Change:** $\frac{dy}{dx}$. - **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: $m_N = -1/m_T$. - Equation of normal: $y - y_1 = m_N(x - x_1)$. - **Increasing/Decreasing Functions:** - $f'(x) > 0$: Increasing. - $f'(x) 0$, local minimum. ### Integration - **Basic Integrals:** - $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$). - $\int \frac{1}{x} dx = \ln|x| + C$. - $\int \sin x dx = -\cos x + C$. - $\int \cos x dx = \sin x + C$. - $\int e^x dx = e^x + C$. - **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:** - $\int_a^b f(x) dx = -\int_b^a f(x) dx$. - $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$. - $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. - $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ 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 the highest order derivative (after simplifying to polynomial form). - **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 $y = vx$. - **Linear Differential Equation:** $\frac{dy}{dx} + Py = Q$. - Integrating Factor (IF): $e^{\int P dx}$. - Solution: $y \cdot \text{IF} = \int (Q \cdot \text{IF}) dx + C$. ### Vector Algebra - **Scalar Product (Dot Product):** $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta = a_1b_1 + a_2b_2 + a_3b_3$. - **Vector Product (Cross Product):** $\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \hat{n}$. - $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$. - **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 Probability:** $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. - **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:** $P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$. - **Independent Events:** $P(A \cap B) = P(A)P(B)$. - **Bayes' Theorem:** $P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^n P(B|A_j)P(A_j)}$. - **Binomial Distribution:** $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$.