}]]}]}]}] 30:T447a,

### 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 $\Delta < 0$, roots are imaginary/complex conjugates. - **Sum of roots ($\alpha + \beta$):** $-\frac{b}{a}$ - **Product of roots ($\alpha \beta$):** $\frac{c}{a}$ - **Formation of quadratic equation:** $x^2 - (\alpha + \beta)x + \alpha\beta = 0$ ### 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 $|r| < 1$ #### Harmonic Progression (HP) - If $a_1, a_2, ..., a_n$ are in HP, then $\frac{1}{a_1}, \frac{1}{a_2}, ..., \frac{1}{a_n}$ are in AP. #### Special Series - $\sum n = \frac{n(n+1)}{2}$ - $\sum n^2 = \frac{n(n+1)(2n+1)}{6}$ - $\sum n^3 = \left(\frac{n(n+1)}{2}\right)^2$ ### 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 $|x|<1$) ### 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^{