Mathematics Formula Sheet - Algebra and Calculus

1. Algebra

Quadratic Equations

• Standard form: $ax^2 + bx + c = 0$
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Discriminant: $\Delta = b^2 - 4ac$
• Sum and product of roots: $\alpha + \beta = -\frac{b}{a}$, $\alpha\beta = \frac{c}{a}$

Progressions

Arithmetic Progression (AP)

• 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)

• n-th term: $a_n = ar^{n-1}$
• Sum of n terms: $S_n = \frac{a(1-r^n)}{1-r}$ (when $r \neq 1$)
• Sum to infinity: $S_{\infty} = \frac{a}{1-r}$ (when $|r| < 1$)

Binomial Theorem

• Binomial expansion: $(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$
• General term: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$
• Middle term: $T_{(n/2)+1}$ (when n is even)

Permutations and Combinations

• Permutations: $^nP_r = \frac{n!}{(n-r)!}$
• Combinations: $^nC_r = \frac{n!}{r!(n-r)!}$
• Properties: $^nC_r = ^nC_{n-r}$, $^nC_0 = ^nC_n = 1$

2. Matrices and Determinants

Matrix Operations

• Addition: $(A + B){ij} = a{ij} + b_{ij}$
• Multiplication: $(AB){ij} = \sum{k} a_{ik}b_{kj}$
• Transpose: $(A^T){ij} = a{ji}$

Determinants

• 2×2 determinant: $\begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc$
• 3×3 determinant: $\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$

3. Trigonometry

Basic Identities

• Pythagorean identities:
  • $\sin^2\theta + \cos^2\theta = 1$
  • $1 + \tan^2\theta = \sec^2\theta$
  • $1 + \cot^2\theta = \csc^2\theta$

Sum and Difference Formulas

• sin(A ± B): $\sin A \cos B \pm \cos A \sin B$
• cos(A ± B): $\cos A \cos B \mp \sin A \sin B$
• tan(A ± B): $\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Multiple Angle Formulas

• sin(2θ): $2\sin\theta\cos\theta$
• cos(2θ): $\cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
• tan(2θ): $\frac{2\tan\theta}{1 - \tan^2\theta}$

4. Coordinate Geometry

Distance and Section Formula

• Distance between two points: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Section formula: $\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$

Straight Lines

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta$
• Equation forms:
  • Slope-intercept: $y = mx + c$
  • Point-slope: $y - y_1 = m(x - x_1)$
  • Two-point: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
  • General: $Ax + By + C = 0$

Circles

• Standard equation: $x^2 + y^2 = r^2$
• Center at (h,k): $(x - h)^2 + (y - k)^2 = r^2$
• General form: $x^2 + y^2 + 2gx + 2fy + c = 0$
• Center: $(-g, -f)$, Radius: $\sqrt{g^2 + f^2 - c}$

Conic Sections

Parabola

• Standard form: $y^2 = 4ax$
• Focus: $(a, 0)$, Directrix: $x = -a$

Ellipse

• Standard form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
• Eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$

Hyperbola

• Standard form: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
• Eccentricity: $e = \sqrt{1 + \frac{b^2}{a^2}}$

5. Differential Calculus

Basic Differentiation

• Power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
• Product rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$
• Quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$
• Chain rule: $\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)$

Derivatives of Functions

• $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$, $\frac{d}{dx}(\cot x) = -\csc^2 x$
• $\frac{d}{dx}(\sec x) = \sec x \tan x$, $\frac{d}{dx}(\csc x) = -\csc x \cot x$
• $\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(a^x) = a^x \ln a$
• $\frac{d}{dx}(\ln x) = \frac{1}{x}$, $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$

Applications of Derivatives

• Rate of change: $\frac{dy}{dx}$ gives the rate of change of y with respect to x
• Increasing/Decreasing:
  • Increasing when $\frac{dy}{dx} > 0$
  • Decreasing when $\frac{dy}{dx} < 0$
• Maxima and Minima:
  • Critical points: $\frac{dy}{dx} = 0$ or undefined
  • Second derivative test: $\frac{d^2y}{dx^2} < 0$ (maximum), $> 0$ (minimum)

6. Integral Calculus

Basic Integration

• Power rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
• $\int \frac{1}{x} dx = \ln|x| + C$
• $\int e^x dx = e^x + C$
• $\int a^x dx = \frac{a^x}{\ln a} + C$

Trigonometric Integrals

• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \sec^2 x dx = \tan x + C$
• $\int \csc^2 x dx = -\cot x + C$

Integration Methods

• Substitution method: If $\int f(g(x))g’(x)dx$, let $u = g(x)$
• Integration by parts: $\int u dv = uv - \int v du$

Definite Integrals

• Fundamental theorem: $\int_a^b f(x)dx = F(b) - F(a)$
• Properties:
  • $\int_a^b f(x)dx = -\int_b^a f(x)dx$
  • $\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$ (where $a < b < c$)

Areas Under Curves

• Area under curve: $A = \int_a^b y dx$
• Area between curves: $A = \int_a^b [f(x) - g(x)] dx$ (where $f(x) > g(x)$)

This formula sheet covers essential algebra and calculus formulas for JEE and NEET preparation. Use this for quick revision during your study sessions.