📐 CUET Maths Formula Sheet-1
Mathematics | FirstInTest
| Formula | Notes |
|---|---|
| 1. Algebra | — |
| Quadratic Equations | — |
| $ax^2 + bx + c = 0$ | Standard Form |
| $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ | Quadratic Formula |
| $D = b^2 - 4ac$ | Discriminant (D) |
| $D > 0$ | Two distinct real roots |
| $D = 0$ | Two equal real roots (repeated root) |
| $D < 0$ | No real roots (complex roots) |
| $\alpha + \beta = -b/a$ | Sum of roots |
| $\alpha\beta = c/a$ | Product of roots |
| $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ | Forming Equation from Roots |
| $x^2 - (\alpha + \beta)x + \alpha\beta = 0$ | Forming Equation from Roots |
| Progressions | — |
| Arithmetic Progression (AP) | — |
| $a_n = a + (n-1)d$ | nth term |
| $S_n = \frac{n}{2} [2a + (n-1)d]$ | Sum of n terms |
| $S_n = \frac{n}{2} (a + l)$ | Sum of n terms, where $l$ = last term |
| Geometric Progression (GP) | — |
| $a_n = ar^{n-1}$ | nth term |
| $S_n = \frac{a (r^n - 1)}{(r - 1)}$ | Sum of n terms, if $r \neq 1$ |
| $S_n = \frac{a (1 - r^n)}{(1 - r)}$ | Sum of n terms, if $r \neq 1$ |
| $S_\infty = \frac{a}{(1-r)}$ | Sum to infinity, if $|r| < 1$ |
| Harmonic Progression (HP) | — |
| If $a, b, c$ are in HP, then $1/a, 1/b, 1/c$ are in AP | Condition for HP |
| $\frac{1}{a_n} = \frac{1}{a} + (n-1)d$ | nth term |
| Matrix Operations | — |
| $(AB)^T = B^T A^T$ | — |
| $(AB)^{-1} = B^{-1} A^{-1}$ | — |
| $(A^T)^{-1} = (A^{-1})^T$ | — |
| 2x2 Determinant | — |
| $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$ | — |
| Inverse of 2x2 Matrix | — |
| $A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ | If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ |
| Properties (Matrices) | — |
| $|A^T| = |A|$ | — |
| $|AB| = |A| \times |B|$ | — |
| $|kA| = k^n |A|$ | For n x n matrix |
| $|A^{-1}| = 1/|A|$ | — |
| 2. Calculus | — |
| Differentiation Formulas | — |
| Basic Derivatives | — |
| $\frac{d}{dx} (x^n) = nx^{n-1}$ | — |
| $\frac{d}{dx} (e^x) = e^x$ | — |
| $\frac{d}{dx} (a^x) = a^x \ln(a)$ | — |
| $\frac{d}{dx} (\ln x) = \frac{1}{x}$ | — |
| $\frac{d}{dx} (\log_a x) = \frac{1}{x \ln a}$ | — |
| Trigonometric Derivatives | — |
| $\frac{d}{dx} (\sin x) = \cos x$ | — |
| $\frac{d}{dx} (\cos x) = -\sin x$ | — |
| $\frac{d}{dx} (\tan x) = \sec^2 x$ | — |
| $\frac{d}{dx} (\cot x) = -\csc^2 x$ | — |
| $\frac{d}{dx} (\sec x) = \sec x \tan x$ | — |
| $\frac{d}{dx} (\csc x) = -\csc x \cot x$ | — |
| Inverse Trigonometric Derivatives | — |
| $\frac{d}{dx} (\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$ | — |
| $\frac{d}{dx} (\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$ | — |
| $\frac{d}{dx} (\tan^{-1} x) = \frac{1}{1+x^2}$ | — |
| $\frac{d}{dx} (\cot^{-1} x) = -\frac{1}{1+x^2}$ | — |
| $\frac{d}{dx} (\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}$ | — |
| $\frac{d}{dx} (\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2-1}}$ | — |
| Product & Quotient Rules | — |
| $\frac{d}{dx} (uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ | Product Rule |
| $\frac{d}{dx} (\frac{u}{v}) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ | Quotient Rule |
| $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$ | Chain Rule |
| Integration Formulas | — |
| Basic Integrals | — |
| $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ | For $n \neq -1$ |
| $\int \frac{1}{x} dx = \ln|x| + C$ | — |
| $\int e^x dx = e^x + C$ | — |
| $\int a^x dx = \frac{a^x}{\ln(a)} + C$ | — |
| $\int \frac{1}{x^2+a^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$ | — |
| $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}(\frac{x}{a}) + C$ | — |
| Trigonometric Integrals | — |
| $\int \sin x dx = -\cos x + C$ | — |
| $\int \cos x dx = \sin x + C$ | — |
| $\int \tan x dx = \ln|\sec x| + C = -\ln|\cos x| + C$ | — |
| $\int \cot x dx = \ln|\sin x| + C$ | — |
| $\int \sec x dx = \ln|\sec x + \tan x| + C$ | — |
| $\int \csc x dx = \ln|\csc x - \cot x| + C$ | — |
| $\int \sec^2 x dx = \tan x + C$ | — |
| $\int \csc^2 x dx = -\cot x + C$ | — |
| Definite Integration Properties | — |
| $\int_a^b f(x)dx = \int_a^b f(t)dt$ | — |
| $\int_a^b f(x)dx = -\int_b^a f(x)dx$ | — |
| $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ | — |
| $\int_0^a f(x)dx = \int_0^a f(a-x)dx$ | — |
| $\int_{-a}^a f(x)dx = 2\int_0^a f(x) dx$ | If $f(x)$ is even |
| $\int_{-a}^a f(x)dx = 0$ | If $f(x)$ is odd |
| 3. 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 Angle Formulas | — |
| $\sin(A + B) = \sin A \cos B + \cos A \sin B$ | — |
| $\cos(A + B) = \cos A \cos B - \sin A \sin B$ | — |
| $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ | — |
| $\sin(A - B) = \sin A \cos B - \cos A \sin B$ | — |
| $\cos(A - B) = \cos A \cos B + \sin A \sin B$ | — |
| $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ | — |
| Double Angle Formulas | — |
| $\sin 2A = 2 \sin A \cos A$ | — |
| $\cos 2A = \cos^2 A - \sin^2 A$ | — |
| $\cos 2A = 2\cos^2 A - 1$ | — |
| $\cos 2A = 1 - 2\sin^2 A$ | — |
| $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$ | — |
| Triple Angle Formulas | — |
| $\sin 3A = 3\sin A - 4\sin^3 A$ | — |
| $\cos 3A = 4\cos^3 A - 3\cos A$ | — |
| $\tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}$ | — |
| Sum to Product Formulas | — |
| $\sin A + \sin B = 2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})$ | — |
| $\sin A - \sin B = 2 \cos(\frac{A+B}{2}) \sin(\frac{A-B}{2})$ | — |
| $\cos A + \cos B = 2 \cos(\frac{A+B}{2}) \cos(\frac{A-B}{2})$ | — |
| $\cos A - \cos B = -2 \sin(\frac{A+B}{2}) \sin(\frac{A-B}{2})$ | — |
| 4. Coordinate Geometry | — |
| Distance Formula | — |
| $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ | Distance between $(x_1, y_1)$ and $(x_2, y_2)$ |
| Section Formula | — |
| $x = \frac{mx_2 + nx_1}{m+n}$ | Internal division (m:n) |
| $y = \frac{my_2 + ny_1}{m+n}$ | Internal division (m:n) |
| $x = \frac{x_1+x_2}{2}$ | Midpoint (m=n=1) |
| $y = \frac{y_1+y_2}{2}$ | Midpoint (m=n=1) |
| Straight Line | — |
| $y = mx + c$ | Slope-intercept form |
| $y - y_1 = m(x - x_1)$ | Point-slope form |
| $\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$ | Two-point form |
| $\frac{x}{a} + \frac{y}{b} = 1$ | Intercept form |
| $Ax + By + C = 0$ | General form |
| $m = -A/B$ | Slope from general form |
| $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ | Distance from point $(x_1, y_1)$ to line $Ax+By+C=0$ |
| Circle | — |
| $(x-h)^2 + (y-k)^2 = r^2$ | Standard form |
| Center: $(h, k)$, Radius: $r$ | For standard form |
| $x^2 + y^2 + 2gx + 2fy + c = 0$ | General form |
| Center: $(-g, -f)$, Radius: $\sqrt{g^2 + f^2 - c}$ | For general form |
| 5. Statistics & Probability | — |
| Mean, Median, Mode | — |
| $\bar{x} = \frac{\sum x}{n}$ | Mean |
| $\bar{x} = \frac{\sum fx}{\sum f}$ | Mean (for grouped data) |
| $\frac{n+1}{2}$ | Median position |
| Value with highest frequency | Mode |
| Standard Deviation & Variance | — |
| $\sigma^2 = \frac{\sum (x - \bar{x})^2}{n}$ | Variance |
| $\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$ | Standard Deviation |
| $\sigma = \sqrt{\frac{\sum x^2}{n} - (\frac{\sum x}{n})^2}$ | Standard Deviation (alternate formula) |
| Probability | — |
| $P(A) = \frac{n(A)}{n(S)}$ | Where $n(S)$ = total outcomes |
| $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ | — |
| $P(A \cap B) = P(A) \times P(B)$ | If independent |
| $P(A') = 1 - P(A)$ | Complement |
| Permutations & Combinations | — |
| $P(n, r) = \frac{n!}{(n-r)!}$ | Permutations |
| $C(n, r) = \frac{n!}{r!(n-r)!}$ | Combinations |
| $C(n, 0) = 1$ | — |
| $C(n, n) = 1$ | — |
| $C(n, r) = C(n, n-r)$ | — |
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