JEE Main Mathematics

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### Sets, Relations, and Functions - **Sets:** Union ($A \cup B$), Intersection ($A \cap B$), Difference ($A - B$), Complement ($A'$). - De Morgan's Laws: $(A \cup B)' = A' \cap B'$, $(A \cap B)' = A' \cup B'$. - **Relations:** - Reflexive: $(a, a) \in R$ - 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. - **Functions:** - One-one (Injective), Onto (Surjective), Bijective. - Composite Function: $(f \circ g)(x) = f(g(x))$. - Inverse Function: $f^{-1}(y) = x \iff f(x) = y$. ### Complex Numbers and Quadratic Equations - **Complex Numbers:** $z = x + iy = r(\cos\theta + i\sin\theta)$ (Polar Form). - Modulus: $|z| = \sqrt{x^2 + y^2}$. - Argument: $\arg(z) = \theta$. - De Moivre's Theorem: $(r(\cos\theta + i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta)$. - Cube roots of unity: $1, \omega, \omega^2$. $1 + \omega + \omega^2 = 0$, $\omega^3 = 1$. - **Quadratic Equation:** $ax^2 + bx + c = 0$. Roots: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. - Discriminant $\Delta = b^2 - 4ac$. - $\Delta > 0$: Real and distinct roots. - $\Delta = 0$: Real and equal roots. - $\Delta ### Matrices and Determinants - **Types of Matrices:** Square, Diagonal, Scalar, Identity, Symmetric ($A = A^T$), Skew-Symmetric ($A = -A^T$). - **Determinant of a Matrix:** For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad - bc$. - Properties: $\det(AB) = \det(A)\det(B)$, $\det(A^T) = \det(A)$, $\det(kA) = k^n \det(A)$. - **Adjoint and Inverse:** - Adjoint: $\text{adj}(A) = (\text{cofactor matrix})^T$. - Inverse: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$. $A^{-1}$ exists if $\det(A) \neq 0$. - **System of Linear Equations:** $AX = B$. - Unique solution: $\det(A) \neq 0$. $X = A^{-1}B$. - Consistent (infinite solutions) or Inconsistent (no solutions): $\det(A) = 0$. ### Permutations and Combinations - **Fundamental Principle of Counting:** If an event can occur in $m$ ways and another independent event in $n$ ways, then both can occur in $mn$ ways. - **Permutations:** Number of ways to arrange $n$ distinct items taken $r$ at a time: $P(n, r) = \frac{n!}{(n-r)!}$. - Permutations with repetitions: $\frac{n!}{p!q!r!...}$ - **Combinations:** Number of ways to select $n$ distinct items taken $r$ at a time: $C(n, r) = \frac{n!}{r!(n-r)!}$. - $C(n, r) = C(n, n-r)$. - Pascal's Identity: $C(n, r) + C(n, r-1) = C(n+1, r)$. ### Binomial Theorem - **Expansion:** $(x+y)^n = \sum_{k=0}^n C(n, k) x^{n-k} y^k$. - **General Term:** $T_{r+1} = C(n, r) x^{n-r} y^r$. - **Middle Term:** - If $n$ is even, $(n/2 + 1)$-th term. - If $n$ is odd, $((n+1)/2)$-th and $((n+3)/2)$-th terms. - **Properties:** - Sum of coefficients: Set $x=1, y=1 \implies (1+1)^n = 2^n$. - $(1+x)^n = C(n,0) + C(n,1)x + ... + C(n,n)x^n$. ### Sequence and 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: $S_n = \frac{a(r^n - 1)}{r-1}$ ($r \neq 1$). - Sum to infinity (if $|r| ### Limits, Continuity, and Differentiability - **Limits:** - $\lim_{x \to a} f(x)$ exists if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$. - Standard Limits: - $\lim_{x \to 0} \frac{\sin x}{x} = 1$. - $\lim_{x \to 0} \frac{\tan x}{x} = 1$. - $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$. - $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$. - $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$. - $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$. - $\lim_{x \to 0} (1 + x)^{1/x} = e$. - 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 $\lim_{x \to a} f(x) = f(a)$. - **Differentiability:** A function $f(x)$ is differentiable at $x=a$ if $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists. - Differentiability implies continuity, but not vice-versa. ### Differentiation - **Basic Derivatives:** - $\frac{d}{dx}(x^n) = nx^{n-1}$. - $\frac{d}{dx}(\sin x) = \cos x$. - $\frac{d}{dx}(\cos x) = -\sin x$. - $\frac{d}{dx}(\tan x) = \sec^2 x$. - $\frac{d}{dx}(e^x) = e^x$. - $\frac{d}{dx}(\ln x) = 1/x$. - **Rules:** - Product Rule: $(uv)' = u'v + uv'$. - Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. - Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$. - **Implicit Differentiation:** Differentiate both sides with respect to $x$, treating $y$ as a function of $x$. - **Parametric Differentiation:** If $x=f(t), y=g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. ### Application of Derivatives - **Rate of Change:** $\frac{dy}{dx}$ is the rate of change of $y$ with respect to $x$. - **Tangents and Normals:** - Slope of tangent at $(x_1, y_1)$ is $m = (\frac{dy}{dx})_{(x_1, y_1)}$. - Equation of tangent: $y - y_1 = m(x - x_1)$. - Slope of normal is $-1/m$. - Equation of normal: $y - y_1 = -\frac{1}{m}(x - x_1)$. - **Increasing/Decreasing Functions:** - $f'(x) > 0$: Increasing. - $f'(x) 0$: Local minimum. - $f''(c) ### Indefinite Integrals - **Basic Formulas:** - $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$). - $\int \frac{1}{x} dx = \ln|x| + C$. - $\int \sin x dx = -\cos x + C$. - $\int \cos x dx = \sin x + C$. - $\int e^x dx = e^x + C$. - $\int a^x dx = \frac{a^x}{\ln a} + C$. - $\int \sec^2 x dx = \tan x + C$. - $\int \csc^2 x dx = -\cot x + C$. - $\int \sec x \tan x dx = \sec x + C$. - $\int \csc x \cot x dx = -\csc x + C$. - **Methods of Integration:** - **Substitution:** $\int f(g(x))g'(x) dx = \int f(u) du$ where $u=g(x)$. - **Integration by Parts:** $\int u dv = uv - \int v du$. (LIATE rule for choosing $u$) - **Partial Fractions:** For rational functions. - **Trigonometric Identities:** Use identities to simplify integrands. ### Definite Integrals - **Fundamental Theorem of Calculus:** $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$. - **Properties:** - $\int_a^b f(x) dx = -\int_b^a f(x) dx$. - $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$. - $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$. - $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. - $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f$ is even ($f(-x)=f(x)$). - $\int_{-a}^a f(x) dx = 0$ if $f$ is odd ($f(-x)=-f(x)$). ### Area Under Curves - Area bounded by $y = f(x)$, x-axis, $x=a$, $x=b$: $\int_a^b |f(x)| dx$. - Area bounded by $x = g(y)$, y-axis, $y=c$, $y=d$: $\int_c^d |g(y)| dy$. - Area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$: $\int_a^b |f(x) - g(x)| dx$. ### Differential Equations - **Order and Degree:** - Order: Highest order derivative present. - Degree: Power of the highest order derivative (after making it a polynomial in derivatives). - **Types of Differential Equations:** - **Variable Separable:** $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x) dx$. - **Homogeneous:** $\frac{dy}{dx} = f(\frac{y}{x})$. Substitute $y=vx$. - **Linear:** $\frac{dy}{dx} + Py = Q$, where $P, Q$ are functions of $x$. - Integrating Factor (IF): $e^{\int P dx}$. - Solution: $y \cdot (\text{IF}) = \int Q \cdot (\text{IF}) dx + C$. ### Coordinate Geometry - **Distance Formula:** $D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. - **Section Formula:** - Internal: $(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n})$. - External: $(\frac{m x_2 - n x_1}{m-n}, \frac{m y_2 - n y_1}{m-n})$. - **Area of Triangle:** $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$. - **Centroid:** $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$. - **Incenter:** $(\frac{ax_1+bx_2+cx_3}{a+b+c}, \frac{ay_1+by_2+cy_3}{a+b+c})$. - **Circumcenter, Orthocenter:** Intersection of perpendicular bisectors and altitudes respectively. ### Straight Lines - **Equation of a Line:** - Slope-intercept form: $y = mx + c$. - Point-slope form: $y - y_1 = m(x - x_1)$. - 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:** $\tan\theta = |\frac{m_1-m_2}{1+m_1m_2}|$. - Parallel lines: $m_1 = m_2$. - Perpendicular lines: $m_1m_2 = -1$. - **Distance of a point $(x_1, y_1)$ from a line $Ax+By+C=0$:** $\frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$. ### Conic Sections - **General Equation:** $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. - Discriminant: $\Delta = B^2 - 4AC$. - $\Delta 0$: Hyperbola. - **Parabola:** $y^2 = 4ax$ (or $x^2 = 4ay$). - Focus $(a,0)$, Directrix $x=-a$. - Eccentricity $e=1$. - **Ellipse:** $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. - Foci $(\pm ae, 0)$, Directrices $x = \pm a/e$. - Eccentricity $e = \sqrt{1 - b^2/a^2}$ ($a>b$). $0 1$. ### 3D Geometry - **Distance Formula:** $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. - **Direction Cosines (DC's):** $l = \cos\alpha, m = \cos\beta, n = \cos\gamma$. $l^2+m^2+n^2=1$. - **Direction Ratios (DR's):** $a, b, c$ proportional to $l, m, n$. - **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: $Ax+By+Cz=D$. - Intercept form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$. - **Shortest Distance between Skew Lines:** - Lines: $\vec{r}_1 = \vec{a}_1 + \lambda\vec{b}_1$, $\vec{r}_2 = \vec{a}_2 + \mu\vec{b}_2$. - SD = $\left|\frac{(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|}\right|$. ### Vector Algebra - **Vector Addition:** Triangle and Parallelogram Law. - **Scalar (Dot) Product:** $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z$. - Perpendicular vectors: $\vec{a} \cdot \vec{b} = 0$. - **Vector (Cross) Product:** $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$. - Magnitude is area of parallelogram. - Parallel vectors: $\vec{a} \times \vec{b} = \vec{0}$. - $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$. - **Scalar Triple Product:** $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$. - Volume of parallelepiped. Coplanar vectors if $[\vec{a} \vec{b} \vec{c}] = 0$. - **Vector Triple Product:** $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$. ### Trigonometry - **Basic Identities:** $\sin^2\theta + \cos^2\theta = 1$, $\sec^2\theta - \tan^2\theta = 1$, $\csc^2\theta - \cot^2\theta = 1$. - **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:** - $\sin 2A = 2\sin A \cos A = \frac{2\tan A}{1+\tan^2 A}$. - $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1-\tan^2 A}{1+\tan^2 A}$. - $\tan 2A = \frac{2\tan A}{1-\tan^2 A}$. - **Inverse Trigonometric Functions:** - $\sin^{-1}x + \cos^{-1}x = \pi/2$. - $\tan^{-1}x + \cot^{-1}x = \pi/2$. - $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$. ### Mathematical Reasoning - **Statements:** True or False, not both. - **Logical Connectives:** - AND ($\land$): True only if both are true. - OR ($\lor$): True if at least one is true. - NOT ($\neg$): Negation. - Implication ($\implies$): $p \implies q$ (if $p$ then $q$). False only if $p$ is true and $q$ is false. - Biconditional ($\iff$): $p \iff q$ ($p$ if and only if $q$). True if both $p, q$ have same truth value. - **Tautology:** Always true. - **Contradiction (Fallacy):** Always false. - **Converse:** $q \implies p$. - **Contrapositive:** $\neg q \implies \neg p$. (Equivalent to original statement) ### Statistics and Probability - **Measures of Central Tendency:** - Mean: $\bar{x} = \frac{\sum x_i}{n}$. - Median: Middle value when data is ordered. - Mode: Most frequent value. - **Measures of Dispersion:** - Variance: $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - (\bar{x})^2$. - Standard Deviation: $\sigma = \sqrt{\sigma^2}$. - **Probability:** $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. - $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. - Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$. - Independent Events: $P(A \cap B) = P(A)P(B)$. - Bayes' Theorem: $P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum P(B|A_j)P(A_j)}$. - **Binomial Distribution:** $P(X=k) = C(n, k) p^k (1-p)^{n-k}$. - Mean $np$, Variance $np(1-p)$.