JEE Advanced Math Cheatsheet

Cheatsheet Content

1. Algebra 1.1 Quadratic Equations General form: $ax^2 + bx + c = 0$, $a \neq 0$ Roots: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Discriminant: $\Delta = b^2 - 4ac$ $\Delta > 0$: Two distinct real roots $\Delta = 0$: Two equal real roots $\Delta Sum of roots: $\alpha + \beta = -b/a$ Product of roots: $\alpha\beta = c/a$ Relation between roots and coefficients: $x^2 - (\alpha+\beta)x + \alpha\beta = 0$ Common roots: One common root: $(\frac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1})^2 = \frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1} \cdot \frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}$ Two common roots: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ Location of roots: Both roots $>\! k$: $\Delta \ge 0$, $a f(k) > 0$, $-b/(2a) > k$ Both roots $ 0$, $-b/(2a) Exactly one root between $k_1, k_2$: $f(k_1)f(k_2) Both roots between $k_1, k_2$: $\Delta \ge 0$, $a f(k_1) > 0$, $a f(k_2) > 0$, $k_1 1.2 Complex Numbers $z = x + iy = r(\cos\theta + i\sin\theta) = re^{i\theta}$ Modulus: $|z| = \sqrt{x^2+y^2} = r$ Argument: $\arg(z) = \theta = \tan^{-1}(y/x)$ Conjugate: $\bar{z} = x - iy = r(\cos\theta - i\sin\theta) = re^{-i\theta}$ Properties: $|z_1z_2| = |z_1||z_2|$, $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$, $\arg(z_1z_2) = \arg(z_1) + \arg(z_2)$, $\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$ De Moivre's Theorem: $(re^{i\theta})^n = r^n e^{in\theta} = r^n(\cos n\theta + i\sin n\theta)$ $n$-th roots of unity: $z^n = 1 \implies z_k = e^{i\frac{2\pi k}{n}} = \cos(\frac{2\pi k}{n}) + i\sin(\frac{2\pi k}{n})$, for $k=0, 1, \dots, n-1$ Cube roots of unity: $1, \omega, \omega^2$ where $\omega = e^{i2\pi/3}$. $1+\omega+\omega^2=0$, $\omega^3=1$. Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$ 1.3 Sequences & Series Arithmetic Progression (AP): $a, a+d, a+2d, \dots$ $n$-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): $a, ar, ar^2, \dots$ $n$-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: $S_\infty = \frac{a}{1-r}$ for $|r| Harmonic Progression (HP): Reciprocals are in AP. Arithmetic-Geometric Progression (AGP): $a, (a+d)r, (a+2d)r^2, \dots$ Sums of special series: $\sum_{k=1}^n k = \frac{n(n+1)}{2}$ $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$ $\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2$ 1.4 Permutations & Combinations Permutations: Arrangement of objects. $P(n,r) = {}^nP_r = \frac{n!}{(n-r)!}$ Combinations: Selection of objects. $C(n,r) = {}^nC_r = \frac{n!}{r!(n-r)!}$ Properties: ${}^nC_r = {}^nC_{n-r}$, ${}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r$ Circular Permutations: $(n-1)!$ for distinct objects. If clockwise/anticlockwise are same, $\frac{(n-1)!}{2}$. 1.5 Binomial Theorem $(x+y)^n = \sum_{r=0}^n {}^nC_r x^{n-r} y^r$ General term $T_{r+1} = {}^nC_r x^{n-r} y^r$ Number of terms: $n+1$ For $(1+x)^n = \sum_{r=0}^n {}^nC_r x^r$ Middle term: If $n$ is even, $(\frac{n}{2}+1)$-th term. If $n$ is odd, $(\frac{n+1}{2})$-th and $(\frac{n+3}{2})$-th terms. Binomial theorem for any index: $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots$ (valid for $|x| 1.6 Matrices & Determinants Determinant of $2 \times 2$: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc$ Determinant of $3 \times 3$: $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg)$ Properties: $|A^T| = |A|$ $|kA| = k^n|A|$ for $n \times n$ matrix $A$ $|AB| = |A||B|$ If rows/columns are identical or proportional, $|A|=0$. If a row/column is all zeros, $|A|=0$. Adjoint of a matrix: $adj(A) = (C_{ij})^T$ where $C_{ij}$ is cofactor of $a_{ij}$. Inverse of a matrix: $A^{-1} = \frac{1}{|A|} adj(A)$ (exists if $|A| \neq 0$) System of linear equations: $AX=B$ Unique solution: $X = A^{-1}B$ if $|A| \neq 0$ (Cramer's Rule) No solution (inconsistent) or infinitely many solutions (consistent): If $|A|=0$. Check $(adj A)B$. If $(adj A)B \neq 0$, no solution. If $(adj A)B = 0$, infinitely many solutions. 1.7 Probability $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ Independent Events: $P(A \cap B) = P(A)P(B)$ Bayes' Theorem: $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{j=1}^n P(A|E_j)P(E_j)}$ Bernoulli Trials & Binomial Distribution: $P(X=k) = {}^nC_k p^k (1-p)^{n-k}$ Mean: $np$, Variance: $np(1-p)$ 2. Trigonometry 2.1 Basic Identities $\sin^2\theta + \cos^2\theta = 1$ $1 + \tan^2\theta = \sec^2\theta$ $1 + \cot^2\theta = \csc^2\theta$ 2.2 Compound Angles $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ 2.3 Multiple & Submultiple Angles $\sin 2A = 2\sin A \cos A = \frac{2\tan A}{1+\tan^2 A}$ $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A = \frac{1-\tan^2 A}{1+\tan^2 A}$ $\tan 2A = \frac{2\tan A}{1-\tan^2 A}$ $\sin 3A = 3\sin A - 4\sin^3 A$ $\cos 3A = 4\cos^3 A - 3\cos A$ $\tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}$ 2.4 Transformations $2\sin A \cos B = \sin(A+B) + \sin(A-B)$ $2\cos A \sin B = \sin(A+B) - \sin(A-B)$ $2\cos A \cos B = \cos(A+B) + \cos(A-B)$ $2\sin A \sin B = \cos(A-B) - \cos(A+B)$ $\sin C + \sin D = 2\sin(\frac{C+D}{2})\cos(\frac{C-D}{2})$ $\sin C - \sin D = 2\cos(\frac{C+D}{2})\sin(\frac{C-D}{2})$ $\cos C + \cos D = 2\cos(\frac{C+D}{2})\cos(\frac{C-D}{2})$ $\cos C - \cos D = -2\sin(\frac{C+D}{2})\sin(\frac{C-D}{2})$ 2.5 Inverse Trigonometric Functions $\sin^{-1}x + \cos^{-1}x = \pi/2$ $\tan^{-1}x + \cot^{-1}x = \pi/2$ $\sec^{-1}x + \csc^{-1}x = \pi/2$ $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$ if $xy $2\tan^{-1}x = \tan^{-1}\left(\frac{2x}{1-x^2}\right) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ 2.6 Solutions of Triangles Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ (R = circumradius) Cosine Rule: $a^2 = b^2+c^2 - 2bc\cos A$ Projection Rule: $a = b\cos C + c\cos B$ Area of triangle: $\Delta = \frac{1}{2}bc\sin A = \sqrt{s(s-a)(s-b)(s-c)}$ (Heron's formula), $s = (a+b+c)/2$ Inradius: $r = \frac{\Delta}{s} = 4R\sin(A/2)\sin(B/2)\sin(C/2)$ Exradius: $r_a = \frac{\Delta}{s-a} = 4R\sin(A/2)\cos(B/2)\cos(C/2)$ 3. Coordinate Geometry 3.1 Straight Lines Distance between $(x_1, y_1)$ and $(x_2, y_2)$: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ Section formula: $(x,y) = \left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$ (internal division) Slope: $m = \tan\theta = \frac{y_2-y_1}{x_2-x_1}$ Equation of 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: $\tan\theta = |\frac{m_1-m_2}{1+m_1m_2}|$ Parallel lines: $m_1=m_2$. Perpendicular lines: $m_1m_2=-1$. Distance from point $(x_1, y_1)$ to line $Ax+By+C=0$: $\frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$ Equation of angle bisectors of $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$: $\frac{A_1x+B_1y+C_1}{\sqrt{A_1^2+B_1^2}} = \pm \frac{A_2x+B_2y+C_2}{\sqrt{A_2^2+B_2^2}}$ 3.2 Circles General equation: $x^2+y^2+2gx+2fy+c=0$ Center: $(-g, -f)$, Radius: $\sqrt{g^2+f^2-c}$ Equation of tangent at $(x_1, y_1)$: $xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$ Length of tangent from $(x_1, y_1)$: $\sqrt{x_1^2+y_1^2+2gx_1+2fy_1+c}$ Condition for orthogonality of two circles: $2g_1g_2+2f_1f_2 = c_1+c_2$ Radical axis of $S_1=0$ and $S_2=0$: $S_1-S_2=0$ 3.3 Parabola Standard equation: $y^2 = 4ax$ Vertex: $(0,0)$, Focus: $(a,0)$, Directrix: $x=-a$ Axis: $y=0$, Latus Rectum: $4a$ Parametric form: $(at^2, 2at)$ Equation of tangent at $(x_1, y_1)$: $yy_1 = 2a(x+x_1)$ Equation of tangent at $(at^2, 2at)$: $yt = x+at^2$ 3.4 Ellipse Standard equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a>b$) Center: $(0,0)$, Vertices: $(\pm a,0)$, Co-vertices: $(0,\pm b)$ Foci: $(\pm ae, 0)$, Directrices: $x = \pm a/e$ Eccentricity: $e = \sqrt{1 - b^2/a^2}$ ($0 Latus Rectum: $2b^2/a$ Parametric form: $(a\cos\theta, b\sin\theta)$ Equation of tangent at $(x_1, y_1)$: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$ 3.5 Hyperbola Standard equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Center: $(0,0)$, Vertices: $(\pm a,0)$ Foci: $(\pm ae, 0)$, Directrices: $x = \pm a/e$ Eccentricity: $e = \sqrt{1 + b^2/a^2}$ ($e > 1$) Latus Rectum: $2b^2/a$ Asymptotes: $y = \pm \frac{b}{a}x$ Conjugate hyperbola: $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ Rectangular hyperbola: $x^2-y^2=a^2$ or $xy=c^2$ 3.6 3D Geometry Distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ Direction Cosines (DC's): $\cos\alpha, \cos\beta, \cos\gamma$. $l^2+m^2+n^2=1$. Direction Ratios (DR's): $a,b,c$. $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$, etc. Angle between two lines with DR's $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$: $\cos\theta = \frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ Equation of plane: Normal form: $lx+my+nz=p$ Intercept form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ Through $(x_1,y_1,z_1)$ with normal DR's $(A,B,C)$: $A(x-x_1)+B(y-y_1)+C(z-z_1)=0$ Distance from point $(x_1, y_1, z_1)$ to plane $Ax+By+Cz+D=0$: $\frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$ Equation of line: Symmetric form: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ Vector form: $\vec{r} = \vec{a} + \lambda\vec{b}$ Shortest distance between skew lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$: $d = \left|\frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|}\right|$ 4. Calculus 4.1 Functions Domain & Range Even/Odd functions: $f(-x)=f(x)$ (even), $f(-x)=-f(x)$ (odd) Periodic functions: $f(x+T)=f(x)$ Composite functions: $f(g(x))$, $g(f(x))$ Inverse functions: $f(f^{-1}(x))=x$, $f^{-1}(f(x))=x$ 4.2 Limits $\lim_{x \to a} f(x)$ exists if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$ L'Hopital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ Standard Limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ $\lim_{x \to 0} \frac{\tan x}{x} = 1$ $\lim_{x \to 0} \frac{e^x-1}{x} = 1$ $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$ $\lim_{x \to 0} \frac{a^x-1}{x} = \ln a$ $\lim_{x \to 0} \frac{(1+x)^n-1}{x} = n$ $\lim_{x \to \infty} (1+\frac{1}{x})^x = e$ $\lim_{x \to 0} (1+x)^{1/x} = e$ 4.3 Continuity & Differentiability Continuity at $x=a$: $\lim_{x \to a} f(x) = f(a)$ Differentiability at $x=a$: $\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ exists (LHD = RHD) If a function is differentiable at a point, it is continuous at that point. The converse is not true. 4.4 Differentiation $(uv)' = u'v + uv'$ $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$ Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$ Implicit Differentiation Derivatives of common functions: $\frac{d}{dx}(x^n) = nx^{n-1}$ $\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(\tan x) = \sec^2 x$ $\frac{d}{dx}(e^x) = e^x$ $\frac{d}{dx}(\ln x) = 1/x$ $\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}$ 4.5 Applications of Derivatives Rate of change: $\frac{dy}{dx}$ Tangents & Normals: Slope of tangent: $\frac{dy}{dx}$ Slope of normal: $-\frac{1}{dy/dx}$ Monotonicity: Increasing: $f'(x) > 0$ Decreasing: $f'(x) Maxima & Minima: First derivative test: $f'(x)$ changes sign. Second derivative test: If $f'(c)=0$, $f''(c)>0 \implies$ local minimum, $f''(c) Rolle's Theorem: If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c \in (a,b)$ such that $f'(c)=0$. Lagrange's Mean Value Theorem: If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$. 4.6 Indefinite Integration $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$) $\int \frac{1}{x} dx = \ln|x| + C$ $\int \sin x dx = -\cos x + C$ $\int \cos x dx = \sin x + C$ $\int e^x dx = e^x + C$ Integration by Parts: $\int u dv = uv - \int v du$ Partial Fractions Trigonometric substitutions (e.g., $\sqrt{a^2-x^2} \implies x=a\sin\theta$) Standard forms: $\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}(\frac{x}{a}) + C$ $\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}(\frac{x}{a}) + C$ $\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\ln|\frac{x-a}{x+a}| + C$ $\int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln|\frac{a+x}{a-x}| + C$ $\int \frac{dx}{\sqrt{x^2 \pm a^2}} = \ln|x + \sqrt{x^2 \pm a^2}| + C$ $\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}(\frac{x}{a}) + C$ $\int \sqrt{x^2 \pm a^2} dx = \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\ln|x + \sqrt{x^2 \pm a^2}| + C$ 4.7 Definite Integration Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x)=f(x)$ Properties: $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f$ is even, $0$ if $f$ is odd. $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ if $f(2a-x)=f(x)$, $0$ if $f(2a-x)=-f(x)$. Wallis' Formula: $\int_0^{\pi/2} \sin^n x \cos^m x dx = \frac{[(n-1)!!][(m-1)!!]}{(n+m)!!} \cdot K$, where $K=\pi/2$ if $n,m$ both even, else $K=1$. 4.8 Area Under Curves Area = $\int_a^b y dx$ or $\int_c^d x dy$ Area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$: $\int_a^b |f(x)-g(x)| dx$ 4.9 Differential Equations Order: Highest derivative present. Degree: Power of highest derivative (after making it polynomial in derivatives). Variable Separable: $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x) dx$ Homogeneous: $\frac{dy}{dx} = f(y/x)$. Substitute $y=vx$. Linear: $\frac{dy}{dx} + P(x)y = Q(x)$. Integrating factor: $IF = e^{\int P(x) dx}$. Solution: $y \cdot IF = \int Q(x) \cdot IF dx + C$. Exact Differential Equations 5. Vectors 5.1 Vector Algebra Position vector of point $P(x,y,z)$: $\vec{p} = x\hat{i} + y\hat{j} + z\hat{k}$ Magnitude: $|\vec{a}| = \sqrt{a_x^2+a_y^2+a_z^2}$ Unit vector: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$ Scalar (Dot) Product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_xb_x+a_yb_y+a_zb_z$ $\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$ Vector (Cross) Product: $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$ $|\vec{a} \times \vec{b}|$ = Area of parallelogram formed by $\vec{a}, \vec{b}$ $\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \parallel \vec{b}$ Scalar Triple Product: $(\vec{a} \times \vec{b}) \cdot \vec{c} = [\vec{a}, \vec{b}, \vec{c}] = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}$ Volume of parallelepiped formed by $\vec{a}, \vec{b}, \vec{c}$ If $[\vec{a}, \vec{b}, \vec{c}]=0$, vectors are coplanar. Vector Triple Product: $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$ 5.2 Lines in 3D Vector equation: $\vec{r} = \vec{a} + \lambda\vec{b}$ (passes through $\vec{a}$, parallel to $\vec{b}$) Cartesian equation: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ Angle between two lines: $\cos\theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}$ 5.3 Planes in 3D Vector equation: $\vec{r} \cdot \hat{n} = d$ (normal form) $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$ (through point $\vec{a}$, normal to $\vec{n}$) $\vec{r} = \vec{a} + \lambda\vec{b} + \mu\vec{c}$ (through point $\vec{a}$, parallel to $\vec{b}, \vec{c}$) Cartesian equation: $Ax+By+Cz+D=0$ Angle between two planes: $\cos\theta = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}||\vec{n_2}|}$ Angle between a line and a plane: $\sin\phi = \frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}$ 6. Miscellaneous 6.1 Principle of Inclusion-Exclusion $|A \cup B| = |A| + |B| - |A \cap B|$ $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$ 6.2 Mathematical Reasoning Statements, Connectives ($\land, \lor, \to, \leftrightarrow, \neg$) Tautology, Contradiction, Contingency Quantifiers ($\forall, \exists$) 6.3 Statistics Mean, Median, Mode Variance: $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - (\bar{x})^2$ Standard Deviation: $\sigma = \sqrt{\text{Variance}}$