JEE Advanced Mathematics 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: $\alpha \beta = c/a$ Nature of roots (rational/irrational): If $a, b, c$ are rational, roots are rational if $\Delta$ is a perfect square, irrational otherwise. Common roots: If $a_1x^2+b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=0$ have a common root, then $(c_1a_2-c_2a_1)^2 = (a_1b_2-a_2b_1)(b_1c_2-b_2c_1)$. 1.2 Complex Numbers $z = x + iy = r(\cos \theta + i \sin \theta) = re^{i\theta}$ Modulus: $|z| = \sqrt{x^2+y^2}$ Argument: $\arg(z) = \theta = \tan^{-1}(y/x)$ Conjugate: $\bar{z} = x - iy$ Properties: $|z_1z_2| = |z_1||z_2|$ $|z_1/z_2| = |z_1|/|z_2|$ $\arg(z_1z_2) = \arg(z_1) + \arg(z_2)$ $\arg(z_1/z_2) = \arg(z_1) - \arg(z_2)$ $z\bar{z} = |z|^2$ De Moivre's Theorem: $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$ Cube roots of unity: $1, \omega, \omega^2$ where $\omega = e^{i2\pi/3}$ $1 + \omega + \omega^2 = 0$ $\omega^3 = 1$ 1.3 Progressions and 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: $S_n = \frac{a(r^n-1)}{r-1}$ ($r \neq 1$) Sum to infinity ($|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: $\sum n = \frac{n(n+1)}{2}$ $\sum n^2 = \frac{n(n+1)(2n+1)}{6}$ $\sum n^3 = \left(\frac{n(n+1)}{2}\right)^2$ 1.4 Permutations and Combinations Permutation: $^nP_r = \frac{n!}{(n-r)!}$ (arrangement) Combination: $^nC_r = \frac{n!}{r!(n-r)!}$ (selection) $^nC_r = ^nC_{n-r}$ $^nC_r + ^nC_{r-1} = ^{n+1}C_r$ 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: $n+1$ If $n$ is even, middle term is $T_{n/2+1}$. If $n$ is odd, middle terms are $T_{(n+1)/2}$ and $T_{(n+3)/2}$. $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots$ (for any real $n$, $|x| 1.6 Matrices and Determinants Determinant of $2 \times 2$: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad-bc$ Determinant of $3 \times 3$: $$ \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}(a_{22}a_{33}-a_{23}a_{32}) - a_{12}(a_{21}a_{33}-a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31}) $$ Properties of Determinants: $\det(A^T) = \det(A)$ $\det(AB) = \det(A)\det(B)$ $\det(kA) = k^n \det(A)$ (for $n \times n$ matrix $A$) If two rows/columns are identical or proportional, $\det(A)=0$. Inverse of a matrix: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$ (exists if $\det(A) \neq 0$) Adjoint matrix: $\text{adj}(A) = (C_{ij})^T$, where $C_{ij}$ is the cofactor of $a_{ij}$. System of linear equations ($AX=B$): Unique solution: $X = A^{-1}B$ (if $\det(A) \neq 0$) No solution or infinite solutions: If $\det(A)=0$. Check $(\text{adj}(A))B$. If $(\text{adj}(A))B \neq O$: No solution (inconsistent) If $(\text{adj}(A))B = O$: Infinite solutions (consistent) 2. Trigonometry 2.1 Trigonometric Ratios and Identities $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ 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: $\sin 2A = 2 \sin A \cos A = \frac{2 \tan A}{1+\tan^2 A}$ $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1-2\sin^2 A = \frac{1-\tan^2 A}{1+\tan^2 A}$ $\tan 2A = \frac{2 \tan A}{1-\tan^2 A}$ Half Angle Formulas: $\sin A = 2 \sin(A/2) \cos(A/2)$ $\cos A = \cos^2(A/2) - \sin^2(A/2)$ Product-to-Sum Formulas: $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$ $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$ $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$ $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$ Sum-to-Product Formulas: $\sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right)$ $\sin C - \sin D = 2 \cos\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right)$ $\cos C + \cos D = 2 \cos\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right)$ $\cos C - \cos D = -2 \sin\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right)$ 2.2 Trigonometric Equations $\sin x = \sin \alpha \implies x = n\pi + (-1)^n \alpha$, $n \in \mathbb{Z}$ $\cos x = \cos \alpha \implies x = 2n\pi \pm \alpha$, $n \in \mathbb{Z}$ $\tan x = \tan \alpha \implies x = n\pi + \alpha$, $n \in \mathbb{Z}$ 2.3 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 $2\tan^{-1} x = \tan^{-1}\left(\frac{2x}{1-x^2}\right) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ 2.4 Properties of Triangle 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: $r = \frac{\Delta}{s} = 4R \sin(A/2)\sin(B/2)\sin(C/2)$ Exradii: $r_a = \frac{\Delta}{s-a}$ 3. Coordinate Geometry 3.1 Straight Lines Distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ Section formula: $(x,y) = \left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$ (internal division) Slope: $m = \tan \theta = \frac{y_2-y_1}{x_2-x_1}$ Equations of a line: Slope-intercept: $y = mx+c$ Point-slope: $y-y_1 = m(x-x_1)$ Two-point: $y-y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1)$ Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ Normal form: $x \cos \alpha + y \sin \alpha = p$ General form: $Ax+By+C=0$ Angle between two lines: $\tan \theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|$ Distance from a point $(x_1, y_1)$ to $Ax+By+C=0$: $d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$ Family of lines: $L_1 + \lambda L_2 = 0$ Area of triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$: $A = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$ 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 $(x_1, y_1)$: $\sqrt{x_1^2+y_1^2+2gx_1+2fy_1+c}$ Radical axis of two circles $S_1=0, S_2=0$: $S_1-S_2=0$ 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 $(x_1, y_1)$: $yy_1=2a(x+x_1)$ In terms of slope $m$: $y=mx+a/m$ In parametric form: $ty=x+at^2$ 3.4 Ellipse Standard equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a>b$) Foci: $(\pm ae, 0)$ Vertices: $(\pm a, 0)$ Directrices: $x = \pm a/e$ Eccentricity: $e = \sqrt{1 - b^2/a^2}$ Major axis length: $2a$, Minor axis length: $2b$ Latus Rectum: $2b^2/a$ Parametric form: $(a \cos \theta, b \sin \theta)$ Equation of tangent: At $(x_1, y_1)$: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$ In terms of slope $m$: $y=mx \pm \sqrt{a^2m^2+b^2}$ 3.5 Hyperbola Standard equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Foci: $(\pm ae, 0)$ Vertices: $(\pm a, 0)$ Directrices: $x = \pm a/e$ Eccentricity: $e = \sqrt{1 + b^2/a^2}$ Transverse axis length: $2a$, Conjugate axis length: $2b$ Latus Rectum: $2b^2/a$ Asymptotes: $\frac{x}{a} \pm \frac{y}{b} = 0$ Parametric form: $(a \sec \theta, b \tan \theta)$ Equation of tangent: At $(x_1, y_1)$: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$ In terms of slope $m$: $y=mx \pm \sqrt{a^2m^2-b^2}$ 3.6 3D Geometry Distance between points $(x_1,y_1,z_1)$ 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): $l=\cos\alpha, m=\cos\beta, n=\cos\gamma$. $l^2+m^2+n^2=1$. Direction Ratios (DR's): $a,b,c$. $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$, etc. Equation of a line: Vector form: $\vec{r} = \vec{a} + \lambda \vec{b}$ Cartesian form: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ Angle between two lines: $\cos \theta = \frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ Equation of a plane: Normal form: $\vec{r} \cdot \hat{n} = d$ Cartesian form: $Ax+By+Cz+D=0$ Intercept form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ Distance from a point $(x_1,y_1,z_1)$ to plane $Ax+By+Cz+D=0$: $\frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$ Angle between two planes: $\cos \theta = \frac{A_1A_2+B_1B_2+C_1C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}$ Shortest distance between two skew lines: $\frac{|(\vec{a_2}-\vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$ 4. Calculus 4.1 Functions Domain and Range Types of functions: One-one (injective), Onto (surjective), Bijective Even function: $f(-x) = f(x)$, Odd function: $f(-x) = -f(x)$ Periodic function: $f(x+T) = f(x)$ for some $T>0$ Composite functions: $(f \circ g)(x) = f(g(x))$ Inverse functions: $f(g(x)) = x$ and $g(f(x)) = x$ for inverse $g=f^{-1}$ 4.2 Limits, Continuity, and Differentiability Limit: $\lim_{x \to a} f(x)$ exists if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$ Standard Limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ $\lim_{x \to 0} \frac{\tan x}{x} = 1$ $\lim_{x \to 0} \frac{e^x-1}{x} = 1$ $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$ $\lim_{x \to 0} \frac{a^x-1}{x} = \ln a$ $\lim_{x \to 0} (1+x)^{1/x} = e$ $\lim_{x \to \infty} (1+1/x)^x = e$ L'Hopital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of form $0/0$ or $\infty/\infty$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ Continuity: $f(x)$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$ Differentiability: $f(x)$ is differentiable at $x=a$ if $\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ exists. (LHD = RHD) 4.3 Differentiation Product Rule: $(uv)' = u'v + uv'$ Quotient Rule: $(u/v)' = \frac{u'v - uv'}{v^2}$ Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$ Derivatives of Standard Functions: $\frac{d}{dx}(x^n) = nx^{n-1}$ $\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(\tan x) = \sec^2 x$ $\frac{d}{dx}(e^x) = e^x$ $\frac{d}{dx}(\ln x) = 1/x$ $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$ 4.4 Applications of Derivatives (AOD) Rate of change: $\frac{dy}{dx}$ Tangents and Normals: Slope of tangent at $(x_1,y_1)$: $m = (dy/dx)_{(x_1,y_1)}$ Equation of tangent: $y-y_1 = m(x-x_1)$ Slope of normal: $-1/m$ Maxima and Minima: First derivative test: Change of sign of $f'(x)$ at critical point. Second derivative test: If $f'(c)=0$, then if $f''(c)>0$ (local min), $f''(c) Monotonicity: $f'(x) > 0$ (increasing), $f'(x) Rolle's Theorem: If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c \in (a,b)$ such that $f'(c)=0$. Mean Value Theorem (Lagrange): If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$. 4.5 Indefinite Integration Standard Integrals: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$) $\int \frac{1}{x} dx = \ln|x| + C$ $\int \sin x dx = -\cos x + C$ $\int \cos x dx = \sin x + C$ $\int e^x dx = e^x + C$ $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}(x/a) + C$ $\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}(x/a) + C$ $\int \frac{1}{x^2-a^2} dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$ $\int \frac{1}{a^2-x^2} dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$ Integration by Parts: $\int u dv = uv - \int v du$ (LIATE rule for choosing $u$) 4.6 Definite Integration Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x)=f(x)$ Properties of Definite Integrals: $\int_a^b f(x) dx = \int_a^b f(t) dt$ $\int_a^b f(x) dx = -\int_b^a f(x) dx$ $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ $\int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a-x) dx$ If $f(2a-x) = f(x)$, then $2\int_0^a f(x) dx$ If $f(2a-x) = -f(x)$, then $0$ $\int_{-a}^a f(x) dx$: If $f(x)$ is even, $2\int_0^a f(x) dx$ If $f(x)$ is odd, $0$ Wallis Formula: $\int_0^{\pi/2} \sin^n x \cos^m x dx = \frac{[(n-1)!!][(m-1)!!]}{(n+m)!!} K$, where $K=\pi/2$ if $m,n$ both even, else $K=1$. 4.7 Area Under Curves Area bounded by $y=f(x)$, x-axis, $x=a, x=b$: $\int_a^b |f(x)| dx$ Area between $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$: $\int_a^b |f(x)-g(x)| dx$ 4.8 Differential Equations Order: highest derivative present. Degree: power of highest derivative (after making it polynomial). Variable Separable: $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x) dx$ Homogeneous: $\frac{dy}{dx} = f(x,y)$ where $f(x,y)$ is homogeneous of degree 0. Substitute $y=vx$. Linear: $\frac{dy}{dx} + P(x)y = Q(x)$ Integrating Factor (IF): $e^{\int P(x) dx}$ Solution: $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) dx + C$ 5. Vectors Dot Product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_xb_x+a_yb_y+a_zb_z$ $\vec{a} \cdot \vec{a} = |\vec{a}|^2$ If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a} \perp \vec{b}$ Cross Product: $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$ $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$ $|\vec{a} \times \vec{b}|$ = Area of parallelogram Scalar Triple Product (Box Product): $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$ 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: $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$ 6. 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: $P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^n P(B|A_j)P(A_j)}$ Binomial Probability Distribution: $P(X=k) = {}^nC_k p^k (1-p)^{n-k}$ Mean: $np$ Variance: $np(1-p)$ 7. Statistics Mean: $\bar{x} = \frac{\sum x_i}{n}$ Variance: $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - (\bar{x})^2$ Standard Deviation: $\sigma = \sqrt{\sigma^2}$ 8. Mathematical Reasoning Negation of statements: $\neg(P \land Q) \equiv \neg P \lor \neg Q$, $\neg(P \lor Q) \equiv \neg P \land \neg Q$ Conditional statement: $P \implies Q$ (If P then Q) Converse: $Q \implies P$ Contrapositive: $\neg Q \implies \neg P$ (equivalent to $P \implies Q$) Biconditional: $P \iff Q$ (P if and only if Q) Tautology: Always true. Contradiction: Always false.