CUET Mathematics Formula Sheet

This formula sheet provides a comprehensive collection of essential mathematical formulas and concepts crucial for success in the CUET Class 12 Mathematics examination. Covering key topics from Algebra, Calculus, and Vectors to Three-Dimensional Geometry and Probability, it aims to be a quick reference guide for last-minute revision and problem-solving.

Key Concepts

Relations and Functions:

• Types of Relations: Reflexive ($aRa$), Symmetric ($aRb \implies bRa$), Transitive ($aRb, bRc \implies aRc$), Equivalence.
• Types of Functions: One-one (injective), Onto (surjective), Bijective.
• Composition of Functions: $(g \circ f)(x) = g(f(x))$.
• Inverse Functions: $f^{-1}(y) = x \iff f(x) = y$.

Inverse Trigonometric Functions:

• Domain and Range of principal value branches.
• Properties involving sum/difference, and reciprocal identities.

Matrices and Determinants:

• Types of Matrices: Square, Diagonal, Scalar, Identity, Symmetric ($A^T = A$), Skew-symmetric ($A^T = -A$).
• Operations: Addition, Subtraction, Multiplication.
• Determinant: Scalar value associated with a square matrix. Used for finding inverse and solving systems of linear equations.
• Adjoint and Inverse: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$. A is invertible if $\det(A) \neq 0$.

Continuity and Differentiability:

• Continuity: A function $f(x)$ is continuous at $x=c$ if $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$.
• Differentiability: Existence of $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Implies continuity.
• Derivatives of Standard Functions: Power rule, trigonometric functions, exponential, logarithmic.
• Chain Rule, Product Rule, Quotient Rule.

Applications of Derivatives:

• Rate of Change: $\frac{dy}{dx}$.
• Increasing/Decreasing Functions: $f'(x) > 0$ for increasing, $f'(x) < 0$ for decreasing.
• Tangents and Normals: Slope of tangent $m_T = f'(x_0)$, slope of normal $m_N = -\frac{1}{f'(x_0)}$.
• Maxima and Minima: First derivative test, Second derivative test.

Integrals:

• Indefinite Integrals: Antiderivatives.
• Methods of Integration: Substitution, Integration by Parts, Partial Fractions.
• Definite Integrals: Area under the curve.
• Properties of Definite Integrals: Linearity, $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$, $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$, etc.

Applications of Integrals:

• Area Under Simple Curves: Area bounded by curves.

Differential Equations:

• Order and Degree: Order of highest order derivative, degree of highest power of highest order derivative.
• Methods of Solving: Variable separable, Homogeneous, Linear differential equation ($\frac{dy}{dx} + Py = Q$).

Vectors:

• Types of Vectors: Zero, Unit, Co-initial, Collinear.
• Scalar (Dot) Product: Proj. of one vector on another.
• Vector (Cross) Product: Perpendicular vector, Area of parallelogram/triangle.
• Scalar Triple Product: Volume of parallelepiped, coplanarity.

Three-Dimensional Geometry:

• Direction Cosines and Ratios: For a line.
• Equation of a Line: Passing through a point and parallel to a vector/two points.
• Equation of a Plane: Normal form, intercept form, passing through a point and perpendicular to a vector, passing through three non-collinear points.
• Shortest Distance: Between two skew lines, parallel lines.
• Angle: Between two lines, two planes, line and plane.

Linear Programming:

• Objective Function: To be maximized or minimized ($Z = ax + by$).
• Constraints: Linear inequalities.
• Feasible Region: Common region satisfying all constraints.
• Optimal Solution: Vertex of the feasible region.

Probability:

• Conditional Probability: $P(E|F) = \frac{P(E \cap F)}{P(F)}$.
• Multiplication Theorem: $P(A \cap B) = P(A)P(B|A)$.
• Independent Events: $P(E \cap F) = P(E)P(F)$.
• Total Probability Theorem:
• Bayes' Theorem: $P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum_{j=1}^n P(E_j)P(A|E_j)}$.
• Random Variables and Probability Distributions: Mean, Variance.

Formulas & Results

Relations & Functions

• If $f: A \to B$ and $g: B \to C$, then $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

Inverse Trigonometric Functions

• $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, |x| \le 1$
• $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}, x \in \mathbb{R}$
• $2\tan^{-1} x = \tan^{-1} \left(\frac{2x}{1-x^2}\right)$
• $2\tan^{-1} x = \sin^{-1} \left(\frac{2x}{1+x^2}\right)$
• $2\tan^{-1} x = \cos^{-1} \left(\frac{1-x^2}{1+x^2}\right)$

Matrices and Determinants

• $\det(AB) = \det(A)\det(B)$
• $\det(kA) = k^n \det(A)$ for $n \times n$ matrix A
• $A (\text{adj} A) = (\text{adj} A) A = \det(A) I$
• System of linear equations: $AX = B$. If $\det(A) \neq 0$, then $X = A^{-1}B$.
• If $\det(A) = 0$ and $(\text{adj} A) B \neq 0$, then inconsistent.
• If $\det(A) = 0$ and $(\text{adj} A) B = 0$, then infinitely many solutions or no solution.

Differentiation

• $\frac{d}{dx} (\sin x) = \cos x$
• $\frac{d}{dx} (\cos x) = -\sin x$
• $\frac{d}{dx} (\tan x) = \sec^2 x$
• $\frac{d}{dx} (e^x) = e^x$
• $\frac{d}{dx} (\log x) = \frac{1}{x}$
• 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: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Integration

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C, n \neq -1$
• $\int \frac{1}{x} dx = \log|x| + C$
• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int e^x dx = e^x + C$
• Integration by Parts: $\int u v' dx = uv - \int u'v dx$ or $\int u dv = uv - \int v du$
• Definite Integral property: $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$
• $\int_0^a f(x)dx = \int_0^a f(a-x)dx$
• $\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx$ if $f$ is even, $0$ if $f$ is odd.
• Standard Integrals:
  • $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$
  • $\int \frac{1}{a^2 + x^2} dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$
  • $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$

Differential Equations

• Linear Differential Equation: $\frac{dy}{dx} + Py = Q$. Integrating factor (I.F.) $= e^{\int P dx}$. Solution: $y (\text{I.F.}) = \int Q (\text{I.F.}) dx + C$.

Vectors

• Scalar Product: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta$. If $\vec{a} = a_i\hat{i} + a_j\hat{j} + a_k\hat{k}$ and $\vec{b} = b_i\hat{i} + b_j\hat{j} + b_k\hat{k}$, then $\vec{a} \cdot \vec{b} = a_ib_i + a_jb_j + a_kb_k$.
• Vector Product: $\vec{a} \times \vec{b} = (|\vec{a}| |\vec{b}| \sin\theta) \hat{n}$. $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_i & a_j & a_k \ b_i & b_j & b_k \end{vmatrix}$. Area of parallelogram $= |\vec{a} \times \vec{b}|$.
• Scalar Triple Product: $[\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$. Volume of parallelepiped $= |[\vec{a}, \vec{b}, \vec{c}]|$.

Three-Dimensional Geometry

• Equation of a Line (vector form): $\vec{r} = \vec{a} + \lambda\vec{b}$.
• Equation of a Plane (normal form): $\vec{r} \cdot \hat{n} = d$.
• Shortest Distance between skew lines: $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ is $D = \left| \frac{(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})}{|\vec{b_1} \times \vec{b_2}|} \right|$.

Probability

• Bayes' Theorem: $P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum_{j=1}^n P(E_j)P(A|E_j)}$
• Variance of a Random Variable: $\text{Var}(X) = E(X^2) - [E(X)]^2$.
• Binomial Distribution: $P(X=k) = C(n, k) p^k (1-p)^{n-k}$. Mean $= np$, Variance $= np(1-p)$.

Worked Examples

Example 1: Inverse Trigonometric Functions Prove that $\tan^{-1} \left(\frac{1}{2}\right) + \tan^{-1} \left(\frac{2}{11}\right) = \tan^{-1} \left(\frac{3}{4}\right)$.

Solution: We use the formula $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left(\frac{x+y}{1-xy}\right)$, provided $xy < 1$. Here, $x = \frac{1}{2}$ and $y = \frac{2}{11}$. $xy = \frac{1}{2} \times \frac{2}{11} = \frac{1}{11} < 1$, so the formula is applicable. LHS = $\tan^{-1} \left(\frac{\frac{1}{2} + \frac{2}{11}}{1 - \frac{1}{2} \times \frac{2}{11}}\right)$ = $\tan^{-1} \left(\frac{\frac{11+4}{22}}{1 - \frac{2}{22}}\right)$ = $\tan^{-1} \left(\frac{\frac{15}{22}}{\frac{20}{22}}\right)$ = $\tan^{-1} \left(\frac{15}{20}\right)$ = $\tan^{-1} \left(\frac{3}{4}\right)$ = RHS. Hence Proved.

Example 2: Differential Equations Find the general solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$.

Solution: This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$, where $P = \frac{1}{x}$ and $Q = x^2$. First, we find the integrating factor (I.F.): I.F. $= e^{\int P dx} = e^{\int \frac{1}{x} dx} = e^{\log|x|} = |x|$. We can take I.F. $= x$ (assuming $x > 0$). The general solution is given by $y (\text{I.F.}) = \int Q (\text{I.F.}) dx + C$. $y \cdot x = \int x^2 \cdot x , dx + C$ $xy = \int x^3 , dx + C$ $xy = \frac{x^4}{4} + C$ So, the general solution is $y = \frac{x^3}{4} + \frac{C}{x}$.

Example 3: Vectors Find the area of the parallelogram whose adjacent sides are given by the vectors $\vec{a} = 3\hat{i} + \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$.

Solution: The area of a parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ is given by $|\vec{a} \times \vec{b}|$. First, calculate the cross product $\vec{a} \times \vec{b}$: $$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & 1 & 4 \ 1 & -1 & 1 \end{vmatrix} $$ $= \hat{i}(1 \cdot 1 - 4 \cdot (-1)) - \hat{j}(3 \cdot 1 - 4 \cdot 1) + \hat{k}(3 \cdot (-1) - 1 \cdot 1)$ $= \hat{i}(1 + 4) - \hat{j}(3 - 4) + \hat{k}(-3 - 1)$ $= 5\hat{i} + \hat{j} - 4\hat{k}$ Next, find the magnitude of this vector: $|\vec{a} \times \vec{b}| = \sqrt{5^2 + 1^2 + (-4)^2}$ $= \sqrt{25 + 1 + 16}$ $= \sqrt{42}$ Therefore, the area of the parallelogram is $\sqrt{42}$ square units.

Common Mistakes

• Forgetting the constant of integration (C) in indefinite integrals.
• Incorrectly applying chain rule, product rule, or quotient rule in differentiation.
• Misinterpreting conditions for continuity and differentiability. A function can be continuous but not differentiable (e.g., $f(x)=|x|$ at $x=0$).
• Errors in matrix multiplication order (non-commutative).
• Confusing scalar and vector products and their properties.
• Incorrectly determining the feasible region in linear programming problems.
• Ignoring the domain and range of inverse trigonometric functions when applying identities.
• Using incorrect limits of integration for definite integrals, especially with properties.
• Sign errors in determinant calculations or vector operations.
• Applying Bayes' Theorem without correctly identifying $P(E_i)$, $P(A|E_i)$, and the total probability in the denominator.

Quick Revision Checklist

• Relations & Functions: Types of relations, one-one, onto, composition, inverse functions.
• Inverse Trigonometry: Principal values, key identities.
• Matrices & Determinants: Matrix operations, determinant properties, adjoint, inverse, solving system of equations.
• Calculus Basics: Continuity, differentiability, derivatives of standard functions, fundamental rules (chain, product, quotient).
• Applications of Derivatives: Tangents/normals, rates of change, increasing/decreasing functions, maxima/minima (first & second derivative tests).
• Integrals: Methods of integration (substitution, parts, partial fractions), standard formulas, properties of definite integrals.
• Applications of Integrals: Area under curves.
• Differential Equations: Order/degree, variable separable, homogeneous, linear differential equations and their solutions.
• Vectors: Dot & cross products, magnitudes, unit vectors, projection, area/volume applications.
• 3D Geometry: Direction cosines/ratios, equations of lines/planes, shortest distance.
• Linear Programming: Objective function, constraints, feasible region, corner point method.
• Probability: Conditional probability, independent events, total probability, Bayes' Theorem, binomial distribution.