CBSE Class 12 Mathematics Formula Sheet All Chapters

Relations and Functions

Types of Relations

In this section we intend to define various types of relations on a given set A.
(i) Void relation: Let A be a set. Then φ ⊆ A × A and so it is a relation on A. This relation is called the void or empty relation on A.
(ii) Universal relation: Let A be a set. Then A × A ⊆ A × A and so it is a relation on A. This relation is called the universal relation on A.
(iii) Identity relation: Let A be a set. Then the relation \( I_A = \{(a, a) : a \in A\} \) on A is called the identity relation on A. In other words, a relation \( I_A \) on A is called the identity relation if every element of A is related to itself only.
(iv) Reflexive relation: A relation R on a set A is said to be reflexive if every element of A is related to itself. Thus, R on a set A is not reflexive if there exists an element a ∈ A such that (a, a) ∉ R.
Note: Every identity relation is reflexive but every reflexive relation is not identity.
(v) Symmetric relation: A relation R on a set A is said to be a symmetric relation iff (a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ A, i.e. a R b ⇒ b R a for all a, b ∈ A.
(vi) Transitive relation: Let A be any set. A relation R on A is said to be a transitive relation iff (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ A, i.e. a R b and b R c ⇒ a R c for all a, b, c ∈ A.
(vii) Equivalence relation: A relation R on a set A is said to be an equivalence relation on A iff
(i) it is reflexive, i.e. (a, a) ∈ R for all a ∈ A
(ii) it is symmetric, i.e. (a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ A
(iii) it is transitive, i.e. (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ A.

Inverse Trigonometric Functions

Principal Values and Domains of Inverse Trigonometric Functions

Values within the principal value branch:
(i) \( \sin^{-1}(\sin x) = x, \quad -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \)
(ii) \( \cos^{-1}(\cos x) = x, \quad 0 \leq x \leq \pi \)
(iii) \( \tan^{-1}(\tan x) = x, \quad -\frac{\pi}{2} \lt x \lt \frac{\pi}{2} \)
(iv) \( \cot^{-1}(\cot x) = x, \quad 0 \lt x \lt \pi \)
(v) \( \sec^{-1}(\sec x) = x, \quad 0 \leq x \leq \pi, \; x \neq \frac{\pi}{2} \)
(vi) \( \text{cosec}^{-1}(\text{cosec}\, x) = x, \quad x \neq 0, \; -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \)

Continuity and Differentiability

1. Differentiation of Some Elementary Functions

2. Basic Theorems

1. \( \frac{d}{dx}(f \pm g) = f'(x) \pm g'(x) \)
2. \( \frac{d}{dx}(k\, f(x)) = k \frac{d}{dx} f(x) \)
3. \( \frac{d}{dx}(f(x) \cdot g(x)) = f(x)\, g'(x) + g(x)\, f'(x) \)
4. \( \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)\, f'(x) - f(x)\, g'(x)}{g^2(x)} \)
5. \( \frac{d}{dx}(f(g(x))) = f'(g(x))\, g'(x) \)

Derivative of Inverse Trigonometric Functions
\( \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}, \quad \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} \), for −1 < x < 1.
\( \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}, \quad \frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2} \) (x ∈ R)
\( \frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x| \sqrt{x^2 - 1}}, \quad \frac{d}{dx}(\text{cosec}^{-1} x) = -\frac{1}{|x| \sqrt{x^2 - 1}} \), for x ∈ (−∞, −1) ∪ (1, ∞)

3. Differentiation Using Substitution

Following substitutions are normally used to simplify these expressions.
(i) \( \sqrt{x^2 + a^2} \) by substituting x = a tan θ, where \( -\frac{\pi}{2} \lt \theta \lt \frac{\pi}{2} \)
(ii) \( \sqrt{a^2 - x^2} \) by substituting x = a sin θ, where \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \)
(iii) \( \sqrt{x^2 - a^2} \) by substituting x = a sec θ, where θ ∈ [0, π], \( \theta \neq \frac{\pi}{2} \)
(iv) \( \sqrt{\frac{x + a}{a - x}} \) by substituting x = a cos θ, where θ ∈ (0, π].

4. Parametric Differentiation

If y = f(θ) and x = g(θ), where θ is a parameter, then \( \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \).

5. Derivative of One Function with Respect to Another

Let y = f(x); z = g(x), then \( \frac{dy}{dz} = \frac{dy/dx}{dz/dx} = \frac{f'(x)}{g'(x)} \).

Application of Derivatives

Useful Formulae of Mensuration to Remember

These formulae are used in problems on rate of change and maxima and minima.
1. Volume of a cuboid = lbh.
2. Surface area of a cuboid = 2(lb + bh + hl).
3. Volume of a cube = a³.
4. Surface area of a cube = 6a².
5. Volume of a cone = \( \frac{1}{3} \pi r^2 h \).
6. Curved surface area of a cone = πrl (l = slant height).
7. Curved surface area of a cylinder = 2πrh.
8. Total surface area of a cylinder = 2πrh + 2πr².
9. Volume of a sphere = \( \frac{4}{3} \pi r^3 \).
10. Surface area of a sphere = 4πr².
11. Area of a circular sector = \( \frac{1}{2} r^2 \theta \), when θ is in radians.
12. Volume of a prism = (area of the base) × (height).
13. Lateral surface area of a prism = (perimeter of the base) × (height).
14. Total surface area of a prism = (lateral surface area) + 2 (area of the base). (Note that lateral surfaces of a prism are all rectangles.)
15. Volume of a pyramid = \( \frac{1}{3} \) (area of the base) × (height).
16. Curved surface area of a pyramid = \( \frac{1}{2} \) (perimeter of the base) × (slant height). (Note that slant surfaces of a pyramid are triangles.)

Integrals

Indefinite Integration

1. If f and g are functions of x such that g′(x) = f(x), then
\( \int f(x)\, dx = g(x) + c \iff \frac{d}{dx}\{g(x) + c\} = f(x) \), where c is called the constant of integration.

2. Standard Formulae
(i) \( \int (ax + b)^n\, dx = \frac{(ax + b)^{n+1}}{a(n + 1)} + c, \; n \neq -1 \)
(ii) \( \int \frac{dx}{ax + b} = \frac{1}{a} \ln |ax + b| + c \)
(iii) \( \int e^{ax + b}\, dx = \frac{1}{a} e^{ax + b} + c \)
(iv) \( \int a^{px + q}\, dx = \frac{1}{p} \cdot \frac{a^{px + q}}{\ln a} + c; \; a \gt 0 \)
(v) \( \int \sin(ax + b)\, dx = -\frac{1}{a} \cos(ax + b) + c \)
(vi) \( \int \cos(ax + b)\, dx = \frac{1}{a} \sin(ax + b) + c \)
(vii) \( \int \tan(ax + b)\, dx = \frac{1}{a} \ln |\sec(ax + b)| + c \)
(viii) \( \int \cot(ax + b)\, dx = \frac{1}{a} \ln |\sin(ax + b)| + c \)
(ix) \( \int \sec^2(ax + b)\, dx = \frac{1}{a} \tan(ax + b) + c \)
(x) \( \int \text{cosec}^2(ax + b)\, dx = -\frac{1}{a} \cot(ax + b) + c \)
(xi) \( \int \sec x\, dx = \ln |\sec x + \tan x| + c \) OR \( \ln \left|\tan\left(\frac{\pi}{4} + \frac{x}{2}\right)\right| + c \)
(xii) \( \int \text{cosec}\, x\, dx = \ln |\text{cosec}\, x - \cot x| + c \) OR \( \ln \left|\tan \frac{x}{2}\right| + c \) OR \( -\ln |\text{cosec}\, x + \cot x| + c \)
(xiii) \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \frac{x}{a} + c \)
(xiv) \( \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1} \frac{x}{a} + c \)
(xv) \( \int \frac{dx}{|x| \sqrt{x^2 - a^2}} = \frac{1}{a} \sec^{-1} \frac{x}{a} + c \)
(xvi) \( \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln \left| x + \sqrt{x^2 + a^2} \right| + c \)
(xvii) \( \int \frac{dx}{\sqrt{x^2 - a^2}} = \ln \left| x + \sqrt{x^2 - a^2} \right| + c \)
(xviii) \( \int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + c \)
(xix) \( \int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln \left| \frac{x - a}{x + a} \right| + c \)
(xx) \( \int \sqrt{a^2 - x^2}\, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c \)
(xxi) \( \int \sqrt{x^2 + a^2}\, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln \left| \frac{x + \sqrt{x^2 + a^2}}{a} \right| + c \)
(xxii) \( \int \sqrt{x^2 - a^2}\, dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \ln \left| \frac{x + \sqrt{x^2 - a^2}}{a} \right| + c \)

3. Integration by Substitution
If we substitute f(x) = t, then f′(x) dx = dt.

4. Integration by Parts
\( \int f(x)\, g(x)\, dx = f(x) \int g(x)\, dx - \int \left( \frac{d}{dx} f(x) \int g(x)\, dx \right) dx \)

5. Integration of the type \( \int \frac{dx}{ax^2 + bx + c}, \; \int \frac{dx}{\sqrt{ax^2 + bx + c}} \)
Make the substitution \( x + \frac{b}{2a} = t \).

6. Integration of the type \( \int \frac{px + q}{ax^2 + bx + c}\, dx, \; \int \frac{px + q}{\sqrt{ax^2 + bx + c}}\, dx \)
Make the substitution \( x + \frac{b}{2a} = t \), then split the integral as the sum of two integrals, one containing the linear term and the other containing the constant term.

Properties of Definite Integrals

1. \( \int_{a}^{b} f(x)\, dx = \int_{a}^{b} f(t)\, dt \)
2. \( \int_{a}^{b} f(x)\, dx = -\int_{b}^{a} f(x)\, dx \)
3. \( \int_{a}^{b} f(x)\, dx = \int_{a}^{c} f(x)\, dx + \int_{c}^{b} f(x)\, dx \)
4. \( \int_{-a}^{a} f(x)\, dx = \int_{0}^{a} (f(x) + f(-x))\, dx = \begin{cases} 2\int_{0}^{a} f(x)\, dx, & f(-x) = f(x) \\ 0, & f(-x) = -f(x) \end{cases} \)
5. \( \int_{a}^{b} f(x)\, dx = \int_{a}^{b} f(a + b - x)\, dx \)
6. \( \int_{0}^{a} f(x)\, dx = \int_{0}^{a} f(a - x)\, dx \)
7. \( \int_{0}^{2a} f(x)\, dx = \int_{0}^{a} (f(x) + f(2a - x))\, dx = \begin{cases} 2\int_{0}^{a} f(x)\, dx, & f(2a - x) = f(x) \\ 0, & f(2a - x) = -f(x) \end{cases} \)

Vector Algebra

I. Position Vector of a Point

Let O be a fixed origin, then the position vector of a point P is the vector \( \overrightarrow{OP} \). If \( \vec{a} \) and \( \vec{b} \) are position vectors of two points A and B, then
\( \overrightarrow{AB} = \vec{b} - \vec{a} \) = p.v. of B − p.v. of A.
Distance formula: Distance between the two points A(\( \vec{a} \)) and B(\( \vec{b} \)) is AB = \( |\vec{a} - \vec{b}| \).
Section formula: \( \vec{r} = \frac{n\vec{a} + m\vec{b}}{m + n} \). Mid point of AB = \( \frac{\vec{a} + \vec{b}}{2} \).
Position vector of point P(x, y, z) is \( x\hat{i} + y\hat{j} + z\hat{k} \).

II. Scalar Product of Two Vectors

\( \vec{a} \cdot \vec{b} = |\vec{a}|\, |\vec{b}| \cos \theta \), where \( |\vec{a}|, |\vec{b}| \) are the magnitudes of \( \vec{a} \) and \( \vec{b} \) respectively and θ is the angle between \( \vec{a} \) and \( \vec{b} \).
1. \( \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1; \; \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0 \)
Projection of \( \vec{a} \) on \( \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \)
2. If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), then \( \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \)
3. The angle φ between \( \vec{a} \) and \( \vec{b} \) is given by \( \cos \phi = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|\, |\vec{b}|}, \; 0 \leq \phi \leq \pi \)
4. \( \vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b} \quad (\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}) \)

III. Vector Product of Two Vectors

1. If \( \vec{a} \) and \( \vec{b} \) are two vectors and θ is the angle between them, then \( \vec{a} \times \vec{b} = |\vec{a}|\, |\vec{b}| \sin \theta\, \hat{n} \), where \( \hat{n} \) is the unit vector perpendicular to both \( \vec{a} \) and \( \vec{b} \) such that \( \vec{a}, \vec{b} \) and \( \hat{n} \) form a right handed screw system.
2. Geometrically \( |\vec{a} \times \vec{b}| \) = area of the parallelogram whose two adjacent sides are represented by \( \vec{a} \) and \( \vec{b} \).
3. \( \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \vec{0}; \; \hat{i} \times \hat{j} = \hat{k}, \; \hat{j} \times \hat{k} = \hat{i}, \; \hat{k} \times \hat{i} = \hat{j} \)
4. If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), then \( \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \)
5. \( \vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \) and \( \vec{b} \) are parallel (collinear) \( (\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}) \), i.e. \( \vec{a} = K\vec{b} \), where K is a scalar.
6. Unit vector perpendicular to the plane of \( \vec{a} \) and \( \vec{b} \) is \( \hat{n} = \pm \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} \)
7. If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are the p.v.'s of 3 points A, B and C, then the vector area of triangle ABC = \( \frac{1}{2}\left[\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}\right] \). The points A, B and C are collinear if \( \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \vec{0} \).

Three Dimensional Geometry

1. Direction Cosines and Direction Ratios

(i) Direction cosines: Let α, β, γ be the angles which a directed line makes with the positive directions of the axes of x, y and z respectively, then cos α, cos β, cos γ are called the direction cosines of the line. The direction cosines are usually denoted by (l, m, n). Thus l = cos α, m = cos β, n = cos γ.
(ii) If l, m, n are the direction cosines of a line, then l² + m² + n² = 1.
(iii) Direction ratios: Let a, b, c be proportional to the direction cosines l, m, n, then a, b, c are called the direction ratios.
(iv) If l, m, n are the direction cosines and a, b, c are the direction ratios of a vector, then
\( l = \pm \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \; m = \pm \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \; n = \pm \frac{c}{\sqrt{a^2 + b^2 + c^2}} \)
(v) If the coordinates of P and Q are (x₁, y₁, z₁) and (x₂, y₂, z₂), then the direction ratios of line PQ are a = x₂ − x₁, b = y₂ − y₁ and c = z₂ − z₁, and the direction cosines of line PQ are \( l = \frac{x_2 - x_1}{|PQ|}, \; m = \frac{y_2 - y_1}{|PQ|} \) and \( n = \frac{z_2 - z_1}{|PQ|} \).

2. Equation of a Line

(i) Symmetric (Cartesian) form: \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = r \)
(ii) Vector equation: \( \vec{r} = \vec{a} + \lambda \vec{b} \)
(iii) Reduction of Cartesian form of equation of a line to vector form and vice versa:
\( \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \iff \vec{r} = (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) + \lambda (a\hat{i} + b\hat{j} + c\hat{k}) \)

3. Angle Between Two Lines

\( \cos \theta = \left| \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\, \sqrt{a_2^2 + b_2^2 + c_2^2}} \right| \)
The lines will be perpendicular if a₁a₂ + b₁b₂ + c₁c₂ = 0, and parallel if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).

4. Skew Lines

(i) The straight lines which are not parallel and non-coplanar, i.e. non-intersecting, are called skew lines.
(ii) Shortest distance between the lines \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu \vec{b}_2 \) is \( d = \left| \frac{(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|} \right| \)
(iii) Shortest distance between the parallel lines \( \vec{r} = \vec{a}_1 + \lambda \vec{b} \) and \( \vec{r} = \vec{a}_2 + \mu \vec{b} \) is \( d = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|} \)

Probability

1. Conditional Probability: \( P(A/B) = \frac{P(A \cap B)}{P(B)} \).
2. Total Probability Theorem: \( P(A) = \sum_{i=1}^{n} P(B_i) \cdot P(A/B_i) \)
3. Bayes' Theorem: If an event A can occur with one of the n mutually exclusive and exhaustive events B₁, B₂, ……, B_n and the probabilities P(A/B₁), P(A/B₂), …, P(A/B_n) are known, then
\( P(B_i / A) = \frac{P(B_i) \cdot P(A/B_i)}{\sum_{i=1}^{n} P(B_i) \cdot P(A/B_i)} \)
\( A = (A \cap B_1) \cup (A \cap B_2) \cup (A \cap B_3) \cup \ldots \cup (A \cap B_n) \)
\( P(A) = P(A \cap B_1) + P(A \cap B_2) + \ldots + P(A \cap B_n) = \sum_{i=1}^{n} P(A \cap B_i) \)