Class 11 Maths Formulas

1. Sets, Relations & Functions

Cardinality Rules

Number of elements in union of sets:

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$
3-Set Union

Union of three non-empty sets:

$$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$$
De Morgan's Laws

Set Complement identities:

$$(A \cup B)' = A' \cap B' \quad \text{and} \quad (A \cap B)' = A' \cup B'$$
Relations

Total number of relations from set A to set B:

$$\text{If } n(A) = p \text{ and } n(B) = q \implies \text{Total Relations} = 2^{pq}$$

2. Trigonometric Functions

Radian Conversion & Arc
$$1\text{ rad} = \frac{180^\circ}{\pi}, \quad 1^\circ = \frac{\pi}{180}\text{ rad}, \quad l = r\theta$$
Compound Angles
$$\sin(X \pm Y) = \sin X \cos Y \pm \cos X \sin Y$$ $$\cos(X \pm Y) = \cos X \cos Y \mp \sin X \sin Y$$ $$\tan(X \pm Y) = \frac{\tan X \pm \tan Y}{1 \mp \tan X \tan Y}$$
Double Angle Formulas
$$\sin 2X = 2\sin X \cos X = \frac{2\tan X}{1 + \tan^2 X}$$ $$\cos 2X = \cos^2 X - \sin^2 X = 1 - 2\sin^2 X = 2\cos^2 X - 1$$ $$\tan 2X = \frac{2\tan X}{1 - \tan^2 X}$$
Triple Angle Formulas
$$\sin 3X = 3\sin X - 4\sin^3 X$$ $$\cos 3X = 4\cos^3 X - 3\cos X$$ $$\tan 3X = \frac{3\tan X - \tan^3 X}{1 - 3\tan^2 X}$$
Sum to Product & Product to Sum Transformations
$$\sin C + \sin D = 2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$$ $$\cos C + \cos D = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$$
$$\cos C - \cos D = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)$$ $$2\sin A \cos B = \sin(A+B) + \sin(A-B)$$

3. Complex Numbers & Quadratic Equations

Modulus & Conjugate

For complex number $z = a + ib$:

$$|z| = \sqrt{a^2 + b^2}, \quad \bar{z} = a - ib, \quad z^{-1} = \frac{\bar{z}}{|z|^2}$$
Polar Form & Roots
$$\text{Polar: } z = r(\cos\theta + i\sin\theta)$$ $$\text{Quadratic Roots: } x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

4. Permutations & Combinations

Permutation & Combination
$$^nP_r = \frac{n!}{(n-r)!}, \quad ^nC_r = \frac{n!}{r!(n-r)!}$$
Combination Identities
$$^nC_r = ^nC_{n-r}$$ $$^nC_r + ^nC_{r-1} = ^{n+1}C_r$$

5. Binomial Theorem

Expansion & General Term
$$(a+b)^n = \sum_{k=0}^{n} {^nC_k} a^{n-k} b^k$$ $$\text{General Term: } T_{r+1} = {^nC_r} a^{n-r} b^r$$
Middle Term

• If $n$ is even $\implies \left(\frac{n}{2} + 1\right)\text{th term}$
• If $n$ is odd $\implies \left(\frac{n+1}{2}\right)\text{th and } \left(\frac{n+3}{2}\right)\text{th terms}$

6. Sequences & Series

Arithmetic Progression (AP)
$$a_n = a + (n-1)d$$ $$S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)$$
Geometric Progression (GP)
$$a_n = a r^{n-1}, \quad S_n = \frac{a(r^n - 1)}{r - 1} \ (r \neq 1)$$ $$\text{Infinite Sum } (|r| < 1): S_\infty = \frac{a}{1 - r}$$

7. Straight Lines

Slope & Line Forms
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta$$ $$\text{Slope-Intercept: } y = mx + c$$ $$\text{Intercept Form: } \frac{x}{a} + \frac{y}{b} = 1$$
Distance Formulas

Point to line distance:

$$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$

Between parallel lines:

$$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$

8. Conic Sections

Conic Standard Equation Key Elements
Circle $(x - h)^2 + (y - k)^2 = r^2$ Center $(h,k)$, Radius $r$
Parabola $y^2 = 4ax$ Focus $(a,0)$, Directrix $x = -a$
Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a>b)$ Eccentricity $e = \sqrt{1 - \frac{b^2}{a^2}}$
Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Eccentricity $e = \sqrt{1 + \frac{b^2}{a^2}}$

9. Three Dimensional Geometry

3D Distance Formula
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
Section Formula
$$P(x,y,z) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)$$

10. Limits & Derivatives

Standard Limits
$$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n \cdot a^{n-1}$$ $$\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0$$
Derivative Rules
$$\text{Product: } (uv)' = u'v + uv'$$ $$\text{Quotient: } \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$

11. Statistics & Probability

Variance & Standard Deviation
$$\text{Variance } (\sigma^2) = \frac{1}{N} \sum (x_i - \bar{x})^2$$ $$\text{Standard Deviation } (\sigma) = \sqrt{\text{Variance}}$$
Probability Rules
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ $$\text{If Mutually Exclusive: } P(A \cup B) = P(A) + P(B)$$
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