All Class 11 Maths formulas

Basic identities

Pythagorean identities

$\displaystyle \sin^2x + \cos^2x = 1, \quad 1 + \tan^2x = \sec^2x, \quad 1 + \cot^2x = \csc^2x$

Radians and degrees

$\displaystyle \pi\ \text{rad} = 180^\circ$

Arc length

$\displaystyle l = r\theta$

θ in radians.

Sum and difference

Sine

$\displaystyle \sin(x \pm y) = \sin x\cos y \pm \cos x\sin y$

Cosine

$\displaystyle \cos(x \pm y) = \cos x\cos y \mp \sin x\sin y$

Tangent

$\displaystyle \tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x\tan y}$

Double and triple angle

sin 2x

$\displaystyle \sin 2x = 2\sin x\cos x = \frac{2\tan x}{1 + \tan^2x}$

cos 2x

$\displaystyle \cos 2x = \cos^2x - \sin^2x = 2\cos^2x - 1 = 1 - 2\sin^2x$

tan 2x

$\displaystyle \tan 2x = \frac{2\tan x}{1 - \tan^2x}$

Triple angle

$\displaystyle \sin 3x = 3\sin x - 4\sin^3x, \quad \cos 3x = 4\cos^3x - 3\cos x$

Sum to product

Sines

$\displaystyle \sin x + \sin y = 2\sin\frac{x + y}{2}\cos\frac{x - y}{2}$

Cosines

$\displaystyle \cos x + \cos y = 2\cos\frac{x + y}{2}\cos\frac{x - y}{2}$

Difference of cosines

$\displaystyle \cos x - \cos y = -2\sin\frac{x + y}{2}\sin\frac{x - y}{2}$

General solutions

sin x = 0 and cos x = 0

$\displaystyle \sin x = 0 \Rightarrow x = n\pi, \quad \cos x = 0 \Rightarrow x = (2n + 1)\frac{\pi}{2}$

sin x = sin y

$\displaystyle x = n\pi + (-1)^n y$

cos x = cos y

$\displaystyle x = 2n\pi \pm y$

tan x = tan y

$\displaystyle x = n\pi + y$

Complex numbers

Powers of i

$\displaystyle i^2 = -1, \quad i^3 = -i, \quad i^4 = 1$

Modulus

$\displaystyle |z| = \sqrt{a^2 + b^2}$

For z = a + ib.

Conjugate

$\displaystyle \bar z = a - ib, \quad z\bar z = |z|^2$

Inverse

$\displaystyle z^{-1} = \frac{\bar z}{|z|^2}$

Polar form

$\displaystyle z = r(\cos\theta + i\sin\theta)$

Modulus rules

$\displaystyle |z_1z_2| = |z_1||z_2|, \quad |z_1 + z_2| \le |z_1| + |z_2|$

Quadratic equations

Roots

$\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Sum and product of roots

$\displaystyle \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}$

Discriminant

$\displaystyle D = b^2 - 4ac$

D < 0 gives complex conjugate roots.

Core formulas

Factorial

$\displaystyle n! = n(n - 1)(n - 2)\cdots 1, \quad 0! = 1$

Permutations

$\displaystyle {}^nP_r = \frac{n!}{(n - r)!}$

When order matters.

Combinations

$\displaystyle {}^nC_r = \frac{n!}{r!(n - r)!}$

When order doesn't matter.

Link between them

$\displaystyle {}^nP_r = r!\;{}^nC_r$

Useful results

Complement

$\displaystyle {}^nC_r = {}^nC_{n - r}$

Pascal's rule

$\displaystyle {}^nC_r + {}^nC_{r - 1} = {}^{n + 1}C_r$

Arrangements with repeats

$\displaystyle \frac{n!}{p_1!\,p_2!\cdots p_k!}$

p₁ things alike, p₂ alike, and so on.

With repetition allowed

$\displaystyle n^r$

Arranging r places from n things.

Formulas

Binomial theorem

$\displaystyle (a + b)^n = \sum_{r=0}^{n} {}^nC_r\,a^{n - r}b^r$

General term

$\displaystyle T_{r + 1} = {}^nC_r\,a^{n - r}b^r$

Number of terms

$\displaystyle n + 1$

Sum of coefficients

$\displaystyle \sum_{r=0}^{n} {}^nC_r = 2^n$

Sum of odd and even coefficients

$\displaystyle {}^nC_0 + {}^nC_2 + \cdots = {}^nC_1 + {}^nC_3 + \cdots = 2^{n - 1}$

Middle term

• If n is even, the one middle term is term number (n/2 + 1).
• If n is odd, there are two middle terms: numbers (n + 1)/2 and (n + 3)/2.

Arithmetic progression

nth term

$\displaystyle a_n = a + (n - 1)d$

Sum of n terms

$\displaystyle S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)$

l is the last term.

Arithmetic mean

$\displaystyle A = \frac{a + b}{2}$

Geometric progression

nth term

$\displaystyle a_n = ar^{n - 1}$

Sum of n terms

$\displaystyle S_n = \frac{a(r^n - 1)}{r - 1}$

For r ≠ 1.

Sum to infinity

$\displaystyle S_\infty = \frac{a}{1 - r}$

Only when |r| < 1.

Geometric mean

$\displaystyle G = \sqrt{ab}$

AM and GM

$\displaystyle A \ge G$

For positive numbers.

Standard sums

First n natural numbers

$\displaystyle \sum k = \frac{n(n + 1)}{2}$

Squares

$\displaystyle \sum k^2 = \frac{n(n + 1)(2n + 1)}{6}$

Cubes

$\displaystyle \sum k^3 = \left[\frac{n(n + 1)}{2}\right]^2$

Basics

Distance between points

$\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Section formula

$\displaystyle \left(\frac{mx_2 + nx_1}{m + n},\; \frac{my_2 + ny_1}{m + n}\right)$

Internal division in ratio m : n.

Slope

$\displaystyle m = \tan\theta = \frac{y_2 - y_1}{x_2 - x_1}$

Angle between lines

$\displaystyle \tan\theta = \left|\frac{m_2 - m_1}{1 + m_1m_2}\right|$

Parallel and perpendicular

$\displaystyle m_1 = m_2, \quad m_1m_2 = -1$

Forms of a line

Point-slope

$\displaystyle y - y_1 = m(x - x_1)$

Slope-intercept

$\displaystyle y = mx + c$

Two-point

$\displaystyle y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$

Intercept

$\displaystyle \frac{x}{a} + \frac{y}{b} = 1$

Normal

$\displaystyle x\cos\omega + y\sin\omega = p$

Distances

Point to line

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

Between parallel lines

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

A and B must match in both lines.

Circle

Equation

$\displaystyle (x - h)^2 + (y - k)^2 = r^2$

General form

$\displaystyle x^2 + y^2 + 2gx + 2fy + c = 0$

Centre (−g, −f), radius √(g² + f² − c).

Parabola

Standard form

$\displaystyle y^2 = 4ax$

Focus (a, 0), directrix x = −a.

Latus rectum

$\displaystyle 4a$

Ellipse

Standard form

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b$

Focal distance

$\displaystyle c^2 = a^2 - b^2$

Eccentricity

$\displaystyle e = \frac{c}{a} < 1$

Latus rectum

$\displaystyle \frac{2b^2}{a}$

Hyperbola

Standard form

$\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Focal distance

$\displaystyle c^2 = a^2 + b^2$

Eccentricity

$\displaystyle e = \frac{c}{a} > 1$

Latus rectum

$\displaystyle \frac{2b^2}{a}$

Standard limits

Polynomial form

$\displaystyle \lim_{x \to a}\frac{x^n - a^n}{x - a} = na^{n - 1}$

Sine

$\displaystyle \lim_{x \to 0}\frac{\sin x}{x} = 1$

x in radians.

Cosine

$\displaystyle \lim_{x \to 0}\frac{1 - \cos x}{x} = 0$

Tangent

$\displaystyle \lim_{x \to 0}\frac{\tan x}{x} = 1$

Derivatives

First principles

$\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

Power

$\displaystyle \frac{d}{dx}x^n = nx^{n - 1}$

Trigonometric

$\displaystyle \frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx}\tan x = \sec^2x$

Product rule

$\displaystyle (uv)' = u'v + uv'$

Quotient rule

$\displaystyle \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

Mean deviation

Ungrouped data

$\displaystyle \text{M.D.}(a) = \frac{\sum|x_i - a|}{n}$

a is the mean or median.

Grouped data

$\displaystyle \text{M.D.}(a) = \frac{\sum f_i|x_i - a|}{N}, \quad N = \sum f_i$

Variance and standard deviation

Variance

$\displaystyle \sigma^2 = \frac{\sum(x_i - \bar x)^2}{n}$

Shortcut form

$\displaystyle \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2$

Grouped data

$\displaystyle \sigma^2 = \frac{\sum f_i(x_i - \bar x)^2}{N}$

Step-deviation method

$\displaystyle \sigma^2 = h^2\left[\frac{\sum f_iy_i^2}{N} - \left(\frac{\sum f_iy_i}{N}\right)^2\right], \quad y_i = \frac{x_i - A}{h}$

Coefficient of variation

$\displaystyle \text{C.V.} = \frac{\sigma}{\bar x}\times100$

Lower means more consistent.

Rules

Probability of an event

$\displaystyle P(E) = \frac{\text{favourable outcomes}}{\text{total outcomes}}$

When outcomes are equally likely.

Range

$\displaystyle 0 \le P(E) \le 1$

Complement

$\displaystyle P(E') = 1 - P(E)$

Addition rule

$\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Mutually exclusive events

$\displaystyle P(A \cup B) = P(A) + P(B)$

Only A, not B

$\displaystyle P(A \cap B') = P(A) - P(A \cap B)$