Class 11 Maths Formula Sheet (CBSE 2026-27)

Cartesian product

• \((a, b) = (x, y) \iff a = x \text{ and } b = y\)
• \(A \times B = \{(a, b) : a \in A,\ b \in B\}\)
• \(n(A) = p,\ n(B) = q \Rightarrow n(A \times B) = pq\)
• \(A \times \varphi = \varphi\); in general \(A \times B \ne B \times A\)
• \(\mathbb{R} \times \mathbb{R} = \{(x, y) : x, y \in \mathbb{R}\}\); \(\ \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \{(x, y, z) : x, y, z \in \mathbb{R}\}\)

Relations

• A relation \(R\) from \(A\) to \(B\) is a subset of \(A \times B\)
• Domain = set of first elements; range = set of second elements; codomain = \(B\)
• Range \(\subseteq\) codomain
• Number of relations from \(A\) to \(B\) \(= 2^{pq}\)

Functions

• \(f : A \to B\) is a relation in which every \(x \in A\) has exactly one image \(f(x) \in B\)
• \(A\) = domain, \(B\) = codomain, range = set of images
• Real function: domain and range are subsets of \(\mathbb{R}\)

Algebra of real functions (\(x \in X\))

• \((f \pm g)(x) = f(x) \pm g(x)\)
• \((fg)(x) = f(x)\,g(x)\); \(\ (kf)(x) = k\,f(x)\)
• \(\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)},\ g(x) \ne 0\)

Laws of logarithms (\(m, n > 0\))

• \(\log_a (mn) = \log_a m + \log_a n\)
• \(\log_a \dfrac{m}{n} = \log_a m - \log_a n\)
• \(\log_a m^k = k \log_a m\)
• \(\log_a x = \dfrac{\log_b x}{\log_b a}\); \(\ \log_a a = 1,\ \log_a 1 = 0\)
• \(a^{\log_a x} = x\) for \(x > 0\)

Angle measure

• Arc length: \(l = r\theta\) (\(\theta\) in radians)
• Radian measure \(= \dfrac{\pi}{180} \times\) degree measure
• Degree measure \(= \dfrac{180}{\pi} \times\) radian measure
• \(1 \text{ rad} \approx 57^\circ 16'\); \(\ 1^\circ \approx 0.01746\) rad; \(\ \pi \text{ rad} = 180^\circ\)

Definitions and identities

• \(\csc x = \dfrac{1}{\sin x},\ \sec x = \dfrac{1}{\cos x},\ \cot x = \dfrac{1}{\tan x}\)
• \(\tan x = \dfrac{\sin x}{\cos x},\ \ \cot x = \dfrac{\cos x}{\sin x}\)
• \(\sin^2 x + \cos^2 x = 1\)
• \(1 + \tan^2 x = \sec^2 x\)
• \(1 + \cot^2 x = \csc^2 x\)

NCERT writes \(\csc x\) as cosec \(x\).

Signs by quadrant

Memory aid: All, Sin, Tan, Cos are positive in quadrants I, II, III, IV.

Domain and range

\(\sin x = 0 \iff x = n\pi\); \(\ \cos x = 0 \iff x = (2n+1)\frac{\pi}{2}\). Period: sin, cos, sec, cosec \(= 2\pi\); tan, cot \(= \pi\).

Sum and difference

• \(\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\)
• \(\cos(x + y) = \cos x \cos y - \sin x \sin y\)
• \(\cos(x - y) = \cos x \cos y + \sin x \sin y\)
• \(\tan(x \pm y) = \dfrac{\tan x \pm \tan y}{1 \mp \tan x \tan y}\)
• \(\cot(x \pm y) = \dfrac{\cot x \cot y \mp 1}{\cot y \pm \cot x}\)

tan formulas need none of \(x, y, x \pm y\) to be an odd multiple of \(\frac{\pi}{2}\); cot formulas need none to be a multiple of \(\pi\).

Multiple angles

• \(\sin 2x = 2\sin x \cos x = \dfrac{2\tan x}{1 + \tan^2 x}\)
• \(\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1\)
• \(\cos 2x = 1 - 2\sin^2 x = \dfrac{1 - \tan^2 x}{1 + \tan^2 x}\)
• \(\tan 2x = \dfrac{2\tan x}{1 - \tan^2 x}\)
• \(\sin 3x = 3\sin x - 4\sin^3 x\)
• \(\cos 3x = 4\cos^3 x - 3\cos x\)
• \(\tan 3x = \dfrac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}\)
• \(2\sin^2 x = 1 - \cos 2x\), \(\ 2\cos^2 x = 1 + \cos 2x\)

Sum to product

• \(\cos x + \cos y = 2\cos\dfrac{x+y}{2}\cos\dfrac{x-y}{2}\)
• \(\cos x - \cos y = -2\sin\dfrac{x+y}{2}\sin\dfrac{x-y}{2}\)
• \(\sin x + \sin y = 2\sin\dfrac{x+y}{2}\cos\dfrac{x-y}{2}\)
• \(\sin x - \sin y = 2\cos\dfrac{x+y}{2}\sin\dfrac{x-y}{2}\)

Product to sum

• \(2\cos x \cos y = \cos(x + y) + \cos(x - y)\)
• \(-2\sin x \sin y = \cos(x + y) - \cos(x - y)\)
• \(2\sin x \cos y = \sin(x + y) + \sin(x - y)\)
• \(2\cos x \sin y = \sin(x + y) - \sin(x - y)\)

Basics

• \(i = \sqrt{-1}\), \(\ i^2 = -1\)
• \(z = a + ib\): \(\ \text{Re}(z) = a\), \(\ \text{Im}(z) = b\)
• \(a + ib = c + id \iff a = c \text{ and } b = d\)
• Powers of \(i\) (\(k \in \mathbb{Z}\)): \(i^{4k} = 1,\ i^{4k+1} = i,\ i^{4k+2} = -1,\ i^{4k+3} = -i\)
• \(\sqrt{-a} = i\sqrt{a}\) for \(a > 0\)
• \(\sqrt{a}\sqrt{b} = \sqrt{ab}\) is false when \(a, b\) are both negative

Algebra (\(z_1 = a + ib,\ z_2 = c + id\))

• \(z_1 \pm z_2 = (a \pm c) + i(b \pm d)\)
• \(z_1 z_2 = (ac - bd) + i(ad + bc)\)
• \(\dfrac{z_1}{z_2} = \dfrac{(ac + bd) + i(bc - ad)}{c^2 + d^2},\ z_2 \ne 0\)
• Inverse: \(z^{-1} = \dfrac{1}{a + ib} = \dfrac{a}{a^2 + b^2} - i\dfrac{b}{a^2 + b^2} = \dfrac{\bar z}{|z|^2}\)

Identities (any complex \(z_1, z_2\))

• \((z_1 \pm z_2)^2 = z_1^2 \pm 2z_1z_2 + z_2^2\)
• \((z_1 \pm z_2)^3 = z_1^3 \pm 3z_1^2z_2 + 3z_1z_2^2 \pm z_2^3\)
• \(z_1^2 - z_2^2 = (z_1 + z_2)(z_1 - z_2)\)

Modulus and conjugate

• \(|z| = \sqrt{a^2 + b^2}\), \(\ \bar z = a - ib\)
• \(z\bar z = |z|^2\)
• \(|z_1 z_2| = |z_1||z_2|\), \(\ \left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}\ (z_2 \ne 0)\)
• \(\overline{z_1 z_2} = \bar z_1 \bar z_2\), \(\ \overline{z_1 \pm z_2} = \bar z_1 \pm \bar z_2\), \(\ \overline{\left(\dfrac{z_1}{z_2}\right)} = \dfrac{\bar z_1}{\bar z_2}\)

Argand plane

• \(z = x + iy\) is the point \(P(x, y)\); x-axis = real axis, y-axis = imaginary axis
• \(|z| = OP\), the distance of \(P\) from the origin
• \(\bar z = x - iy\) is the point \((x, -y)\), the mirror image of \(P\) in the real axis

Rules for solving

• The same number may be added to or subtracted from both sides
• Multiplying or dividing both sides by a positive number keeps the sign
• Multiplying or dividing both sides by a negative number reverses the sign: \(a < b \Rightarrow -a > -b\)
• Solutions = all values of \(x\) that make the inequality true

Number line and interval form

• \(x < a\) or \(x > a\): open (hollow) circle at \(a\)
• \(x \le a\) or \(x \ge a\): dark (filled) circle at \(a\)
• \(x < a \iff x \in (-\infty, a)\); \(\ x \le a \iff x \in (-\infty, a]\)
• \(x > a \iff x \in (a, \infty)\); \(\ x \ge a \iff x \in [a, \infty)\)
• \(a < x \le b \iff x \in (a, b]\)

Counting and factorials

• Fundamental principle: if one event can occur in \(m\) ways and then another in \(n\) ways, both occur in that order in \(m \times n\) ways
• \(n! = 1 \times 2 \times 3 \times \dots \times n\)
• \(n! = n \times (n - 1)!\); \(\ 0! = 1\)

Permutations (arrangements)

• \({}^nP_r = \dfrac{n!}{(n - r)!},\ 0 \le r \le n\)
• \({}^nP_n = n!\), \(\ {}^nP_0 = 1\)
• With repetition allowed: \(n^r\)
• \(n\) objects with \(p_1\) alike of one kind, \(p_2\) of another, …, \(p_k\) of a \(k\)th kind: \(\dfrac{n!}{p_1!\,p_2! \cdots p_k!}\)

Combinations (selections)

• \({}^nC_r = \dfrac{n!}{r!\,(n - r)!},\ 0 \le r \le n\)
• \({}^nP_r = {}^nC_r \times r!\)
• \({}^nC_0 = {}^nC_n = 1\), \(\ {}^nC_1 = n\)

Properties of \({}^nC_r\)

• \({}^nC_r = {}^nC_{n-r}\)
• \({}^nC_a = {}^nC_b \Rightarrow a = b \text{ or } a + b = n\)
• \({}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r\)

Special cases

• \((a - b)^n = \displaystyle\sum_{k=0}^{n} (-1)^k\,{}^nC_k\,a^{n-k}b^k\)
• \((1 + x)^n = {}^nC_0 + {}^nC_1 x + {}^nC_2 x^2 + \dots + {}^nC_n x^n\)
• \((1 - x)^n = {}^nC_0 - {}^nC_1 x + {}^nC_2 x^2 - \dots + (-1)^n\,{}^nC_n x^n\)
• \({}^nC_0 + {}^nC_1 + \dots + {}^nC_n = 2^n\)

Pascal's triangle

Each entry is the sum of the two entries above it (that is \({}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r\)).

A.P. (recall from Class 10) and A.M.

• \(n\)th term: \(a_n = a + (n - 1)d\)
• \(S_n = \dfrac{n}{2}\left[2a + (n - 1)d\right] = \dfrac{n}{2}(a + l)\), \(l\) = last term
• A.M. of \(a\) and \(b\): \(\ A = \dfrac{a + b}{2}\)
• \(n\) A.M.s between \(a\) and \(b\): common difference \(d = \dfrac{b - a}{n + 1}\)

Geometric progression

• \(a, ar, ar^2, \dots\) with common ratio \(r = \dfrac{a_{k+1}}{a_k}\)
• \(n\)th term: \(a_n = ar^{n-1}\)
• \(S_n = \dfrac{a(r^n - 1)}{r - 1} = \dfrac{a(1 - r^n)}{1 - r},\ r \ne 1\)
• If \(r = 1\): \(S_n = na\)
• Infinite G.P., \(|r| < 1\): \(\ S_\infty = \dfrac{a}{1 - r}\)

Geometric mean

• G.M. of positive \(a, b\): \(\ G = \sqrt{ab}\) (so \(a, G, b\) is a G.P.)
• \(n\) G.M.s between \(a\) and \(b\): \(r = \left(\dfrac{b}{a}\right)^{\frac{1}{n+1}}\), and \(G_k = ar^k\)
• Three numbers in A.P.: \(a - d,\ a,\ a + d\). In G.P.: \(\frac{a}{r},\ a,\ ar\)

A.M. and G.M. relation

• For positive \(a, b\): \(\ A \ge G\), i.e. \(\dfrac{a + b}{2} \ge \sqrt{ab}\)
• \(A - G = \dfrac{(\sqrt a - \sqrt b)^2}{2}\); equality only when \(a = b\)
• Numbers with A.M. \(A\) and G.M. \(G\) are the roots of \(x^2 - 2Ax + G^2 = 0\), i.e. \(A \pm \sqrt{A^2 - G^2}\)

Slope

• \(m = \tan\alpha,\ \alpha \ne 90^\circ\) (\(\alpha\) = inclination with the positive x-axis)
• \(m = \dfrac{y_2 - y_1}{x_2 - x_1},\ x_1 \ne x_2\)
• Horizontal line: \(m = 0\); vertical line: slope not defined
• Parallel: \(m_1 = m_2\); perpendicular: \(m_1 m_2 = -1\)
• \(A, B, C\) collinear \(\iff\) slope \(AB\) = slope \(BC\)

Angle between two lines

This gives the acute angle \(\theta\); the obtuse angle is \(180^\circ - \theta\).

Distances

• Point \((x_1, y_1)\) to line \(Ax + By + C = 0\): $$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$
• Between parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\): $$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$

Sections of a cone

\(\alpha\) = semi-vertical angle of the cone, \(\beta\) = angle between the cutting plane and the axis.

• Plane not through vertex: \(\beta = 90^\circ\): circle; \(\alpha < \beta < 90^\circ\): ellipse; \(\beta = \alpha\): parabola; \(0 \le \beta < \alpha\): hyperbola
• Plane through vertex (degenerate): \(\alpha < \beta \le 90^\circ\): a point; \(\beta = \alpha\): a straight line; \(0 \le \beta < \alpha\): a pair of intersecting lines

Circle

• Centre \((h, k)\), radius \(r\): \((x - h)^2 + (y - k)^2 = r^2\)
• Centre at origin: \(x^2 + y^2 = r^2\)
• \(x^2 + y^2 + 2gx + 2fy + c = 0\): centre \((-g, -f)\), radius \(\sqrt{g^2 + f^2 - c}\)

Axes, planes and points

• Three mutually perpendicular axes; coordinate planes XY, YZ, ZX divide space into 8 octants
• In \(P(x, y, z)\): \(x, y, z\) are the distances from the YZ, ZX and XY planes
• On x-axis \((x, 0, 0)\); y-axis \((0, y, 0)\); z-axis \((0, 0, z)\)
• In XY-plane \((x, y, 0)\); YZ-plane \((0, y, z)\); ZX-plane \((x, 0, z)\)

Signs in the octants

Distance formula

• Distance from origin: \(OP = \sqrt{x^2 + y^2 + z^2}\)

Limits: basics

• \(\lim_{x \to a} f(x)\) exists \(\iff\) LHL \(= \lim_{x \to a^-} f(x)\) equals RHL \(= \lim_{x \to a^+} f(x)\)
• \(\lim_{x \to a} f(x)\) and \(f(a)\) may differ (one may exist without the other)
• Polynomial: \(\lim_{x \to a} f(x) = f(a)\)
• Rational: \(\lim_{x \to a} \dfrac{g(x)}{h(x)} = \dfrac{g(a)}{h(a)}\) if \(h(a) \ne 0\)

Algebra of limits

• \(\lim [f(x) \pm g(x)] = \lim f(x) \pm \lim g(x)\)
• \(\lim [f(x)\,g(x)] = \lim f(x) \cdot \lim g(x)\)
• \(\lim \dfrac{f(x)}{g(x)} = \dfrac{\lim f(x)}{\lim g(x)}\), if \(\lim g(x) \ne 0\)
• \(\lim [k\,f(x)] = k \lim f(x)\)
• Sandwich theorem: if \(f \le g \le h\) and \(\lim f = \lim h = l\), then \(\lim g = l\)

Derivative: definition

• At a point: \(f'(a) = \lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h}\)
• First principle: $$\frac{d}{dx}f(x) = f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
• \(f'(a)\) = slope of the tangent to \(y = f(x)\) at \((a, f(a))\) = rate of change of \(f\) at \(a\)

Rules of differentiation

• \((u \pm v)' = u' \pm v'\)
• Product: \((uv)' = u'v + uv'\)
• Quotient: \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2},\ v \ne 0\)
• \((k\,u)' = k\,u'\); derivative of a constant \(= 0\)

Central tendency (recall)

• Mean: \(\bar x = \dfrac{\sum x_i}{n}\); for frequencies \(\bar x = \dfrac{\sum f_i x_i}{N}\), \(N = \sum f_i\)
• Step deviation: \(u_i = \dfrac{x_i - A}{h}\), \(\ \bar x = A + \dfrac{\sum f_i u_i}{N} \times h\)
• Median (ungrouped, sorted): \(n\) odd: \(\left(\frac{n+1}{2}\right)\)th value; \(n\) even: mean of the \(\frac{n}{2}\)th and \(\left(\frac{n}{2} + 1\right)\)th values
• Median (grouped): \(\ M = l + \dfrac{\frac{N}{2} - C}{f} \times h\)

\(l\) = lower limit of the median class, \(C\) = cumulative frequency before it, \(f\) = its frequency, \(h\) = class width.

Range and mean deviation

• Range \(=\) maximum value \(-\) minimum value
• Ungrouped, about \(a\): \(\ \text{M.D.}(a) = \dfrac{\sum |x_i - a|}{n}\)
• \(\text{M.D.}(\bar x) = \dfrac{\sum |x_i - \bar x|}{n}\), \(\ \text{M.D.}(M) = \dfrac{\sum |x_i - M|}{n}\)
• Grouped: \(\ \text{M.D.}(\bar x) = \dfrac{\sum f_i |x_i - \bar x|}{N}\)
• Grouped: \(\ \text{M.D.}(M) = \dfrac{\sum f_i |x_i - M|}{N}\)

For continuous data, \(x_i\) is the mid-point of each class.

Variance and SD: ungrouped data

• \(\sigma^2 = \dfrac{1}{n}\sum (x_i - \bar x)^2\), \(\ \sigma = \sqrt{\dfrac{1}{n}\sum (x_i - \bar x)^2}\)
• Equivalent: \(\sigma^2 = \dfrac{\sum x_i^2}{n} - (\bar x)^2\)

Variance and SD: frequency distributions

• \(\sigma^2 = \dfrac{1}{N}\sum f_i (x_i - \bar x)^2\), \(\ \sigma = \sqrt{\dfrac{1}{N}\sum f_i (x_i - \bar x)^2}\)
• \(\sigma = \dfrac{1}{N}\sqrt{N\sum f_i x_i^2 - \left(\sum f_i x_i\right)^2}\)

Events (\(S\) = sample space)

• Mutually exclusive: \(A \cap B = \varphi\)
• Exhaustive: \(E_1 \cup E_2 \cup \dots \cup E_n = S\); mutually exclusive and exhaustive if also \(E_i \cap E_j = \varphi\) for \(i \ne j\)

Axiomatic probability

• \(P(E) \ge 0\) for every event \(E\); \(\ P(S) = 1\)
• \(E, F\) mutually exclusive \(\Rightarrow P(E \cup F) = P(E) + P(F)\)
• For outcomes \(\omega_i\): \(0 \le P(\omega_i) \le 1\), \(\ \sum P(\omega_i) = 1\), \(\ P(A) = \sum_{\omega_i \in A} P(\omega_i)\)
• \(P(\varphi) = 0\); \(\ 0 \le P(A) \le 1\)
• Equally likely outcomes: \(\ P(A) = \dfrac{n(A)}{n(S)}\)