Class 12 Maths Formula Sheet (CBSE 2026-27)

Domains and principal value branches

Key facts

• \(\sin^{-1} x \ne (\sin x)^{-1}\); in fact \((\sin x)^{-1} = \dfrac{1}{\sin x}\)
• \(y = \sin^{-1} x \iff x = \sin y\) (\(y\) in the principal branch)
• Principal value = the value lying in the principal branch
• \(\sin(\sin^{-1} x) = x\) for \(x \in [-1, 1]\)
• \(\sin^{-1}(\sin x) = x\) for \(x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
• \(\cos^{-1}(\cos x) = x\) for \(x \in [0, \pi]\); \(\ \tan^{-1}(\tan x) = x\) for \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
• The graph of \(y = \sin^{-1} x\) is the mirror image of the principal branch of \(y = \sin x\) in the line \(y = x\) (same for the others)

Types of matrices

• Order \(m \times n\): \(m\) rows, \(n\) columns; \(A = [a_{ij}]_{m \times n}\)
• Column \([a_{ij}]_{m \times 1}\); row \([a_{ij}]_{1 \times n}\); square: \(m = n\)
• Diagonal: \(a_{ij} = 0\) for \(i \ne j\)
• Scalar: diagonal with all \(a_{ii} = k\)
• Identity \(I\): \(a_{ii} = 1\), \(a_{ij} = 0\) for \(i \ne j\)
• Zero matrix \(O\): all entries 0
• \(A = B\) iff same order and \(a_{ij} = b_{ij}\) for all \(i, j\)

Addition and scalar multiplication

• \(A + B = [a_{ij} + b_{ij}]\) (same order only); \(\ kA = [k\,a_{ij}]\)
• \(-A = (-1)A\), \(\ A - B = A + (-1)B\)
• \(A + B = B + A\); \(\ (A + B) + C = A + (B + C)\)
• \(A + O = A\); \(\ A + (-A) = O\)
• \(k(A + B) = kA + kB\); \(\ (k + l)A = kA + lA\)

Multiplication

• \(AB\) is defined only if columns of \(A\) = rows of \(B\): \(A_{m \times n} B_{n \times p} = C_{m \times p}\)
• \(c_{ik} = \displaystyle\sum_{j=1}^{n} a_{ij} b_{jk}\) (row \(i\) of \(A\) · column \(k\) of \(B\))
• \(A(BC) = (AB)C\); \(\ A(B + C) = AB + AC\); \(\ (A + B)C = AC + BC\)
• \(AI = IA = A\)
• In general \(AB \ne BA\); and \(AB = O\) is possible with \(A \ne O\), \(B \ne O\)

Transpose, symmetric, skew symmetric

• \(A' = A^T = [a_{ji}]_{n \times m}\)
• \((A')' = A\); \(\ (kA)' = kA'\); \(\ (A + B)' = A' + B'\); \(\ (AB)' = B'A'\)
• Symmetric: \(A' = A\); skew symmetric: \(A' = -A\) (diagonal entries are 0)
• \(A + A'\) is symmetric, \(A - A'\) is skew symmetric
• Every square matrix: \(A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')\)

Minors and cofactors

• Minor \(M_{ij}\): determinant left after deleting row \(i\) and column \(j\)
• Cofactor: \(A_{ij} = (-1)^{i+j} M_{ij}\)
• \(|A|\) = sum of (elements of any row or column) × (their cofactors), e.g. \(|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}\)
• Elements of one row × cofactors of another row sum to 0, e.g. \(a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0\)

Area of a triangle

• Area is positive, so take the absolute value
• Points are collinear \(\iff \Delta = 0\)

Adjoint and inverse

• \(\text{adj}\,A\) = transpose of the matrix of cofactors: \(\ \text{adj}\,A = [A_{ij}]'\)
• \(A(\text{adj}\,A) = (\text{adj}\,A)A = |A|\,I\)
• Singular if \(|A| = 0\); non-singular if \(|A| \ne 0\)
• \(A\) is invertible \(\iff |A| \ne 0\); \(\ A^{-1} = \dfrac{1}{|A|}\,\text{adj}\,A\)

Useful results (\(A\) of order \(n\))

• \(|kA| = k^n|A|\)
• \(|\text{adj}\,A| = |A|^{n-1}\)
• \(|AB| = |A||B|\); \(\ |A'| = |A|\); \(\ |A^{-1}| = \dfrac{1}{|A|}\)

Continuity

• \(f\) is continuous at \(c\) if \(\lim_{x \to c} f(x) = f(c)\), i.e. LHL = RHL = \(f(c)\)
• Sum, difference, product and quotient (where \(g(x) \ne 0\)) of continuous functions are continuous
• Composition of continuous functions is continuous
• Polynomials, \(\sin x\), \(\cos x\), \(|x|\), \(e^x\) are continuous everywhere; \(\log x\) for \(x > 0\)

Differentiability

• \(f\) is differentiable at \(c\) if LHD = RHD: $$\lim_{h \to 0^-}\frac{f(c + h) - f(c)}{h} = \lim_{h \to 0^+}\frac{f(c + h) - f(c)}{h}$$
• Differentiable \(\Rightarrow\) continuous; the converse is false (e.g. \(|x|\) at \(x = 0\))

Rules

• \((u \pm v)' = u' \pm v'\); \(\ (uv)' = u'v + uv'\)
• \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2},\ v \ne 0\)
• Chain rule: \(\dfrac{dy}{dx} = \dfrac{dy}{dt}\cdot\dfrac{dt}{dx}\)
• \(\dfrac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)\)
• Implicit: differentiate both sides w.r.t. \(x\), treating \(y\) as a function of \(x\) (e.g. \(\frac{d}{dx}y^2 = 2y\frac{dy}{dx}\))
• Parametric \(x = f(t), y = g(t)\): \(\ \dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt},\ \dfrac{dx}{dt} \ne 0\)
• Second order: \(\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)\), also written \(y''\) or \(y_2\)

Logarithms and log differentiation

• \(\log_b a = x \iff b^x = a\); \(\ \log_a p = \dfrac{\log_b p}{\log_b a}\)
• \(\log(xy) = \log x + \log y\); \(\ \log\dfrac{x}{y} = \log x - \log y\); \(\ \log x^n = n\log x\)
• \(e^{\log x} = x\ (x > 0)\); \(\ a^x = e^{x\log a}\)
• For \(y = [u(x)]^{v(x)}\): take \(\log y = v\log u\), then \(\dfrac{1}{y}\dfrac{dy}{dx} = v'\log u + v\dfrac{u'}{u}\) (needs \(u > 0\))

Rate of change

• \(\dfrac{dy}{dx}\) = rate of change of \(y\) w.r.t. \(x\); \(\left.\dfrac{dy}{dx}\right|_{x = x_0}\) = rate at \(x = x_0\)
• If \(x = f(t), y = g(t)\): \(\ \dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt},\ \dfrac{dx}{dt} \ne 0\)
• Marginal cost \(= \dfrac{dC}{dx}\), marginal revenue \(= \dfrac{dR}{dx}\)

Increasing and decreasing

• Increasing on \(I\): \(x_1 < x_2 \Rightarrow f(x_1) \le f(x_2)\); strictly: \(f(x_1) < f(x_2)\)
• \(f'(x) > 0\) on \((a, b)\) \(\Rightarrow\) strictly increasing on \([a, b]\)
• \(f'(x) < 0\) on \((a, b)\) \(\Rightarrow\) strictly decreasing on \([a, b]\)
• \(f'(x) = 0\) on \((a, b)\) \(\Rightarrow\) constant on \([a, b]\)
• (\(f\) continuous on \([a, b]\), differentiable on \((a, b)\))

Critical points and first derivative test

• Critical point \(c\): \(f'(c) = 0\) or \(f\) not differentiable at \(c\)
• \(f'\) changes \(+ \to -\) through \(c\): local maximum
• \(f'\) changes \(- \to +\) through \(c\): local minimum
• No sign change: neither (point of inflexion)

Second derivative test (\(f'(c) = 0\))

• \(f''(c) < 0\): local maximum, value \(f(c)\)
• \(f''(c) > 0\): local minimum, value \(f(c)\)
• \(f''(c) = 0\): test fails, so use the first derivative test

Basics

• If \(\dfrac{d}{dx}F(x) = f(x)\), then \(\displaystyle\int f(x)\,dx = F(x) + C\)
• \(\displaystyle\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx\)
• \(\displaystyle\int k\,f(x)\,dx = k\int f(x)\,dx\)
• \(\dfrac{d}{dx}\displaystyle\int f(x)\,dx = f(x)\)

Integrals via trig identities

• \(\sin^2 x = \dfrac{1 - \cos 2x}{2}\), \(\ \cos^2 x = \dfrac{1 + \cos 2x}{2}\)
• \(\sin^3 x = \dfrac{3\sin x - \sin 3x}{4}\), \(\ \cos^3 x = \dfrac{3\cos x + \cos 3x}{4}\)
• \(2\sin A\cos B = \sin(A + B) + \sin(A - B)\), and the other product-to-sum formulas

Quadratic denominators

• For \(\displaystyle\int\frac{dx}{ax^2 + bx + c}\), \(\displaystyle\int\frac{dx}{\sqrt{ax^2 + bx + c}}\), \(\displaystyle\int\sqrt{ax^2 + bx + c}\,dx\), complete the square: $$ax^2 + bx + c = a\left[\left(x + \frac{b}{2a}\right)^2 + \left(\frac{c}{a} - \frac{b^2}{4a^2}\right)\right]$$
• For \(\displaystyle\int\frac{px + q}{ax^2 + bx + c}dx\) or \(\displaystyle\int\frac{px + q}{\sqrt{ax^2 + bx + c}}dx\): write \(px + q = A(2ax + b) + B\) and find \(A, B\) by comparing coefficients

Integration by parts

• Choose \(f_2\) as the function whose integral you know. Common order for \(f_1\) (ILATE): Inverse trig, Log, Algebraic, Trig, Exponential
• \(\displaystyle\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C\)

Definite integrals

• \(\displaystyle\int_a^b f(x)\,dx\) = area bounded by \(y = f(x)\), the x-axis and \(x = a\), \(x = b\)
• First fundamental theorem: if \(A(x) = \displaystyle\int_a^x f(t)\,dt\), then \(A'(x) = f(x)\)
• Second fundamental theorem: \(\displaystyle\int_a^b f(x)\,dx = \Big[F(x)\Big]_a^b = F(b) - F(a)\), where \(F' = f\)
• With substitution, change the limits too

Properties of definite integrals

• \(P_0\): \(\int_a^b f(x)\,dx = \int_a^b f(t)\,dt\)
• \(P_1\): \(\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx\); \(\ \int_a^a f(x)\,dx = 0\)
• \(P_2\): \(\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx\)
• \(P_3\): \(\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx\)
• \(P_4\): \(\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx\)
• \(P_5\): \(\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a - x)\,dx\)
• \(P_6\): \(\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx\) if \(f(2a - x) = f(x)\); \(= 0\) if \(f(2a - x) = -f(x)\)
• \(P_7\): \(\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx\) if \(f\) is even; \(= 0\) if \(f\) is odd

Area under a curve

• Bounded by \(y = f(x)\), the x-axis, \(x = a\), \(x = b\) \((b > a)\): \(\ \text{Area} = \displaystyle\int_a^b y\,dx = \int_a^b f(x)\,dx\)
• Bounded by \(x = \varphi(y)\), the y-axis, \(y = c\), \(y = d\): \(\ \text{Area} = \displaystyle\int_c^d x\,dy = \int_c^d \varphi(y)\,dy\)
• If the curve is below the x-axis, the integral is negative: take \(\left|\displaystyle\int_a^b f(x)\,dx\right|\)
• If the curve crosses the axis, split the interval and add the absolute values

Useful results (standard curves)

• Circle \(x^2 + y^2 = a^2\): area \(= \pi a^2\)
• Ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\): area \(= \pi ab\)
• Parabola \(y^2 = 4ax\) and its latus rectum \(x = a\): area \(= \dfrac{8a^2}{3}\)
• Use symmetry: find the area in one quadrant and multiply

Basics

• Order = order of the highest derivative present
• Degree (defined only if the equation is a polynomial in its derivatives) = highest power of the highest-order derivative
• General solution: as many arbitrary constants as the order
• Particular solution: no arbitrary constants (found using given conditions)

Variables separable

• \(\dfrac{dy}{dx} = g(x)\,h(y)\): write \(\dfrac{dy}{h(y)} = g(x)\,dx\), then \(\displaystyle\int\frac{dy}{h(y)} = \int g(x)\,dx + C\)

Homogeneous equations

• \(F(x, y)\) is homogeneous of degree \(n\) if \(F(\lambda x, \lambda y) = \lambda^n F(x, y)\)
• \(\dfrac{dy}{dx} = F(x, y)\) is homogeneous if \(F\) has degree 0
• Put \(y = vx\): \(\ \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}\), then separate variables
• For \(\dfrac{dx}{dy} = G(x, y)\), put \(x = vy\): \(\ \dfrac{dx}{dy} = v + y\dfrac{dv}{dy}\)

Linear equations

• \(\dfrac{dy}{dx} + Py = Q\) (\(P, Q\) functions of \(x\) or constants): $$\text{I.F.} = e^{\int P\,dx},\qquad y\cdot\text{I.F.} = \int (Q \times \text{I.F.})\,dx + C$$
• \(\dfrac{dx}{dy} + P_1 x = Q_1\) (\(P_1, Q_1\) functions of \(y\) or constants): $$\text{I.F.} = e^{\int P_1\,dy},\qquad x\cdot\text{I.F.} = \int (Q_1 \times \text{I.F.})\,dy + C$$

Useful: \(e^{\log f(x)} = f(x)\).

Basics

• Position vector of \(P(x, y, z)\): \(\vec r = x\hat i + y\hat j + z\hat k\), \(\ |\vec r| = \sqrt{x^2 + y^2 + z^2}\)
• \(\overrightarrow{AB} = (x_2 - x_1)\hat i + (y_2 - y_1)\hat j + (z_2 - z_1)\hat k\)
• Unit vector along \(\vec a\): \(\hat a = \dfrac{\vec a}{|\vec a|}\)
• Direction cosines of \(\vec r = a\hat i + b\hat j + c\hat k\): \(l = \dfrac{a}{r},\ m = \dfrac{b}{r},\ n = \dfrac{c}{r}\); \(a, b, c\) are direction ratios
• \(l^2 + m^2 + n^2 = 1\); \(\ \vec r = r(l\hat i + m\hat j + n\hat k)\)

Types and operations

• Zero, unit, coinitial, collinear (parallel to the same line), equal (same magnitude and direction), negative vectors
• Triangle law: \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\); sides of a triangle taken in order add to \(\vec 0\)
• Parallelogram law: the sum of two coinitial vectors is the diagonal
• \(\vec a + \vec b = \vec b + \vec a\); \(\ (\vec a + \vec b) + \vec c = \vec a + (\vec b + \vec c)\)
• \(\lambda\vec a\) has magnitude \(|\lambda||\vec a|\); same direction if \(\lambda > 0\), opposite if \(\lambda < 0\)
• \(\vec a, \vec b\) collinear \(\iff \vec b = \lambda\vec a\) \(\iff \dfrac{b_1}{a_1} = \dfrac{b_2}{a_2} = \dfrac{b_3}{a_3}\)

Components

• For \(\vec a = a_1\hat i + a_2\hat j + a_3\hat k\), \(\vec b = b_1\hat i + b_2\hat j + b_3\hat k\):
• \(\vec a \pm \vec b = (a_1 \pm b_1)\hat i + (a_2 \pm b_2)\hat j + (a_3 \pm b_3)\hat k\)
• \(\lambda\vec a = \lambda a_1\hat i + \lambda a_2\hat j + \lambda a_3\hat k\)
• \(\vec a = \vec b \iff a_1 = b_1,\ a_2 = b_2,\ a_3 = b_3\)

Section formula (position vectors \(\vec a, \vec b\))

• Internal, ratio \(m : n\): \(\ \vec r = \dfrac{m\vec b + n\vec a}{m + n}\)
• External, ratio \(m : n\): \(\ \vec r = \dfrac{m\vec b - n\vec a}{m - n}\)
• Mid-point: \(\ \vec r = \dfrac{\vec a + \vec b}{2}\)

Scalar (dot) product

• \(\vec a \cdot \vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3\)
• \(\cos\theta = \dfrac{\vec a \cdot \vec b}{|\vec a||\vec b|}\)
• \(\vec a \perp \vec b \iff \vec a \cdot \vec b = 0\) (non-zero vectors)
• \(\hat i \cdot \hat i = \hat j \cdot \hat j = \hat k \cdot \hat k = 1\); \(\ \hat i \cdot \hat j = \hat j \cdot \hat k = \hat k \cdot \hat i = 0\)
• \(\vec a \cdot \vec a = |\vec a|^2\); \(\ \vec a \cdot \vec b = \vec b \cdot \vec a\); \(\ \vec a \cdot (\vec b + \vec c) = \vec a \cdot \vec b + \vec a \cdot \vec c\)
• Projection of \(\vec a\) on \(\vec b\): \(\ \dfrac{\vec a \cdot \vec b}{|\vec b|} = \vec a \cdot \hat b\)

Vector (cross) product

• \(\vec a \times \vec b = |\vec a||\vec b|\sin\theta\,\hat n\), \(\hat n \perp\) both, right-handed
• \(\vec a \times \vec b = \begin{vmatrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\)
• \(\vec a \times \vec b = -\,\vec b \times \vec a\); \(\ \vec a \parallel \vec b \iff \vec a \times \vec b = \vec 0\)
• \(\hat i \times \hat j = \hat k,\ \hat j \times \hat k = \hat i,\ \hat k \times \hat i = \hat j\); \(\ \hat i \times \hat i = \vec 0\)
• \(\sin\theta = \dfrac{|\vec a \times \vec b|}{|\vec a||\vec b|}\)
• Unit vector perpendicular to both: \(\ \pm\dfrac{\vec a \times \vec b}{|\vec a \times \vec b|}\)

Direction cosines and ratios

• DCs \(l, m, n\) = cosines of the angles with the positive x, y, z axes; \(\ l^2 + m^2 + n^2 = 1\)
• DRs \(a, b, c\) are proportional to the DCs: $$l = \frac{\pm a}{\sqrt{a^2 + b^2 + c^2}},\ m = \frac{\pm b}{\sqrt{a^2 + b^2 + c^2}},\ n = \frac{\pm c}{\sqrt{a^2 + b^2 + c^2}}$$
• Line through \(P(x_1, y_1, z_1)\), \(Q(x_2, y_2, z_2)\): DRs \(x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1\); DCs are these divided by \(PQ\)

Equation of a line

• Through \(\vec a\), parallel to \(\vec b\): \(\ \vec r = \vec a + \lambda\vec b\)
• Cartesian (through \((x_1, y_1, z_1)\), DRs \(a, b, c\)): $$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$ (with DCs: replace \(a, b, c\) by \(l, m, n\))
• Through two points \(\vec a, \vec b\): \(\ \vec r = \vec a + \lambda(\vec b - \vec a)\); Cartesian: \(\dfrac{x - x_1}{x_2 - x_1} = \dfrac{y - y_1}{y_2 - y_1} = \dfrac{z - z_1}{z_2 - z_1}\)

Terminology

• Objective function: \(Z = ax + by\) (to be maximised or minimised)
• Constraints: linear inequalities; non-negativity: \(x \ge 0,\ y \ge 0\)
• Feasible region: the common region of all constraints; each point in it is a feasible solution
• Optimal solution: the feasible point giving the best value of \(Z\)
• Infeasible: no point satisfies all constraints

Corner point theorems

• If an optimal value exists, it occurs at a corner point (vertex) of the feasible region
• If the feasible region is bounded, \(Z\) has both a maximum and a minimum, each at a corner point
• An unbounded region may have no maximum or minimum

Conditional probability

• \(P(E \mid F) = \dfrac{P(E \cap F)}{P(F)},\ P(F) \ne 0\)
• Equally likely outcomes: \(P(E \mid F) = \dfrac{n(E \cap F)}{n(F)}\)
• \(P(S \mid F) = P(F \mid F) = 1\); \(\ 0 \le P(E \mid F) \le 1\)
• \(P(E' \mid F) = 1 - P(E \mid F)\)
• \(P((A \cup B) \mid F) = P(A \mid F) + P(B \mid F) - P((A \cap B) \mid F)\)

Multiplication theorem

• \(P(E \cap F) = P(E)\,P(F \mid E),\ P(E) \ne 0\)
• \(P(E \cap F) = P(F)\,P(E \mid F),\ P(F) \ne 0\)
• \(P(E \cap F \cap G) = P(E)\,P(F \mid E)\,P(G \mid E \cap F)\)

Independent events

• \(E, F\) independent \(\iff P(E \cap F) = P(E)\,P(F)\)
• Then \(P(E \mid F) = P(E)\) and \(P(F \mid E) = P(F)\)
• If \(E, F\) are independent, so are \(E\) and \(F'\), \(E'\) and \(F\), \(E'\) and \(F'\)
• \(P(\text{at least one of } E, F) = 1 - P(E')\,P(F')\)
• Three events: mutually independent if every pair is independent and \(P(E \cap F \cap G) = P(E)P(F)P(G)\)
• Independent is not the same as mutually exclusive.

Total probability and Bayes' theorem

• Partition: \(E_1, \dots, E_n\) pairwise disjoint, \(E_1 \cup \dots \cup E_n = S\), each \(P(E_i) > 0\)
• Total probability: $$P(A) = \sum_{j=1}^{n} P(E_j)\,P(A \mid E_j)$$
• Bayes' theorem: $$P(E_i \mid A) = \frac{P(E_i)\,P(A \mid E_i)}{\sum_{j=1}^{n} P(E_j)\,P(A \mid E_j)}$$
• \(P(E_i)\) are priors, \(P(E_i \mid A)\) are posteriors.