Class 12 Maths Notes

Class 12 Mathematics Notes

Complete, exam-ready notes for Class 12 mathematics: relations and functions, inverse trigonometric functions, matrices and determinants, continuity and differentiability, integrals, vectors and probability — with key formulas written for CBSE boards and JEE revision.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is the central idea of Class 12 Maths?

Class 12 Maths builds calculus and algebra foundations: limits, continuity, differentiation and integration, plus matrices, determinants, vectors, 3D geometry and probability — the core of the CBSE syllabus and a large share of JEE Main.

Relations, Functions and Inverse Trigonometry

A relation R from A to B is a subset of A × B; its domain and range are defined accordingly.
A function f: A → B must give exactly one output per input (well-defined).
One-one (injective): distinct inputs give distinct outputs; onto (surjective): every element of B is used.
A function is invertible iff it is both one-one and onto (bijective).
Principal values of inverse trig functions are chosen for uniqueness, e.g.
$\sin^{-1} x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$
.

\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2},\quad \tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}

Inverse trig identities

Matrices and Determinants

Matrix

A rectangular array of numbers arranged in rows and columns. For matrix multiplication, the number of columns of the first must equal the number of rows of the second: $(m\times n)(n\times p) = (m\times p)$ .

Identity matrix I: the multiplicative identity (AI = IA = A).
A square matrix is invertible if its determinant is non-zero;
$A^{-1} = \frac{\text{adj}(A)}{|A|}$
.
Determinant: |AB| = |A||B|.
Elementary row operations are used in Gauss-Jordan inversion.

A^{-1} = \frac{\text{adj}(A)}{|A|}

Inverse of a matrix

Continuity and Differentiability

Continuity

\lim_{x\to a}f(x) = f(a)

A function is continuous at x = a when the limit equals the function value. Differentiability implies continuity, but the converse is not true (e.g. |x| is continuous but not differentiable at 0).

Chain rule:
$\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$
.
Product rule:
$(uv)' = u'v + uv'$
.
Quotient rule:
$(u/v)' = \frac{u'v - uv'}{v^2}$
.
Logarithmic differentiation for complex powers and products.

\frac{d}{dx}(uv) = u'v+uv',\quad \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}

Product and quotient rules

Applications of Derivatives

Rate of change:
$\frac{dy}{dx}$
gives the rate.
Increasing when f'(x) > 0, decreasing when f'(x) < 0.
Maxima/minima: critical points where f'(x) = 0; use the second derivative test:
$f''(a)<0$
→ max,
$f''(a)>0$
→ min.
Tangents and normals: slope of tangent = f'(x).

Integrals

Definite integral

\int_a^b f(x)\,dx

The area under the curve from x = a to x = b. The Fundamental Theorem of Calculus links it to the antiderivative: $\int_a^b f(x)\,dx = F(b) - F(a)$ .

Integration by parts:
$\int u\,dv = uv - \int v\,du$
(choose u by ILATE).
Substitution:
$\int f(g(x))g'(x)\,dx = \int f(u)\,du$
.
Standard integrals: trigonometric, exponential, logarithmic.
Properties of definite integrals:
$\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$
(useful for symmetric limits).

Vectors, 3D Geometry and Probability

Dot product:
$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$
(scalar).
Cross product:
$|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$
(vector, perpendicular to both).
Direction cosines relate to angles a line makes with the axes.
Bayes' theorem:
$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
.
Mean and variance of a binomial distribution:
$\mu = np,\ \sigma^2 = npq$
.

\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta,\quad P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Vector dot product and Bayes' theorem

Solved Examples

Example: Find the maximum value of $f(x) = -x^2 + 4x + 1$ on the real line.

Solution: $f'(x) = -2x + 4$ . Set $f'(x)=0 \Rightarrow x = 2$ . $f''(x) = -2 < 0$ , so it is a maximum. $f(2) = -4 + 8 + 1 = 5$ .

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Inverse of a matrix

A^{-1} = \frac{\text{adj}(A)}{|A|}

Product rule

(uv)' = u'v + uv'

Integration by parts

\int u\,dv = uv - \int v\,du

Dot product

\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta

Bayes' theorem

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Binomial mean/variance

\mu = np,\ \sigma^2 = npq

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

Differentiability implies continuity, but not vice versa.
A function is invertible iff bijective.
A⁻¹ exists iff |A| ≠ 0.
Second derivative test: f''<0 → max, f''>0 → min.
Use ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx for symmetric limits.

FAQ

Common questions

When is a matrix invertible?

A square matrix is invertible when its determinant is non-zero, and then A⁻¹ = adj(A)/|A|.

What is the Fundamental Theorem of Calculus?

It links definite integrals to antiderivatives: ∫ₐᵇ f(x)dx = F(b) − F(a), where F is an antiderivative of f.

What is the second derivative test?

At a critical point where f'(x) = 0, if f''(x) < 0 it is a local maximum, and if f''(x) > 0 it is a local minimum.

What does the ILATE rule help with?

It selects u in integration by parts, preferring Inverse, Logarithmic, Algebraic, Trigonometric, then Exponential functions.

Continue learning

Mastering this chapter with live help

Notes help, but doubts clear fastest in a live class. Narayan Gurukul Academy (ClassApna) runs small-batch CBSE, JEE and NEET coaching from our Mohali centre and online — with daily doubt support and mock tests.

Book a Free Demo See plans & pricing

One-on-one guidance available · Live online classes across India