CBSE Maths — Class 12 Maths Board Complete Chapter Guide

Chapter Overview & Weightage

Class 12 Maths board exam is 80 marks (written) + 20 marks (internal). The written paper covers six units, and the weightage has been fairly consistent over the last five years.

CBSE Class 12 Maths is 3 hours, 80 marks. Questions follow this pattern: Section A (18 MCQs + 2 Assertion-Reason = 20 marks), Section B (5 × 2 = 10 marks), Section C (6 × 3 = 18 marks), Section D (4 × 5 = 20 marks), Section E (3 case-based × 4 = 12 marks).

Unit	Topics	Marks
Unit I: Relations & Functions	Relations, Functions, Inverse Trig	8
Unit II: Algebra	Matrices, Determinants	10
Unit III: Calculus	Continuity, Differentiability, Applications, Integrals, Differential Equations	35
Unit IV: Vectors & 3D	Vectors, 3D Geometry	14
Unit V: Linear Programming	LPP	5
Unit VI: Probability	Conditional Probability, Bayes’ Theorem, Distributions	8

Calculus alone is 35 marks — nearly 44% of the paper. If you do nothing else, own Calculus.

Year	Calculus Marks	Algebra Marks	Vectors+3D Marks
2024	35	10	14
2023	35	10	14
2022	35	10	14
2021	35	10	14

The weightage is remarkably stable — CBSE rarely surprises here. This is good news: you can plan your time with confidence.

Key Concepts You Must Know

Ranked by exam frequency and marks potential:

Calculus (35 marks — highest priority)

Differentiation using chain rule, product rule, quotient rule
Implicit differentiation and logarithmic differentiation
Applications: increasing/decreasing functions, maxima/minima (both first and second derivative tests)
Integration: by substitution, by parts, by partial fractions
Definite integrals using properties (especially $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$ )
Area under curves using definite integrals
Differential equations: variable separable, homogeneous, linear first-order

Vectors & 3D (14 marks)

Dot product and cross product — know the geometric interpretation, not just the formula
Scalar triple product for volume/coplanarity
Equation of a line in vector and Cartesian form
Equation of a plane — three forms (normal, intercept, passing through a point)
Angle between line and plane, shortest distance between skew lines

Algebra (10 marks)

Matrix operations: addition, multiplication, transpose
Finding inverse using adjoint method: $A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)$
Solving systems of equations using matrices (Cramer’s Rule or matrix method)
Determinant properties — row operations, cofactor expansion

Probability (8 marks)

Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
Bayes’ Theorem — standard template problems appear every year
Binomial distribution: mean $= np$ , variance $= npq$

Relations & Functions + Inverse Trig (8 marks)

Equivalence relations (reflexive, symmetric, transitive)
One-one and onto functions — proof technique matters
Principal value branch of inverse trig functions
Simplification formulas: $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$ , $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$

Linear Programming (5 marks)

Graphical method only at this level
Corner point theorem — evaluate objective function at all vertices of feasible region
Identify bounded vs. unbounded regions

Important Formulas

\frac{d}{dx}[\sin^{-1}x] = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}[\tan^{-1}x] = \frac{1}{1+x^2}

\frac{d}{dx}[x^x] = x^x(1 + \ln x) \quad \text{(use logarithmic differentiation)}

For implicit: differentiate both sides w.r.t. $x$ , treating $y$ as a function of $x$ .

When to use: Any question asking you to find $\frac{dy}{dx}$ where the function involves inverse trig, $x^x$ , or $x^{f(x)}$ .

\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C

\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C

$\int u \cdot v\,dx = u\int v\,dx - \int\left(\frac{du}{dx}\int v\,dx\right)dx \quad \text{(by parts --- ILATE order)}$

ILATE order for integration by parts: Inverse trig → Logarithmic → Algebraic → Trigonometric → Exponential. Pick $u$ as whichever appears first.

\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx \quad \text{if } f(2a-x) = f(x)

\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f(-x) = f(x) \\ 0 & \text{if } f(-x) = -f(x) \end{cases}

When to use: These properties save 5–7 minutes per question. Whenever limits are symmetric (like $-\pi$ to $\pi$ , or $0$ to $2\pi$ ), check these first.

P(E_i | A) = \frac{P(E_i) \cdot P(A|E_i)}{\sum_{j=1}^{n} P(E_j) \cdot P(A|E_j)}

When to use: Any “find the probability that event $E_i$ caused outcome $A$ ” type question. Draw a tree diagram first — it prevents formula errors.

d = \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}

Solved Previous Year Questions

PYQ 1 — Integration by Parts (2024 Board, Section C, 3 marks)

Question: Evaluate $\int x \cdot e^x\,dx$ .

Here we have Algebraic ( $x$ ) and Exponential ( $e^x$ ). ILATE says Algebraic comes before Exponential, so $u = x$ and $dv = e^x\,dx$ .

\int x \cdot e^x\,dx = x \cdot e^x - \int 1 \cdot e^x\,dx

= x e^x - e^x + C = e^x(x - 1) + C

Many students apply ILATE in reverse order, setting $u = e^x$ and $dv = x\,dx$ . This technically works but creates a circular integral that goes nowhere. Always follow ILATE.

PYQ 2 — Maxima/Minima Application (2023 Board, Section D, 5 marks)

Question: A square piece of tin of side 18 cm is to be made into a box (without top) by cutting a square from each corner. Find the side of the square cut so that the volume is maximum.

Let the side of the square cut = $x$ cm. Then base of box = $(18 - 2x)$ cm, height = $x$ cm.

V = x(18-2x)^2

\frac{dV}{dx} = (18-2x)^2 + x \cdot 2(18-2x)(-2)

= (18-2x)[(18-2x) - 4x] = (18-2x)(18-6x)

Setting $\frac{dV}{dx} = 0$ : either $x = 9$ or $x = 3$ .

$x = 9$ means the base becomes zero — not physical. So $x = 3$ .

\frac{d^2V}{dx^2} = -4(18-6x) + (18-2x)(-6)

At $x = 3$ : $\frac{d^2V}{dx^2} = -4(0) + (12)(-6) = -72 < 0$ . Maximum confirmed.

Maximum volume $= 3(12)^2 = 432$ cm³.

PYQ 3 — Bayes’ Theorem (2024 Board, Section D, 5 marks)

Question: A bag I contains 3 red and 4 black balls. Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.

Let $E_1$ = ball drawn from Bag I, $E_2$ = ball drawn from Bag II, $A$ = ball is red.

$P(E_1) = P(E_2) = \frac{1}{2}$ , $P(A|E_1) = \frac{3}{7}$ , $P(A|E_2) = \frac{5}{11}$ .

P(E_2|A) = \frac{P(E_2) \cdot P(A|E_2)}{P(E_1) \cdot P(A|E_1) + P(E_2) \cdot P(A|E_2)}

= \frac{\frac{1}{2} \cdot \frac{5}{11}}{\frac{1}{2} \cdot \frac{3}{7} + \frac{1}{2} \cdot \frac{5}{11}} = \frac{\frac{5}{11}}{\frac{3}{7} + \frac{5}{11}} = \frac{\frac{5}{11}}{\frac{33+35}{77}} = \frac{\frac{5}{11}}{\frac{68}{77}}

= \frac{5}{11} \times \frac{77}{68} = \frac{5 \times 7}{68} = \frac{35}{68}

Difficulty Distribution

Difficulty	% of Paper	Where it appears
Easy	~40%	Section A MCQs, standard derivative/integral evaluations, straightforward matrix problems
Medium	~40%	Section B and C — applications of derivatives, definite integrals with properties, 3D geometry
Hard	~20%	Section D 5-mark questions — optimization problems, complex integrals, area between curves

A student targeting 70+ should solve all Easy + Medium questions correctly. Hard questions (Section D) are where students waste time — attempt these only after securing the rest.

The Case-Based questions (Section E, 12 marks) are considered Easy-Medium but students lose marks by misreading what’s asked. Read each sub-part carefully.

Expert Strategy

Three weeks before the exam:

The toppers who score 95+ don’t study everything — they study the right things more deeply. Here’s what they do differently.

Week 1 priority: Calculus only. Spend the first week mastering integration techniques. Every day, solve 3 integrals from each type: substitution, by parts, partial fractions. Calculus is 35 marks — you cannot afford gaps here.

For definite integrals, before integrating, always check if a symmetry property applies. In at least one 2024 paper question, applying the property $\int_0^a f(a-x)\,dx = \int_0^a f(x)\,dx$ reduces a 10-minute calculation to 2 minutes.

Week 2 priority: Vectors + 3D, then Matrices. These are formula-heavy but once the formulas are memorised and practised, they’re very reliable marks. The “shortest distance between skew lines” question appears almost every year in Section D.

Week 3: Probability and past papers. Solve at least 3 full previous year papers under timed conditions. The 2024, 2023, and 2022 papers are most representative of current patterns.

During the exam: Attempt Section A (MCQs) first — 20 marks in 20-25 minutes. Don’t spend more than 2 minutes per MCQ. Move to Section E (case-based) next while your mind is fresh — these are guaranteed marks if you read carefully. Then Section D (5-mark), Section C (3-mark), Section B (2-mark).

Show all steps in Section C and D, even when using a formula directly. CBSE awards step marks. A student who writes the formula, substitutes correctly, and gets a wrong final answer can still score 2/3 or 3/5.

Common Traps

Determinant sign error in cofactor expansion: When expanding along a row or column, the signs alternate starting with $+$ . The sign of element $a_{ij}$ is $(-1)^{i+j}$ . Students forget this for the (1,2) and (2,1) positions, which should be negative.

Missing the modulus in logarithmic integrals: $\int \frac{1}{x}\,dx = \ln|x| + C$ , not $\ln x + C$ . In CBSE board marking, leaving out the modulus loses a half-mark in integration questions. Habit-forming mistake — fix it now.

In LPP, checking only the corner points and missing the unbounded region check: When the feasible region is unbounded and you’re minimising, the minimum may not exist. CBSE asks you to “verify” this — draw the line $Z = k$ (where $k$ is your candidate minimum) and check that no part of the feasible region satisfies $Z < k$ . Students skip this step and lose marks.

Differential equations — forgetting the constant of integration: When solving $\frac{dy}{dx} = f(x)$ , writing $y = F(x)$ instead of $y = F(x) + C$ loses 1 mark in every such question. The constant appears at the separation step, not the end.

In 3D geometry, confusing direction ratios with direction cosines: Direction cosines satisfy $l^2 + m^2 + n^2 = 1$ . Direction ratios don’t. When the question asks for the “angle between two lines,” you need direction cosines (or the dot product formula directly). Plugging in un-normalised ratios into $\cos\theta = l_1l_2 + m_1m_2 + n_1n_2$ gives wrong answers.

Inverse trig range errors: $\sin^{-1}$ returns values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ , while $\cos^{-1}$ returns values in $[0, \pi]$ . A common MCQ trap: “find $\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$ ” — the answer is NOT $\frac{7\pi}{6}$ (outside range). You must find the equivalent angle in $[0, \pi]$ , which is $\frac{5\pi}{6}$ .

The board exam rewards students who know the pattern and practise within it. With Calculus locked down and a systematic approach to each section, 85+ is very much within reach.