CBSE Mathematics Study Guide

Overview

This guide covers the CBSE Class 11 and 12 Mathematics syllabus (NCERT). It is structured by topic with definitions, key results, worked examples, and exam-focused advice.

The CBSE Class 12 board exam carries 80 marks (theory) + 20 marks (internal assessment). The paper consists of objective questions (MCQs), short-answer questions (2-3 marks), and long-answer questions (5-6 marks).

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1. Sets and Functions

1.1 Sets and Set Notation

A set is a well-defined collection of distinct objects. Sets are denoted by capital letters ($A, B, C$) and elements by lowercase letters ($a, b, c$).

Notation.

• $a \in A$: $a$ is an element of $A$
• $a \notin A$: $a$ is not an element of $A$
• $A \subseteq B$: $A$ is a subset of $B$ (every element of $A$ is also in $B$)
• $A \subset B$: $A$ is a proper subset of $A \neq B$
• $\emptyset$: the empty set
• $|A|$ or $n(A)$: the cardinality (number of elements) of $A$

Set operations.

Operation	Notation	Meaning
Union	$A \cup B$	All elements in $A$ or $B$ or both
Intersection	$A \cap B$	All elements in both $A$ and $B$
Complement	$A'$ or $\bar{A}$	All elements in the universal set $U$ not in $A$
Difference	$A - B$	Elements in $A$ but not in $B$

Laws.

• Commutative: $A \cup B = B \cup A$; $A \cap B = B \cap A$
• Associative: $(A \cup B) \cup C = A \cup (B \cup C)$
• Distributive: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
• De Morgan: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$

1.2 Venn Diagrams

Venn diagrams represent sets as overlapping circles inside a rectangle (the universal set).

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

For three sets:

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$

1.3 Relations

A relation $R$ from set $A$ to set $B$ is a subset of $A \times B$.

Types of relations:

• Reflexive: $(a, a) \in R$ for all $a \in A$
• Symmetric: $(a, b) \in R \Rightarrow (b, a) \in R$
• Transitive: $(a, b) \in R$ and $(b, c) \in R \Rightarrow (a, c) \in R$
• Equivalence relation: reflexive, symmetric, and transitive

1.4 Functions

A function $f: A \to B$ maps each element of $A$ to exactly one element of $B$.

Key terms:

• Domain: the set of all valid inputs
• Codomain: the set $B$ into which $f$ maps
• Range: the set of actual outputs $\{f(a) : a \in A\}$

Types of functions:

• One-one (injective): $f(a_1) = f(a_2) \Rightarrow a_1 = a_2$
• Onto (surjective): range = codomain (every element of $B$ is mapped to)
• Bijective: both one-one and onto

Composition. If $f: A \to B$ and $g: B \to C$, then $(g \circ f)(x) = g(f(x))$.

Inverse function. If $f$ is bijective, then $f^{-1}$ exists and $f^{-1}(f(x)) = x$.

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2. Algebra

2.1 Matrices

A matrix is a rectangular array of numbers. An $m \times n$ matrix has $m$ rows and $n$ columns.

Operations.

• Addition: $A + B$ is defined only when $A$ and $B$ have the same order
• Scalar multiplication: $kA$ multiplies each entry by $k$
• Matrix multiplication: If $A$ is $m \times p$ and $B$ is $p \times n$, then $AB$ is $m \times n$

Transpose. $A^T$ is obtained by interchanging rows and columns. $(AB)^T = B^T A^T$.

2.2 Determinants

The determinant of a $2 \times 2$ matrix:

$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

For a $3 \times 3$ matrix, expand by cofactors along any row or column:

$|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$

where $C_{ij} = (-1)^{i+j} M_{ij}$ and $M_{ij}$ is the minor (determinant after removing row $i$, column $j$).

Properties:

• $|AB| = |A| \cdot |B|$
• $|A^T| = |A|$
• $|kA| = k^n |A|$ for an $n \times n$ matrix
• If any two rows or columns are identical, $|A| = 0$

2.3 Inverse of a Matrix

For a non-singular square matrix $A$ ($|A| \neq 0$):

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

where $\text{adj}(A)$ is the adjoint (transpose of the cofactor matrix).

Solving systems of linear equations. For $AX = B$ where $A$ is $n \times n$:

• If $|A| \neq 0$: unique solution $X = A^{-1}B$
• If $|A| = 0$ and $(\text{adj } A)B \neq 0$: no solution (inconsistent)
• If $|A| = 0$ and $(\text{adj } A)B = 0$: infinitely many solutions

2.4 Complex Numbers

A complex number is $z = a + bi$ where $a$ is the real part, $b$ is the imaginary part, and $i^2 = -1$.

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

Conjugate: $\bar{z} = a - bi$; $z\bar{z} = a^2 + b^2 = |z|^2$

Polar form: $z = r(\cos\theta + i\sin\theta)$ where $r = |z|$ and $\theta = \arg(z)$.

2.5 Quadratic Equations

The general quadratic $ax^2 + bx + c = 0$ has roots:

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

Discriminant $\Delta = b^2 - 4ac$:

• $\Delta > 0$: two distinct real roots
• $\Delta = 0$: one repeated real root
• $\Delta < 0$: two complex conjugate roots

Relations between roots $\alpha, \beta$:

• $\alpha + \beta = -b/a$
• $\alpha\beta = c/a$

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3. Calculus

3.1 Limits

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

means $f(x)$ approaches $L$ as $x$ approaches $a$.

Standard limits:

$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \qquad \lim_{x \to \infty}\left(1 + \frac{1}{x}\right)^x = e$

L’Hôpital’s Rule. If $\lim_{x \to a}\frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:

$\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}$

provided the right-hand limit exists.

3.2 Derivatives

From first principles:

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

Standard derivatives:

$f(x)$	$f'(x)$
$x^n$	$nx^{n-1}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$e^x$	$e^x$
$\ln x$	$1/x$
$a^x$	$a^x \ln a$

Rules:

• Sum: $(f + g)' = f' + g'$
• Product: $(fg)' = f'g + fg'$
• Quotient: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$
• Chain: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

3.3 Applications of Derivatives

Rate of change. If $y = f(x)$, then $\dfrac{dy}{dx}$ is the instantaneous rate of change.

Increasing/Decreasing. $f$ is increasing where $f'(x) > 0$ and decreasing where $f'(x) < 0$.

Maxima and Minima. At a critical point $f'(x) = 0$:

• Second derivative test: $f''(x) < 0 \Rightarrow$ local max; $f''(x) > 0 \Rightarrow$ local min
• If $f''(x) = 0$, use the first derivative test

Tangent and Normal. At a point $(x_0, y_0)$ on $y = f(x)$:

• Tangent gradient: $f'(x_0)$
• Tangent equation: $y - y_0 = f'(x_0)(x - x_0)$
• Normal gradient: $-1/f'(x_0)$

3.4 Integration

Integration is the reverse of differentiation.

Indefinite integrals (antiderivatives):

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \qquad \int \frac{1}{x}\,dx = \ln|x| + C$

Methods:

• Substitution: $\int f(g(x))g'(x)\,dx = \int f(u)\,du$ where $u = g(x)$
• Integration by parts: $\int u\,dv = uv - \int v\,du$

Definite integrals:

$\int_a^b f(x)\,dx = F(b) - F(a)$

Area under a curve:

$\text{Area between } f(x) \text{ and } x\text{-axis from } a \text{ to } b = \left|\int_a^b f(x)\,dx\right|$

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4. Coordinate Geometry

4.1 Straight Lines

Slope-intercept form: $y = mx + c$

Point-slope form: $y - y_1 = m(x - x_1)$

Two-point form: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$

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

Distance from a point to a line $Ax + By + C = 0$:

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

Angle between two lines with slopes $m_1$ and $m_2$:

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

4.2 Conic Sections

Circle. Centre $(h, k)$, radius $r$:

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

General form: $x^2 + y^2 + 2gx + 2fy + c = 0$; centre $(-g, -f)$, radius $\sqrt{g^2 + f^2 - c}$.

Parabola. Focus-directrix form $y^2 = 4ax$:

• Focus: $(a, 0)$; Directrix: $x = -a$; Axis: $y = 0$; Latus rectum: $4a$

Ellipse. $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ where $a > b$:

• Foci: $(\pm c, 0)$ where $c^2 = a^2 - b^2$
• Eccentricity: $e = c/a < 1$
• Latus rectum length: $2b^2/a$

Hyperbola. $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$:

• Foci: $(\pm c, 0)$ where $c^2 = a^2 + b^2$
• Eccentricity: $e = c/a > 1$
• Asymptotes: $y = \pm \dfrac{b}{a}x$
• Latus rectum length: $2b^2/a$

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5. Trigonometry

5.1 Identities

Pythagorean identities:

$\sin^2\theta + \cos^2\theta = 1 \qquad 1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta$

Compound angle:

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$ $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$ $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$

Double angle:

$\sin 2A = 2\sin A\cos A \qquad \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$

5.2 Trigonometric Equations

General solutions:

• $\sin\theta = \sin\alpha \Rightarrow \theta = n\pi + (-1)^n\alpha$
• $\cos\theta = \cos\alpha \Rightarrow \theta = 2n\pi \pm \alpha$
• $\tan\theta = \tan\alpha \Rightarrow \theta = n\pi + \alpha$

5.3 Inverse Trigonometric Functions

Function	Domain	Range
$\sin^{-1}x$	$[-1, 1]$	$[-\pi/2, \pi/2]$
$\cos^{-1}x$	$[-1, 1]$	$[0, \pi]$
$\tan^{-1}x$	$\mathbb{R}$	$(-\pi/2, \pi/2)$

Key identities:

$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \qquad \tan^{-1}x + \tan^{-1}\frac{1}{x} = \frac{\pi}{2}\text{ for } x > 0$

5.4 Properties of Triangles

Sine rule: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R$

Cosine rule: $a^2 = b^2 + c^2 - 2bc\cos A$

Area: $\Delta = \frac{1}{2}ab\sin C = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$

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6. Probability and Statistics

6.1 Measures of Central Tendency

Mean (arithmetic): $\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$

Median: the middle value when data is arranged in order

Mode: the most frequently occurring value

For grouped data, the median and mode use interpolation formulas from the cumulative frequency distribution.

6.2 Variance and Standard Deviation

Variance:

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

Standard deviation: $\sigma = \sqrt{\text{variance}}$

For combined data from two groups of sizes $n_1, n_2$ with means $\bar{x}_1, \bar{x}_2$ and variances $\sigma_1^2, \sigma_2^2$:

$\bar{x}_{\text{combined}} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$

6.3 Probability

Rules:

• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• $P(A') = 1 - P(A)$
• $P(A \cap B) = P(A) \cdot P(B|A)$ (multiplication rule)
• Independent events: $P(A \cap B) = P(A) \cdot P(B)$

Conditional probability: $P(A|B) = \dfrac{P(A \cap B)}{P(B)}$

Bayes’ theorem: $P(A|B) = \dfrac{P(B|A) \cdot P(A)}{P(B)}$

6.4 Binomial Distribution

A binomial experiment has $n$ independent trials, each with probability $p$ of success.

$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$

Mean: $\mu = np$

Variance: $\sigma^2 = np(1-p)$

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7. Key Formulas

Topic	Formula
Quadratic roots	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Distance between points	$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Sum of AP	$S_n = \frac{n}{2}[2a + (n-1)d]$
Sum of GP	$S_n = \frac{a(r^n - 1)}{r - 1}$
$n$th derivative of $e^{ax}$	$a^n e^{ax}$
$\int e^{ax}\,dx$	$\frac{1}{a}e^{ax} + C$
$\int \frac{1}{1+x^2}\,dx$	$\tan^{-1}x + C$
Binomial coefficient	$\binom{n}{r} = \frac{n!}{r!(n-r)!}$
Circle equation	$(x-h)^2 + (y-k)^2 = r^2$
Euler’s formula	$e^{i\theta} = \cos\theta + i\sin\theta$

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8. Exam Tips

1. Show all working. CBSE awards method marks even if the final answer is wrong. Never skip steps.
2. Manage time carefully. The paper is 3 hours; spend roughly 1 minute per mark. Attempt all questions.
3. Memorise the NCERT formulas. Most exam questions are directly based on NCERT textbook derivations and formulas.
4. Draw clean diagrams. In coordinate geometry and trigonometry, a well-labelled diagram often earns partial marks.
5. Check the domain of inverse trigonometric functions. Marks are frequently lost by giving values outside the principal range.
6. Use step-marking strategy. For 6-mark questions, write each step on a new line. Examiners look for specific intermediate results.
7. Practise previous-year papers. CBSE tends to repeat question patterns. At minimum, solve the last 5 years’ papers.

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Common Pitfalls

1. Forgetting the constant of integration $+C$ in indefinite integrals — this is penalised in almost every paper.
2. Incorrect domain for inverse trig functions. Remember: $\sin^{-1}$ maps to $[-\pi/2, \pi/2]$; $\cos^{-1}$ maps to $[0, \pi]$.
3. Confusing range and codomain. The range is the set of actual outputs, not the entire codomain.
4. Arithmetic errors in determinants. Sign errors in cofactor expansion are extremely common — double-check $(-1)^{i+j}$.
5. Applying L’Hôpital’s rule to non-indeterminate forms. Always verify the limit is $0/0$ or $\infty/\infty$ first.
6. Missing absolute values in $\int \frac{1}{x}\,dx = \ln|x| + C$ and in area calculations.
7. Incorrectly applying the second derivative test. When $f''(x_0) = 0$, you must use the first derivative test — do not conclude “no max/min”.

Worked Examples

Example 1: Finding the Area Under a Curve

Problem: Find the area enclosed by y = x^2, the x-axis, and the lines x = 0 and x = 2. Solution: Area = integral from 0 to 2 of x^2 dx = [x^3/3] from 0 to 2 = 8/3 - 0 = 8/3 square units. Since y >= 0 on this interval, no absolute value is needed.

Example 2: Solving a System Using Matrices

Problem: Solve: 2x + y = 5, x - y = 1. Solution: In matrix form AX = B, where A = [[2,1],[1,-1]], B = [[5],[1]]. |A| = (2)(-1) - (1)(1) = -3. Since |A| != 0, unique solution exists. A^{-1} = (1/-3)[[-1,-1],[-1,2]] = (1/3)[[1,1],[1,-2]]. X = A^{-1}B = (1/3)[[1,1],[1,-2]][[5],[1]] = (1/3)[[6],[3]] = [[2],[1]]. So x = 2, y = 1.

Example 3: Probability Using Bayes’ Theorem

Problem: A disease affects 1% of a population. A test has 99% sensitivity (true positive rate) and 95% specificity (true negative rate). A person tests positive. What is the probability they actually have the disease? Solution: Let D = disease, T = positive test. P(D) = 0.01, P(D’) = 0.99. P(T|D) = 0.99, P(T|D’) = 1 - 0.95 = 0.05. P(T) = P(T|D)P(D) + P(T|D’)P(D’) = 0.99(0.01) + 0.05(0.99) = 0.0099 + 0.0495 = 0.0594. P(D|T) = P(T|D)P(D)/P(T) = 0.0099/0.0594 = 0.1667 or approximately 16.7%. Despite a positive test, the probability of having the disease is only about 1/6 due to the low prior probability.

Summary

CBSE Mathematics covers sets and functions, algebra (matrices, determinants, complex numbers, quadratics), calculus (limits, derivatives, integration, applications), coordinate geometry (straight lines, conic sections), trigonometry (identities, equations, inverse functions, properties of triangles), and probability and statistics (measures of central tendency, variance, binomial distribution, Bayes’ theorem). The theory paper is 80 marks and the internal assessment is 20 marks. Key exam strategies include showing all working, managing time, memorising NCERT formulas, and practising previous-year papers.