**CBSE Sample Papers**

# (Syllabus) CBSE Class 12th - Mathematics: Year 2013

## Syllabus CBSE Class 12th

Mathematics

## Units & Marks:

- RELATIONS AND FUNCTIONS - 10 Marks
- ALGEBRA - 13 Marks
- CALCULUS - 44 Marks
- VECTORS AND THREE - DIMENSIONAL GEOMETRY - 17 Marks
- LINEAR PROGRAMMING - 06 Marks
- PROBABILITY - 10 Marks

**Total : 100 Marks**

## UNIT I. RELATIONS AND FUNCTIONS

**1. Relations and Functions : (10) Periods
**Types of relations: reflexive, symmetric, transitive and equivalence
relations. One to one and onto functions, composite functions, inverse of a
function. Binary operations.

**2. Inverse Trigonometric Functions: (12) Periods
**Definition, range, domain, principal value branches. Graphs of inverse
trigonometric functions. Elementary properties of inverse trigonometric
functions.

## UNIT-II: ALGEBRA

**1. Matrices: (18) Period
**Concept, notation, order, equality, types of matrices, zero matrix,
transpose of a matrix, symmetric and skew symmetric matrices. Addition,
multiplication and scalar multiplication of matrices, simple properties of
addition, multiplication and scalar multiplication. Non-commutativity of
multiplication of matrices and existence of non-zero matrices whose product is
the zero matrix (restrict to square matrices of order 2). Concept of elementary
row and column operations. Invertible matrices and proof of the uniqueness of
inverse, if it exists; (Here all matrices will have real entries).

**2. Determinants: (20) Periods
**Determinant of a square matrix (up to 3 x 3 matrices), properties of
determinants, minors, cofactors and applications of determinants in finding the
area of a triangle. Adjoint and inverse of a square matrix. Consistency,
inconsistency and number of solutions of system of linear equations by examples,
solving system of linear equations in two or three variables (having unique
solution) using inverse of a matrix.

## UNIT-III: CALCULUS

**1. Continuity and Differentiability: (18) Periods
**Continuity and differentiability, derivative of composite functions, chain
rule, derivatives of inverse trigonometric functions, derivative of implicit
functions.Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.

**2. Applications of Derivatives: (10) Periods
**Applications of derivatives: rate of change of bodies, increasing/decreasing
functions, tangents and normals, use of derivatives in approximation, maxima and
minima (first derivative test motivated geometrically and second derivative test
given as a provable tool). Simple problems (that illustrate basic principles and
understanding of the subject as well as real-life situations).

**3. Integrals: (20) Periods
**Integration as inverse process of differentiation. Integration of a variety
of functions by substitution, by partial fractions and by parts, simple
integrals to be evaluated.

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

**4. Applications of the Integrals: (10) Periods
**Applications in finding the area under simple curves, especially lines,
circles/parabolas/ ellipses (in standard form only), Area between the two above
said curves (the region should be clearly identifiable).

**5. Differential Equations: (10) Periods
**Definition, order and degree, general and particular solutions of a
differential equation. Formation of differential equation whose general solution
is given. Solution of differential equations by method of separation of
variables, homogeneous differential equations of first order and first degree.
Solutions of linear differential equation of the type:

**dy/dx + py = q where p and q are functions of x
or constant
+ px = q, where p and q are functions of y or constant**

## UNIT-IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY

**1. Vectors: (12) Periods
**Vectors and scalars, magnitude and direction of a vector. Direction cosines
and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel
and collinear vectors), position vector of a point, negative of a vector,
components of a vector, addition of vectors, multiplication of a vector by a
scalar, position vector of a point dividing a line segment in a given ratio.
Scalar (dot) product of vectors, projection of a vector on a line. Vector
(cross) product of vectors. Scalar triple product of vectors.

**2. Three - dimensional Geometry: (12) Periods
**Direction cosines and direction ratios of a line joining two points.
Cartesian and vector equation of a line, coplanar and skew lines, shortest
distance between two lines. Cartesian and vector equation of a plane. Angle
between (i) two lines, (ii) two planes. (iii) a line and a plane. Distance of a
point from a plane.

## UNIT-V: LINEAR PROGRAMMING

**1. Linear Programming: (12) Periods
**Introduction, related terminology such as constraints, objective function,
optimization, different types of linear programming (L.P.) problems,
mathematical formulation of L.P. problems, graphical method of solution for
problems in two variables, feasible and infeasible regions, feasible and
infeasible solutions, optimal feasible solutions (up to three non-trivial
constraints).

## UNIT-VI: PROBABILITY

**1. Probability: (18) Periods
**Conditional probability, multiplication theorem on probability. independent
events, total probability, Baye's theorem, Random variable and its probability
distribution, mean and variance of random variable. Repeated independent
(Bernoulli) trials and Binomial distribution.

## Recommended Textbooks:

1) Mathematics Part I - Textbook for Class XI, NCERT Publication

2) Mathematics Part II - Textbook for Class XII, NCERT Publication