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MATHEMATICS

CBSE

SAMPLEPAPERSSAMPLESAMPLEPAPERSPAPERS

COVERS

2016

BOARD PAPERS

(SOLVED)

320.00

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One Paper Three Hours Marks : 100

Units No. of Periods Marks

I. RELATIONS AND FUNCTIONS 30 10

II. ALGEBRA 50 13

III. CALCULUS 80 44

IV. VECTORS AND THREE-DIMENSIONAL GEOMETRY 30 17

V. LINEAR PROGRAMMING 20 06

VI. PROBABILITY 30 10

TOTAL 240 100

UNIT I : RELATIONS AND FUNCTIONS

1. Relations and Functions : (15 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 : (15 Periods)

Definition, range, domain, principal value branch. Graphs of inverse

trigonometric functions. Elementary properties of inverse trigonometric functions.

UNIT II : ALGEBRA

1. Matrices : (25 Periods)

Concept, notation, order, equality, types of matrices, zero and identity matrix,

transpose of a matrix, symmetric and skew-symmetric matrices. Operation on

matrices: addition and multiplication and multiplication with scalar. 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 : (25 Periods)

Determinant of a square matrix (up to 3 × 3 matrices), properties of

determinants, minors, co-factors and applications of determinants in finding

MATHEMATICS—XII

Syllabus 2015.pmd 7/13/2016, 6:39 PM1

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the area of a triangle. Adjoint and inverse of a square matrix. Consistency,inconsistency and number of solutions of system of linear equations byexamples, solving system of linear equations in two or three variables (havingunique solution) using inverse of a matrix.

UNIT III : CALCULUS

1. Continuity and Differentiability : (20 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, maximaand minima (first derivative test motivated geometrically and second derivativetest given as a provable tool). Simple problems (that illustrate basic principlesand 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. Evaluation of simpleintegrals of the following types and problems based on them:

2 2 22 2 2 2 2

, , , ,

dx dx dx dx dx

x a ax bx cx a a x ax bx c± + +± − + +∫ ∫ ∫ ∫ ∫ ,

2 2 2 2

2 2

, , ,

px q px qdx dx a x dx x a dx

ax bx c ax bx c

+ +± −

+ + + +∫ ∫ ∫ ∫

2 2, ( )ax bx c dx px q ax bx c dx+ + + + +∫ ∫ .

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (withoutproof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals : (15 Periods)

Applications in finding the area under simple curves, especially lines, circles/

parabolas/ellipses (in standard form only). Area between the two above saidcurves (the region should be clearly identifiable).

Syllabus 2015.pmd 7/13/2016, 6:39 PM2

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5. Differential Equations : (15 Periods)

Definition, order and degree, general and particular solutions of a differential

equation. Formation of differential equation whose general solution is given. Solutionof differential equations by method of separation of variables, solution ofhomogeneous differential equations of first order and first degree. Solutions oflinear differential equation of the type :

dypy q

dx+ = , where p and q are functions of x or constants.

dx

dy + px = q, where p and q are functions of y or constants.

UNIT IV : VECTORS AND THREE-DIMENSIONAL GEOMETRY

1. Vectors : (15 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 andcollinear vectors), position vector of a point, negative of a vector, componentsof a vector, addition of vectors, multiplication of a vector by a scalar, positionvector of a point dividing a line segment in a given ratio.

Definition, Geometrical Interpretation, properties and applications of scalar (dot)product of vectors, vector (cross) product of vectors, scalar triple product ofvectors.

2. Three-dimensional Geometry : (15 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 betweentwo 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 : (20 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 forproblems in two variables, feasible and infeasible regions (bounded andunbounded), feasible and infeasible solutions, optimal feasible solutions (up tothree non-trivial constraints).

UNIT VI : PROBABILITY

1. Probability : (30 Periods)

Conditional probability, multiplication theorem on probability, Independent

events, total probability, Baye’s theorem, Random variable and its probabilitydistribution, mean and variance of a random variable. Repeated independent(Bernoulli) trials and Binomial distribution.

Syllabus 2015.pmd 7/13/2016, 6:39 PM3

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Questions Paper Designs 2014-15(Maths-XII).pmd 7/13/2016, 6:51 PM1

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CONTENTS●●●●● CBSE Question Papers 2016 (Delhi) Solved 1—23

●●●●● CBSE Question Papers 2016 (Outside Delhi) Solved 24—47

●●●●● CBSE Question Papers 2015 (Delhi) Solved 1—25


●●●●● CBSE Question Papers 2014 (Delhi) Solved 1—35


Fast Track Revision F-1—F-22

Other Sample Papers (Solved) 1—118

MBD Sample Question Paper–1 1—24





Model Question Papers for Practice

Model Question Paper—1 M-1—M-4












Contents 2015.pmd 7/13/2016, 6:17 PM1

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Contents 2015.pmd 7/13/2016, 6:17 PM2

C.B.S.E. QUESTION PAPERS 2016 (SOLVED)

MATHEMATICSCLASS–XII (DELHI)

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SET—I

SECTION–A

(Question numbers 1 to 6 carry 1 mark each)

1. Find the maximum value of

1 1 1

1 1 + sin 1

1 1 1 + cos

θθ

.

Solution. Operating R2 → R2 – R1 and R3 → R3 – R1, we get:

Δ =

1 1 1

0 sin 0

0 0 cos

θθ

= sin θ cos θ = 1

2sin 2 θ.

∴ Max. value of Δ = 1

2(1) = 1

2.

2. If A is a square matrix such that A2 = I, then find the simplified value of(A – I)3 + (A + I)3 – 7A.

Solution. (A – I)3 + (A + I)3 – 7A= A3 – I3 – 3 A2I + 3A I2 + A3 + I3 + 3A2I + 3AI2 –7A= 2A3 + 6AI2 – 7A = 2AA2 + 6A I – 7A= 2AI + 6A – 7A = 2A + 6A – 7A = A. [�A2 = I]

3. Matrix A =

−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦

0 2 2

3 1 3

3 3 1

b

a

is given to be symmetric, find values of a and b.

Code No. 65/1/1/DSeries : ONS/1

1

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CBSE Board 2016 (Delhi).pmd 7/13/16, 6:17 AM1

MBD Sure Shot CBSE Sample PapersSolved Class 12 Mathematics 2017

Publisher : MBD GroupPublishers

ISBN : 9789351850892 Author : Panel Of Experts

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