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MATHEMATICS
EXEMPLAR PROBLEMS
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Class XI
FOREWORD
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The National Curriculum Framework (NCF) 2005 initiated a new phase of development
of syllabi and textbooks for all stages of school ed
Chapter
5
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COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
5.1 Overview
We know that the square of a real number is always non-negative e.g. (4)2 = 16 and
( 4)2 = 16. Therefore, square root of 16 is 4. What about the square root
Chapter
7
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PERMUTATIONS AND
COMBINATIONS
7.1 Overview
The study of permutations and combinations is concerned with determining the number
of different ways of arranging and selecting objects out of a given number of objec
Chapter
3
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TRIGONOMETRIC FUNCTIONS
3.1 Overview
3.1.1 The word trigonometry is derived from the Greek words trigon and metron
which means measuring the sides of a triangle. An angle is the amount of rotation of a
revolvin
Chapter
16
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PROBABILITY
Where a mathematical reasoning can be had, it is as great a folly to
16.1 Introduction
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make use of any other, as to grope for a thing in the dark, when
you have a candle in your hand. JOHN ARBUTHNOT
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Chapter
13
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LIMITS AND DERIVATIVES
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13.1 Introduction
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With the Calculus as a key, Mathematics can be successfully applied to the
explanation of the course of Nature WHITEHEAD
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This chapter is an introduction to Calculus.
Chapter
10
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STRAIGHT LINES
10.1 Overview
10.1.1 Slope of a line
If is the angle made by a line with positive direction of x-axis in anticlockwise direction,
then the value of tan is called the slope of the line and is den
Chapter
9
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SEQUENCE AND SERIES
9.1 Overview
By a sequence, we mean an arrangement of numbers in a definite order according
to some rule. We denote the terms of a sequence by a1 , a 2, a3 , . , etc., the subscript
denotes
Chapter
4
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PRINCIPLE OF MATHEMATICAL
INDUCTION
4.1 Overview
Mathematical induction is one of the techniques which can be used to prove variety
of mathematical statements which are formulated in terms of n, where n is a
po
Chapter
2
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RELATIONS AND FUNCTIONS
2.1 Overview
This chapter deals with linking pair of elements from two sets and then introduce
relations between the two elements in the pair. Practically in every day of our lives, we
p
Chapter
12
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INTRODUCTION TO THREE
DIMENSIONAL GEOMETRY
12.1 Overview
12.1.1 Coordinate axes and coordinate planes Let XOX, YOY, ZOZ be three
mutually perpendicular lines that pass through a point O such that XOX and YOY l
Chapter
11
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CONIC SECTIONS
11.1 Overview
11.1.1 Sections of a cone Let l be a fixed vertical line and m be another line intersecting
it at a fixed point V and inclined to it at an angle (Fig. 11.1).
Fig. 11.1
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Suppose w
Chapter
8
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BINOMIAL THEOREM
8.1 Overview:
8.1.1 An expression consisting of two terms, connected by + or sign is called a
binomial expression. For example, x + a, 2x 3y,
1 1
4
, etc., are all binomial
3 , 7x
5y
x x
expr
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ANSWERS
1.3 EXERCISE
1. (i) cfw_2
(ii) cfw_0, 1
(iii) cfw_1, p
2. (i) cfw_0, 1, 1
11
(ii)
3
(iii)
3.
cfw_
3, 2, 2, 3
cfw_1, 2, 22, 23, .2 P 1,(2p 1
4. (i) True
(ii) False
7. (i) cfw_2, 4, 6, 8, . , 98
8. (i) cfw_4,
6
Chapter
LINEAR INEQUALITIES
v Mathematics is the art of saying many things in many
different ways. MAXWELLv
6.1 Introduction
In earlier classes, we have studied equations in one variable and two variables and also
solved some statement problems by trans
ANSWERS
EXERCISE 1.1
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1. (i), (iv), (v), (vi), (vii) and (viii) are sets.
2. (i) (ii) (iii)
(vi) (v) (vi)
3. (i) A = cfw_3, 2, 1, 0, 1, 2, 3, 4, 5, 6 (ii) B = cfw_1, 2, 3, 4, 5
(iii) C = cfw_17, 26, 35, 44, 53, 62, 71
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CHAPTER 1
REAL NUMBERS
(A) Main Concepts and Results
Euclids Division Lemma : Given two positive integers a and b, there exist unique
integers q and r satisfying a = bq + r, 0 r < b.
Euclids Division Algorithm to obtain
Chapter
1
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SETS
1.1 Overview
This chapter deals with the concept of a set, operations on sets.Concept of sets will be
useful in studying the relations and functions.
1.1.1 Set and their representations A set is a well-def
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CHAPTER 8
INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS
(A) Main Concepts and Results
Trigonometric Ratios of the angle A in a triangle ABC right angled at B
are defined as:
sine of A = sin A =
side opposite to A BC
Chapter
6
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LINEAR INEQUALITIES
6.1 Overview
6.1.1 A statement involving the symbols >, <, , is called an inequality. For
example 5 > 3, x 4, x + y 9.
(i) Inequalities which do not involve variables are called numerical in
SUPPLEMENTARY MATERIAL
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SUPPLEMENTARY MATERIAL
( px + q )
a x 2 + bx + c dx .
We choose constants A and B such that
d
2
A
( a x + bx + c ) + B
dx
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px + q =
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CHAPTER 7
=
A(2ax + b) + B
Comparing the coefficien
Chapter
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MATHEMATICAL REASONING
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There are few things which we know which are not capable of
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mathematical reasoning and when these can not, it is a sign that our
knowledge of them is very small and confused and where a
MATHEMATICS
PART II
Textbook for Class XII
MATHEMATICS
PART II
Textbook for Class XII
Foreword
The National Curriculum Framework, 2005, recommends that childrens life at school
must be linked to their life outside the school. This principle marks a depart
Chapter
4
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DETERMINANTS
4.1 Introduction
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a1 x + b1 y = c 1
a2 x + b2 y = c 2
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In the previous chapter, we have studied about matrices
and algebra of matrices. We have also learnt that a system
of algebraic equations can be e
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CHAPTER 12
SURFACE AREAS AND VOLUMES
(A) Main Concepts and Results
The surface area of an object formed by combining any two of the basic solids,
namely, cuboid, cone, cylinder, sphere and hemisphere.
The volume of an ob
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CHAPTER 5
ARITHMETIC PROGRESSIONS
(A) Main Concepts and Results
An arithmetic progression (AP) is a list of numbers in which each term is
obtained by adding a fixed number d to the preceding term, except the first term
a
APPLICATION OF INTEGRALS
Chapter
359
8
APPLICATION OF INTEGRALS
v One should study Mathematics because it is only through Mathematics that
nature can be conceived in harmonious form. BIRKHOFF v
8.1 Introduction
In geometry, we have learnt formulae to calc
Chapter
13
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LIMITS AND DERIVATIVES
13.1 Overview
13.1.1 Limits of a function
Let f be a function defined in a domain which we take to be an interval, say, I. We shall
study the concept of limit of f at a point a in I.
We
Chapter
11
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CONIC SECTIONS
Let the relation of knowledge to real life be very visible to your pupils
11.1 Introduction
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and let them understand how by knowledge the world could be
transformed. BERTRAND RUSSELL
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In the pre