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7
lekdyu Integrals
v Just as a mountaineer climbs a mountain because it is there, so
a good mathematics student studies new material because
it is there. JAMES B. BRISTOL v
7.1 Hkwfedk (Introduction)
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Basic Properties & Facts
Arithmetic Operations
Properties of Inequalities
If a < b then a + c < b + c and a c < b c
a b
If a < b and c > 0 then ac < bc and <
c c
a b
If a < b and c < 0 then ac > bc and >
c c
b ab
a =
c c
a
a
b =
c
bc
a c ad + bc
+ =
b d
LOCUS
1
Definite
Integration
CONCEPT NO
TES
NOTES
01.
Basic Properties
02.
More Properties
03.
Integration as Limit of a Sum
Maths / Definite Integration
LOCUS
2
Definite Integration
As explained in the chapter titled Integration Basics, the fundamental t
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1
Applications of
Derivatives
CONCEPT NOTES
01.
Tangents and Normals
02.
Monotonicity
03.
Maxima and Minima
04.
Mean Value Theorems And Other Applications
05.
Graphs - II
Maths / Applications of Derivatives
LOCUS
2
Applications of Derivatives
This c
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1
Section - 1
INTRODUCTION TO LIMITS
The concept of limits forms the basis of calculus and is a very powerful one. Both differential and integral calculus
are based on this concept and as such, limits need to be studied in good detail.
This section
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1
Continuity and Differentiability
This chapter requires a good understanding of limits. The concepts of continuity and differentiability are more or
less obvious extensions of the concept of limits.
Section - 1
INTRODUCTION TO CONTINUITY
We start w
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1
Permutations and
Combinations
This chapter is one of the most interesting chapters that well study at this level. The beauty and challenge of
this branch of mathematics lies in the innumerous tricks and mathematical artifices that abound in this s
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1
Indefinite
Integration
CONCEPT NO
TES
NOTES
01.
Basic Rules and Formulae
02.
Integration By Simple Rearrangements
03.
Integration By Substitution
04.
Expansion Using Partial Fractions
05.
Integration By Parts
06.
Miscellaneous Expressions / Substi
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1
Integration:Basics
Section - 1
INTRODUCTION
Instead of starting right away with a list of (scary-looking !) integration formulae, we will first try to understand the
physical significance of integration and why it is required at all; this will the
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1
Differentiation
CONCEPT NO
TES
NOTES
01.
Introduction to differentiation
02.
Differentiation of Standard Functions
03.
Rules for differentiation
04.
Differentiation of Parametric / Implicit Functions
05.
L' Hospital' Rule
Maths / Differentiation
L
ANSWERS
287
ANSWERS
1.3 EXERCISE
1. (b,b), (c,c), (a,c)
2. [-5,5]
3. 4 x 2
4x 1
1
x
x 3
2
4.
f
5.
f 1 cfw_( b, a ) , ( d , b ) , ( a, c ) ,(c, d )
6.
f f x
2,
7.
x 4 6 x3 10 x 2 3x
1
8. (i) represents function which is surjective but not injective
(ii) do
Set-II
DESIGN OF THE QUESTION
PAPER
MATHEMATICS CLASS XII
Time : 3 Hours
Max. Marks : 100
The weightage of marks over different dimensions of the question paper shall
be as follows:
(A)Weightage to different topics/content units
S.No.
1.
2.
3.
4.
5.
6.
To
Set-I
DESIGN OF THE QUESTION
PAPER
MATHEMATICS - CLASS XII
Time : 3 Hours
Max. Marks : 100
The weightage of marks over different dimensions of the question paper shall be as
follows:
(A)
Weightage to different topics/content units
S.No.
Topic
1. Relations
Chapter
13
PROBABILITY
13.1
Overview
13.1.1 Conditional Probability
If E and F are two events associated with the same sample space of a random
experiment, then the conditional probability of the event E under the condition that the
event F has occurred,
Chapter
12
LINEAR PROGRAMMING
12.1 Overview
12.1.1 An Optimisation Problem A problem which seeks to maximise or minimise a
function is called an optimisation problem. An optimisation problem may
involve maximisation of profit, production etc or minimisati
Chapter
11
THREE DIMENSIONAL GEOMETRY
11.1 Overview
11.1.1 Direction cosines of a line are the cosines of the angles made by the line with
positive directions of the co-ordinate axes.
11.1.2 If l, m, n are the direction cosines of a line, then l2 + m2 + n
Chapter
10
VECTOR ALGEBRA
10.1 Overview
10.1.1 A quantity that has magnitude as well as direction is called a vector.
a
10.1.2 The unit vector in the direction of a is given by | a | and is represented by a .
10.1.3 Position vector of a point P (x, y, z)
Chapter
9
DIFFERENTIAL EQUATIONS
9.1 Overview
(i)
An equation involving derivative (derivatives) of the dependent variable with
respect to independent variable (variables) is called a differential equation.
(ii)
A differential equation involving derivativ
Chapter
8
APPLICATION OF INTEGRALS
8.1 Overview
This chapter deals with a specific application of integrals to find the area under simple
curves, area between lines and arcs of circles, parabolas and ellipses, and finding the
area bounded by the above sai
Chapter
7
INTEGRALS
7.1 Overview
d
F (x) = f (x). Then, we write f ( x ) dx = F (x) + C. These integrals are
dx
called indefinite integrals or general integrals, C is called a constant of integration. All
these integrals differ by a constant.
7.1.1
Let
7.
Chapter
6
APPLICATION OF DERIVATIVES
6.1 Overview
6.1.1 Rate of change of quantities
For the function y = f (x),
d
(f (x) represents the rate of change of y with respect to x.
dx
Thus if s represents the distance and t the time, then
ds
represents the rat
Chapter
5
CONTINUITY AND
DIFFERENTIABILITY
5.1
Overview
5.1.1 Continuity of a function at a point
Let f be a real function on a subset of the real numbers and let c be a point in the
domain of f. Then f is continuous at c if
lim f ( x) = f (c)
x c
More el
Chapter
2
INVERSE TRIGONOMETRIC
FUNCTIONS
2.1 Overview
2.1.1 Inverse function
Inverse of a function f exists, if the function is one-one and onto, i.e, bijective.
Since trigonometric functions are many-one over their domains, we restrict their
domains and
Chapter
3
Matrices
3.1 Overview
3.1.1
A matrix is an ordered rectangular array of numbers (or functions). For example,
x 4 3
A= 4 3 x
3 x 4
The numbers (or functions) are called the elements or the entries of the matrix.
The horizontal lines of elements a
Chapter
1
RELATIONS AND FUNCTIONS
1.1 Overview
1.1.1 Relation
A relation R from a non-empty set A to a non empty set B is a subset of the Cartesian
product A B. The set of all first elements of the ordered pairs in a relation R from a
set A to a set B is
Chapter
4
DETERMINANTS
4.1
Overview
To every square matrix A = [aij] of order n, we can associate a number (real or complex)
called determinant of the matrix A, written as det A, where aij is the (i, j)th element of A.
If A
a b
, then determinant of A, de