Download NCERT Handbook Term 1 Mathematics Class 9

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Index
PART A : CHAPTER-WISE QUICK REVISION NOTES,
NCERT SOLUTIONS AND PRACTICE QUESTION BANK
1.
Number System
2.
Polynomials
A-27-A-50
3.
Introduction to Euclid's Geometry
A-51-A-66
4.
Lines and Angles
A-67-A-94
5.
Triangles
6.
Coordinate Geometry
A-129-A-142
7.
Heron’s Formula
A-143-A-166
A-1-A-26
A-95-A-128
PART B : SAMPLE PAPERS WITH SOLUTIONS FOR SA1
•
SAMPLE PAPER-1
B- 1- B- 12
•
SAMPLE PAPER-2
B- 13- B- 22
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SAMPLE PAPER-3
B- 23- B- 34
•
SAMPLE PAPER-4
B- 35- B- 48
•
SAMPLE PAPER-5
B- 49- B- 60
Number System
1
Key
Concepts
NUMBER LINE
Representation of various types of numbers on the
number line.
–3
–2
–1
–1
2
0 1 1
4 2
1
2
3
RATIONAL NUMBERS BETWEEN ANY TWO
GIVEN RATIONAL NUMBERS
In general, there are infinitely many rational numbers
between any two given rational numbers. To find a
rational number between s and t and divide by 2,
VARIOUS TYPES OF NUMBERS
1.
Set of Natural Numbers, N = {1, 2, 3, …}
Representation of N on number line
2.
3.
4.
1 2 3 4
Set of whole numbers, W = {0, 1, 2, 3, …}
number line of W
0 1 2 3 4
Set of integers,
Z = {…, –3, –2, –1, 0, 1, 2, 3, … }
number line of Z (integers) :
Rational numbers : A number ‘r’ is called a
rational number, if it can be written in the form
p
, where p and q are integers and q ¹ 0.
q
ìp
ü
Q = í : p, q Î I , q ¹ 0 ý
q
î
þ
Note: The rational numbers also include the natural
numbers, whole numbers and integers.
Rational numbers,
EQUIVALENT RATIONAL NUMBERS
The rational numbers do not have a unique
p
representation in the form
, where p and q are
q
integers and q ¹ 0.
s+t
lies between s and t. Proceeding in
2
this manner, we may find more rational numbers
between s and t.
that is,
IRRATIONAL NUMBERS
A number ‘s’ is called irrational, if it cannot be
written in the form
p
, where p and q are integers and
q
q ¹ 0.
Examples are :
2, 3, 15, Õ, 0.10110111011110...
Note: when we use the symbol
, we assume that
it is the positive square root of the number. So
4
= 2, though both 2 and – 2 are square roots of 4.
REAL NUMBERS
The set of rational numbers and irrational
numbers form a set of real numbers which is denoted
by R.
Remark: Every real number is represented by a
unique point on the number line. Also, every
point on the number line represents a unique real
number.
A-2
MATHEMATICS
REAL NUMBER & THEIR DECIMAL EXPANSIONS
1.
Decimal Expansions of Real Numbers : The
decimal expansions of real numbers can be
used to distinguish between rationals and
irrationals.
For instance: (i) Decimal expansion of 10
3
3.33333.......
3 10
9
10
9
10
9
10
9
1
Remainders : 1, 1, 1, 1, 1,...
Divisor : 3
Note:(i)
The remainders either become 0 after a
certain stage, or start repeating
themselves.
(ii) The number of entries in the repeating
string of remainders is less than the
1
divisor (in , one number repeats itself
3
and the divisor is 3)
(iii) If the remainders repeat, then we get a
repeating block of digits in the quotient
Remark: On division of p by q, two main
things happen-either the remainder becomes
zero or never becomes zero and we get a
repeating string of remainders.
2.
Terminating Decimal Expansions : In this
case, the decimal expansion terminates or ends
after a finite number of steps. We call such
a decimal expansion as terminating.
3.
Non-terminating Recurring Expansions : In
this case we have a repeating block of digits
in the quotient. We say that this expansion
is non-terminating recurring.
4.
How to Write Non-terminating Recurring
Expansions in Short :
The usual way of showing that 3 repeats in
1
the quotient of
is to write it as 0.3 .
3
Similarly, since the block of digits 142857
1
1
repeats in the quotient of , we write
as
7
7
0.142857, where the bar above the digits
indicates the block of digits that repeats.
5.
Result 1 : The decimal expansion of a rational
number is either terminating or nonterminating recurring. Moreover, a number
whose decimal expansion is terminating or
non-terminating recurring is rational.
Result 2: The decimal expansion of an
irrational number is non- terminating nonrecurring. Moreover, a number whose decimal
expansion is non-terminating non- recurring
is irrational.
Indentification of the nature of real numbers
from their decimal expansions
Case I. When the decimal expansion is
terminating.
Consider the real number 3.142678, whose
decimal expansion is terminating.
3142678 p
º ,q¹0
We have 3.142678 =
1000000 q
Where,
p = 3142678
q = 1000000 (¹ 0)
Hence 3. 142678 is a rational number.
So, every number with a terminating decimal
expansion can be expressed in the form
p
q
(q ¹ 0) , where p and q are integer and hence
such a number is a rational number.
Case II. When the decimal expansion is nonterminating.
Let us take a example :
Ex.
Consider the real number 0.3333....... (or 0.3)
Let x = 0.3333......(= 0. 3 )
Since one digit repeats, we multiply x by 10
to get
10x =10 × (0.3333.........) = 3.333.....
10x = 3 + 0.333.........
Þ
10x = 3 + x Þ 9x = 3
Þ
Þ
x=
3 Þ x=1
3
9
Hence 0.3333....... (or 0. 3 ) is a rational number..
[We may check the reverse that = 0. 3 ].
Similarly, for the number 1.272727......... two
digits are repeating we have to multiply it by
100.
Number System
A-3
RATIONAL NUMBERS
Representation of rational numbers on the number
line through successive magnification
Let us try to repersent 3.47 on the number line.
We know that 3.47 lies between 3 and 4. We
divide the portion between 3 and 4 into 10 equal
parts as below:
4.
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
3
4
Now, 3.47 lies between 3.4 and 3.5. Again we
divide the portion between 3.4 and 3.5 into 10 equal
parts.
Then QD =
3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49
3.4
Representation of x for any Given
Positive Real Number x Geometrically :
Method. Let P be a fixed point on a given line.
Let y be any positive real number. Mark a
point Q on the given line such that PQ = x.
Again mark a new point R on the given
line such that QR = 1. Find the midpoint of
PR and mark that point as O. Draw a semicircle with centre O and radius OR. Draw a
line perpendicular to PR passing through Q
and intersecting the semi-circle at D.
y.
3.5
Now, we can easily locate 3.47 on the number line.
In the above method, we have successively
magnified different portions to represent 3.47 on the
number line.
This method of representation of real number
on the number line is known as method of successive
magnification.
1.
Operations on Rational Numbers : Rational
numbers satisfy the commutative, associative
and distributive laws for addition and
multiplication.
2.
Operations on Irrational Numbers : Irrational
numbers also satisfy the commutative,
associative and distributive laws for addition
and multiplication.
Some useful facts :
(i) The sum or difference of a rational
number and an irrational number is
irrational.
(ii) The product or quotient of a non-zero
rational number with an irrational number
is irrational.
(iii) If we add, subtract, multiply or divide
two irrationals, the result may be rational
or irrational.
3.
Operation of Taking Square Root of Real
Numbers : We know that, if a is a natural
number, then a = b means b2 = a and
b ³ 0. The same definition can be extended
for positive real numbers.
Let a > 0 be a real number. Then a = b
means b2 = a and b ³ 0.
D
Öx
P
5.
Q 1 R
O
nth Root of a Real Number : Let a > 0 be a
real number and n be a positive integer.
Then
6.
y
n
n
a = b, if b = a and b > 0.
Some Identities Related to Square Roots :
Let a and b be positive real numbers.
Then
(i)
ab =
a
b
(ii)
a
=
b
a
b
(iii) ( a + b ) ( a - b ) = a - b
(iv) (a + b ) (a - b ) = a 2 - b
(v)
( a + b)( c + d ) =
ac + ad + bc + bd
(vi) ( a + b ) 2 = a + 2 ab + b.
7.
Rationalisation : When the denominator of
an expression contains a term with a square
root, the procedure of converting it to an
equivalent expression whose denominator is
a rational number is called rationalising the
denominator.
A-4
I.
MATHEMATICS
(i)
(ii)
(iii)
m
n
m+n
a .a =a
(am)n = amn
am
= am-n , m > n
an
(iv) am bm = (ab)m where a is called the
base and m and n are the exponents.
(v) Value of (a)0 :
We have (a)0 = 1.
(vi) a–n =
II.
Let a> 0 be a real number and p and q be
rational numbers. Then we have
Laws of Exponents for Real Numbers
p
. a q= a
(i)
a
(iii)
ap
= a p-q
aq
p+q
(ii) (a p) q = a
(iv) a
p
pq
b p = (ab) p.
III.
Meaning of n a in the Language of
Exponents : we define n a = a1/n.
IV.
Definition : Let a > 0 be a real number. Let
m and n be integers such that m and n have
no common factors other than 1, and n > 0.
Then,
1
.
an
Extended Laws of Exponents : These extended
laws of exponents are as follows :
am/n = ( n a ) m =
n
am .
TEXTBOOK SOLUTI ONS
EXERCISE 1.1
1.
Sol.
2.
Sol.
Is zero a rational number? Can you write it in
p
the form , where p and q are integers and
q
q ¹ 0?
Yes! zero is a rational number. We can write
p
zero in the form q , as follows :
0 0 0
0 = = = ......so on., q can be negative
1 2 3
integer also.
Find six rational number between 3 and 4.
A rational number between ‘a’ and ‘b’ is given
a+b
as
2
7
3+
3+ 4 7
2 = 13
\
= Þ
2
2
2
4
25
13
3+
3+
8 = 49
4 = 25 Þ
Þ
2
8
2
16
49
97
3+
97
16
32 = 103 .
Þ
Þ
=
2
32
2
64
Thus, six rational numbers between 3 and 4 are
7 13 25 49 97
193
, , , ,
and
.
2 4 8 16 32
64
3.
Sol.
4.
Sol.
Find five rational numbers between
3
4
and .
5
5
3 3 ´10 30 4 4 ´10 40
, =
=
=
=
, therefore,
5 5 ´10 50 5 5 ´10 50
3
4
five rational numbers between and
are
5
5
31 32 33 34 35
, , , , .
50 50 50 50 50
State whether the following statements are
true or false? Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number
(iii) Every rational number is a whole number.
(i) True, since the collection of whole
numbers contains all natural numbers.
(ii) False, because, – 3 is not a whole number.
1
(iii) False, Q is not a whole number..
2
EXERCISE 1.2
1.
3+
Sol.
State whether the following statements are
true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the
form m , where m is a natural number..
(iii) Every real number is an irrational
number.
(i) True, (Q real numbers are collection of
rational and irrational numbers.)
Number System
2.
Sol.
3.
Sol.
A-5
manner, you can get the line segment Pn–1 Pn
by drawing a line segment of unit length
perpendicular to OPn–1. In this manner, you
will have created the created the points: P1, P2,
P3 ......, Pn ......, and joined them to create a spiral
(ii) False, because on negative number can
be the square root of any natural number.
(iii) False, (2 is real but not irrational.)
Are the square roots of all positive integers
irrational ? If not, give an example of the
square roots of a number that is a rational
number.
No, the square roots of all positive integers are
depicting
2, 3, 4,....
not irrational. For example, 16 = 4 is a
rational number.
Show how 5 can be represented on the
number line.
Consider a unit square OABC onto the number
line with the vertex O which coincides with zero.
Then OB = 12 + 12 = 2
Construct BD of unit length perpendicular to
OB. Then OD = ( 2) 2 + 12 = 3
Construct DE of unit length perpendicular to
OD. Then OE =
EXERCISE 1.3
( 3)2 + 12 = 4 = 2
Similarly, Construct EF of unit length ^ OE.
1.
Then OF = 22 + 12 = 5
Using a compass, with centre O and radius OF,
draw an arc which intersects the number line in
the point R. Then R corresponds to
5.
Sol.
Write the following in decimal form and say
what kind of decimal expansion each has :
(i)
36
100
(ii)
(iv)
3
13
(v)
(i)
Given
4.
Sol.
5
Classroom activity (Constructing the ‘square
root spiral’):
Take a large sheet of paper and construct the
square root spiral in the following manner. Start
with a point O and draw a line segment OP1 of
unit length. Draw a line segment P 1 P 2
perpendicular to OP 1 of unit length
[see figure]. Now draw a line segment P2P3
perpendicular to OP2.Then draw a line segment
P3P4 perpendicular to OP3. Continuing in this
2
11
(iii) 4
(vi)
1
8
329
400
36
= 0.36 Hence, The decimal
100
expansion is terminating.
(ii) Given
Representation of
1
11
1
11
11 ) 1.000000 (0.090909......
99
100
99
100
99
1
Division shows that
1
= 0.090909...... = 0.09
11
Hence, The decimal expansion of
non-terminating repeating
.
1
is
11
A-6
MATHEMATICS
1
1 33
(iii) Consider 4 = 4 + =
8
8 8
\
(v) Consider
11 ) 2.0000 (0.1818......
11
90
88
20
11
90
88
2
On dividing by 11, we get
8 ) 33.000 (4.125
32
10
8
20
16
40
40
x
1
Thus, from the divison, we get 4 = 4.125
8
2
= 0.1818..... = 0.18 .
11
1
Hence, The decimal expansion of 4 is
8
terminating.
So, The decimal expansion of
terminating repeating.
329
(vi)
400
By Dividing, we get
400 ) 329.0000 (0.8225
3200
900
800
1000
800
2000
2000
x
329
\
= 0.8225
400
3
(iv) Given number is
13
13) 3.00000000000 (0.230769230769......
26
40
39
100
91
90
78
120
117
30
26
40
39
100
91
90
78
120
117
3
\
3
Hence, The decimal expansion of
is
13
non-terminating repeating.
2
is non11
Hence, The decimal expansion of
terminating.
2.
3
= 0.230769230769.......
13
= 0.230769
2
11
Sol.
329
is
400
1
=
. Can you predict
7 0.142857
2 3 4 5 6
what the decimal expansions of , , , ,
7 7 7 7 7
are, without actually doing the long division?
If so, how?
[Hint : Study the remainders while finding the
1
value of carefully.]
7
Yes! We can predict the decimal expansions of
You know that
2 3 4 5 6
, , , , without actually doing the long
7 7 7 7 7
division as follows :
NCERT Handbook Term 1 Mathematics
Class 9 (NCERT Solutions + FA
activities + SA Practice Questions & 5
Sample Papers)
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