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1
Number System
CHAPTER
We are Starting from a Point but want to Make it a Circle of Infinite Radius
We use numbers in our daily life for counting, measuring, etc. Let us recall the types of numbers.
Natural Numbers:
Counting numbers are called Natural numbers, e.g., 1, 2, 3, 4, ...  are Natural Numbers. It is
denoted by N. There are infinite natural numbers and the smallest natural number is 1.
Whole Numbers:
The natural numbers along with zero form the system of whole numbers, e.g., 0, 1, 2, 3, 4, …  . It
is denoted W. There is no largest whole number and the smallest whole number is 0.
Integers:
The number system consisting of natural numbers, their negative and zero is called integers, e.g., 
, … - 2, - 1, 0, 1, 2, 3, … +  . It is denoted by Z or I. The smallest and the largest integers cannot
be determined.
Even numbers:
Natural numbers which are divisible by 2 are called even numbers, e.g., 2, 4, 6, 8 …  . Smallest
even number is 2. There is no largest even number.
Odd Numbers:
Natural numbers which are not divisible by 2 are called odd number. e.g, 1, 3, 5, 7, …  . Smallest
odd number is 1. There is no largest odd number
Prime Numbers:
Natural numbers which have exactly two factors, i.e., 1 and the number itself are called prime
numbers, e.g., 2, 3, 5, 7, ...  .
Rational numbers:
If a number can be expressed in the form of
p
, where q  0 , where p and q are integers, then the
q
number is called rational number. It is denoted by Q. For Example, 
9 16 8 27
, , ,
etc.
25 7 1
51
Irrational Numbers:
The numbers which cannot be expressed in the form of
p
, where p and q are integers and q  0 , is
q
called irrational number. It is denoted by I. For example,
irrational numbers?
2, 3 , 5 ,2  3 , 3  5 , 3 3 are
Important Point:
Every positive irrational number has a negative irrational number corresponding to it.
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2  3  5,
5  3  2,
3  2  3  2  6,
6 2
6
 3
2
Sometimes, product of two irrational numbers is a rational number
For example:
2 2 
(i)

2 2  2


(ii) 2  3  2  3  (2) 2 
 3
2
 43 1
The Number Line:
The number line is a straight line between negative infinity on the left to positive infinity on the
right. On number line integers are placed at a equal distance.
 ...
-4 -3 -2 -1
0
1
2
3

4...
Real Numbers:
A set of rational and irrational numbers is called the set of real numbers. It is denoted by R.
Therefore, Real numbers = Rational numbers + Irrational numbers. In other words, all numbers that
can be represented on the number line are called real numbers.
R+: Positive real numbers and
R–: Negative real numbers.
TYPES OF NUMBERS
Real numbers (R)
Rational numbers (Q)
Imaginary numbers
Z  (a  ib), a, b  R
Irrational numbers (T)
The number which cannot be
Expressed as
Natural numbers(N)
Whole numbers (W)
Integer(I)
Negative
Integers
Positive
Integers
Even
Odd
number number
Prime
numbers
Composite
numbers
{- 1, - 2, - 3, - 4…}
{1, 2, 3, 4…}
Important Points:
1 is neither prime nor composite.
1 is an odd integer.
0 is neither positive nor negative.
2 is smallest even prime number.
All prime numbers (except 2) are odd.
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Non-negative
Integers
{0, 1, 2, 3,…}
Non-positive
Integers
{0, - 1, - 2, - 3, …}
Fraction:
A fraction is a quantity which expresses a part of a whole.
Numerator
Fraction =
Deno min ator
Types of Fractions:
(a)
Proper Fraction: If numerator is less than its denominator, then it is called a proper fraction.
2 6
For example: ,
.
5 18
(b)
Improper Fraction : If numerator is greater than or equal to its denominator, then it is called an
improper fraction.
For example:
(c)
5 18 13
, ,
.
2 7 13
1 2 5
Mixed fraction: It consists of an integer a proper fraction. For example: 1 , 3 ,7 .
2 3 9
Mixed fraction can always be changed into improper fraction advice versa.
For example: 7
(d)
19 9  2  1
1
1
5 7  9  5 63  5 68

 9  9



and
2
2
2
2
9
9
9
9
Equivalent fractions / Equal fractions: Fractions with the same value.
For example:
2 4 6 8  2
, , ,  
3 6 9 12  3 
Important Points:
Value of fraction is not change by multiplying or dividing the numerator or denominator by the same
number.
For example:
(i)
2 2  5 10


5 5  5 25
So,
2 10

5 25
(ii)
36 36  4 9


16 16  4 4
So,
36 9

16 4
If in a fraction, its numerator and denominator are of equal value then fraction is equal to unity i.e. 1.
(e)
Like fractions: Fraction with same denominators.
For example:
(f)
2 3 9 11
, , ,
7 7 7 7
Unlike Fractions: Fractions with different denominators.
For example:
2 4 9 9
, , ,
5 7 8 2
Important Point:
Unlike fractions can be converted into like fractions.
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For example:
3
4
and
5
7
3 7 21
4 5 20
 
and  
5 7 35
7 5 35
(g)
Decimal fraction: The fractions whose denominators are of the powers of 10.
For example:
(h)
2
9
 0.2,
 0.09
10
100
Vulgar fraction: Denominators are not the power of 10.
For example:
3 9 5
, ,
7 2 193
We can also study real numbers by studying classification of decimal expansion.
Decimal Expansion of real numbers
Terminating
Non-Terminating
Recurring
Pure
Recurring
Non-recurring
Mixed
Recurring
(a) Terminating (or finite decimal fractions) :
For example:
7
21
 4.2
 0.875 ,
5
8
(b) Non-terminating decimal fractions : There are two types of Non-terminating decimal fractions :
(i) Non-terminating periodic fractions or non-terminating recurring (repeating) decimal factions :
For example:
10
 3.333...  3. 3
3
1
 0.142857142857...  0.142857
7
(ii) Non-terminating non-periodic fraction or non-terminating non-recurring fractions :
For example 15.2731259629
Important point:
The decimal expansion of a rational number is either terminating or non-terminating recurring.
Moreover, a number whose decimal expansion is terminating or non-terminating recurring is
rational.
The decimal expansion of an irrational number is non-terminating non recurring. Moreover, a
number whose decimal expansion is non-terminating non recurring is irrational.
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2 = 1.41421356237309504880…
For example:
 = 3.1415926535897932384626433…
We often take
22
22
as an approximate value of  , but  
.
7
7
Rational Numbers
To find out rational numbers between two Rational
If r and s be two rational numbers then
rs
is between r and s.
2
rs
Then find the rational number between r and
, i.e.
2
r
rs
2 .
2
Example 1:
Find two rational numbers between 1 and 2.
A rational number between 1 and 2 =
A rational number between 1 and
1 2 3

2
2
1 3 1 5 5
3
= 1      .
2
2 2 2 2 4
So, two rational numbers between 1 and 2 is
3
5
and .
2
4
Example 2:
Write five rational numbers between 3 and 4.
3
3 10 30

10
10
4
4 10 40

\
10
10
Five rational numbers between 3 and 4 are
31 32 33 34
35
, , , , and
.
10 10 10 10
10
Example 3:
Find four rational numbers between
2
5
and .
3
3
2 10 20
5 10 50
 
and  
3 10 30
3 10 30
Four rational number between
2
5
21 22 23
24
, ,
and .
and are
30 30 30
30
3
3
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IX ACADEMIC QUESTIONS
Subjective
Assignment-1
1.
Is zero a rational number? If yes, write it in the form of p/q.
2.
Find six rational numbers between
(i) 3 and 4
(ii) 2 and 3
Find 5 rational numbers between
4.
5.
Find five rational numbers between 1 and 2.
Find six rational numbers between -1 and 3.
2
5
Find four rational numbers between and .
3
3
3
7
Find six numbers between and
4
4
7.
8.
12.
Find nine rational numbers between 0 and 0.1.
1
1
Find three rational numbers lying between and
5
4
2
3
Find five rational numbers lying between and
5
4
1
Find a rational numbers lying between 1and
2
State true or false for the following. Justify your answer
(i)
Every whole number is a natural number.
(ii)
Every integer is a rational number.
(iii)
Every rational number is an integer.
(iv)
Every integer is a whole number.
(v)
Every natural number is a whole number.
13.
State true or false. Justify by giving example.
(i)
Every real number is an irrational number.
(ii)
Every real no. is either rational or irrational
(iii)
 is an irrational number
(iv)
Irrational numbers cannot be represented by points on the number line.
(v)
The sum of two rational numbers is rational.
(vi)
The sum of two irrational numbers is an irrational.
9.
10.
11.
(iv) 15 and 16
1
2
and .
2
5
3.
6.
(iii) 9 and 10
(vii) The product of two rational numbers is rational.
(viii) The product of two irrational numbers is an irrational.
(ix)
The sum of a rational number and an irrational number is an irrational.
(x)
The product of a non-zero rational number and an irrational number is a rational number.
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14.
Represent
6 on the number line.
15.
Represent
7 on the number line.
16.
Represent 3 on the number line.
CONVERSION OF A RATIONAL NUMBER INTO DECIMAL
Terminating Decimals (Finite Decimals)
3
Let us convert into decimal from.
4
3
(Rational number)  0.75 (Terminating decimal)
4
This is a terminating decimal form. Hence, we conclude that terminating decimals are rational numbers,
since the process of division terminates (comes to an end). Here, the division stops at a point, where
there is no remainder.
Non-Terminating and repeating decimals (Recurring Decimals)
2
Let us convert into decimal form
3
2
 0.6666....  0.6
3
0.6666…. is a non-terminating and repeating decimal and is represented as 0.6 . Non terminating and
repeating decimals are also known as recurring decimals.

2
is a rational number whose decimal expansion is a non-terminating and repeating decimal (recurring
3
decimal).
Q.1 Write the following in decimal form and say what kind of decimal expansion each has:
36
1
1
(i)
(NCERT) (ii)
(NCERT)
(iii) 4 (NCERT)
11
100
8
3
145
2
(iv)
(NCERT) (v)
(NCERT)
(vi)
(NCERT)
11
8
13
3
874
(vii)
(viii)
40
999
p
Conversion of terminating decimals into
form
q
Steps:
1.
Assume the given recurring decimal as x.
2.
In the given number, check the number of digits which have bar on their heads.
3.
If only one digit has bar on its head, multiply the number by 10; if two digits have bar on its head,
multiply the number by 100; similarly if three digits have bars on its head, multiply the number by
100; similarly if three digits have bars on its head, multiply the number by 1000 and so on. When
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we multiple the number by 10, the decimal will shift one place to the right i.e., if x = 0.6 then 10x
= 6.6
4.
Subtract the numbers obtained in step 1 and that obtained in step 3.
5.
Simplify the equation obtained in step 4 and calculate x.
6.
Write the value of x in simplest form.
p
Example: 1.8: Express each of the following in form.
q
(i) 0.7
(ii) 0.32
(iii) 0.459
Solution:
(i) Let x = 0.7
Multiplying both sides by 10, we get
10x = 7.7
Subtracting (i) and (ii), we get
10x  x  7.7  0.7
9x = 7
7
 x
9
7
0.7 

9
(ii) Let
x = 0.32
Multiplying both sides by 100, we get
100x = 32. 32
99x = 32
32
 x=
99
32
0.32 =

99
…(i)
…(ii)
…(i)
…(ii)
(iii) Let x  0.459
Multiplying both sides by 1000, we get
1000x = 459. 459
Subtracting (i) and (ii), we get
1000x – x = 459.459 − 0.459
999x = 459
459 51

 x=
999 111
51

0.459 
111
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…(ii)
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IX ACADEMIC QUESTIONS
Subjective
Assignment-2
Convert the following decimal numbers in the form
p
.
q
1.
(i) 0.15
(ii) 0.00026
(iii) 8.0025
2.
(i) 0.6 (NCERT)
(ii) 0.47 (NCERT)
(iii) 0.001 (NCERT)
3.
(i) 0.25
(ii) 0.1532
(iii) 2.2612
4.
Express (2.23  0.32) in
5.
If 10x  0.3  0.2, find the value of x.
6.
If
p
form
q
x
 0.6  0.4, find the value of x.
9
Irrational Number
A number which cannot be expressed in the form of
p
, where p and q are integers and q  0 , is known as
q
irrational number.
Example:
2, 3, 5, 6, 21 are irrational numbers.
A non terminating and non-repeating decimal is known as irrational numbers.
Examples:
1.4123567…………….
2.101001000………….
0.121221222………….
All the above decimal numbers are non-terminating and non-repeating decimal and therefore these are
irrational numbers.
Properties of Irrational Numbers:
(i)
The sum of two irrational numbers need not be an irrational number.
Example:
Sum of
3 and  3  3  ( 3)  3  3  0 , which is a rational numbers
Sum of (3  5) and (2  5)  3  5  2  5  3  2  5, which is a rational number.
(ii)
The difference of two irrational numbers need not be an irrational number.
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Example:
Difference of (2  6) and 6  2  6  6  2 , which is a rational number.
(iii) The product of two irrational numbers need not be an irrational number.
Examples:
Product of
5 and  5  5   5  5 is a rational number.
(iv) The division of two irrational numbers need not be an irrational number.
Examples:
Division of 3 2 and
Division of
(v)
8 and
2
2
3 2
2
8
2
= 3, is a rational number.
 4 = 2, is a rational number.
The sum of a rational and irrational number is irrational number.
Examples: Sum of 2 and
3 = (2  3) , which is an irrational number.
(vi) The difference of a rational and an irrational number is irrational number.
Examples: Difference of 2 and
3  (2  3) , which is an irrational number.
(vii) The product of a rational and an irrational number is irrational number.
Examples: The product of 2 and
5  2  5  2 5, which is an irrational number.
(viii) The division of a rational and an irrational number is irrational numbers.
Examples: The division of 5 and
5
5
5
 5 , which is an irrational number.
Addition and Subtraction of Irrational Numbers:
Example: Add (3 2  5 3) and ( 2  3) .
Solution:
(3 2  5 3)  ( 2  3) = (3 2  2)  (5 3  3)
= (3  1) 2  (5  1) 3  4 2  6 3
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IX ACADEMIC QUESTIONS
Subjective
Assignment-3
1.
Add (4 5  3 7  6) and (2 5  4 6)
2. Add (7 7  4 3) and (4 7  3 3)
3.
Add (2 3  5 2) and ( 3  2 2)
4. Add (2 2  5 3  7 5)and (3 3  2  5)
5.
1
3
2

1

7
2  6 11  and  7 
2  11  6.

2
2
3

3

7.
27  3. 12
8.
9.
50  98  162
10.
3
a 4 b  3 ab4
12.
4
81  8. 3 216  15. 3 32  225
11.
13.
5  20  45
8  6.
1
2
147  108
3
16  3 54  3 192  3 375
Multiplication of Irrational Numbers
To multiply irrational numbers, multiply the rational parts and irrational parts separately.
For example:
Multiply 2 2 and 5 3 , write (2 × 5) and ( 2  3) separately.
2 2  5 3  (2  5)  ( 2  3)  10 6
Division of Irrational Numbers:
To divide irrational numbers, divide the rational parts and irrational parts separately.
For example:
Divide 10 6 and 5 2 , write (10  5) and ( 6  2)
10 6  5 2 
10 6 10
6
 
2 3
5
5 2
2
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IX ACADEMIC QUESTIONS
Subjective
Assignment-4
1.
Multiply (3  3) by (2  2) .
2. Simplify ( 5  2)2
3.
Simplify ( 5  2)( 5  2) 4. 2. 3 4  3. 3 16
5.
2.4 3  5. 4 81
6.
7.
3. 3 32  3. 3 4
8. 10  4
6 3
9.
4
43 4
10.
11.
21. 384  8. 96
13.
( 6  7)( 6  7)
5  3 10
12. ( 7  5)( 7  5)
14. 4 28  3 7
Rationalisation of the Denominator of an Irrational number having two terms in the denominator, multiply
the numerator and denominator of the number by the conjugate of its denominator.
For example: To rationalize
1
, multiply the numerator and denominator by ( a  b) .
a b
1
1
a b
a b
a b




2
2
ab
a b
a b
a  b ( a )  ( b)
Example: Rationalise the denominator of
1
.
3 2
Solution: Multiplying numerator and denominator by ( 3  2)
1
3 2
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=
1
3 2

3 2
3 2
=
3 2
( 3) 2  ( 2) 2
=
3 2
=
3 2
3 2.
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IX ACADEMIC QUESTIONS
Subjective
Assignment-5
1.
2.
Rationalise the denominators of the following real numbers:
(i)
7
2
(v)
1
1
(vi)
3 7 5
2 3
(ii)
2
3
(iii)
(vii)
4
3 5
1
3 5
5
3 5 4 3
Simplify the following by rationalizing the denominator:
(i)
5 6
3 5
(ii)
5 6
7 6
3  1.732 and
(iii)
2 6 5
5 3 5 2
(iv)
3 5 2 6
48  18
5  2.236, find the value of
3.
If
4.
Rationalise the denominator of the following:
(i)
7 2
9  2 14
Rationalise the denominator of
6.
Rationalise the denominator of
7.
Simplify each of the following:
(i)
3
2

5 3 5 3
(iii)
1
4 3 3 5
y2
(ii)
5.
8.
(iv)
x 2  y2  x
4
.
2 3  7
1
5 7 2
(ii)
3 2 2 3 3 2 2 3

3 2 2 3 3 2 2 3
7 1
7 1

7 1
7 1
Find the values of ‘a’ and ‘b’ in each of the following:
(i)
(ii)
(v)
3 7
ab 7
3 7
11  7
 a  b 77
11  7
5  27
ab 3
7  48
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(ii)
5 3
 a  b 15
5 3
(iv)
2 3
 a b 6
18  12
(vi)
73 5 a b 5
 
2
7 3 5 2
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9.
Find the values of a and b
5 2
52

 a b 5
52
5 2
1
1
1

 13  4
15  14
14  13

10.
Show that
11.
Prove that
12.
If x = 2  3, find the value of x 
13.
If x = 3  5, find the value of x 2 
4  15
1
1 2
1

2 3
1

3 4
1

4 5
 5 1  0
1
.
x
1
x2
.
Laws of Exponents for Real Numbers
If a, n and m are natural numbers, then
(i) am × an = am × n
(iii)
am
n
a
(ii) (am)n = amn
 a m  n , where m > n. If a , b, m are natural numbers then
Examples:
(i) 23 × 24 = 23 + 4 = 27
(ii)
55
(ii) (32)3 = 32 × 3 = 36
 55 2  53
52
(iv) 23  33  (2  3)3  63
Laws of Radicals
If n is a positive integer and a and b are positive rational numbers, then
n
(i)
( a)
(ii)
n
(iii)
(iv)
(v)
n
 a 
1 n
n
 nn
1
1
1
n
a
a  n b  a n  b n  (ab)1/n  n ab
mn
n
a
n
b
n m
a  n a 

a1/n
b1/n
1
m

 
1
1 m
an
1 1

m
 an
1
 a mn  mn a
1
a
 a n
  n
b
b
(a p ) m 
1
pm n
m a 

 a
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pm

1
1 n
m
1
 (a p ) n  n a p
NTSE, NSO
Diploma, XI Entrance
14
(iv) am × bm = (ab)m
Examples:
(i)
4
(iii)
(v)
3  31/4
5
(ii)
2  5 3  21/5  31/5  (2  3)1/5  5 6
3 2
5  ( 2 5)1/3  51/6  6 5
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 4 3 4  (31/4 )4  34/4  31  3
3
(iv)
(vi)
12
3
3
3 5

121/3
31/3
(a 2 )5 
1/3
 12 
 
3
1
10 3
5 a 
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Diploma, XI Entrance
15
 41/3  3 4

 
1
10 3
a5
1
 (a 2 ) 3  3 a 2
IX ACADEMIC QUESTIONS
Subjective
Assignment-6
1
1
1
1. Find: (i) 64 2
(ii) 32 5
(iii) 1253
(NCERT)
3
2
3
1
2. Find: (i) 9 2
(ii) 32 5
(iii) 16 4
(iv) 1253
3. Find: (i)
2
23
1
 25
 1 
(ii)  3 
3 
7
(iii)
1
112
1
114
1
(NCERT)
1
(iv) 7 2  8 2
(NCERT)
4. Find the value of x in each of the following:
(ii) 5x  2  32x 3  135
(i) 25x  2x  220
5
3
5
5. If 52x 1  25x 1  2500, find the value of x.
6. Prove that
7. If
528  527  526
5 5 5
29
28
27

31
.
145
1 274  272

, then find the value of x.
x 272  270
8. Prove that
2
1 x
2a  2b
 xa 
9. Prove that  b 
x 
a b

2
1  x 2b 2a
 xb 
 c 
x 
bc
10. If 3x = 5y = 15−z, then show that
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2
 xc 
 a 
x 
x
5
 3
(iii)     
ca
1
1 1 1
  0.
x y z
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Diploma, XI Entrance
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2x

125
27
XI SCIENCE & DIP. ENTRANCE
Multiple Choice Questions
Assignment – 7
1. 1.272727 ………. can be expressed in rational form as
(a)
14
99
(b)
14
11
(c)
11
14
(d)
2. The sum of two rational numbers is………. number
(a) Rational
(b) Irrational
(c) Either a& or
3.
99
14
(d) Natural
0.123 can be expressed in rational form as
(a)
900
111
(b)
2( 2  6)
4. The fraction
(a)
3( 2  3)
2 2
3
111
900
(c)
123
10
(d)
121
900
(c)
2 3
3
(d)
4
3
(d)
3
8
4
is equal to
(b) 1
5. Which of the following is a pure surd?
(a)
6. If
x  0,
then
(a) x x
7.
 256x16 

4 
 81y 
(a)
3y
4x 4
33 5
(c) 12
(b)
x4 x
(c)
(b)
2y
6x 4
(c)
(b)
3
x x x 
8
x
(d)
3y
8x 4
(d)
8
x7
1/ 4

8. Set of natural numbers is a subset of
(a) Set of even numbers
(c) Set of composite numbers
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(b) Set of odd numbers
(d) Set of real numbers
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17
4y
5x 4
9. For an integer n, a student states the following:
I. If n is odd, (n + 1)2 is even.
II. If n is even, (n – 1)2 is odd.
(n  1) is irrational
III. If n is even
Which of the above statements would be true?
(a) I and III
(b) I and II
(c) I, II and III
(d) II and III
10. The irrational number between 2 and 3 is
(a)
11. If
(b)
2
7 3
( 10  3)
(b) 4.398

2 5
( 6  5)
(a) 1

(a)
15
( 15  3 2)
5
a 3 b2 c
(b)
(c)
(a) 2 , 6
1
4
(b)
(d) 6.398
1
2
(d) 3
a 2 b3c4 is
4
(c) 3 a3b2c
a 3 b2 c
14. Two irrational numbers between
1
2
is

(b) 2
5
10  20  40  5  80
(c) 3.398
3 2
13. The rationalizing factor of
(d) 11
5
5 = 2.236 and 10 = 3.162, then the value of
(a) 5.398
12.
(c)
3
1
4
3 ,3
2 and
1
6
(d) a3b2c
3 are
1
1
1
(c) 6 8 , 3 4
1
(d) 3 2 , 38
15. Which of the following sets of fraction is in ascending order?
(a)
7 9 11
, ,
9 11 13
(b)
11 9 7
, ,
13 11 9
(c)
11 7 9
, ,
13 9 11
(d)
9 7 11
, ,
11 9 13
16. 1/ ( 3  2) is not equal to
(a)
3 2
17. The value of
(b)
2 / ( 6  2)
(c) ( 3  2) / (5  2 6) (d) 3 / (9  6)
1
1
1
1


 ..... 
is
1 2 2  3 3  4
99 100
(a) Less than
99
100
(c) Greater than
99
100
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(b) Equal to
99
100
(d) Equal to
99
100
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18. The numerator of
a  a 2  b2

a  a 2  b2
(a) a2
a  a 2  b2
(c) a2 – b2
(b) b2
19. The ascending order of the surds
(a)
9
4, 6 3, 3 2
9
(b)
20. Rational number between
2 3
2
(a)
is
a  a 2  b2
3
2, 6 3, 9 4 is
4, 3 2, 6 3
2 and
(c)
3
2, 6 3, 9 4
(d) 6 3, 9 4, 3 2
3 is
2 3
2
(b)
(d) 4a2 – 2b2
(c) 1.5
(d) 1.8
21. The greater between 17  12 and 11  6 is
17  12
(a)
11  6 (c) Both are equal
(b)
22. Which of the expressions is the same as
(a)
3
2 1
23. If m 
(a)
3
(b)
1
( 2  1)
3
4 1
(c)
(d) Cannot compare
?
3
4  3 2 1
(d)
3
4  23 2 1
cad
, then b equals
ab
m(a  b)
ca
1
cab  ma
(c)
m
1 c
(b)
1
1
(d)
ma
m  ca
1
 x q  qr  x r  rp  x p  pq
24. The value of  r    p    q  is equal to
x  x  x 
1 1 1
 
q r
(a) x p
(b) 0

3
4
(c)
x pq  qr  rp
(d) 1
2
25. The value of  6 27  6  equals

3
2
(a)
(b)
26. The rationalizing factor of
(a)
3
5
(b)
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3
2
(c)
23
3
3
4
(d)
3
4
5 is
52
(c) 52
(d) 53
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Diploma, XI Entrance
19
3 1
1
then x 3  3 
2
x
27. If x =
(a)
28 3  15
8
(b)
27 3  35
4
28. The rational number between
(a)
2
5
28 3  15
8
(d)
27 3  35
4
1
1
and is
2
3
(b)
1
5
(c)
3
5
(d)
4
5
(b)
2n 1
(c) 1 – 2n
(d)
7
8
(c) 18
(d) 7 7
2n  4  2(2n )
2(2n 3 )
29. Simplify :
(a) 2n 1 
(c)
1
8
2
1
1

30. If  a    9 , then a 3  3 equals
a
a

(a)
10 3
3
(b)
3 3
31. If both ‘a’ and ‘b’ are rational numbers, then ‘a’ and ‘b’ from
(a) a 
32.
9
19
,b 
11
11
(b)
a
19
9
,b 
11
11
(c) a 
3 5
3 2 5
2
8
,b 
11
11
 a 5  b , are
(d) a 
10
21
,b 
11
11
a  b  c  d where d, c, b, a are consecutive natural numbers. Then which of the
following is true?
33.
(a)
a b  c d
(b)
a b c d
(c)
a c b d
(d)
c d  a b
2 6
2 3 5
equals
(a)
2 3 5
(b) 4  2  3
(c)
2  3  6 5
(d)
34. The value of
(a) 527
1
( 2  5  3)
2
1
of 1527 is
3
(b) 159
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(c) 5 1526
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(d) 5  39
35. Which one of the following is correct?
(a)
(c)
(

7  7 3)( 7 3  7 3  2)  24
3
 ( 3 7  7 3  4)( 3 7  7 3  6)
(
3
(
(b)

7  7 3)( 3 7  7 3  2)  24

7  7 3)( 7 3  3 7  2)  24
 ( 3 7  7 3  4)( 3 7  7 3  6)
(
(d)
 ( 3 7  7 3  4)( 3 7  7 3  6)
3
3
 ( 3 7  7 3  4)( 3 7  7 3  6)
36. If 25x – 1 = 52x – 1 – 100, then the value of x is
(a) 3
(b)
2
(c) 4
37. If x = 2  3 , then the value of x2 +
(a) 14, 8 3
(b)
10
39. The 100th root of 10(10
10
(a) 108
)
(d) 1
1
1
and x 2  2 is
2
x
x
14, 8 3
38. If 444 + 444 + 444 + 444 = 4x, then x is
(a) 45
(b) 44

7  7 3)( 3 7  7 3  2)  24
(c) 14, 8 3
(d) 14, 8 3
(c) 176
(d) 11
is
8
(b) 1010
(c) ( 10)(
10 )10
(d) 10( (10))
40. Which of the following numbers has the terminal decimal representation?
(a)
1
7
(b)
1
3
41. 2.6  0.82 is equal to…………
(a) 82/99
(b) 182/99
(c)
3
5
(d)
(c) 24/99
17
3
(d) 3/99
42. If x  2 5  3 and y  2 5  3 , then x4 + y4 =
(a) 1538
(b) 1200
(c) 1048
(d) 149
43. If x  3  3 2 and y  3  3 2, then x 4  y4  8x 2 y2 
(a) 3914
(b) 3010
(c) –486
(d) –856
44. If a  1  2  3 and b  1  2  3, then a 2  b2  2a  2b 
(a) 11
(b) 8
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(c) 152
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(d) 15
10
45. If x 
1
3 2 2
and y 
(a) 4
46. If x 
3 2 2
, then value of xy2 + x2y is
(b) 12
5 3
80  48  45  27
(a) 15
47. If x 
1
(c) 6
, then value of 4x 2  3x  5
(b) 2
7 3
10  3

(a) 2
3 2
15  3 2
(d) 9

(c) 12
2 5
6 5
(b) 1
(d) 5
, then value of x 4  x 2 is
(c) 0
(d) 12
Passage: Given that 2 = 1.414, 3 = 1.732, 5 = 2.236, 10 =3.162 and  = 3.141
48. Evaluate
10  2
5
(a) 5.141
49. Evaluate
(b) 0.823
(c) 3.084
(d) 0.915
(c) 12.413
(d) 0.674
(c) 3.146
(d) 6.813
10  2 5  3 5
8
(a) 1.529
(b) 8.236
50. Find the value of
(a) 12.479
2  3  2
(b) 9.428
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ANSWER
Assignment – 1:
0 0 0
22 23 24 25 26 27
15 16 17 18 19 20
2.(i) , , , , ,
, , , , ,
1. Yes, , , , etc.
(ii)
1 2 3
7 7 7 7 7 7
7 7 7 7 7 7
64 65 66 67 68 69
106 107 108 109 110 111
25 26 27 28 29
, , , , ,
,
,
,
,
,
, , , ,
(iii)
(iv)
3.
7 7 7 7 7 7
7 7 7 7 7 7
60 60 60 60 60
7 8 9 10 11
6 5 4 3 2
11 12 13 14
, , , ,
, , ,
4. , , , ,
5.
6.
6 6 6 6 6
7 7 7 7 7
15 15 15 15
22 23 24 25 26 27
1 2 3 4 5 6 7 8 9
, , , , ,
,
,
,
,
,
,
,
,
7.
8.
28 28 28 28 28 28
100 100 100 100 100 100 100 100 100
17 18 19
9 10 11 12 13
1
, ,
, , , ,
9.
10.
11.
4
80 80 80
20 20 20 20 20
12. (i) (iii) (iv) are false and (ii) and (v) is True
13. (ii), (iii) (iv) (v) (vi) and (ix) are True and (i) (viii) (x) are false.
Assignment – 2
1. (i) 3/20
2. (i) 2/3
3. (i) 25/99
4. 23/9
(ii) 13/5000
(ii) 43/90
(ii) 1517/9900
5. 1/90
(iii) 3201/400
(iii) 1/999
(iii) 5599/2475
6. 10
Assignment – 3
1. 6 5  3 7  5 6
5.
7  2  5 11
9. 7 2
13. 30
2. 11 7  3
3. 3 3  3 2
4.
6. 6 5
7. 9 3
8.  2
3
3
10.
2
11. ab a b
2 8 3 6 5
12. 5 3 2  3 3
2
Assignment – 4
1. 6  3 2  2 3  6
2. 7  2 10
5. 30 4 3
6.
9.
2
13. –1
12
10.
4
1
2
14. 8/3
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3. 3
4. 24
7. 36 3 2
8.
21
11.
4
12. 2
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5
2
Assignment – 5
7 2
2
(ii)
2 3
3
(iii)
4 5
15
(v) 2  3
(vi)
3 7 5
38
(vii)
5(3 5  4 3)
3
1. (i)
2. (i)
31  10 6
19
(ii)
3. 4.54
6.
4. (i)
2 5  70  5 2
20
8. a = 8, b = 3
39  8 30
21
21  3 2  42  6 (iii)
7. (i)
7 2
5
x 2  y2  x 5.
(ii)
25  3
22
(ii) 10
3 5
4
(iv)
6 6
6
3  2 3  21
3
2 7
3
5
(iv) a = 2, b =
6
(iii)
9
1
(iii) a  , b 
2
2
(ii) a = 4, b = 1
(iv)
(v) a = –1, b = 1 (vi) a = 47, b = 21
9. a = 0, b = 8
12. 4
13.
119  45 5
8
Assignment – 6
1. (i) 8
2. (i) 27
13
3. (i) 2
15
4. (i) x = 1
5. x = 3 7. x 
(ii) 2
(ii) 4
(iii) 5
(iii) 8
(iv) 5
1
4
1
(ii) 3–21
(iii) 11
(ii) x = 3
(iii) x = 3
(iv) 56 2
1
4
Assignment – 7
1.b
2.c
3.b
4.d
5.a
6.d
7.a
8.d
9.b
10.c
11.a
12.a
13.a
14.c
15.a
16.d
17.b
18.d
19.a
20.c
21.b
22.c
23.d
24.d
25.d
26.b
27.b
28.a
29.d
30.c
31.a
32.b
33.a
34.c
35.a
36.b
37.c
38.a
39.
b 40.c
41.b
42.a
43.c
44.b
45.c
46.c
47.a
48.d
49.d
50.b
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