# R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 1 – Number Systems

## Chapter 1 – Number Systems Exercise Ex. 1A

Question 1

Is zero a rational number? Justify.

Solution 1

A
number which can be expressed as , where ‘a’ and ‘b’ both are integers and b ≠ 0, is called
a rational number.

Since,
0 can be expressed as , it is a rational number.

Question 2

Represent each of the following rational numbers on the number line:

(i)

Solution 2

(i)

Question 3

Represent each of the following rational numbers on the number line:

(ii)

Solution 3

(ii)

Question 4

Represent each of the following rational numbers on the
number line:

Solution 4

Question 5

Represent each of the following rational numbers on the number line:

(iv) 1.3

Solution 5

(iv) 1.3

Question 6

Represent each of the following rational numbers on the number line:

(v) -2.4

Solution 6

(v) -2.4

Question 7

Find a rational number lying between

Solution 7

Question 8

Find a rational number lying between

1.3 and 1.4

Solution 8

Question 9

Find a rational number lying between

-1 and

Solution 9

Question 10

Find a rational number lying between

Solution 10

Question 11

Find a rational number between

Solution 11

Question 12

Find three rational numbers lying between

How many rational numbers can be determined between these
two numbers?

Solution 12

Infinite rational
numbers can be determined between given two rational numbers.

Question 13

Find four rational numbers between

Solution 13

We have

We know that 9 < 10 < 11 < 12 < 13 <
14 < 15

Question 14

Find six rational numbers between 2 and 3.

Solution 14

2 and 3 can be
represented asrespectively.

Now six rational numbers
between 2 and 3 are

.

Question 15

Find five rational numbers between

Solution 15

Question 16

Insert 16 rational numbers between 2.1 and 2.2.

Solution 16

Let x = 2.1 and y = 2.2

Then, x < y because 2.1 < 2.2

Or we can say that,

Or,

That is, we have,

We know that,

Therefore, we can have,

Therefore, 16 rational numbers between, 2.1 and 2.2 are:

So, 16 rational numbers between 2.1 and 2.2 are:

2.105, 2.11, 2.115, 2.12, 2.125, 2.13, 2.135, 2.14, 2.145, 2.15, 2.155, 2.16, 2.165, 2.17, 2.175, 2.18

Question 17

State whether the given statement is true or false. Give reasons. for your answer.

Every natural number is a whole number.

Solution 17

True. Since the collection of natural number is a sub collection of whole numbers, and every element of natural numbers is an element of whole numbers

Question 18

Write, whether the given statement is true or false. Give reasons.

Every whole number is a natural number.

Solution 18

False. Since 0 is whole number but it is not a natural number.

Question 19

State whether the following statements are true or false.

Every integer is a whole number.

Solution 19

False,
integers include negative of natural numbers as well, which are clearly not
whole numbers. For example -1 is an integer but not a whole number.

Question 20

Write, whether the given statement is true or false. Give reasons.

Ever integer is a rational number.

Solution 20

True. Every integer can be represented in a fraction form with denominator 1.

Question 21

State whether the following statements are true or false.

Every rational number is an integer.

Solution 21

False,
integers are counting numbers on both sides of the number line i.e. they are
both positive and negative while rational numbers are of the form . Hence, Every rational number is not an integer but every
integer is a rational number.

Question 22

Write, whether the given statement is true or false. Give reasons.

Every rational number is a whole number.

Solution 22

False. Since division of whole numbers is not closed under division, the value of , may not be a whole number.

## Chapter 1 – Number Systems Exercise Ex. 1B

Question 1

Without actual division, find which of the following rationals are terminating decimals.

Solution 1

If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.

Since, 80 has prime factors 2 and 5,  is a terminating decimal.

Question 2

Without actual division, find which of the following rationals are terminating decimals.

Solution 2

If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.

Since, 24 has prime factors 2 and 3 and 3 is different from 2 and 5,

is not a terminating decimal.

Question 3

Without actual division, find which of the following rationals are terminating decimals.

Solution 3

If the prime factors of the denominators of the fraction are other than 2 and 5, then the rational number is not a terminating decimal.

Since 12 has prime factors 2 and 3  and 3 is different from 2 and 5,

is not a terminating decimal.

Question 4

Without actual division, find which of the following
rational numbers are terminating decimals.

Solution 4

Since the denominator of a given rational number is
not of the form 2m × 2n, where m and n are whole
numbers, it has non-terminating decimal.

Question 5

Write
each of the following in decimal form and say what kind of decimal expansion
each has.

Solution 5

Hence, it has
terminating decimal expansion.

Question 6

Write
each of the following in decimal form and say what kind of decimal expansion each
has.

Solution 6

Hence, it has
terminating decimal expansion.

Question 7

Write
each of the following in decimal form and say what kind of decimal expansion
each has.

Solution 7

Hence, it has non-terminating
recurring decimal expansion.

Question 8

Write
each of the following in decimal form and say what kind of decimal expansion
each has.

Solution 8

Hence, it has
non-terminating recurring decimal expansion.

Question 9

Write
each of the following in decimal form and say what kind of decimal expansion
each has.

Solution 9

Hence, it has
non-terminating recurring decimal expansion.

Question 10

Write
each of the following in decimal form and say what kind of decimal expansion
each has.

Solution 10

Hence, it has
terminating decimal expansion.

Question 11

Write
each of the following in decimal form and say what kind of decimal expansion
each has.

Solution 11

Hence, it has
terminating decimal expansion.

Question 12

Write
each of the following in decimal form and say what kind of decimal expansion
each has.

Solution 12

Hence, it has
non-terminating recurring decimal expansion.

Question 13

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 13

Let x =

i.e. x = 0.2222…. ….(i)

10x =
2.2222…. ….(ii)

On subtracting (i) from (ii), we get

9x = 2

Question 14

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 14

Let x =

i.e. x = 0.5353….  ….(i)

100x =
53.535353…. ….(ii)

On subtracting (i) from (ii), we get

99x = 53

Question 15

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 15

Let x =

i.e. x = 2.9393….  ….(i)

100x =
293.939……. ….(ii)

On subtracting (i) from (ii), we get

99x = 291

Question 16

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 16

Let x =

i.e. x = 18.4848….  ….(i)

100x =
1848.4848……. ….(ii)

On subtracting (i) from (ii), we get

99x = 1830

Question 17

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 17

Let x =

i.e. x = 32.123535..…

100x =
3212.3535…… ….(i)

10000x =
321235.3535……. ….(ii)

On subtracting (i) from (ii), we get

9900x = 318023

Question 18

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 18

Let x =

i.e. x = 0.235235..…   ….(i)

1000x =
235.235235……. ….(ii)

On subtracting (i) from (ii), we get

999x = 235

Question 19

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 19

Let x =

i.e. x = 0.003232..…

100x =
0.323232……. ….(i)

10000x =
32.3232…. ….(ii)

On subtracting (i) from (ii), we get

9900x = 32

Question 20

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 20

Let x =

i.e. x = 1.3232323..… ….(i)

100x =
132.323232……. ….(ii)

On subtracting (i) from (ii), we get

99x = 131

Question 21

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 21

Let x =

i.e. x = 0.3178178..…

10x =
3.178178…… ….(i)

10000x =
3178.178……. ….(ii)

On subtracting (i) from (ii), we get

9990x = 3175

Question 22

Express each of the following decimals in the form , where p, q are integers and q
0.

Solution 22

Let x =

i.e. x = 0.40777..…

100x =
40.777…… ….(i)

1000x =
407.777……. ….(ii)

On subtracting (i) from (ii), we get

900x = 367

Question 23

Express as a fraction in simplest form.

Solution 23

Let x =

i.e. x = 2.3636….  ….(i)

100x =
236.3636……. ….(ii)

On subtracting (i) from (ii), we get

99x = 234

Let y =

i.e. y = 0.2323….  ….(iii)

100y =
23.2323…. ….(iv)

On subtracting (iii) from (iv), we get

99y = 23

Question 24

Express in the form of

Solution 24

Let x =

i.e. x = 0.3838….  ….(i)

100x = 38.3838….  ….(ii)

On subtracting (i) from (ii), we get

99x = 38

Let y =

i.e. y = 1.2727….  ….(iii)

100y = 127.2727…….  ….(iv)

On subtracting (iii) from (iv), we get

99y = 126

Question 25

Without actual division, find which of the following rationals are terminating decimals.

Solution 25

If the prime factors of the denominator are 2 and/or 5 then the rational number is a terminating decimal.

Since 125 has prime factor 5 only

is a terminating decimal.

## Chapter 1 – Number Systems Exercise Ex. 1C

Question 1

What are irrational numbers? How do they differ from rational numbers? Give examples.

Solution 1

Irrational number: A number which cannot be expressed either as a terminating decimal or a repeating decimal is known as irrational number. Rather irrational numbers cannot be expressed in the fraction form,

For example, 0.101001000100001 is neither a terminating nor a repeating decimal and so is an irrational number.

Also, etc. are examples of irrational numbers.

Question 2

Find two rational and two irrational numbers between 0.5
and 0.55.

Solution 2

Since 0.5 < 0.55

Let x = 0.5, y = 0.55
and y = 2

Two irrational numbers
between 0.5 and 0.55 are 0.5151151115……. and 0.5353553555….

Question 3

Find three different irrational numbers between the
rational numbers .

Solution 3

Thus, three different irrational
numbers between the rational numbers  are as follows:

0.727227222…..,
0.757557555….. and 0.808008000…..

Question 4

Find two rational numbers of the form between the numbers 0.2121121112… and  0.2020020002……

Solution 4

Let a and
b be two rational numbers between the numbers 0.2121121112… and 0.2020020002……

Now, 0.2020020002…… <0.2121121112…

Then, 0.2020020002…… < a
< b < 0.2121121112…

Question 5

Find two irrational numbers between 0.16 and 0.17.

Solution 5

Two irrational numbers between
0.16 and 0.17 are as follows:

0.1611161111611111611111…… and
0.169669666…….

Question 6

State in each case, whether the given statement is true or false.

The sum of two rational numbers is rational.

Solution 6

True

Question 7

State in each case, whether the given statement is true or false.

The sum of two irrational numbers is irrational.

Solution 7

False

Question 8

State in each case, whether the given statement is true or false.

The product of two rational numbers is rational.

Solution 8

True

Question 9

State in each case, whether the given statement is true or false.

The product of two irrational numbers is irrational.

Solution 9

False

Question 10

State in each case, whether the given statement is true or false.

is irrational and is rational.

Solution 10

True

Question 11

State in each case, whether the given statement is true or false.

The sum of a rational number and an irrational number is irrational.

Solution 11

True

Question 12

State in each case, whether the given statement is true or false.

The product of a nonzero rational number and an irrational number is a rational number.

Solution 12

False

Question 13

State in each case, whether the given statement is true or false.

Every real number is rational.

Solution 13

False

Question 14

State in each case, whether the given statement is true or false.

Every real number is either rational or irrational.

Solution 14

True

Question 15

Classify the following numbers as rational or irrational.

Solution 15

Since quotient of a rational and an irrational is
irrational, the given number is irrational.

Question 16

Classify the following numbers as rational or irrational.

Solution 16

Question 17

Classify the following numbers as rational or irrational. Give reasons to support you answer.

Solution 17

We know that, if n is a not a perfect square, then is an irrational number.

Here,  is a not a perfect square number.

So, is irrational.

Question 18

Classify the following numbers as rational or irrational.

Solution 18

Question 19

Classify the following numbers as rational or irrational.

3.040040004…..

Solution 19

The given number 3.040040004….. has non-terminating and non-recurring decimal expansion.

Hence, it is an irrational number.

Question 20

Classify the following numbers as rational or irrational. Give reasons to support you answer.

Solution 20

is the product of a rational number and an irrational number .

Theorem: The product of a non-zero rational number and an irrational number is an irrational number.

Thus, by the above theorem, is an irrational number.

So, is an irrational number.

Question 21

Classify the following numbers as rational or irrational.

4.1276

Solution 21

The given number 4.1276 has terminating
decimal expansion.

Hence, it is a rational number.

Question 22

Classify the following numbers as rational or irrational.

Solution 22

Since the given number has non-terminating recurring
decimal expansion, it is a rational number.

Question 23

Classify the following numbers as rational or irrational.

1.232332333….

Solution 23

The given number 1.232332333…. has non-terminating and non-recurring decimal expansion.

Hence, it is an irrational number.

Question 24

Classify the following numbers as rational or irrational.

2.356565656…..

Solution 24

The given number 2.356565656….. has non-terminating recurring decimal expansion.

Hence, it is a rational number.

Question 25

Classify the following numbers as rational or irrational.

6.834834….

Solution 25

The given number 6.834834…. has non-terminating recurring decimal expansion.

Hence, it is a rational number.

Question 26

Let x be a rational number and y be an irrational number.
Is x + y necessarily an irrational number? Give an example in support of your

Solution 26

We
know that the sum of a rational and an irrational is irrational.

Hence,
if x is rational and y is irrational, then x + y is necessarily an irrational
number.

For
example,

Question 27

Let a be a rational number and b
be an irrational number. Is ab necessarily an

Solution 27

We
know that the product of a rational and an irrational is irrational.

Hence,
if a is rational and b is irrational, then ab is necessarily an irrational number.

For
example,

Question 28

Is the product of two irrationals always irrational?

Solution 28

No, the product of two irrationals
need not be an irrational.

For example,

Question 29

Give an example of two irrational numbers whose

(i) difference is an irrational number.

(ii) difference is a rational number.

(iii) sum is an irrational number.

(iv) sum is an rational number.

(v) product is an irrational number.

(vi) product is a rational number.

(vii) quotient is an irrational number.

(viii) quotient is a rational number.

Solution 29

(i) Difference is an irrational number:

(ii) Difference is a rational number:

(iii) Sum is an irrational number:

(iv) Sum is an rational number:

(v) Product is an irrational number:

(vi) Product is a rational number:

(vii) Quotient is an irrational number:

(viii) Quotient is a rational number:

Question 30

Examine whether the following numbers are rational or
irrational.

Solution 30

Question 31

Insert a rational and an irrational number between 2 and
2.5

Solution 31

Rational
number between 2 and 2.5 =

Irrational
number between 2 and 2.5 =

Question 32

How many irrational numbers lie between? Find any three irrational numbers lying between .

Solution 32

There are infinite irrational
numbers between.

We have

Hence, three irrational numbers lying between  are as follows:

1.5010010001……., 1.6010010001…… and 1.7010010001…….

## Chapter 1 – Number Systems Exercise Ex. 1D

Question 1

Solution 1

We have:

Question 2

Solution 2

We have:

Question 3

Solution 3

Question 4

Multiply:

Solution 4

Question 5

Multiply:

Solution 5

Question 6

Multiply:

Solution 6

Question 7

Multiply:

Solution 7

Question 8

Multiply:

Solution 8

Question 9

Multiply:

Solution 9

Question 10

Divide:

Solution 10

Question 11

Divide:

Solution 11

Question 12

Divide:

Solution 12

Question 13

Simplify

Solution 13

= 9 – 11

= -2

Question 14

Simplify

Solution 14

= 9 – 5

= 4

Question 15

Simplify:

Solution 15

Question 16

Simplify:

Solution 16

Question 17

Simplify

Solution 17

Question 18

Simplify:

Solution 18

Question 19

Simplify

Solution 19

Question 20

Examine whether the following numbers are rational or
irrational:

Solution 20

Thus, the given number is rational.

Question 21

Examine whether the following numbers are rational or
irrational:

Solution 21

Clearly, the given number is irrational.

Question 22

Examine whether the following numbers are rational or
irrational:

Solution 22

Thus, the given number is rational.

Question 23

Examine whether the following numbers are rational or
irrational:

Solution 23

Thus, the given number is irrational.

Question 24

On her birthday Reema distributed chocolates in an orphanage. The total
number of chocolates she distributed is given by .

(i) Find the number of chocolates distributed by her.

(ii) Write the moral values depicted here by Reema.

Solution 24

(i) Number of chocolates distributed
by Reema

(ii) Loving, helping and caring
attitude towards poor and needy children.

Question 25

Simplify

Solution 25

Question 26

Simplify

Solution 26

Question 27

Simplify

Solution 27

## Chapter 1 – Number Systems Exercise Ex. 1E

Question 1

Represent on the number line.

Solution 1

Draw a number line as
shown.

On the number line, take
point O corresponding to zero.

Now take point A on number
line such that OA = 2 units.

Draw perpendicular AZ at
A on the number line and cut-off arc AB = 1 unit.

By Pythagoras Theorem,

OB2 = OA2
+ AB2 = 22 + 12 = 4 + 1 = 5

OB =

Taking O as centre and OB =  as radius draw an
arc cutting real line at C.

Clearly, OC = OB =

Hence, C represents  on the number line.

Question 2

Visualize the representation of  on the number line
up to 4 decimal places.

Solution 2

Question 3

Locate on the number line.

Solution 3

Draw a number line as
shown.

On the number line, take
point O corresponding to zero.

Now take point A on number
line such that OA = 1 unit.

Draw perpendicular AZ at
A on the number line and cut-off arc AB = 1 unit.

By Pythagoras Theorem,

OB2 = OA2
+ AB2 = 12 + 12 = 1 + 1 = 2

OB =

Taking O as centre and OB =  as radius draw an
arc cutting real line at C.

Clearly, OC = OB =

Thus, C represents  on the number line.

Now, draw perpendicular
CY at C on the number line and cut-off arc CE = 1 unit.

By Pythagoras Theorem,

OE2 = OC2
+ CE2 = 2 + 12 = 2 + 1 = 3

OE =

Taking O as centre and OE =  as radius draw an
arc cutting real line at D.

Clearly, OD = OE =

Hence, D represents  on the number line.

Question 4

Locate on the number line.

Solution 4

Draw a number line as
shown.

On the number line, take
point O corresponding to zero.

Now take point A on number
line such that OA = 3 units.

Draw perpendicular AZ at
A on the number line and cut-off arc AB = 1 unit.

By Pythagoras Theorem,

OB2 = OA2
+ AB2 = 32 + 12 = 9 + 1 = 10

OB =

Taking O as centre and OB =  as radius draw an
arc cutting real line at C.

Clearly, OC = OB =

Hence,
C represents  on the number line.

Question 5

Locate on the number line.

Solution 5

Draw a number line as
shown.

On the number line, take
point O corresponding to zero.

Now take point A on number
line such that OA = 2 units.

Draw perpendicular AZ at
A on the number line and cut-off arc AB = 2 units.

By Pythagoras Theorem,

OB2 = OA2
+ AB2 = 22 + 22 = 4 + 4 = 8

OB =

Taking O as centre and OB =  as radius draw an
arc cutting real line at C.

Clearly, OC = OB =

Hence,
C represents  on the number line.

Question 6

Represent  geometrically on the
number line.

Solution 6

Draw
a line segment AB = 4.7 units and extend it to C such that BC = 1 unit.

Find
the midpoint O of AC.

With
O as centre and OA as radius, draw a semicircle.

Now,
draw BD
⊥ AC, intersecting the semicircle at
D.

Then,
BD =
units.

With
B as centre and BD as radius, draw an arc, meeting
AC produced at E.

Then, BE = BD =  units.

Question 7

Represent on the number line.

Solution 7

Draw
a line segment OB = 10.5 units and extend it to C such that BC = 1 unit.

Find
the midpoint D of OC.

With
D as centre and DO as radius, draw a semicircle.

Now,
draw BE
⊥ AC, intersecting the semicircle at
E.

Then,
BE =
units.

With
B as centre and BE as radius, draw an arc, meeting
AC produced at F.

Then, BF = BE =  units.

Question 8

Represent geometrically on the number line.

Solution 8

Draw a line segment AB = 7.28 units and extend it to C such that BC = 1 unit.

Find the midpoint O of AC.

With O as centre and OA as radius, draw a semicircle.

Now, draw BD AC, intersecting the semicircle at D.

Then, BD = units.

With D as centre and BD as radius, draw an arc, meeting AC produced at E.

Then, BE = BD = units.

Question 9

Represent  on the number line.

Solution 9

Draw
a line segment OB = 9.5 units and extend it to C such that BC = 1 unit.

Find
the midpoint D of OC.

With
D as centre and DO as radius, draw a semicircle.

Now,
draw BE
⊥ AC, intersecting the semicircle
at E.

Then,
BE =
units.

With
B as centre and BE as radius, draw an arc, meeting
AC produced at F.

Then,
BF = BE =
units.

Extend
BF to G such that FG = 1 unit.

Then,
BG =

Question 10

Visualize the representation of
3.765 on the number line using successive magnification.

Solution 10

## Chapter 1 – Number Systems Exercise Ex. 1F

Question 1

Write the rationalising factor of the denominator in .

Solution 1

The rationalising factor of the denominator in  is

Question 2

Simplify

Solution 2

Question 3

Solution 3

Thus, the given number is rational.

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

*Question
modified

Solution 10

Question 11

Solution 11

Question 12

Rationalise the denominator of following:

Solution 12

On multiplying the numerator and denominator of the given number by , we get

Question 13

Rationalise the denominator of following:

Solution 13

On multiplying the numerator and denominator of the given number by , we get

Question 14

Rationalise the denominator of following:

Solution 14

Question 15

Rationalise the denominator of following:

Solution 15

Question 16

Rationalise the denominator of each of the following.

Solution 16

Question 17

Rationalise the denominator of following:

Solution 17

Question 18

Rationalise the denominator of each of the following.

Solution 18

Question 19

Rationalise the denominator of each of the following.

Solution 19

Question 20

Rationalise the denominator of each of the following.

Solution 20

Question 21

Solution 21

Question 22

.

Solution 22

Question 23

Rationalise the denominator of each of the following.

Solution 23

Question 24

Rationalise the denominator of each of the following.

Solution 24

Question 25

Rationalise the denominator of each of the following.

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

find
the value to three places of decimals, of each of the following.

Solution 29

Question 30

find
the value to three places of decimals, of each of the following.

Solution 30

Question 31

find
the value to three places of decimals, of each of the following.

Solution 31

Question 32

Find rational numbers a and b such that

Solution 32

Question 33

Find rational numbers a and b such that

Solution 33

Question 34

Find rational numbers a and b such that

Solution 34

Question 35

Find rational numbers a and b such that

Solution 35

Question 36

find
to three places of decimals, the value of each of the following.

Solution 36

Question 37

find
to three places of decimals, the value of each of the following.

Solution 37

Question 38

find
to three places of decimals, the value of each of the following.

Solution 38

Question 39

find
to three places of decimals, the value of each of the following.

Solution 39

Question 40

find
to three places of decimals, the value of each of the following.

Solution 40

Question 41

find
to three places of decimals, the value of each of the following.

Solution 41

Question 42

Simplify by rationalising the
denominator.

Solution 42

Question 43

Simplify by rationalising the
denominator.

Solution 43

Question 44

Simplify:

Solution 44

Question 45

Simplify

Solution 45

Question 46

Simplify

Solution 46

Question 47

Simplify

Solution 47

Question 48

Prove that

Solution 48

Question 49

Prove that

Solution 49

Question 50

Find the values of a and b if

Solution 50

## Chapter 1 – Number Systems Exercise Ex. 1G

Question 1

Simplify

Solution 1

Question 2

Simplify

Solution 2

Question 3

Simplify:

Solution 3

Question 4

Simplify

Solution 4

Question 5

Prove that

Solution 5

Question 6

Prove that

Solution 6

Question 7

Prove that

Solution 7

Question 8

Simplify and express the result in the exponential form of x.

Solution 8

Question 9

Simplify the product

Solution 9

Question 10

Simplify

Solution 10

Question 11

Simplify

Solution 11

Question 12

Simplify

Solution 12

Question 13

Find the value of x in each of the following.

Solution 13

Question 14

Find the value of x in each of the following.

Solution 14

Question 15

Find the value of x in each of the following.

Solution 15

Question 16

Find the value of x in each of the following.

5x – 3× 32x – 8 = 225

Solution 16

5x – 3 × 32x – 8
= 225

5x – 3× 32x – 8 = 52 × 32

x – 3 = 2 and 2x – 8 = 2

x = 5 and 2x = 10

x = 5

Question 17

Find the value of x in each of the following.

Solution 17

Question 18

Prove that

Solution 18

Question 19

Prove that

Solution 19

Question 20

Prove that

Solution 20

Question 21

Prove that

Solution 21

Question 22

If x is a positive real number and exponents are rational
numbers, simplify

Solution 22

Question 23

If prove that m – n = 1.

Solution 23

Question 24

Write the following in ascending order of magnitude.

Solution 24

Question 25

Simplify:

Solution 25

Question 26

Simplify:

Solution 26

Question 27

Simplify:

Solution 27

Question 28

Simplify:

Solution 28

Question 29

Simplify:

Solution 29

Question 30

Simplify:

Solution 30

Question 31

Simplify:

Solution 31

Question 32

Simplify:

Solution 32

Question 33

Simplify:

Solution 33

Question 34

Evaluate:

Solution 34

Question 35

Evaluate:

Solution 35

Question 36

Evaluate:

Solution 36

Question 37

Evaluate:

Solution 37

Question 38

Evaluate:

Solution 38

Question 39

Evaluate:

Solution 39

Question 40

If a = 2, b = 3, find the value of (ab
+ ba)-1

Solution 40

Given,
a = 2 and b = 3

Question 41

If a = 2, b = 3, find the value of (aa
+ bb)-1

Solution 41

Given,
a = 2 and b = 3

Question 42

Simplify

Solution 42

Question 43

Simplify

(14641)0.25

Solution 43

(14641)0.25

Question 44

Simplify

Solution 44

Question 45

Simplify

Solution 45

Question 46

Evaluate

Solution 46

Question 47

Evaluate

Solution 47

Question 48

Evaluate

Solution 48

Question 49

Evaluate

Solution 49

Question 50

Evaluate

Solution 50

Question 51

Evaluate

Solution 51

Question 52

Evaluate

Solution 52

Question 53

Evaluate

Solution 53

## Chapter 1 – Number Systems Exercise MCQ

Question 1

Which of the following is a rational number?

(a)

(b) π

(c)

(d) 0

Solution 1

Correct
option: (d)

0
can be written as  where p and q are
integers and q ≠ 0.

Question 2

The decimal expansion that a rational number cannot have
is

(a) 0.25

(b)

(c)

(d) 0.5030030003….

Solution 2

Correct
option: (d)

The decimal expansion of a rational number is either
terminating or non-terminating recurring.

The decimal expansion of 0.5030030003…. is non-terminating
non-recurring, which is not a property of a rational number.

Question 3

Which of the following is an irrational number?

(a) 3.14

(b) 3.141414….

(c) 3.14444…..

(d) 3.141141114….

Solution 3

Correct
option: (d)

The decimal expansion of an irrational number is
non-terminating non-recurring.

Hence, 3.141141114….. is an
irrational number.

Question 4

A rational number equivalent to  is

(a)

(b)

(c)

(d)

Solution 4

Correct
option: (d)

Question 5

Choose the rational number which does not lie between

(a)

(b)

(c)

(d)

Solution 5

Correct
option: (b)

Given
two rational numbers are negative and
is a positive
rational number.

So,
it does not lie between

Question 6

Π is

(a) a
rational number

(b) an
integer

(c) an
irrational number

(d) a
whole number

Solution 6

Correct
option: (c)

Π = 3.14159265359…….., which is
non-terminating non-recurring.

Hence,
it is an irrational number.

Question 7

The decimal expansion of  is

(a) finite
decimal

(b) 1.4121

(c) nonterminating
recurring

(d) nonterminating, nonrecurring

Solution 7

Correct
option: (d)

The decimal expansion of , which is non-terminating, non-recurring.

Question 8

Which of the following is an irrational number?

(a)

(b)

(c) 0.3799

(d)

Solution 8

Correct
option: (a)

The decimal expansion of , which is non-terminating, non-recurring.

Hence, it is an irrational number.

Question 9

Hoe many digits are there in the repeating block of digits
in the decimal expansion of

(a) 16

(b) 6

(c) 26

(d) 7

Solution 9

Correct
option: (b)

Question 10

Which of the following numbers is irrational?

(a)

(b)

(c)

(d)

Solution 10

Correct
option: (c)

The decimal expansion of , which is non-terminating, non-recurring.

Hence, it is an irrational number.

Question 11

The product of two irrational numbers is

(a) always
irrational

(b) always rational

(c) always an
integer

(d)sometimes
rational and sometimes irrational

Solution 11

Question 12

A rational number between -3 and 3 is

(a) 0

(b) -4.3

(c) -3.4

(d) 1.101100110001….

Solution 12

Correct
option: (a)

On
a number line, 0 is a rational number that lies between -3 and 3.

Question 13

Which of the following is a true
statement?

(a) The sum of two
irrational numbers is an irrational number

(b) The product of
two irrational numbers is an irrational number

(c) Every real
number is always rational

(d) Every real
number is either rational or irrational

Solution 13

Question 14

Which of the following is a true
statement?

(a)

(b)

(c)

(d)

Solution 14

Question 15

A rational number lying between  is

(a)

(b)

(c) 1.6

(d) 1.9

Solution 15

Correct option: (c)

Question 16

Which of the following is a rational number?

(a)

(b) 0.101001000100001…

(c) π

(d) 0.853853853…

Solution 16

Correct
option: (d)

The decimal expansion of a rational number is either
terminating or non-terminating recurring.

Hence, 0.853853853… is a
rational number.

Question 17

The product of a nonzero rational number with an
irrational number is always a/an

(a) irrational
number

(b) rational
number

(c) whole
number

(d) natural
number

Solution 17

Correct option: (a)

The product of a non-zero rational number with an
irrational number is always an irrational number.

Question 18

The value of , where p and q are integers and q
0, is

(a)

(b)

(c)

(d)

Solution 18

Correct
option: (b)

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Two rational numbers between  are

(a)

(b)

(c)

(d)

Solution 23

Correct
option: (c)

Two
rational numbers between

Question 24

An irrational number
between 5 and 6 is

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

The sum of

(a)

(b)

(c)

(d)

Solution 27

Correct option: (b)

Let x =

i.e. x = 0.3333…. ….(i)

10x =
3.3333…. ….(ii)

On subtracting (i) from (ii), we get

9x = 3

Let y =

i.e. y = 0.4444…. ….(i)

10y =
4.4444…. ….(ii)

On subtracting (i) from (ii), we get

9y = 4

Question 28

The value of

(a)

(b)

(c)

(d)

Solution 28

Correct option: (c)

Let x =

i.e. x = 2.4545….  ….(i)

100x = 245.4545…….  ….(ii)

On subtracting (i) from (ii), we get

99x = 243

Let y =

i.e. y = 0.3636….  ….(iii)

100y = 36.3636….  ….(iv)

On subtracting (iii) from (iv), we get

99y = 36

Question 29

Which of the following is the value of ?

(a) -4

(b) 4

(c)

(d)

Solution 29

Correct
option: (b)

Question 30

when simplified is

(a) positive
and irrational

(b) positive
and rational

(c) negative
and irrational

(d) negative
and rational

Solution 30

Correct
option: (b)

Which is positive and rational number.

Question 31

when simplified is

(a) positive
and irrational

(b) positive
and rational

(c) negative
and irrational

(d) negative
and rational

Solution 31

Correct
option: (b)

Which is positive and rational number.

Question 32

When  is divided by , the quotient is

(a)

(b)

(c)

(d)

Solution 32

Correct
option: (c)

Question 33

The value of  is

(a) 10

(b)

(c)

(d)

Solution 33

Correct
option: (a)

Question 34

Every point on number line represents

(a) a
rational number

(b) a
natural number

(c) an
irrational number

(d) a
unique number

Solution 34

Correct
option: (d)

Every point on number line represents a unique number.

Question 35

The value of  is

(a)

(b)

(c)

(d)

Solution 35

Correct
option: (b)

Question 36

= ?

(a)

(b)

(c)

(d) None
of these

Solution 36

Correct
option: (b)

Question 37

=?

(a)

(b) 2

(c) 4

(d) 8

Solution 37

Correct
option: (b)

Question 38

(125)-1/3 = ?

(a) 5

(b) -5

(c)

(d)

Solution 38

Correct
option: (c)

Question 39

The value of 71/2
81/2 is

(a) (28)1/2

(b) (56)1/2

(c) (14)1/2

(d) (42)1/2

Solution 39

Correct
option: (b)

Question 40

After simplification,  is

(a) 132/15

(b) 138/15

(c) 131/3

(d) 13-2/15

Solution 40

Correct
option: (d)

Question 41

The value of is

(a)

(b)

(c) 8

(d)

Solution 41

Correct
option: (a)

Question 42

The value of is

(a) 0

(b) 2

(c)

(d)

Solution 42

Correct
option: (b)

Question 43

The value of (243)1/5 is

(a) 3

(b) -3

(c) 5

(d)

Solution 43

Correct
option: (a)

Question 44

93
+ (-3)3 – 63 = ?

(a) 432

(b) 270

(c) 486

(d) 540

Solution 44

Correct
option: (c)

93 + (-3)3
63 = 729 – 27 – 216 = 486

Question 45

Which of the following is a rational
number?

Solution 45

Question 46

Simplified value of  is

(a) 0

(b) 1

(c) 4

(d) 16

Solution 46

Correct
option: (b)

Question 47

The value of is

(a) 2-1/6

(b) 2-6

(c) 21/6

(d) 26

Solution 47

Correct
option: (c)

Question 48

Simplified value of (25)1/3×
51/3 is

(a) 25

(b) 3

(c) 1

(d) 5

Solution 48

Correct
option: (d)

Question 49

The value of is

(a) 3

(b) -3

(c) 9

(d)

Solution 49

Correct
option: (a)

Question 50

There is a number x such that x2 is irrational
but x4 is rational. Then, x can be

(a)

(b)

(c)

(d)

Solution 50

Correct
option: (d)

Question 51

If  then value of p is

(a)

(b)

(c)

(d)

Solution 51

Correct
option: (b)

Question 52

The value of is

(a)

(b)

(c)

(d)

Solution 52

Correct
option: (b)

Question 53

The value of xp-q
xq – r⋅ xr – p is equal to

(a) 0

(b) 1

(c) x

(d) xpqr

Solution 53

Correct
option: (b)

xp-q
xq – r⋅ xr – p

= xp – q + q – r + r – p

= x0

= 1

Question 54

The value of  is

(a) -1

(b) 0

(c) 1

(d) 2

Solution 54

Correct
option: (c)

Question 55

= ?

(a) 2

(b)

(c)

(d)

Solution 55

Correct
option: (a)

Question 56

Every rational number is

(a) a natural
number

(b) a whole number

(c) an integer

(d)a real number

Solution 56

Question 57

If  then x = ?

(a) 1

(b) 2

(c) 3

(d) 4

Solution 57

Correct
option: (d)

Question 58

If (33)2 = 9x then 5x
= ?

(a) 1

(b) 5

(c) 25

(d) 125

Solution 58

Correct
option: (d)

(33)2 = 9x

(32)3 = (32)x

x = 3

Then 5x = 53
= 125

Question 59

On simplification, the expression  equals

(a)

(b)

(c)

(d)

Solution 59

Correct
option: (b)

Question 60

The simplest rationalisation factor of  is

(a)

(b)

(c)

(d)

Solution 60

Correct
option: (d)

Thus, the simplest rationalisation factr of

Question 61

The simplest rationalisation factor of  is

(a)

(b)

(c)

(d)

Solution 61

Correct
option: (b)

The simplest rationalisation factor of  is

Question 62

The rationalisation factor of  is

(a)

(b)

(c)

(d)

Solution 62

Correct
option: (d)

Question 63

Rationalisation of the denominator of  gives

(a)

(b)

(c)

(d)

Solution 63

Correct
option: (d)

Question 64

(a)

(b) 2

(c) 4

(d)

Solution 64

Correct
option: (c)

Question 65

(a)

(b)

(c)

(d) None
of these

Solution 65

Correct
option: (c)

Question 66

(a)

(b) 14

(c) 49

(d) 48

Solution 66

Correct option: (b)

Question 67

Between any two rational numbers there

(a) is no rational
number

(b) is exactly one
rational number

(c) are infinitely
many rational numbers

(d)is no
irrational number

Solution 67

Question 68

(a) 0.075

(b) 0.75

(c) 0.705

(d) 7.05

Solution 68

Correct
option: (c)

Question 69

(a) 0.375

(b) 0.378

(c) 0.441

(d) None
of these

Solution 69

Correct
option: (b)

Question 70

The value of  is

(a)

(b)

(c)

(d)

Solution 70

Correct
option: (d)

Question 71

The value of  is

(a)

(b)

(c)

(d)

Solution 71

Correct
option: (c)

Question 72

(a) 0.207

(b) 2.414

(c) 0.414

(d) 0.621

Solution 72

Correct
option: (c)

Question 73

= ?

(a) 34

(b) 56

(c) 28

(d) 63

Solution 73

Correct
option: (a)

Question 74

Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:

(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).

(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).

(c) Assertion (A)
is true and Reason (R) is false.

(d) Assertion (A)
is false and Reason (R) is true.

 Assertion (A) Reason (R) A rational number between two rational numbers p and q is .

(a)/(b)/(c)/(d).

Solution 74

Question 75

Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:

(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).

(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).

(c) Assertion (A)
is true and Reason (R) is false.

(d) Assertion (A)
is false and Reason (R) is true.

 Assertion (A) Reason (R) Square root of a positive integer which is not a perfect square is an irrational number.

(a)/(b)/(c)/(d).

Solution 75

Question 76

Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:

(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).

(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).

(c) Assertion (A)
is true and Reason (R) is false.

(d) Assertion (A)
is false and Reason (R) is true.

 Assertion (A) Reason (R) e is an irrational number. Π is an irrational number.

(a)/(b)/(c)/(d).

Solution 76

Question 77

Each question consists of two statements,
namely, Assertion (A) and Reason (R). For selecting the correct answer, use
the following code:

(a) Both Assertion
(A) and Reason (R) are true and Reason (R) is a correct explanation of
Assertion (A).

(b) Both Assertion
(A) and Reason (R) are true but Reason (R) is not a correct explanation of
Assertion (A).

(c) Assertion (A)
is true and Reason (R) is false.

(d) Assertion (A)
is false and Reason (R) is true.

 Assertion (A) Reason (R) The sum of a rational number and an irrational number is an irrational number.

(a)/(b)/(c)/(d).

Solution 77

Question 78

The decimal representation of a rational
number is

(a) always
terminating

(b) either
terminating or repeating

(c) either
terminating or non-repeating

(d)neither
terminating nor repeating

Solution 78

Question 79

Match the following columns:

 Column I Column II (p) 14(q) 6(r) a rational number(s) an irrational number

(a)-…….,

(b)-…….,

(c)-…….,

(d)-…….,

Solution 79

Question 80

Match the following columns:

 Column I Column II

(a)-…….,

(b)-…….,

(c)-…….,

(d)-…….,

Solution 80

Question 81

The decimal representation of an
irrational number is

(a) always
terminating

(b) either
terminating or repeating

(c) either terminating
or non-repeating

(d)neither
terminating nor repeating

Solution 81

## Chapter 1 – Number Systems Exercise VSAQ

Question 1

What can you say about the sum of a rational number and an
irrational number?

Solution 1

The sum of a rational number and
an irrational number is irrational.

Example: 5 +  is irrational.

Question 2

Simplify (32)1/5 + (-7)0 + (64)1/2.

Solution 2

Question 3

Evaluate .

Solution 3

Question 4

Simplify .

Solution 4

Question 5

If a = 1, b = 2 then find the value of (ab + ba)-1.

Solution 5

Given,
a = 1 and b = 2

Question 6

Simplify .

Solution 6

Question 7

Give an example of two irrational numbers whose sum as
well as product is rational.

Solution 7

Question 8

Is the product of a rational and irrational numbers always
irrational? Give an example.

Solution 8

Yes, the product of a rational and
irrational numbers is always irrational.

For
example,

Question 9

Give an example of a number x such that x2 is
an irrational number and x3 is a rational number.

Solution 9

Question 10

Write the reciprocal of ().

Solution 10

The reciprocal of ()

Question 11

Solution 11

Question 12

Solve .

Solution 12

Question 13

Simplify

Solution 13

Question 14

If 10x = 64, find the value of .

Solution 14

Question 15

Evaluate

Solution 15

Question 16

Simplify .

Solution 16

Question 17

The number  will terminate after
how many decimal places?

Solution 17

Thus, the given number
will terminate after 3 decimal places.

Question 18

Find the value of (1296)0.17× (1296)0.08.

Solution 18

(1296)0.17× (1296)0.08

Question 19

Simplify .

Solution 19

Question 20

Find an irrational number between 5 and 6.

Solution 20

An irrational number between 5 and
6 =

Question 21

Find the value of .

Solution 21

Question 22

Rationalise

Solution 22

Question 23

Solve for x: .

Solution 23

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