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#### Page No 1.13:

#### Question 1:

Express the following rational numbers as decimals:

(i) $\frac{42}{100}$

(ii) $\frac{327}{500}$

(iii) $\frac{15}{4}$

#### Answer:

(i) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method as below.

Hence,

(ii) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method as below.

Hence,

(iii) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method as below.

Hence, $\frac{15}{4}=3.75$

#### Page No 1.13:

#### Question 2:

Express the following rational numbers as decimals:

(i) $\frac{2}{3}$

(ii) $-\frac{4}{9}$

(iii) $\frac{-2}{15}$

(iv) $-\frac{22}{13}$

(v) $\frac{437}{999}$

(vi) $\frac{33}{26}$

#### Answer:

(i) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(ii) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(iii) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(iv) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(v) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(vi) Given rational number is

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

#### Page No 1.13:

#### Question 3:

Look at several examples of rational numbers in the form $\frac{p}{q}(q\ne 0),$ where *p* and *q* are integers with no common factors other than 1 and having terminating decimal representations. Can you guess what property *q* must satisfy?

#### Answer:

Prime factorization is the process of finding which prime numbers you need to multiply together to get a certain number. So prime factorization of denominators (*q*) must have only the power of 2 or 5 or both.

#### Page No 1.22:

#### Question 1:

Express each of the following decimals in the form $\frac{p}{q}:$

(i) 0.39

(ii) 0.750

(iii) 2.15

(iv) 7.010

(v) 9.90

(vi) 1.0001

#### Answer:

(i)** **Given decimal is

Now we have to convert given decimal number into the form

Let

Hence,

(ii) Given decimal is

Now we have to convert given decimal number into form

Let

Hence,

(iii) Given decimal is

Now we have to express the given decimal number into form

Let

Hence,

(iv) Given decimal is 7.010

Now we have to express the given decimal number into form

Let

Hence,

(v) Given decimal is

Now we have to find given decimal number into form

Let

Hence, =$\frac{99}{10}$

(vi) Given decimal is

Now we have to find given decimal number into form

Hence,

#### Page No 1.22:

#### Question 2:

Express each of the following decimals in the form $\frac{p}{q}:$

(i) $0.\overline{)4}$

(ii) $0.\overline{37}$

(iii) $0.\overline{54}$

(iv) $0.\overline{621}$

(v) $125.\overline{3}$

(vi) $4.\overline{7}$

(vii) $0.\overline{47}$

#### Answer:

(i) Let

Hence,

(ii) Let

Hence,

(iii) Let

Hence,

(iv) Let

Hence,

(v) Let

Hence,

(vi) Let

Let

Therefore,

Hence,

(vii) Let

Since,

Therefore,

Hence,

#### Page No 1.31:

#### Question 1:

Define an irrational number.

#### Answer:

An irrational number is a real number that cannot be reduced to any ratio between an integer *p* and a natural number *q*.

If the decimal representation of an irrational number is non-terminating and non-repeating, then it is called irrational number. For example

#### Page No 1.31:

#### Question 2:

Explain, how irrational numbers differ from rational numbers?

#### Answer:

Every rational number must have either terminating or non-terminating but irrational number must have non- terminating and non-repeating decimal representation.

A rational number is a number that can be written as simple fraction (ratio) and denominator is not equal to zero while an irrational is a number that cannot be written as a ratio.

#### Page No 1.31:

#### Question 3:

Examine, whether the following numbers are rational or irrational:

(i) $\sqrt{7}$

(ii) $\sqrt{4}$

(iii) 2 + $\sqrt{3}$

(iv) $\sqrt{3}+\sqrt{2}$

(v) $\sqrt{3}+\sqrt{5}$

(vi) ($\sqrt{2}-2{)}^{2}$

(vii) $(2-\sqrt{2})(2+\sqrt{2})$

(viii) $(\sqrt{2}+\sqrt{3}{)}^{2}$

(ix) $\sqrt{5}-2$

(x) $\sqrt{23}$

(xi) $\sqrt{225}$

(xii) 0.3796

(xiii) 7.478478

(xiv) 1.101001000100001

#### Answer:

(i) Let

Therefore,

It is non-terminating and non-repeating

Hence is an irrational number

(ii) Let

Therefore,

It is terminating.

Hence is a rational number.

(iii) Let be the rational

Squaring on both sides

Since, *x* is rational

is rational

is rational

is rational

is rational

But, is irrational

So, we arrive at a contradiction.

Hence is an irrational number

(iv) Let be the rational number

Squaring on both sides, we get

Since, *x* is a rational number

is rational number

is rational number

is rational number

is rational number

But is an irrational number

So, we arrive at contradiction

Hence is an irrational number

(v) Let be the rational number

Squaring on both sides, we get

Now, *x* is rational number

is rational number

is rational number

is rational number

is rational number

But is an irrational number

So, we arrive at a contradiction

Hence is an irrational number

(vi) Let be a rational number.

Since, *x* is rational number,

⇒ *x* – 6 is a rational nu8mber

⇒is a rational number

⇒is a rational number

But we know thatis an irrational number, which is a contradiction

So is an irrational number

(vii) Let

So is a rational number

(viii) Let be rational number

Using the formula

⇒is a rational number

⇒is a rational number

But we know thatis an irrational number

So, we arrive at a contradiction

So is an irrational number.

(ix) Let $x=\sqrt{5}-2$ be the rational number

Squaring on both sides, we get

$x=\sqrt{5}-2\phantom{\rule{0ex}{0ex}}{x}^{2}={\left(\sqrt{5}-2\right)}^{2}\phantom{\rule{0ex}{0ex}}{x}^{2}=25+4-4\sqrt{5}\phantom{\rule{0ex}{0ex}}{x}^{2}-29=-4\sqrt{5}\phantom{\rule{0ex}{0ex}}\frac{{x}^{2}-29}{-4}=\sqrt{5}$

Now, *x* is rational

${x}^{2}\mathrm{is}\mathrm{rational}.\phantom{\rule{0ex}{0ex}}\mathrm{So},{\mathrm{x}}^{2}-29\mathrm{is}\mathrm{rational}\phantom{\rule{0ex}{0ex}}\frac{{\mathrm{x}}^{2}-29}{-4}=\sqrt{5}\mathrm{is}\mathrm{rational}.$

But, $\sqrt{5}$ is irrational. So we arrive at contradiction

Hence $x=\sqrt{5}-2$ is an irrational number

(x) Let

It is non-terminating or non-repeating

Hence is an irrational number

(xi) Let

Hence is a rational number

(xii) Given $0.3796$.

It is terminating

Hence it is a rational number

(xiii) Given number

It is repeating

Hence it is a rational number

(xiv) Given number is

It is non-terminating or non-repeating

Hence it is an irrational number

#### Page No 1.31:

#### Question 4:

Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:

(i) ($\sqrt{4}$)

(ii) $3\sqrt{18}$

(iii) $\sqrt{1.44}$

(iv) $\sqrt{\frac{9}{27}}$

(v) $-\sqrt{64}$

(vi) $\sqrt{100}$

#### Answer:

(i) Given number is *x* =

*x *= 2, which is a rational number

(ii) Given number is

$3\sqrt{18}=3\sqrt{3\times 3\times 2}=3\times 3\sqrt{2}=9\sqrt{2}$

So it is an irrational number

(iii) Given number is

Now we have to check whether it is rational or irrational

So it is a rational

(iv) Given that

Now we have to check whether it is rational or irrational

So it is an irrational number

(v) Given that

Now we have to check whether it is rational or irrational

Since,

So it is a rational number

(vi) Given that

Now we have to check whether it is rational or irrational

Since,

So it is rational number

#### Page No 1.31:

#### Question 5:

In the following equations, find which variables *x, y, z* etc. represent rational or irrational numbers:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

#### Answer:

(i) Given that

Now we have to find the value of *x*

So it *x* is an irrational number

(ii) Given that

Now we have to find the value of *y*

So *y* is a rational number

(iii) Given that

Now we have to find the value of *z*

So it is rational number

(iv) Given that

Now we have to find the value of *u*

So it is an irrational number

(v) Given that

Now we have to find the value of *v*

So it is an irrational number

(vi) Given that

Now we have to find the value of *w*

So it is an irrational number

(vii) Given that

Now we have to find the value of *t*

So it is an irrational number

#### Page No 1.31:

#### Question 6:

Give two rational numbers lying between 0.232332333233332... and 0.212112111211112.

#### Answer:

Let

Here the decimal representation of *a* and *b* are non-terminating and non-repeating. So we observe that in first decimal place of *a* and *b* have the same digit but digit in the second place of their decimal representation are distinct. And the number *a* has 3 and *b* has 1. So *a* > *b*.

Hence two rational numbers are lying between and

#### Page No 1.31:

#### Question 7:

Give two rational numbers lying between 0.515115111511115...0.5353353335...

#### Answer:

Let and

Here the decimal representation of *a* and *b *are non-terminating and non-repeating. So we observe that in first decimal place *a* and *b* have the same digit but digit in the second place of their decimal representation are distinct. And the number *a* has 1 and *b* has 3. So *a* < *b*.

Hence two rational numbers are lying between and

#### Page No 1.31:

#### Question 8:

Find one irrational number between 0.2101 and 0.222... = $0.\overline{2}$.

#### Answer:

Let

Here *a* and *b *are rational numbers .Since *a* has terminating and *b* has repeating decimal. We observe that in second decimal place *a* has 1 and *b* has 2. So *a* < *b*.

Hence one irrational number is lying between and

#### Page No 1.32:

#### Question 9:

Find a rational number and also an irrational number lying between the numbers 0.3030030003 ... and 0.3010010001 ...

#### Answer:

Let

Here decimal representation of *a* and *b* are non-terminating and non-repeating. So *a* and *b *are irrational numbers. We observe that in first two decimal place of *a* and *b* have the same digit but digit in the third place of their decimal representation is distinct.

Therefore, *a* > *b*.

Hence one rational number is lying between and

And irrational number is lying between and

#### Page No 1.32:

#### Question 10:

Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.

#### Answer:

Let and

Here we observe that in the first decimal *x* has digit 7 and *y* has 8. So *x* < *y*. In the second decimal place *x* has digit 1. So, if we considering irrational numbers

$a=0.72072007200072..\phantom{\rule{0ex}{0ex}}b=0.73073007300073..\phantom{\rule{0ex}{0ex}}c=0.74074007400074....$

We find that

Hence are required irrational numbers.

#### Page No 1.32:

#### Question 11:

Give an example of each, of two irrational numbers whose:

(i) difference is a rational number.

(ii) difference is an irrational number.

(iii) sum is a rational number.

(iv) sum is an irrational number.

(v) product is an rational number.

(vi) product is an irrational number.

(vii) quotient is a rational number.

(viii) quotient is an irrational number.

#### Answer:

(i) Let

And, so

Therefore, andare two irrational numbers and their difference is a rational number

(ii) Let are two irrational numbers and their difference is an irrational number

Because is an irrational number

(iii) Let are two irrational numbers and their sum is a rational number

That is

(iv) Let are two irrational numbers and their sum is an irrational number

That is

(v) Let are two irrational numbers and their product is a rational number

That is

(vi) Let are two irrational numbers and their product is an irrational number

That is

(vii) Let are two irrational numbers and their quotient is a rational number

That is

(viii) Let are two irrational numbers and their quotient is an irrational number

That is

#### Page No 1.32:

#### Question 12:

Find two irrational numbers between 0.5 and 0.55.

#### Answer:

Let

Here *a* and *b* are rational number. So we observe that in first decimal place *a* and *b* have same digit .So *a* < *b*.

Hence two irrational numbers are and lying between 0.5 and 0.55.

#### Page No 1.32:

#### Question 13:

Find two irrational numbers lying between 0.1 and 0.12.

#### Answer:

Let

Here *a *and *b* are rational number. So we observe that in first decimal place *a* and *b* have same digit. So *a* < *b*.

Hence two irrational numbers are and lying between 0.1 and 0.12.

#### Page No 1.32:

#### Question 14:

Prove that $\sqrt{3}+\sqrt{5}$ is an irrational number.

#### Answer:

Given that is an irrational number

Now we have to prove is an irrational number

Let is a rational

Squaring on both sides

Now is rational

is rational

is rational

is rational

But, is an irrational

Thus we arrive at contradiction thatis a rational which is wrong.

Hence is an irrational

#### Page No 1.36:

#### Question 1:

Complete the following sentences:

(i) Every point on the number line corresponds to a .... number which many be either ... or ...

(ii) The decimal form of an irrational number is neither ... nor ...

(iii) The decimal representation of a rational number is either ... or ...

(iv) Every real number is either ... number or ... number.

#### Answer:

(i) Every point on the number line corresponds to a __real__ number which may be either __rational__ or an __irrational__ number.

(ii) The decimal form of an irrational number is neither __terminating__ nor __repeating__.

(iii) The decimal representation of rational number is either __terminating,__ __recurring__.

(iv) Every real number is either __rational__ number or an __irrational__ number because rational or an irrational number is a family of real number.

#### Page No 1.36:

#### Question 2:

Find whether the following statement are true or false.

(i) Every real number is either rational or irrational.

(ii) $\pi $ is an irrational number.

(iii) Irrational numbers cannot be represented by points on the number line.

#### Answer:

(i) True, because rational or an irrational number is a family of real number. So every real number is either rational or an irrational number.

(ii) True, because the decimal representation of an irrational is always non-terminating or non-repeating. So is an irrational number.

(iii) False, because we can represent irrational numbers by points on the number line.

#### Page No 1.36:

#### Question 3:

Represent $\sqrt{6},\sqrt{7},\sqrt{8}$ on the number line.

#### Answer:

We are asked to represent on the number line

We will follow certain algorithm to represent these numbers on real line

We will consider point *A* as reference point to measure the distance

(1) First of all draw a line *AX* and *YY*^{’}^{ }perpendicular to *AX*

(2) Consider , so

(3) Take *A* as center and *AB* as radius, draw an arc which cuts line *AX* at *A*_{1}

(4) Draw a perpendicular line *A*_{1}*B*_{1} to *AX* such that and

(5) Take *A* as center and *AB*_{1} as radius and draw an arc which cuts the line *AX* at *A*_{2}_{.}

Here

So

So *A*_{2} is the representation for

(1) Draw line *A*_{2}*B*_{2} perpendicular to *AX*

(2) Take *A* center and *AB*_{2} as radius and draw an arc which cuts the horizontal line at *A*_{3} such that

So point *A*_{3} is the representation of

(3) Again draw the perpendicular lineto *AX*

(4) Take *A* as center and *AB*_{3} as radius and draw an arc which cuts the horizontal line at *A*_{4}

Here;

*A*_{4} is basically the representation of

#### Page No 1.36:

#### Question 4:

Represent $\sqrt{3.5},\sqrt{9.4},\sqrt{10.5}$ on the real number line.

#### Answer:

We are asked to represent the real numbers on the real number line

We will follow a certain algorithm to represent these numbers on real number line

_{(a) }

We will take *A* as reference point to measure the distance

(1) Draw a sufficiently large line and mark a point *A* on it

(2) Take a point *B* on the line such that

(3) Mark a point *C* on the line such that

(4) Find mid point of *AB* and let it be *O*

(5) Take *O* as center and *OC* as radius and draw a semi circle. Draw a perpendicular *BD* which cuts the semi circle at *D*

(6) Take *B* as the center and *BD* as radius, draw an arc which cuts the horizontal line at *E*

(7) Point *E* is the representation of

(b)

We will take *A* as reference point to measure the distance. We will follow the same figure in the part (a)

(1) Draw a sufficiently large line and mark a point *A* on it

(2) Take a point *B* on the line such that

(3) Mark a point *C* on the line such that

(4) Find mid point of *AB* and let it be *O*

(5) Take *O* as center and *OC* as radius and draw a semi circle. Draw a perpendicular *BC* which cuts the semi circle at *D*

(6) Take *B* as the center and *BD* as radius, draw an arc which cuts the horizontal line at *E*

(7) Point *E* is the representation of

(c)

We will take *A* as reference point to measure the distance. We will follow the same figure in the part (a)

(1) Draw a sufficiently large line and mark a point *A* on

(2) Take a point *B* on the line such that

(3) Mark a point *C* on the line such that

(4) Find mid point of *AB* and let it be *O*

(5) Take *O* as center and *OC* as radius and draw a semi circle. Draw a perpendicular *BC* which cuts the semi circle at *D*

(6) Take *B* as the center and *BD* as radius, draw an arc which cuts the horizontal line at *E*

(7) Point *E* is the representation of

#### Page No 1.40:

#### Question 1:

Visualise 2.665 on the number line, using successive magnification.

#### Answer:

We know that 2.665 lies between 2 and 3. So, we divide the number line into 10 equal parts

and mark each point of division. The first mark on the right of 2 will be 2.1 followed by 2.2 and so on.

The point left of 3 will be 2.9. Now, the magnified view of this will show that 2.665 lies between 2.6 and

2.7. So, our focus will be now 2.6 and 2.7. We divide this again into 10 equal parts. The first part will be

2.61 followed by 2.62 and so on.

We now magnify this again and find that 2.665 lies between 2.66 and 2.67. So, we magnify this portion

and divide it again into 10 equal parts. The first part will represent 2.661, next will be 2.662 and so on.

So, 2.665 will be 5th mark in this subdivision as shown in the figure.

#### Page No 1.40:

#### Question 2:

Visualise the representation of $5.3\overline{7}$ on the number line upto 5 decimal places, that is upto 5.37777.

#### Answer:

We know that $5.3\overline{7}$ will lie between 5 and 6. So, we locate $5.3\overline{7}$ between 5 and 6. We divide this portion of the number line

between 5 and 6 into 10 equal parts and use a magnifying glass to visualize $5.3\overline{7}$ .

$5.3\overline{7}$ lies between 5.37 and 5.38. To visualize $5.3\overline{7}$ more accurately we use a magnifying glass to visualize between 5.377 and 5.378.

Again, we divide the portion between 5.377 and 5.378 into 10 equal parts and visualize more closely to represent

$5.3\overline{7}$ as given in the figure. This is located between 5.3778 and 5.3777.

#### Page No 1.40:

#### Question 1:

Which one of the following is a correct statement?

(a) Decimal expansion of a rational number is terminating

(b) Decimal expansion of a rational number is non-terminating

(c) Decimal expansion of an irrational number is terminating

(d) Decimal expansion of an irrational number is non-terminating and non-repeating

#### Answer:

The decimal expansion of an irrational number is non-terminating and non- repeating. Thus, we can say that a number, whose decimal expansion is non-terminating and non- repeating, called irrational number. And the decimal expansion of rational number is either terminating or repeating. Thus, we can say that a number, whose decimal expansion is either terminating or repeating, is called a rational number.

Hence the correct option is .

#### Page No 1.40:

#### Question 2:

Which one of the following statements is true?

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

(b) The sum of two irrational numbers is always a rational number

(c) The sum of two irrational numbers may be a rational number or an irrational number

(d) The sum of two irrational numbers is always an integer

#### Answer:

Since, and are two irrational number and

Therefore, sum of two irrational numbers may be rational

Now, letandbe two irrational numbers and

Therefore, sum of two irrational number may be irrational

Hence the correct option is .

#### Page No 1.40:

#### Question 3:

Which of the following is a correct statement?

(a) Sum of two irrational numbers is always irrational

(b) Sum of a rational and irrational number is always an irrational number

(c) Square of an irrational number is always a rational number

(d) Sum of two rational numbers can never be an integer

#### Answer:

The sum of irrational number and rational number is always irrational number.

Let *a *be a rational number and *b *be an irrational number.

Then,

As 2*ab** *is irrational therefore is irrational.

Hence is irrational.

Therefore answer is .

#### Page No 1.40:

#### Question 4:

Which of the following statements is true?

(a) Product of two irrational numbers is always irrational

(b) Product of a non-zero rational and an irrational number is always irrational

(c) Sum of two irrational numbers can never be irrational

(d) Sum of an integer and a rational number can never be an integer

#### Answer:

Since we know that the product of rational and irrational numbers is always irrational.

For example: Let $\frac{1}{2}\mathrm{and}\sqrt{3}$ are rational and irrational numbers respectively and their product is $\frac{\sqrt{3}}{2}$.

Hence the correct is option (b).

#### Page No 1.40:

#### Question 5:

Which of the following is irrational?

(a)$\sqrt{\frac{4}{9}}$

(b) $\frac{4}{5}$

(c) $\sqrt{7}$

(d) $\sqrt{81}$

#### Answer:

Given that

And 7 is not a perfect square.

Hence the correct option is.

#### Page No 1.40:

#### Question 6:

Which of the following is irrational?

(i) 0.14

(ii) $0.14\overline{16}$

(iii) $0.\overline{1416}$

(iv) 0.1014001400014...

#### Answer:

Given that

Here is non-terminating or non-repeating. So it is an irrational number.

Hence the correct option is.

#### Page No 1.40:

#### Question 7:

Which of the following is rational?

(a) $\sqrt{3}$

(b) $\mathrm{\pi}$

(c) $\frac{4}{0}$

(d) $\frac{0}{4}$

#### Answer:

Given that

Here,, this is the form of . So this is a rational number

Hence the correct option is.

#### Page No 1.40:

#### Question 8:

The number 0.318564318564318564 ........ is:

(a) a natural number

(b) an integer

(c) a rational number

(d) an irrational number

#### Answer:

Since the given number is repeating, so it is rational number because rational number is always either terminating or repeating

Hence the correct option is.

#### Page No 1.40:

#### Question 9:

If *n* is a natural number, then $\sqrt{n}$ is

(a) always a natural number

(b) always an irrational number

(c) always an irrational number

(d) sometimes a natural number and sometimes an irrational number

#### Answer:

The term “natural number” refers either to a member of the set of positive integer.

And natural number starts from one of counting digit .Thus, if *n* is a natural number then sometimes *n* is a perfect square and sometimes it is not.

Therefore, sometimesis a natural number and sometimes it is an irrational number

Hence the correct option is.

#### Page No 1.41:

#### Question 10:

Which of the following numbers can be represented as non-terminating, repeating decimals?

(a) $\frac{39}{24}$

(b) $\frac{3}{16}$

(c) $\frac{3}{11}$

(d) $\frac{137}{25}$

#### Answer:

Given that

Here is repeating but non-terminating.

Hence the correct option is.

#### Page No 1.41:

#### Question 11:

Every point on a number line represents

(a) a unique real number

(b) a natural number

(c) a rational number

(d) an irrational number

#### Answer:

In basic mathematics, number line is a picture of straight line on which every point is assumed to correspond to real number.

Hence the correct option is.

#### Page No 1.41:

#### Question 12:

Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c) $0.\overline{1516}$

(d) 0.5015001500015.

#### Answer:

Given decimal numbers are

Here the number is non terminating or non-repeating.

Hence the correct option is.

#### Page No 1.41:

#### Question 13:

The number in the form , where *p* and *q* are integers and *q* ≠ 0, is

(a)

(b)

(c)

(d)

#### Answer:

Given that

The correct option is.

#### Page No 1.41:

#### Question 14:

The number $0.\overline{3}$ in the form $\frac{p}{q}$, where *p* and *q* are integers and *q *≠ 0, is

(a) $\frac{33}{100}$

(b) $\frac{3}{10}$

(c) $\frac{1}{3}$

(d) $\frac{3}{100}$

#### Answer:

Given number is

The correct option is

#### Page No 1.41:

#### Question 15:

$0.3\overline{2}$ when expressed in the form $\frac{p}{q}$ (*p, q* are integers *q* ≠ 0), is

(a) $\frac{8}{25}$

(b) $\frac{29}{90}$

(c) $\frac{32}{99}$

(d) $\frac{32}{199}$

#### Answer:

Given that

Now we have to express this number into form

Let *X* =$0.3\overline{2}$

The correct option is

#### Page No 1.41:

#### Question 16:

$23.\overline{43}$ when expressed in the form $\frac{p}{q}$(*p, q* are integers *q* ≠ 0), is

(a) $\frac{2320}{99}$

(b) $\frac{2343}{100}$

(c) $\frac{2343}{999}$

(d) $\frac{2320}{199}$

#### Answer:

Given that

Now we have to express this number into the form of

Let

$x=23.43\phantom{\rule{0ex}{0ex}}x=23+0.4343...\phantom{\rule{0ex}{0ex}}x=23+\frac{43}{99}\phantom{\rule{0ex}{0ex}}x=\frac{2277+43}{99}=\frac{2320}{99}$

The correct option is

#### Page No 1.41:

#### Question 17:

$0.\overline{001}$ when expressed in the form $\frac{p}{q}$ (*p, q* are integers, *q* ≠ 0), is

(a) $\frac{1}{1000}$

(b) $\frac{1}{100}$

(c) $\frac{1}{1999}$

(d) $\frac{1}{999}$

#### Answer:

Given that

Now we have to express this number into form

The correct option is

#### Page No 1.41:

#### Question 18:

The value of $0.\overline{23}$ + $0.\overline{22}$ is

(a) $0.\overline{45}$

(b) $0.\overline{43}$

(c) $0.\overline{45}$

(d) $0.45$

#### Answer:

Given that

Let

Now we have to find the value of

The correct option is

#### Page No 1.41:

#### Question 19:

An irrational number between 2 and 2.5 is

(a) $\sqrt{11}$

(b) $\sqrt{5}$

(c) $\sqrt{22.5}$

(d) $\sqrt{12.5}$

#### Answer:

Let

Here *a* and *b* are rational numbers. So we observe that in first decimal place *a* and *b* have distinct. According to question *a* < *b*.so an irrational number between 2 and 2.5 is OR

Hence the correct answer is.

#### Page No 1.41:

#### Question 20:

The number of consecutive zeros in ${2}^{3}\times {3}^{4}\times {5}^{4}\times 7$, is

(a) 3

(b) 2

(c) 4

(d) 5

#### Answer:

We are given the following expression and asked to find out the number of consecutive zeros

We basically, will focus on the powers of 2 and 5 because the multiplication of these two number gives one zero. So

${2}^{3}\times {3}^{4}\times {5}^{4}\times 7={2}^{3}\times {5}^{4}\times {3}^{4}\times 7\phantom{\rule{0ex}{0ex}}={2}^{3}\times {5}^{3}\times 5\times {3}^{4}\times 7\phantom{\rule{0ex}{0ex}}={\left(2\times 5\right)}^{3}\times 5\times {3}^{4}\times 7\phantom{\rule{0ex}{0ex}}={10}^{3}\times 5\times {3}^{4}\times 7\phantom{\rule{0ex}{0ex}}=5\times 81\times 7\times 1000\phantom{\rule{0ex}{0ex}}=2835000$

Therefore the consecutive zeros at the last is 3

So the option (a) is correct

#### Page No 1.42:

#### Question 21:

The smallest rational number by which $\frac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is

(a) $\frac{1}{10}$

(b) $\frac{3}{10}$

(c) 3

(d) 30

#### Answer:

Give number is. Now multiplying by in the given number, we have

Hence the correct option is

#### Page No 1.42:

#### Question 1:

The decimal expansion of a rational number is either ______ or _______.

#### Answer:

The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

Hence, the decimal expansion of a rational number is either __terminating__ or __recurring__.

#### Page No 1.42:

#### Question 2:

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

#### Answer:

The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

In case of irrational number, the decimal expansion neither terminates nor repeats after finitely many digits.

Hence, the decimal expansion of an irrational number is non-terminating and __non-repeating__.

#### Page No 1.42:

#### Question 3:

The decimal expansion of $\sqrt{2}$ is _______ and _________.

#### Answer:

$\sqrt{2}$ is an irrational number.

The decimal expansion of an irrational number neither terminates nor repeats after finitely many digits.

Hence, the decimal expansion of $\sqrt{2}$ is __non-terminating__ and __non-repeating__.

#### Page No 1.42:

#### Question 4:

The value of 1.999. in the form of $\frac{m}{n}$, where *m *and *n *are integers and* n* ≠ 0, is _______.

#### Answer:

$\mathrm{Let}x=1.999........\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{Multiply}\left(1\right)\mathrm{by}10\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10x=19.999........\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10x-x=19.999....-1.999....\phantom{\rule{0ex}{0ex}}\Rightarrow 9x=18\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{18}{9}\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{2}{1}$

Hence, the value of 1.999... in the form of $\frac{m}{n}$, where *m *and *n *are integers and* n* ≠ 0, is $\overline{)\frac{2}{1}}$.

#### Page No 1.42:

#### Question 5:

Every recurring decimal is a _________ number.

#### Answer:

The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

Hence, every recurring decimal is a __rational__ number.

#### Page No 1.42:

#### Question 6:

π is an _______ number.

#### Answer:

The decimal expansion of π neither terminates nor repeats after finitely many digits.

Therefore, it is an irrational number.

Hence, π is an __irrational__ number.

#### Page No 1.42:

#### Question 7:

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

#### Answer:

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

Hence, the product of a non-zero rational number with an irrational number is always an __irrational__ number.

#### Page No 1.42:

#### Question 8:

The simplest form of $1.\overline{)6}$ is _______.

#### Answer:

$\mathrm{Let}x=1.6666........\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{Multiply}\left(1\right)\mathrm{by}10\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10x=16.6666........\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10x-x=16.6666....-1.6666....\phantom{\rule{0ex}{0ex}}\Rightarrow 9x=15\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{15}{9}\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{5}{3}$

Hence, the simplest form of $1.\overline{)6}$ is $\overline{)\frac{5}{3}}$.

#### Page No 1.42:

#### Question 9:

$0.\overline{)3}+0.\overline{)4}$ is equal to _________.

#### Answer:

$\mathrm{Let}x=0.3333........\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{Multiply}\left(1\right)\mathrm{by}10\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10x=3.3333........\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10x-x=3.333....-0.333....\phantom{\rule{0ex}{0ex}}\Rightarrow 9x=3\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{3}{9}...\left(3\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Let}y=0.4444........\left(4\right)\phantom{\rule{0ex}{0ex}}\mathrm{Multiply}\left(1\right)\mathrm{by}10\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10y=4.4444........\left(5\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Subtracting}\left(4\right)\mathrm{from}\left(5\right),\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}10y-y=4.4444....-0.4444....\phantom{\rule{0ex}{0ex}}\Rightarrow 9y=4\phantom{\rule{0ex}{0ex}}\Rightarrow y=\frac{4}{9}...\left(6\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\phantom{\rule{0ex}{0ex}}0.\overline{)3}+0.\overline{)4}=x+y\phantom{\rule{0ex}{0ex}}=\frac{3}{9}+\frac{4}{9}\left(\mathrm{From}\left(3\right)\mathrm{and}\left(6\right)\right)\phantom{\rule{0ex}{0ex}}=\frac{7}{9}$

Hence, $0.\overline{)3}+0.\overline{)4}$ is equal to $\overline{)\frac{7}{9}}$.

#### Page No 1.42:

#### Question 10:

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

#### Answer:

The sum of a rational number and an irrational number always results in an irrational number.

Hence, the sum of a rational number and an irrational number is an __irrational__ number.

#### Page No 1.42:

#### Question 11:

Every real number is either ________ or _______ number.

#### Answer:

The real number includes all the rational as well as irrational numbers.

Hence, every real number is either __rational__ or __irrational__ number.

#### Page No 1.5:

#### Question 1:

Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where *p* and *q* are integers and *q ≠ *0?

#### Answer:

Yes, zero is a rational number because it is either terminating or non-terminating so we can write in the form of , where *p *and *q *are natural numbers and *q* is not equal to zero.

So,

Therefore,

#### Page No 1.5:

#### Question 2:

Find five rational numbers between 1 and 2.

#### Answer:

We need to find 5 rational numbers between 1 and 2.

Consider,

And

So, five rational numbers between $\frac{6}{6}\mathrm{and}\frac{12}{6}$will be $\frac{7}{6},\frac{8}{6},\frac{9}{6},\frac{10}{6},\frac{11}{6}$.

Hence 5 rational numbers between 1 and 2 are: OR .

#### Page No 1.5:

#### Question 3:

Find six rational numbers between 3 and 4.

#### Answer:

We need to find 6 rational numbers between 3 and 4.

Consider,

And

So, six rational numbers between $\frac{21}{7}\mathrm{and}\frac{28}{7}$ will be .

Hence 6 rational numbers between 3 and 4 are .

#### Page No 1.5:

#### Question 4:

Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.

#### Answer:

We need to find 5 rational numbers betweenand .

Since, LCM of denominators

So, consider

And,

Hence 5 rational numbers between and are: OR

#### Page No 1.5:

#### Question 5:

Are the following statements true or false? Give reasons for 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 natural number is a whole number.

(v) Every integer is a whole number.

(vi) Every rational number is a whole number.

#### Answer:

(i) False, because whole numbers start from zero and natural numbers start from one

(ii) True, because it can be written in the form of a fraction with denominator 1

(iii) False, rational numbers are represented in the form of fractions. Integers can be represented in the form of fractions but all fractions are not integers. for example: $\frac{3}{4}$ is a rational number but not an integer.

(iv) True, because natural numbers belong to whole numbers

(v) False, because set of whole numbers contains only zero and set of positive integers, whereas set of integers is the collection of zero and all positive and negative integers.

(vi) False, because rational numbers include fractions but set of whole number does not include fractions.

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