1 Marks Questions

1. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Ans. We know that square root of every positive integer will not yield an integer.

We know that is 2, which is an integer. But,will give an irrational number.

Therefore, we conclude that square root of every positive integer is not an irrational number.

2. Write three numbers whose decimal expansions are non-terminating non-recurring.

Ans. The three numbers that have their expansions as non-terminating on recurring decimal are given below.

0.04004000400004….

0.07007000700007….

0.013001300013000013….

3. Find three different irrational numbers between the rational numbersand.

Ans. Let us convertinto decimal form, to get

.

Three irrational numbers that lie betweenare:

0.73073007300073….

0.74074007400074….

0.76076007600076….

4. Which of the following rational numbers have terminating decimal representation

Ans.

5. How many rational numbers can be found between two distinct rational numbers?

(i)Two

(ii) Ten

(iii) Zero

(iv) Infinite

Ans. (iv) Infinite

6. The value of in

(i)1

(ii)-1

(iii) 2

(iv) none of these

Ans. (i) 1

7. (27)^-2/3 is equal to

(i) 9

(ii) 1/9

(iii) 3

(iv) none of these

Ans. (ii) 1/9

8. Every natural number is

(i) not an integer

(ii) always a whole number

(iii) an irrational number

(iv) not a fraction

Ans. (ii) always a whole number

9. Select the correct statement from the following

Ans.

10. is equal to

Ans.

11. 0.83458456……………is

(i) an irrational number

(ii) rational number

(iii) a natural number

(iv) a whole number.

Ans. (i) an irrational number

12. A terminating decimal is

(i) a natural number

(ii) a rational number

(iii) a whole number

(iv) an integer.

Ans. (ii) a rational number

13. The form of the number 0.8 is

Ans.

14. The value of is

(i) 1

(ii) 10

(iii) 3

(iv) 0

Ans. (ii) 10

15. The sum of rational and an irrational number

(i) may be natural

(ii) may be irrational

(iii) is always irrational

(iv) is always rational

Ans. (iii) is always irrational

16. The rational number not lying between and is

Ans.

17. is equal to

(a)

(b)

(c)

(d) None of these

Ans. (a)

18. The number is

(a) natural number

(b) irrational number

(c) rational number

(d) integer

Ans. (b) irrational number

19. The simplest form of is

Ans.

20. The value of + is

Ans. (A)

21. The value of is

Ans. (B)

=

22. is equal to

Ans.

=

23. is an

(A) natural number

(B) rational number

(C) integer

(D) irrational number

Ans. (D) is an irrational number

2 Marks Questions

1. Is zero a rational number? Can you write it in the form, where p and q are integers and?

Ans. Consider the definition of a rational number.

A rational number is the one that can be written in the form of, where p and q are integers and.

Zero can be written as.

So, we arrive at the conclusion that 0 can be written in the form of, where q is any integer.

Therefore, zero is a rational number.

2. Find six rational numbers between 3 and 4.

Ans. We know that there are infinite rational numbers between any two numbers.

A rational number is the one that can be written in the form of, where p and q are integer sand.

We know that the numbers all lie between 3 and 4.

We need to rewrite the numbers in form to get the rational numbers between 3 and 4.

So, after converting, we get.

We can further convert the rational numbers into lowest fractions.

On converting the fractions into lowest fractions, we get.

Therefore, six rational numbers between 3 and 4 are.

3. Find five rational numbers between.

Ans. We know that there are infinite rational numbers between any two numbers.

A rational number is the one that can be written in the form of, where p and q are

Integers and.

We know that the numbers can also be written as.

We can conclude that the numbers all lie between

We need to rewrite the numbers in form to get the rational numbers between 3 and 4.

So, after converting, we get.

We can further convert the rational numbers into lowest fractions.

On converting the fractions, we get.

Therefore, six rational numbers between 3 and 4 are.

4. Show how can be represented on the number line.

Ans. According to the Pythagoras theorem, we can conclude that

.

We need to draw a line segment AB of 1 unit on the number line. Then draw a straight line segment BC of 2 units. Then join the points C and A, to form a line segment BC.

Then draw the arc ACD, to get the numberon the number line.

5. You know that. Can you predict what the decimal expansions ofare, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of carefully.]

Ans. We are given that.

We need to find the values of, without performing long division.

We know that, can be rewritten as.

On substituting value of as, we get

Therefore, we conclude that, we can predict the values of, without performing long division, to get

6. Express in the form. Are you surprised by your answer? Discuss why the answer makes sense with your teacher and classmates.

Ans.

We need to multiply both sides by 10 to get

We need to subtract (a)from (b), to get

We can also write as.

Therefore, on converting in theform, we get the answer as.

Yes, at a glance we are surprised at our answer.

But the answer makes sense when we observe that 0.9999……… goes on forever. SO there is not gap between 1 and 0.9999……. and hence they are equal.

7. Visualize 3.765 on the number line using successive magnification.

Ans. We know that the number 3.765 will lie between 3.764 and 3.766.

We know that the numbers 3.764 and 3.766 will lie between 3.76 and 3.77.

We know that the numbers 3.76 and 3.77 will lie between 3.7 and 3.8.

We know that the numbers 3.7 and 3.8 will lie between 3 and 4.

Therefore, we can conclude that we need to use the successive magnification, after locating numbers 3 and 4 on the number line

8. Visualizeon the number line, up to 4 decimal places.

Ans. We know that the numbercan also be written as.

We know that the numberwill lie between 4.261 and 4.263.

We know that the numbers 4.261 and 4.263 will lie between 4.26 and 4.27.

We know that the numbers 4.26 and 4.27 will lie between 4.2 and 4.3.

We know that the numbers 4.2 and 4.3 will lie between 4 and 5.

Therefore, we can conclude that we need to use the successive magnification, after locating numbers 4 and 5 on the number line.

9. Recall, is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is,. This seems to contradict the fact that is irrational. How will you resolve this contradiction?

Ans. We know that when we measure the length of a line or a figure by using a scale or any device, we do not get an exact measurement. In fact, we get an approximate rational value. So, we are not able to realize that either circumference (c) or diameter(d) of a circle is irrational.

Therefore, we can conclude that as such there is not any contradiction regarding the value of and we realize that the value ofis irrational.

10. Represent 9.3 on the number line.

Ans. Mark the distance 9.3 units from a fixed point A on a given line to obtain a point B such that AB = 9.3 units. From B mark a distance of 1 unit and call the new point as C. Find the mid-point of AC and call that point as O. Draw a semi-circle with centre O and radius OC = 5.15 units. Draw a line perpendicular to AC passing through B cutting the semi-circle at D. Then BD =

11. Find: (i)(ii) (iii)

Ans. (i)

We know that

We conclude thatcan also be written as

Therefore, the value ofwill be 8.

(ii)

We know that

We conclude thatcan also be written as

Therefore, the value ofwill be 2.

(iii)

We know that

We conclude thatcan also be written as

Therefore, the value ofwill be 5.

12. Simplify:

Ans.

The LCM of 3 and 4 is 12

13. Find the two rational numbers betweenand .

Ans. First rational number between and

14. Find two irrational numbers between 2 and 3.

Ans. Irrational number between 2 and 3 is

Irrational number between 2 and is

and are two irrational numbers between 2 and 3.

15. Multiply

Ans.

16. Evaluate

Ans. (i)

(ii)

17. Find rationalizing factor of

Ans.

Rationalizing factor is

18. Rationalize the denominator and subtract it from

Ans.

Difference between and

=

=

=

19. Shove that is irrational

Ans.

(x is a rational number)

X is a rational number 3 is also rational number

but is irrational number which is contradiction

20. Find two rational numbers between 7 and 5.

Ans. First rational number =

Second rational number =

Two rational numbers between 7 and 5 are 6 and

21. Show that is not a rational number.

Ans. let is rational number.

Say è

is a rational number 5 is also rational number

is also rational number

But is irrational number which is a contradiction

is irrational number

22. Simplify

Ans.

23. Evaluate

Ans.

24. Find four rational numbers between

Ans.

and

Take any four rational numbers between and i.e. rational numbers between and are

25. Writ the following in decimal form

Ans.(i)

(ii)

26. Express in the form

Ans.

______(1) [Multiplying both sides by 10]

[Multiplying both sides by 1000]

_______(2)

Subtracting (1) from (2)

27. Multiply by

Ans. and

Or and

LCM of 2 and 3 is 6

=

28. Find the value of if and

Ans.=

29. Convert into rational number

Ans. Let

Multiplying both sides by 100

100=25.252525…

100=

Subtracting (i) from (ii)

30. Simplify

Ans.

=

=

=

31. Simplify

Ans.

3 Marks Questions

1. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.

(ii) Every integer is a whole number.

(iii) Every rational number is a whole number.

Ans. (i) Consider the whole numbers and natural numbers separately.

We know that whole number series is.

We know that natural number series is.

So, we can conclude that every number of the natural number series lie in the whole number series.

Therefore, we conclude that, yes every natural number is a whole number.

(ii) Consider the integers and whole numbers separately.

We know that integers are those numbers that can be written in the form of, where.

Now, considering the series of integers, we have.

We know that whole number series is.

We can conclude that all the numbers of whole number series lie in the series of integers. But every number of series of integers does not appear in the whole number series.

Therefore, we conclude that every integer is not a whole number.

(iii) Consider the rational numbers and whole numbers separately.

We know that rational numbers are the numbers that can be written in the form, where.

We know that whole number series is.

We know that every number of whole number series can be written in the form of as.

We conclude that every number of the whole number series is a rational number. But, every rational number does not appear in the whole number series.

Therefore, we conclude that every rational number is not a whole number.

2. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form, where m is a natural number.

(iii) Every real number is an irrational number.

Ans.

(i)Consider the irrational numbers and the real numbers separately.

We know that irrational numbers are the numbers that cannot be converted in the form, where p and q are integers and.

We know that a real number is the collection of rational numbers and irrational numbers.

Therefore, we conclude that, yes every irrational number is a real number.

(ii) Consider a number line. We know that on a number line, we can represent negative as well as positive numbers.

We know that we cannot get a negative number after taking square root of any number.

Therefore, we conclude that not every number point on the number line is of the form, where m is a natural number.

(iii) Consider the irrational numbers and the real numbers separately.

We know that irrational numbers are the numbers that cannot be converted in the form, where p and q are integers and.

We know that a real number is the collection of rational numbers and irrational numbers.

So, we can conclude that every irrational number is a real number. But every real number is not an irrational number.

Therefore, we conclude that, every real number is not a rational number.

3. Express the following in the form, where p and q are integers and q0.

(i)

(ii)

(iii)

Ans. (i)

We need to multiply both sides by 10 to get

We need to subtract (a)from (b), to get

We can also writeas or.

Therefore, on converting in theform, we get the answer as.

(ii)

We need to multiply both sides by 10 to get

We need to subtract (a)from (b), to get

We can also writeas or.

Therefore, on converting in theform, we get the answer as.

(iii)

We need to multiply both sides by 1000 to get

We need to subtract (a)from (b), to get

We can also write as.

Therefore, on converting in theform, we get the answer as.

4. What can the maximum number of digits be in the recurring block of digits in the decimal expansion of? Perform the division to check your answer.

Ans. We need to find the number of digits in the recurring block of.

Let us perform the long division to get the recurring block of.

We need to divide 1 by 17, to get

We can observe that while dividing 1 by 17 we got the remainder as 1, which will continue to be 1 after carrying out 16 continuous divisions.

Therefore, we conclude that, which is a non-terminating decimal and recurring decimal.

5. Look at several examples of rational numbers in the form (q 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Ans. Let us consider the examples of the form that are terminating decimals.

We can observe that the denominators of the above rational numbers have powers of 2, 5 or both.

Therefore, we can conclude that the property, which q must satisfy in, so that the rational numberis a terminating decimal is that q must have powers of 2, 5 or both.

6. Classify the following numbers as rational or irrational:

(i)

(ii)

(iii)

(iv)

(v)

Ans. (i)

We know that

which is also an irrational number.

Therefore, we conclude thatis an irrational number.

(ii)

= 3.

Therefore, we conclude thatis a rational number.

(iii)

We can cancelin the numerator and denominator, asis the common number in numerator as well as denominator, to get

Therefore, we conclude thatis a rational number.

(iv)

We know that.

We can conclude that, when 1 is divided by, we will get an irrational number.

Therefore, we conclude thatis an irrational number.

(v)

We know that

We can conclude thatwill also be an irrational number.

Therefore, we conclude thatis an irrational number.

7. Simplify each of the following expressions:

(i)

(ii)

(iii)

(iv)

Ans. (i)

We need to apply distributive law to find value of.

Therefore, on simplifying, we get.

(ii)

We need to apply distributive law to find value of.

= 6

Therefore, on simplifying, we get 6.

(iii)

We need to apply the formula to find value of.

Therefore, on simplifying, we get.

(iv)

We need to apply the formula to find value of.

Therefore, on simplifying, we get 3.

8. Find:

(i)

(ii)

(iii)

(iv)

Ans.

(i)

We know that

We conclude thatcan also be written as

= 27

Therefore, the value ofwill be 27.

(ii)

We know that

We conclude thatcan also be written as

= 4

Therefore, the value ofwill be 4.

(iii)

We know that

We conclude thatcan also be written as

= 8

Therefore, the value ofwill be 8.

(iv)

We know that

We conclude thatcan also be written as, or.

We know that

We know thatcan also be written as

Therefore, the value ofwill be.

9. Simplify:

(i)

(ii)

(iii)

(iv)

Ans.

(i)

We know that.

We can conclude that

Therefore, the value ofwill be.

(ii)

We know that

We conclude that can also be written as.

(iii)

We know that

We conclude that.

Therefore, the value ofwill be.

(iv)

We know that.

We can conclude that

Therefore, the value of will be

10. Express 0.8888… in the form p/q.

Ans.

(multiplying both sides by 10)

[subtracting (1) from (2)]

11. Simply by rationalizing denominator

Ans. (Rationalizing by denominator)

12. Simplify

Ans.

13. Visualize 3.76 on the number line using successive magnification.

Ans.

14. Prove that

Ans.

⁺⁺

15. Represent on number line

Ans. in same line

.

16. Simplify

Ans.

=

=

=

=

17. Express in the from

Ans. Let

Multiplying by 10

Multiplying by 1000

18. Simplify

Ans.

19. Find three rational numbers between

Ans. Three rational numbers between and are and

20. Give an example of two irrational numbers whose

(i) sum is a rational number

(ii) product is a rational number

(iii) Quotient is a rational number.

Ans. (i) and

Sum which is a rational number

(ii) and

Product = rational

(iii) and

Quotient = =

21. If

Ans. (Rationalizing denominator)

=

=

=

22. Visualize 2.4646 on the number line using successive magnification.

Ans.

23. Rationalize the denominator of

Ans.

=

=

=

24. Visualize the representation of on the number line up to 3 decimal places.

Ans.

25. Shone that is not rational number

Ans. let is rational number

(is rational)

is rational number and 5 is also rational number.

is rational number

But and is irrational number in same line which is a contradiction

is irrational number

26. Simplify

Ans.

=

=

=

=

27. Simplify

Ans.

=

=

=

=

28. If find the value of

Ans.

29. Find 6 rational numbers between and

Ans. is divided into 10 equal parts than the value of each part

6 rational numbers between and

30. Show how can be represented on the number line

Ans. Take unit

And

In

Take

And

In

Take unit

4 Marks Questions

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Ans.

(i)

On dividing 36 by 100, we get

Therefore, we conclude that, which is a terminating decimal.

(ii)

On dividing 1 by 11, we get

We can observe that while dividing 1 by 11, we got the remainder as 1, which will continue to be 1.

Therefore, we conclude that, which is a non-terminating decimal and recurring decimal.

(iii)

On dividing 33 by 8, we get

We can observe that while dividing 33 by 8, we got the remainder as 0.

Therefore, we conclude that, which is a terminating decimal.

(iv)

On dividing 3 by 13, we get

We can observe that while dividing 3 by 13 we got the remainder as 3, which will continue to be 3 after carrying out 6 continuous divisions.

Therefore, we conclude that, which is a non-terminating decimal and recurring decimal.

(v)

On dividing 2 by 11, we get

We can observe that while dividing 2 by 11, first we got the remainder as 2 and then 9, which will continue to be 2 and 9 alternately.

Therefore, we conclude that, which is a non-terminating decimal and recurring decimal.

(vi)

On dividing 329 by 400, we get

We can observe that while dividing 329 by 400, we got the remainder as 0.

Therefore, we conclude that, which is a terminating decimal.

2. Classify the following numbers as rational or irrational:

(i) 23

(ii) 225

(iii) 0.3796

(iv) 7.478478...

(v) 1.101001000100001...

Ans.

(i)

We know that on finding the square root of 23, we will not get an integer.

Therefore, we conclude thatis an irrational number.

(ii)

We know that on finding the square root of 225, we get 15, which is an integer.

Therefore, we conclude thatis a rational number.

(iii)0.3796

We know that 0.3796 can be converted into.

While, converting 0.3796 intoform, we get

.

The rational numbercan be converted into lowest fractions, to get.

We can observe that 0.3796 can be converted into a rational number.

Therefore, we conclude that 0.3796 is a rational number.

(iv)7.478478….

We know that 7.478478…. is a non-terminating recurring decimal, which can be converted intoform.

While, converting 7.478478…. intoform, we get

While, subtracting (a) from (b), we get

We know thatcan also be written as.

Therefore, we conclude that 7.478478…. is a rational number.

(v)1.101001000100001....

We can observe that the number 1.101001000100001.... is a non-terminating on recurring decimal.

We know that non terminating and non-recurring decimals cannot be converted intoform.

Therefore, we conclude that 1.101001000100001.... is an irrational number.

3. Rationalize the denominators of the following:

(i)

(ii)

(iii)

(iv)

Ans.

(i)

We need to multiply the numerator and denominator ofby, to get

.

Therefore, we conclude that on rationalizing the denominator of, we get.

(ii)

We need to multiply the numerator and denominator ofby, to get

.

We need to apply the formulain the denominator to get

Therefore, we conclude that on rationalizing the denominator of, we get.

(iii)

We need to multiply the numerator and denominator ofby, to get

.

We need to apply the formulain the denominator to get

Therefore, we conclude that on rationalizing the denominator of, we get.

(iv)

We need to multiply the numerator and denominator ofby, to get

.

We need to apply the formulain the denominator to get

Therefore, we conclude that on rationalizing the denominator of, we get.

5 Marks Questions

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Ans.

(i)

On dividing 36 by 100, we get

Therefore, we conclude that, which is a terminating decimal.

(ii)

On dividing 1 by 11, we get

We can observe that while dividing 1 by 11, we got the remainder as 1, which will continue to be 1.

Therefore, we conclude that, which is a non-terminating decimal and recurring decimal.

(iii)

On dividing 33 by 8, we get

We can observe that while dividing 33 by 8, we got the remainder as 0.

Therefore, we conclude that, which is a terminating decimal.

(iv)

On dividing 3 by 13, we get

We can observe that while dividing 3 by 13 we got the remainder as 3, which will continue to be 3 after carrying out 6 continuous divisions.

Therefore, we conclude that, which is a non-terminating decimal and recurring decimal.

(v)

On dividing 2 by 11, we get

We can observe that while dividing 2 by 11, first we got the remainder as 2 and then 9, which will continue to be 2 and 9 alternately.

Therefore, we conclude that, which is a non-terminating decimal and recurring decimal.

(vi)

On dividing 329 by 400, we get

We can observe that while dividing 329 by 400, we got the remainder as 0.

Therefore, we conclude that, which is a terminating decimal.

2. Classify the following numbers as rational or irrational:

(i) 23

(ii) 225

(iii) 0.3796

(iv) 7.478478...

(v) 1.101001000100001...

Ans.

(i)

We know that on finding the square root of 23, we will not get an integer.

Therefore, we conclude thatis an irrational number.

(ii)

We know that on finding the square root of 225, we get 15, which is an integer.

Therefore, we conclude thatis a rational number.

(iii)0.3796

We know that 0.3796 can be converted into.

While, converting 0.3796 intoform, we get

.

The rational numbercan be converted into lowest fractions, to get.

We can observe that 0.3796 can be converted into a rational number.

Therefore, we conclude that 0.3796 is a rational number.

(iv)7.478478….

We know that 7.478478…. is a non-terminating recurring decimal, which can be converted intoform.

While, converting 7.478478…. intoform, we get

While, converting 7.478478 ….. into p/q form, we get

We know thatcan also be written as.

Therefore, we conclude that 7.478478…. is a rational number.

(v) 1.101001000100001....

We can observe that the number 1.101001000100001.... is a non-terminating on recurring decimal.

We know that non terminating and non-recurring decimals cannot be converted intoform.

Therefore, we conclude that 1.101001000100001.... is an irrational number.

3. Rationalize the denominators of the following:

(i)

(ii)

(iii)

(iv)

Ans.

(i)

We need to multiply the numerator and denominator ofby, to get

.

Therefore, we conclude that on rationalizing the denominator of, we get.

(ii)

We need to multiply the numerator and denominator ofby, to get

.

We need to apply the formulain the denominator to get

Therefore, we conclude that on rationalizing the denominator of, we get.

(iii)

We need to multiply the numerator and denominator ofby, to get

.

We need to apply the formulain the denominator to get

Therefore, we conclude that on rationalizing the denominator of, we get.

(iv)

We need to multiply the numerator and denominator ofby, to get

.

We need to apply the formulain the denominator to get

Therefore, we conclude that on rationalizing the denominator of, we get.

4. It and Find the value of

Ans.

5. Find the value of

Ans.

=
=
=
=
=
=

6. Simplify

Ans.

= -18

7. Find a and b if

Ans.
Rationalizing denominator of L.H.S.






Comparing coefficient on both sides, we get
and b =