Loading....
Coupon Accepted Successfully!

 

Question-1

Is zero a rational number? Can you write it in the form, Where p and q are integers and q ≠ 0?

Solution:
Yes.  

0 =  etc.

Also denominator q can be taken as a negative integer

Question-2

Find six rational numbers between 3 and 4.

Solution:
Let us take the 3 and 4 as rational numbers with denominator 6 + 1 = 7
Then the 6 rational numbers between 3 and 4 are

Question-3

Find five rational numbers between and .
Solution:
3/5=0.6 , 4/5=0.8
It can be observed that 0.6<0.61<0.65<0.69<0.7<0.75<0.8
Scince the decimals 0.61,0.65,0.69,0.7 and 0.75are terminating, they are ratinal numbers.
0.61,0.65,0.69,0.7 and 0.75 are five rational numbers lying between 3/5 and 4/5

The five rational numbers between and are  

Question-4

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.

Solution:
(i) True, since the collection of whole number contains all the natural numbers.
(ii) False, for example
– 2 is not a whole number.
(iii) False, for example is a rational number but not a whole number.

Question-5

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.

Solution:
(i) True, since collection of real numbers is made up of rational and irrational numbers.
(ii) False, no negative number can be the square root of any natural number.
(iii)False, For example 2 is real but not irrational.

Question-6

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.

Solution:
No, For example is a rational number.

Question-7

Show how can be represented on the number line.

Solution:
Consider a unit square OABC and transfer in onto the number line making sure that the vertex O coincides with zero.
Then OB = =

Construct BD of unit length perpendicular to OB.

Then OD = =

Construct DE of unit length perpendicular to OD.

Then OE = = = 2

Construct EF of unit length perpendicular to OE.

Then OF = =

Using a compass, with centre O and radius OF, draw an arc which intersects the number line in the point R. Then R corresponds to .


Representation of

Question-8

Write the following in decimal form and say what kind of decimal expansion each has:(i) (ii) (iii)4 (iv) (v) (vi)

Solution:
(i) 0.36. It is a terminating decimal.
(ii) . It is a non terminating repeating decimal.
(iii) 4.125. It is a terminating decimal.
(iv) is a non terminating repeating decimal.
(v) is a non terminating repeating decimal.
(vi) 0.8225 is a terminating decimal.

Question-9

You know that = . Can you predict what the decimal expansions of ,,,, are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of carefully.]

Solution:
Yes, it can be done as follows


 

Question-10

Express the following in the form , Where p and q are integers and q0.
(i) 0.6 (ii)0.47 (iii)0.001

Solution:
  (i) Let x = 0.666 ...........(1)

     .............(2)
               
      (2) - (1) 
    ⇒ 9x = 6

    x =
(ii)
Let x = 
 = 0.47777…..  ...........(1)

Multiplying both sides by 10 (since one digit is repeating), 

we get 10x = 4.7777…….      ............(2)
(2)-(1)  
10x – x = 4.3
9x = 4.3            
x = =

Thus, 
 

Here p = 43, q = 90( 0)

(iii) 
 

Let x = = 0.001001001…    ...........(1)

Multiplying both sides by 1000(since three digits are repeating), 

we get 1000x = 1.001001…        ............(2)       
 (2)-(1)
1000x – x = 1
999x = 1
            
x =

Thus, 
  

Here P = 1, q = 999 ( 0)

Question-11

Express 0.99999…… the following in the form. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Solution:
Let x = 0.99999…        ...............(1)

Multiplying both sides by 10(since one digit it repeating ), we get

10x = 9.9999….            .............(2)

 
(2) - (1) 10x – x = 9 9x = 9x = = 1

Thus 0.99999 …….. = 1 =

Here p = 1, q = 1

Since 0.99999….. goes on for ever, so there is no gap between 1 and 0.99999….. and hence they are equal

Question-12

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

Solution:
The maximum number of digits in the repeating block of digits in the decimal expansion of can be 16.


             
Thus, =
By long Division, the number of digits in the repeating block of digits in the decimal expansion of = 16.
The answer is verified

Question-13

Look at several examples of rational numbers in the form, 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?

Solution:
The property that q must satisfy in order that the rational numbers in the form , where p and q are integers with no common factors other than 1, have terminating decimal representation (expansions) is that the prime factorization of q has only powers of 2 or powers of 5 or both, i.e., q must be of the form 2m × 5n ; m = 0,1,2,3,…….., n = 0,1,2,3,………

Question-14

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

Solution:
0.01001000100001...
0.20200220003200002...
0.003000300003...

Question-15

Find three different irrational numbers between the rational numbers  and .

Solution:


Thus, = 0.714285 …….. =

Thus, = 0.8181…….. =
Three different irrational numbers between the rational numbers and can be taken as

0.75075007500075000075…….
0.767076700767…….
0.808008000800008…..

Question-16

Classify the following numbers as rational or irrational:
(i)  (ii) (iii)0.3796 (iv) 7.478478… (v) 1.101001000100001…

Solution:

      

Thus, = 4.795831523…. The decimal expansion is non terminating non-recurring. is an irrational number.

(ii)


= 15 =


is a rational number.
 
(iii) 0.3796 The decimal expansion is terminating 0.3796 is a rational number

(iv) 7.478478…….
7.478478 …….. = 7. The decimal expansion is non-terminating recurring. 7.478478……….. is a rational number.


(v) 1.101001000100001…………… The decimal expansion is non-terminating non-recurring. 1.101001000100001…… is an irrational number.

Question-17

Visualise 3.765 on the number line, using successive magnification.

Solution:

Question-18

Visualise on the number line, up to 4 decimal places.

Solution:

Question-19

Classify the following numbers as rational or irrational:
(i) 2
(ii) (3 + (iii)  (iv)  (v) 2π

Solution:
(i) 2 - 2 is a rational number and is an irrational number.2 is an irrational number.  The difference of a rational number and an irrational number
is irrational.


(ii) (3 +
(3 +
= 3 + = 3 which is a rational number.

(iii)
= = 0.285714285714...
 which is a rational number.

(iv) 1( 0) is a rational and is an irrational number. is an irrational number
The quotient of a non-zero rational number
with an irrational number is irrational.

(v) 2π 2 is a rational number and π is an irrational number.  The product of a non zero with an irrational number is irrational 2π is an irrational number

Question-20

Simplify each of the following expressions:
(i) (3 + ) (2 +) (ii) (3 + ) (3
) (iii) ( + )2
(iv) ( - )( + )

Solution:
(i) (3 +)(2 +)

(3 +) (2 +) = 3(2 +) + | Left Distributive law of multiplication over
addition
                            = (3)(2) + 3+((2))+ ()()
                            = 6 + 3+2 +        |
  =
                            = 6 + 3+ 2+

(ii) (3+) (3-)(3 +)(3
                            = (3)2 
 ()2
                            = 9
 3 = 6

(iii) ( + )2= ( + )(+
                            = ()2 + 2+()2
                            = 5 + 2 + 2             |
=
                                                  
                            = 7 + 2


(iv) ( -)(+(  (+
                            = ()2 
– (2
                            = 5
 2 = 3

Question-21

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

Solution:
Actually = is only an approximate value of π and also a non-terminating decimal.

Question-22

Represent  on the number line.

Solution:


Mark the distance 9.3 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 mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Then BD = .

Question-23

Rationalise the denominators of the following:
(i) (ii) (iii) (iv)

Solution:
(i) = ×
 = |Multiplying and dividing by


(ii)  
=
| Multiplying and dividing by
= = +


(iii) =× | Multiplying and dividing by
= =

(iv) = × | Multiplying and dividing by 
= =

Question-24

Find:

Solution:
 i)  (64)1/2=(82)1/2

    = 82 ´ ½

    = 81

    = 8


(ii) (32)1/5  = (25)1/5 
              = 25
´ 1/5

    = 21

              = 2

(iii) (125)1/3 = (53)1/3
                = 53
´ 1/3

       = 51

       = 5

Question-25

(i) (ii) (iii) (iv)

Solution:
 
(i) = ×
          =                     |Multiplying and dividing by


(ii) =          | Multiplying and dividing by
                    = = +


(iii)  =× | Multiplying and dividing by
                      = =

(iv) = × | Multiplying and dividing by 
                  = =

Question-26

Represent on the number line. 

Solution:

 

Mark the distance 9.3 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 mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Then BD =   .




Test Your Skills Now!
Take a Quiz now
Reviewer Name