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Question-1

Write the following rational numbers in decimal form:
(i)    

(ii)     

(iii) 3    

(iv)     

(v)     

(vi)     

(vii)     

(viii)

Solution:
(i) = 0.42
    

(ii) = 0.654
   

(iii) 3== 3.375
     

(iv)  = 0.833… = 0.8333
     

(v)  = 0.2
    

(vi)  
    

(vii)  =
     

(viii) =

 

Question-2

If a is a positive rational number and n is a positive integer greater than 1, prove that an is a rational number.

Solution:
We know that product of two rational number is always a rational number. Hence if a is a rational number then
a2 = a x a is a rational number,
a3 = a2 x a is a rational number,

a4 = a3x a is a rational number,
...
... 
an = an-1 x a is a rational number.

Question-3

Find three rational numbers lying between 0 and 0.1. Find twenty rational numbers between 0 and 0.1. Give a method to determine any number of rational numbers between 0 and 0.1.

Solution:
The three rational numbers lying between 0 and 0.1 are 0.01, 0.05, 0.09. 

The twenty rational numbers between 0 and 0.1 are 0.001, 0.002, 0.003, 0.004, … 0.011, 0.012, … 0.099.

To determine any number of rational numbers between 0 and 0.1 insert 0 after the decimal.

Question-4

Complete the following:

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

(ii) The decimal form of an irrational number is neither _____________  or _____________.

(iii) The decimal representation of the rational number is _____________.

(iv) 0 is _____________  number. [Hint: a rational /an irrational]

Solution:
(i) Every point on the number line corresponds to a real number which may be either rational or irrational.

(ii) The decimal form of an irrational number is neither recurring or terminating.

(iii) The decimal representation of the rational number is 0.296

(iv) 0 is a rational number.

Question-5

Which of the following rational numbers have the terminating decimal representation?
(i) 3/5              (ii)7/20                    (iii)2/13
(iv) 27/40        (v) 133/125            (vi) 23/7


[Making use of the result that a rational number p/q where p and q have no common factor(s) will have a terminating representation if and only if the prime factors of q are 2's or 5's or both.]

Solution:
(i) The prime factor of 5 is 5. Hence 3/5 has a terminating decimal representation.

(ii) 20 = 4
× 5 = 22 × 5.
     The prime factors of 20 are both 2's and 5's. Hence 7/20 has a terminating decimal.

(iii) The prime factor of 13 is 13. Hence 2/13 has non-terminating decimal.

(iv) 40 = 23
× 5.
     The prime factors of 40 are both 2's and 5's. Hence 27/40 has a terminating decimal.

(v) 125 = 53
     The prime factor of 125 is 5's. Hence 13/125 has a terminating decimal.

(vi) The prime factor of 7 is 7. Hence 23/7 has a non-terminating decimal representation.

Question-6

You have seen that is not a rational number. Show that 2 +   is not a rational number.

Solution:
Let 2 + be a rational number say r.
Then 2 +
 = r
 = r – 2
But,  is an irrational number.
Therefore, r
 2 is also an irrational number.
 => r is an irrational number.
Hence our assumption r is a rational number is wrong.

Hence, 2 + is not a rational number.

Question-7

Prove that 3 is not a rational number.

Solution:
Let 3be a rational number say r.
Then 3
= r
= (1/3)r
(1/3) r is a rational number because product of two rational number is a rational number.
=>
is a rational number but  is not a rational number.
Therefore our assumption that 3
 is a rational number is wrong.

Question-8

Show that is not a rational number.

Solution:
Let  be a rational number, say  where q 0.
Then =
Since 13 = 1 , and 23 = 8, it follows that 1 <  < 2
Then q > 1 because if q = 1 then  will be an integer, and there is no integer between 1 and 2.
Now, 6 = 
6 = 
6q2
q being an integer, 6q2 is an integer, and since q > 1 and q does not have a common factor with p and consequently with p3.

So, is a fraction different from an integer.
Thus 6q2 .
This contradiction proves the result.

Question-9

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

(i)

(ii) 3

(iii)

(iv)

(v) -

(vi)

Solution:
(i) = 2 is rational.

(ii) 3= 3=3
× 3 9is irrational.
(iii) === 1.2 is rational.
(iv) = ==is irrational.
(v) -- 0.8 is rational.
(vi) = 10 is rational.

Question-10

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

(i) x2 = 5

(ii) y2 = 9
(iii) z2 = 0.04
(iv) u2 = 17/4
(v) v2 = 3
(vi) w3 = 27
(vii) t2 = 0.4

 

Solution:
(i) x2 = 5
x = is irrational.

(ii) y2 = 9

y = 3 is rational.

(iii) z2 = 0.04

z = 0.2 is rational.

(iv) u2 =

u =
      == is irrational.

(v) v2 = 3

v = is irrational.

(vi) w3 = 27
   w = = 3 is rational.

(vii) t2 = 0.4 t = = = is irrational.

Question-11

Give an example to show that the product of a rational number and an irrational number may be a rational number.

Solution:
A rational number 0 multiplied by an irrational number gives the rational number 0.

Question-12

State with reason which of the following are surds and which are not.

(i)
×

(ii) ×

(iii) ×

(iv)
×

(v) 5
× 2

(vi)
×

(vii)
×

(viii) 6
× 9

(ix)
×

(x)
×
.

Solution:
(i) × = × =× × = 5 is a surd.
(ii) × × =2× × × = 4is a surd.
(iii) × × =3× = 9 is not a surd.
(iv) × = 4 × 2 = 8 is not a surd.
(v) 5× 2= 5× 2 = 5 × 2× 2× = 5 × 2 × 2 × 2 × = 40is a surd.
(vi) × = × = 5 × = 5 × 5 = 25 is not a surd.
(vii) × = 10 is a surd.
(viii) 6× 9= 54 is a surd.
(ix) × = ×  = 2 × 3= 2 × × × 3 × = 30 is a surd.

(x)
× = × = × × ×  = 3 × is a surd.

Question-13

Give two examples to show that the product of two irrational numbers may be a rational number.

Solution:
Take a = (2+) and b =(2-); a and b are irrational numbers, but their product
       
(2+)(2-) = 4 - 3 = 1  is a rational number.
Take c = and d = -; c and d are irrational numbers, but their product = -3,
is a rational number.

Question-14

Find the value of correct to two places of decimal.

Solution:
We know that 22 = 4 < 5 < 9 = 32
Taking positive square roots we get
2 <
  < 3.

Next, (2.2)2 = 4.84 < 5 < 5.29 = (2.3)2
Taking positive square roots, we have
2.2 <
  < 2.3

Again, (2.23)2 = 4.9729< 5< 5.0176 = ( 2.24)2
Taking positive square roots, we obtain
2.23 <
 < 2.24
Hence the required approximation is 2.24 as (2.24)2 is nearest to 5 than (2.23)2.

Question-15

Prove that  -  is irrational.

Solution:
Let  -  be a rational number, say r
Then
  -  = r
On squaring both sides we have
   (
 - )2 = r2
  3 - 2
 + 2 = r2 
        5 - 2
 = r2
           - 2
 = r2 - 5
                 
  = -(r2 - 5)/2
Now -(r2 - 5)/2 is a rational number and
 is an irrational number. 
Since a rational number cannot be equal to an irrational number. Our assumption that

  is rational is wrong.

Question-16

Prove that +  is an irrational number.

Solution:
Let  +  be a rational number, say r
Then
 +  = r
On squaring both sides, 
 (
 +  )2 = r2
 3 + 2
 + 5 = r2
       8 + 2
 = r2
            2
 = r2 - 8
              
 = (r2 - 8)/2
Now (r2 - 8)/2 is a rational number and
15 is an irrational number. 
Since a rational number cannot be equal to an irrational number. Our assumption that

 +  is rational is wrong.

Question-17

Examine, whether the following numbers are rational or irrational:
(i) (
2 + 2)2
(ii) (2 -
2) 
× (2 + 2) 
(iii) (2 + 3)2
(iv)

Solution:
(i) (2 + 2)2 = (2)2 + 22x2 + (2)2 = 2 + 42 + 4 = 6 + 42 .
It is an irrational number.

(ii) (2 -
2) × (2 + 2) = (2)2 - (2)2 = 4 - 2 = 2.
It is a rational number.

(iii) (
2 + 3)2 = (2)2 + 22 × 3 + (3)2 = 2 + 26 + 3 = 5 + 26
It is an irrational number.

(iv) = =
2
It is an irrational number.

Question-18

Prove that
(a) 2 +is not a rational number and
(b)
 is not a rational number.

Solution:
(a) If possible, let 2 += a , where a is rational.
Then, (2 + )2 = a2

7 + 4= a2

=  -------(i)
Now, a is rational
is rational.
is rational [from (i)]

This is a contradiction.
Hence, 2 + is not a rational number.

(b) If possible, let = , where p and q are integers,
having no common factors and q 0.
Then, ()3 = ()3
7q3 = p3 ------(i)
p is a multiple of 7
p is multiple of 7.
Let p = 7m, where m is an integer.
Then, p3 = 343 m3 ------(ii)

7q3 = 343 m3 [from (i) and (ii)]
q3 = 49 m3 q3 is a multiple of 7.
q is a multiple of 7.
Thus, p and q are both multiples of 7, or 7 is a factor of p and q.
This contradicts our assumption that p and q have no common factors.

Hence
 is not a rational number.

Question-19

Examine whether the following numbers are rational or irrational:
(i) (3 +)

(ii) (3-)(3+)

(iii)

Solution:
(i) (3 +) = 9 + 2 + 6 = 11 + 6, which is irrational.

(ii) (3 -)(3 + ) = (3) - () = 9 – 3 = 6, which is rational.

(iii)= × = = , which is irrational.

Question-20

Find two irrational numbers lying between  and .

Solution:
Irrational numbers lying between and is ,i.e = 6(1/4)
Irrational numbers lying between and 6(1/4) is = 2(1/4)× 6(1/8)
Hence two irrational numbers lying between and are 6(1/4) and 2(1/4)× 6(1/8).

Question-21

Express as a decimal fraction.

Solution:


Therefore = 0.109375

Question-22

Express as a decimal fraction.

Solution:


Therefore = 0.096.

Question-23

  Express as a decimal fraction.

Solution:
    

Therefore 34.692307




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