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Question-1

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.
... an = an-1 x a is a rational number.

Question-2

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…=

     

(v) = 0.2

    

(vi) =

    

(vii) =

     

(viii) =

       

Question-3

Give three rational numbers lying between and .

Solution:
The rational number lying between is and .

= = =

Therefore, < < 
Now, the rational number lying between and is

= =

Therefore, < < 

The rational number lying between and is

===

Therefore, < < 

Hence the three rational numbers lying between and  are .
 

Question-4

Insert a rational and an irrational number between 2 and 3.

Solution:
A rational number between 2 and 3 = = 2.5

An irrational number between 2 and 3 is .

Question-5

How many rational numbers and irrational numbers can be inserted between 2 and 3 ?

Solution:
There are infinite number of rational and irrational numbers between 2 and 3.

Question-6

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 the square root of its product.

i.e. The rational numbers between a and b is

Question-7

Find three rational numbers lying between

Solution:
The rational number lying between - and -

==

Therefore, -<- <- 

The rational number lying between -   and-

 ===

Therefore, -< <- 

The rational number lying between - and -

===

Therefore, -< <- 

Therefore the three rational numbers are

Question-8

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 ……………… nor ……………………

(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 nor
terminating.

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

(iv) 0 is a rational number. 

Question-9

Give an example for each, if two irrational numbers, whose

        (i) difference is a rational number.
        (ii) difference is a irrational number.
        (iii) sum is a rational number.
        (iv) sum is an irrational number.
        (v) product is an irrational number,
        (vi) product is an irrational number.
        (vii) quotient is a rational number.
        (viii) quotient is an irrational number.

Solution:

Question-10

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 x 5 = 22 x 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 x 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 133/125 has a terminating decimal.

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

Question-11

Find the decimal representation of 1/7, 2/7. Deduce from the decimal representation of 1/7, without actual calculation, the decimal representation of 3/7, 4/7, 5/7 and 6/7.

Solution:

Question-12

Express 0.6666………in the form of

Solution:
Let x = 0.6666……….                          …(i)

10 x =                  ...(ii)

(ii) – (i), 9x = 6                  x ==

 
0.6666 =

Question-13

If a and b are two rational numbers, prove that a + b, a - b, ab are rational numbers. If b 0, show that a/b is also a rational number.

Solution:
Let a = p/q where q 0 and b = r/s where s 0 be the rational numbers then

(i) a + b ==  

where q s
0, since q 0 and s 0.

Also p s + r q is an integer.

Hence a + b is a rational number.

(ii) a - b =
=

where q s 0, since q 0 and s 0

Also p s - r q is an integer

Hence a - b is a rational number.

(iii) ab =

where q s 0, since q 0 and s 0.

Also pr is an integer.
Hence ab is a rational number.


(iv) Since b
0, we have r/s 0 thus r 0 and s 0.


where q
0 and r 0.
Also ps is an integer
Hence a/b is a rational number.

Question-14

Express 0.272727………in the form of .

Solution:
Let x = 0.272727……….             …(i) 100x = 27.2727.....         …(ii)

(ii) – (i), 99x = 27                x = =
 
0.272727………=

Question-15

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

Solution:
Let 2 + 2 be a rational number say r.
Then 2 +
2 =  r
2 =  r -2
But,
2 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.

Question-16

Express 3.7777………. in the form of .

Solution:
      Let x =  3.7777.……….              …(i)

  10x = 37.               …(ii)

(ii) – (i), 9x = 34             x = 3.7777……… =

Question-17

Prove that 33 is not a rational number.

Solution:
Let 33 be a rational number say r .
Then 3
3 =  r
3 = (1/3) r
(1/3) r is a rational number because product of two rational number is a rational number.
=>
3 is a rational number but 3 is not a rational number.
Therefore our assumption 3
3 is a rational number is wrong.

Question-18

Express 18.484848…………….in the form of .

Solution:
            Let x   =  18.484848.……….........................…(i)      100 x  = 1848.4848………....................(ii)

(ii) – (i), 99 x = 1830  x = =   18.484848………=

Question-19

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-20

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
× 39 is irrational.

(iii) === 1.2 is rational.

(iv) = ==is irrational.

(v) - = -0.8 is rational.

(vi)= 10 is rational.

Question-21

Find two irrational numbers between 2 and 2.5.

Solution:
The two irrational numbers between 2 and 2.5 are 2.101001000100001….. and 2.201001000100001…..

Question-22

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
         (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-23

Find two irrational numbers between 0.1 and 0.12.

Solution:
The two irrational numbers between 0.1 and 0.12 are 0.1010010001… and 0.1101001000100001……

Question-24

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-25

Give two irrational number lying between 2 and 3.

Solution:
The two irrational number lying between 2 and 3 are 2.1 and 2.2

Question-26

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) × = × =× × =5is a surd.

(ii) × =× = 2 ×××
                                         = 4 is 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 ×
                                                = 40 is 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-27

 Express the following recurring decimals into vulgar fractions :

Solution:
(a) = 0.666….      .........(1)
10 ×   = 6.666…    ........ (2)
     (2) - (1) ⇒9 ×  = 6
  = 6/9 = 2/3  


(b) = 0.161616….. ---(1)
100 ×   = 16.1616….. -----(2) 
Subtracting (1) from (2) , we get
      99 ´   = 16
   = 16/99  


(c) 0.  = 0.234234….. ----(i)
1000 × 0. = 234.234234… -----(ii) 
Subtracting (i) from (ii) , we get
999 × 0.  = 234
0.  = 234/999.  


(d)   = 0.125454….                  -----(i)
100 ×   = 12.545454….        -----(ii)
And, 10000 ´   = 1254.5454… -----(iii)
Subtracting (ii) from (iii) , we get
9900 ×   = 1242
  = 1242/9900 = 69/550.

Question-28

Find the values of a and b if = a + b

Solution:

= × [Multiplying the numerator and the denominator by (-1)]

          = == 2 -

It is given that 2 -= a + b

a = 2 and b = -1.

Question-29

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 = 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-30

Find the values of a and b if = a + b

Solution:

= ×    [Multiplying the numerator and the denominator by (3 +)]

        =

         
=

It is given that = a + b
       

a = and b =

Question-31

Find the value of 5 correct to two places of decimal.

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

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

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

Question-32

Find the values of a and b if = a + b

Solution:
=× [Multiplying the numerator and the denominator by 7 - 4]

            =

            =

It is given that = a + b
         

a = 11 and b = -6.

Question-33

Prove that 3 - 2 is irrational.

Solution:
Let 3 - 2 be a rational number, say r

Then
3 - 2 =   r 

On squaring both sides we have
    (
3 - 2)2  r 2
  3 - 2
6 + 2 =   r 2 
        5 - 2
6 =  r 2
           - 2
6 =  r 2 - 5
              
6 = -(  r 2 - 5)/2

Now -(
 r 2 - 5)/2 is a rational number and 6 is an irrational number. 
Since a rational number cannot be equal to an irrational number. Our assumption that

3 - 2 is rational is wrong. 

Question-34

Find the values of a and b if = a + b

Solution:

= × [Multiplying the numerator and the denominator by 5 +]

          =

          = 

          =


It is given that = a + b

a = and b =

Question-35

Prove that 3 + 5 is an irrational number.

Solution:
Let 3 + 5 be a rational number, say r
Then
3 + 5 =   r 
On squaring both sides, 
     (
3 + 5)2  r 2
 3 + 2
15 + 5 =  r 2 
       8 + 2
15 =   r 2
             2
15 =  r 2 - 8
              
15 = (  r 2 - 8)/2
Now (
 r 2 - 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

3 + 5 is rational is wrong. 

Question-36

Find the values of a and b if = a + b

Solution:
   = × [Multiplying the numerator and the denominator by 3 + 4]

            =

            =


It is given that = a + b

                        = a+ b

                              a = and b =
 

Question-37

If -= a + b, find the values of a and b.

Solution:
-=

                     =

                     =

It is given that -= a + b

                     = a + b

                     a 0 and b

Question-38

Give two rational numbers lying between 0.232332333233332… and 0.21211211121111…..

Solution:
The two rational numbers are 0.222. and 0.221
 

Question-39

Simplify by rationalising the denominator:

Solution:
=× [Rationalising the denominator]

            ==

Question-40

Examine, whether the following numbers are rational or irrational:
          (i) (
2 + 2)2
       (ii) (2-
2)x(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)x(2 + 2) = (2)2 - (2)2 = 4 - 2 = 2 .
      ...  It is a rational number.

(iii) (
2 + 3)2 = (2)2 + 22x3 + (3)2 = 2 + 26 + 3 = 5 + 26
      ...  It is an irrational number.

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

Question-41

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

Solution:
If possible, let 2 + = a , where a is rational.
         Then, (2 + ) = a
                    7 + 4 = a
                           =-------(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 = p/q , where p and q are integers, having no common factors and q 0.
Then, () = (p/q
)
7q = p------(i)
p is a multiple of 7
p is multiple of 7.
Let p = 7m, where m is an integer.
Then, p = 343 m ------(ii)
   7q = 343 m [from (i) and (ii)]
    q = 49 m
q 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-42

Simplify by rationalising the denominator:

Solution:
=× [Rationalising the denominator]

            ===

Question-43

Simplify by rationalising the denominator:

Solution:
=× [Rationalisation the denominator]
 

               =

 

               =
 

               =
 

               =

Question-44


Examine whether the following numbers are rational or irrational:
(i)     ,(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-45

Simplify by rationalising the denominator:

Solution:
=×    [Rationalising the denominator]

            = =

            =
            = 17 - 12

Question-46

Find three irrational numbers between 2 and 2.5 .

Solution:
If a and b are any two distinct positive rational numbers such that ab is not a perfect square, then the irrational number lies between a and b.
Irrational number between 2 and 2.5 is , i.e

        Irrational number between 2 and is =  2(1/2) × 5(1/4)

 Irrational number between and 2.5 is   =  
Thus, the three irrational numbers between 2 and 2.5 are , 2(1/2) × 5(1/4)and (1/2) × 5× 2.

Question-47

Simplify by rationalising the denominator:  

Solution:
=× [Rationalising the denominator]

             =

            =

            =

            = ()

Question-48

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-49

Simplify by rationalising the denominator: +

Solution:
+=

 

                      =
 

                      =

Question-50

Express as a decimal fraction.

Solution:
0.109375

    
600
     576
     240
     192
     480
     448
     320
     320  
       0

Therefore = 0.109375

Question-51

Simplify by rationalising the denominator: -

Solution:
-=

                      =

                       =
                              =

Question-52

Express as a decimal fraction.  

Solution:
  

Therefore = 0.096  

Question-53

Express as a decimal fraction.

Solution:
0.111

   10
    9
   10
    9
    1

Therefore = 0.111

Question-54

Simplify by rationalising the denominator: +

Solution:
+=

                      =

                      =

                      =

                      =

Question-55

Represent 0. in the form of .

Solution:
     = 0. ……………(i)

100 = 57. ……………(ii)

(ii) – (i)
99 = 57
Therefore = 57/99

Question-56

Simplify by rationalising the denominator: -

Solution:
-=

 

                      =
 

                      =

                      = -8

Question-57

Simplify by rationalising the denominator:

Solution:
       = ×

 

                  =
 

                  =
 

                  =
 

                  =[Rationalising the denominator]
 

                  =
 

                  = ()

Question-58

 Represent 0.2 in the form of .

Solution:
= 0.2 ............(1)

10000= 2341.2 ..............(2)

(2)-(1) ⇒ 9999= 2341
Therefore = 2341/9999  

Question-59

Simplify by rationalising the denominator:

Solution:
=×
 

                  =
 

                  =
 

                  =

                  =
×
 

                  =
 

                  = (+ +)
 

                  = (3+2+)

Question-60

Which of the following is surds:
          (i)
          (ii)
          (iii) 3÷ 6
         

Solution:
 
 

Question-61

Express as a pure surd:
          (i)
(ii) (iii) 

          (iv) (v) (vi)

Solution:
(i) = = =

(ii) =

(iii) =

(iv) ===  

(v) = ==

(vi) == =

Question-62

Express as a mixed surd in its simplest form:
          (i) (ii) (iii)

          (iv) (v) (vi)

Solution:
(i) = = =2x2x = 4

(ii) = = =

(iii) = = =

(iv) = = = 3x5x=15 

(v) == =  

(vi) = = = 5x3 = 15.

 

Question-63

Which is greater?
         
(i) or 

          (ii) or

          (iii) or

          (iv) or

          (v) or

          (vi) or

Solution:
(i) L.C.M. of 2 and 3 is 6.

Thus, = =

And = = 

Therefore >

Hence, >

(ii) L.C.M. of 1 and 4 is 4.

Thus, = 

and = 

Therefore >

Hence, >

(iii) L.C.M. of 4 and 3 is 12.

Thus, = =

and ==

Therefore >

Hence, >

(iv) L.C.M. of 3 and 4 is 12.

Thus, == 

and = =

Therefore >

Hence, >

(v) L.C.M. of 8 and 4 is 8.

Thus, =

and = =

Therefore >

Hence, >

(vi) L.C.M. of 3 and 4 is 12

Thus, = =

and = =

Therefore >

Hence, >.

Question-64

Arrange in descending order of magnitude
          (i)

          (ii) ,,

          (iii) , ,

Solution:
(i) L.C.M. of 3, 4 and 2 is 12

Thus, = =

=  

= =

Therefore   >

Hence, > > 

(ii) L.C.M of 3,4 and 3 is 12
Thus,= = 

==  

= 

.. > >

Hence, > >

(iii) L.C.M. of 4,3 and 2 is 12

Thus, == 

= 

=   =   

Therefore > >

Hence, >>.

Question-65

Simplify by combining similar terms:
          5
+20

Solution:
 5+20 = (5+20) = 25

Question-66

Simplify by combining similar terms:
         
+

Solution:
 Reducing into simplest form
==
3

Therefore 2 += 2+3=(2+3) = 5.

Question-67

Simplify by combining similar terms:  
           4-3+2

Solution:
 Reducing into simplest form
= =2


= = 5

Therefore 4- 3 + 2= 4- 3 x 2 + 2 x 5 
                                            =
4-6 + 10
                                            = (4-6+10)
                                            = 8 .

Question-68

Simplify by combining similar terms:
          +-

Solution:
Reducing into simplest form.
= = 2

  = = 2 x 2
= 4

..+ - =2 +4 - =(2 + 4 - 1) =5.

Question-69

Simplify by combining similar terms:
          
- 3 +4

Solution:
Reducing into simplest form
= = 3

= = 2


... - 3 + 4= 3 - 3 x 2 + 4 

                                = 3
- 6 + 4 

                                =

Question-70

Simplify by combining similar terms:
          4--7

Solution:
Reducing into simplest form
= = 2

= =5

= = 2x2 = 4

..
4- - 7 = 4x 2 -5 - 7 x 4

                            = 8 - 5 -28 

                            =  (8-28) - 5

                            = -20 - 5

Question-71

Simplify by combining similar terms:
          +-

Solution:
Reducing into simplest form
= =

= = 

... + -= 2+ 7x2 -  

                                 = 2+ 14-  

                                 = (2+14 -5)

                                 = 11

Question-72

Simplify by combining similar terms:
         
+ -

Solution:
Reducing into simplest form
==2 

== 5

= =2x2=4 

... 2 +3 - = 2+3x5-4x4 

                                       = 4 +15-16

                                       = (4 +15 -16)

                                       = 3

Question-73

Simplify by combining similar terms:
          3 - +

Solution:
Reducing into simplest form
= = 7 

   = = 

... 3 - + = 3 x 7 - + 7 x

                                     = 21-

                                     = 

                                     =

                                     = .

Question-74

Simplify and express the result in its simplest form:
          x

Solution:
x = ==
                                                    = .

Question-75

Simplify and express the result in its simplest form:
          x

Solution:
x ==.

Question-76

Simplify and express the result in its simplest form:
          x

Solution:
x= ===2.

Question-77

Simplify and express the result in its simplest form:
          x

Solution:
x=28=28= 28x6
                                                  = 168.

Question-78

Simplify and express the result in its simplest form:
         
x

Solution:
L.C.M. of 3, 2 is 6
==

==

x=x= =

 

Question-79

Simplify and express the result in its simplest form:
         
x

Solution:
L.C.M. of 3, 4 is 12

= =

= =

x = x = =

Question-80

Simplify and express the result in its simplest form:
          ÷ 3

Solution:
÷ 3  =  

= = =
.

Question-81

Simplify and express the result in its simplest form: 
          (1) ÷ x
          (2) x x

Solution:
1) L.C.M of 2 and 3 is 6
          = =

          = =

        
x x ==

         ÷ (x )=÷ = = =

      
2) L.C.M. of 2,3 and 4 is 12
          =     

          =  

          = ==

         
x x x x =

                                 = = = =    

Question-82

Write the simplest rationalising factor of   
          (i)          (ii)       (iii)         (iv)        (v)

Solution:
(i) 2 x = 2 x = 4

      ... is the simplest rationalising factor of 2

(ii) x = = = 10

        ..  is the simplest rationalising factor of

(iii) = ==

         x = = 5x3 = 15
          ... is the simplest rationalising factor of

(iv) x = =

                         = 2x5=10
         ... is the simplest rationalising factor of  

(v) =

        Now x == 6
        ...is the simplest rationalising factor of .

Question-83

Express with a rational denominator the following surds:
          (i)       (ii)      (iii)    (iv)     (v) (vi) 

Solution:
(i)

The simplest Rational factor of isitself.


... x

(ii)

The simplest R.F of is itself.

...x =

(iii)

=

   The simplest R.F of is itself.


..=

(iv)
   The simplest R.F of  is itself.


... x

 (v) The simplest R.F of is itself.
   


 (vi) The simplest R.F of is itself as

... = x=

Question-84

(a) Which is greater or   ? 
(b) Arrange in ascending order of magnitude , , .

Solution:
(a) L.C.M of 2 and 3 is 6
      Thus
=

            

           Now      

         

 (b) 
L.C.M of 4,3, and 2 is 12
       Thus, 
==

              ==

              = =

       
Now,  < <

                 ...
< < .

Question-85

Find the value to three places of decimals, of each of the following. It is given that= 1.414, = 1.732,  = 3.162 and = 2.236 (approx.)   (i) (ii)   (iii)   (iv)   (v)    (vi)

Solution:
(i)
    = = = 0.707

(ii)
      = x = 0.5773= 0.577

(iii)
      = = = 0.3162

(iv)
     = = = = =1.0796 = 1.080

(v)
     = x = = == = 0.1546= 0.155

(vi)  = x =

              = = == 0.655

 

Question-86

If both of a and b are rational numbers, find the values of a and b in each of the following equalities:
          (i) 
    (ii) (iii)

       
(iv) 

Solution:
(i) -1 is the conjugate surd of  +1
  \

         

       

        


         2 - a + b
 

      Hence, a = 2 , b = -1

 (ii)  3 +
is the conjugate of 3-

   
\    =

                        =

                        =

On comparing both sides
         


We have a =  and b =

 
(iii) 7-4
is the conjugate of 7 + 4

           
     


                  =    

                                   = 

                           
   = 35 -20

On comparing both sides
        
11-6

We have,  a
= 11, b = -6.

(iv)  is the conjugate of the denominator 3√2 - 2√3

       \

                                 =


                         
=

                        
= 2 +
On comparing both sides
       2 +

we have,  a
= 2 and b = -

Question-87

Simplify each of the following by rationalizing the denominator:
           (i) (ii)   (iii)    

Solution:
 (i) \

        5+ is the rationalizing factor of 5 -  

                  =

 
(ii) is the rationalizing factor of

        \

                       = .
(iii) 2 is the rationalizing factor of
                    

                     


                     
      =

Question-88

Rationalize the denominator of  
          (i)
(ii)  (iii)  

Solution:
(i) The denominator is a trinomial surd.  
      We proceed as with a binomial surd by grouping two of the terms. Thus,


         

                            
                            
  = 9 + 6 5 + 5 - 8
                               = 6 + 6
5
                               = 6(1+
5)

                            
    \   
  

                  

                  


                  
=

                   =


                  
=

 
(ii) The denominator is a trinomial surd.  
    
We proceed as with a binomial surd by grouping two of the terms. 

    Thus,  


                                    
= 3              

                

                                   

                                  
  
                     
                                   
          

 (iii) The denominator is a trinomial surd.  
      
We proceed as with a trinomial surd, by grouping two of the terms.

   Thus, 

                               
     
 
                                   

    \  

                        


                            

 

Question-89

Simplify each of the following:
          (i)     (ii)

Solution:
(i) Let us rationalise the denominator of each term:
   
   

                
                                                                            

 

                        \

(ii) Let us rationalize the denominator of each term:
 

    

        

        


   \

Question-90

Taking  and  (approx.). Find the value to three places of decimals of each of the following: (i)  (ii)     

Solution:
(i)  =

                                 =

                                 =

                                 =  

                                 =

                                 = (2.236 - 2.449)

                                 = (-0.213)

                                 = - 0.213

(ii)   
   
                                 
       

                            
                            

   
                 
           = 14.268
 
              
          
                                             = 16 - 1.732

                                             = 14.268.

Question-91

Simplify (a) + -   (b)

Solution:
(a) = =
        
= =
   
... + - = +-    = (1+2-3)   = 0

(b) = =
      = =
   
  ... = x   = = 10 x 2 =20  

Question-92

Given that = 1.7321, find correct to 3 places of decimals, the value of  

Solution:
  = = 2 x 2 x 2= 8
   = = 2 x =
    =   =
  
  = x =
                               = (8-2-5)
= = 1.732.

Question-93

Simplify by rationalizing the denominator :
          (i)               (ii)

Solution:
 (i)

(ii)  =
                                   

Question-94

If   = 2.236 (approx.), evaluate  correct to three places of decimals.

Solution:

Question-95

Simplify :

Solution:

Question-96

 Simplify : (i)   (ii)

Solution:

(ii) =

                             =

Question-97

Express each of the following as a mixed surd in the simplest form:

 

Solution:
 

Question-98

Express each of the following as a mixed surd in the simplest form:
          (i) (ii) 5

Solution:
(i) = = × = =

(ii) 5= 5
              = 5 × 3 × = 15 ×

Question-99

Arrange in descending order of magnitude:
          , and .    

Solution:
The given surds are of order 3,6 and 9 respectively.
The L.C.M. of 3, 6 and 9 is 18.
Reducing each surd to a surd of order 18,

= = =

= = =   

= = =

Hence , > >

              > > .

Question-100

Simplify : +             

Solution:
Reducing to the simplest form, we get
= = a.

= = b.

+ = a + b
                      
                       = (a+b)

Question-101

Simplify :
            - 8+ 15+  

Solution:
- 8 + 15 +
 

= = 3 ;
 

= = 6 ;
 

= = 2 ;
 

= = 15 ;
 

- 8 + 15 + = (3 – 8 × 6 + 15 × 2 + 15)

                                                = (3 – 48 + 30 + 15) = 0

Question-102

Simplify:
           (3 - 5)(4 + 3)

Solution:

 

Question-103

Simplify: 
           ( ÷ )

Solution:
L.C.M of 6,2 and 3 is 6. 
( ÷ ) = 

Question-104

Find the rationalising factor of

Solution:
= ´ ´ =
() ´ () = abc
Hence the required R.F = =

Question-105

Find the rationalising factor of

Solution:
= = × = 2´ = 2 × 5(1/5)
2 × 5(1/5) × 5(4/5) = 2 × 5 = 10
Hence the required R.F is 5(1/5) = =

Question-106

Convert and into surds of the same but smallest order.

Solution:
Since = 31/4 and = 21/6, it follows that the given surds are of order 4 and 6 respectively. The L.C.M. of 4 and 6 is 12.So,we shall convert each one of the given surds into a surd of order 12.

 = 31/4 = 3(1/4) (3/3) = 3(3/12) = ( 33)1/12 = (27)1/12 =

= 21/6 = 2(1/6) (2/2) = 2(2/12) = (22)1/12 = (4)1/12 =

Question-107

If = 2.236 and = 3.162, evaluate

Solution:
The rationalising factor of is .
= × =

                                         =

                                         - (1/2) = 2.236 –(1/2)(3.162)

                                         = (2.236 – 1.581) = 0.655

Question-108

If = a - b, find the values of a and b.

Solution:
= ×

                =

           =

           =      

Question-109

Express with a rational denominator.

Solution:
= ×

                     = × =
 

                     =× =
 

                     = =

Question-110

Find the value of correct to three places of decimal, it being given that = 1.4142.

Solution:
= × =

                    = =

                   = ×

                   = =

                  =




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