TOPICS
Exercise - 1.1

Question-1 :-  Using appropriate properties find.
(i) -2/3 x 3/5 + 5/2 - 3/5 x 1/6
(ii) 2/5 x (-3/7) - 1/6 x 3/2 + 1/14 x 2/5

Solution :-
(i) -2/3 x 3/5 + 5/2 - 3/5 x 1/6
=   -2/3 x 3/5 - 3/5 x 1/6 + 5/2        [by commutativity]
=   (-2/3 x 3/5) - (3/5 x 1/6) + 5/2
=   3/5 [-2/3 - 1/6] + 5/2              [by distributivity]
=   -3/5 [2/3 + 1/6] + 5/2
=   -3/5 [(4 + 1)/6] + 5/2
=   -3/5 x 5/6 + 5/2
=   -3/6 + 5/2
=   -1/2 + 5/2
=   (-1 + 5)/2
=   4/2
=   2
(ii) 2/5 x (-3/7) - 1/6 x 3/2 + 1/14 x 2/5
=   2/5 x (-3/7) + 1/14 x 2/5 - 1/6 x 3/2       [by commutativity]
=   [2/5 x (-3/7)] + [1/14 x 2/5] - 1/6 x 3/2 
=   2/5 [-3/7 + 1/14] - 3/12                    [by distributivity]
=   2/5 [(-6 + 1)/14] - 1/4
=   2/5 x (-5/14) - 1/4
=   -2/14 - 1/4
=   (-4 - 7)/28
=   -11/28
    

Question-2 :-  Write the additive inverse of each of the following.
(i) 2/8,     (ii) -5/9,     (iii) -6/(-5),     (iv) 2/(-9),     (v) 19/(-6)

Solution :-
(i) 2/8,  
    Here, -2/8 is the additive inverse of 2/8 because -2/8 + 2/8 = 0

(ii) -5/9, 
    Here, 5/9 is the additive inverse of -5/9 because -5/9 + 5/9 = 0
            
(iii) -6/(-5), 
    Here, 6/(-5) is the additive inverse of -6/(-5) because -6/(-5) + [-6/(-5)] = 0
            
(iv) 2/(-9), 
    Here, -2/(-9) is the additive inverse of 2/(-9) because -2/(-9) + 2/(-9) = 0
            
(v) 19/(-6)
    Here, -19/(-6) is the additive inverse of 19/(-6) because -19/(-6) + 19/(-6) = 0
    

Question-3 :-  Verify that – (– x) = x for.
(i) x = 11/15,     (ii) x = -13/17

Solution :-
(i) x = 11/15 
    The additive inverse of x = 11/15 is -x = -11/15
    Then -11/15 + 11/15 = 0
    Now, - (- x) = - (- 11/15) = 11/15

(ii) x = -13/17
    The additive inverse of x = -13/17 is -x = 13/17
    Then -13/17 + 13/17 = 0
    Now, - (- x) = - [- (-13/17)] = -13/17
    

Question-4 :-  Find the multiplicative inverse of the following.
(i) -13,    (ii) -13/19,    (iii) 1/5,    (iv) -5/8 × (-3/7),    (v) -1 × (-2/5),    (vi) -1

Solution :-
(i) -13    
    Here, 1/(-13) is the mutiplicative inverse of -13 because -13 x 1/(-13) = 1.    [a/b x c/d = 1]     

(ii) -13/19    
    Here, 19/(-13) is the mutiplicative inverse of -13/19 because -13/19 x 19/(-13) = 1.    [a/b x c/d = 1]    
        
(iii) 1/5    
    Here, 5/1 is the mutiplicative inverse of 1/5 because 1/5 x 5/1 = 1.    [a/b x c/d = 1]     
        
(iv) -5/8 × (-3/7) = 15/56  
    Here, 56/15 is the mutiplicative inverse of 15/56  because 15/56 x 56/15 = 1.    [a/b x c/d = 1] 
        
(v) -1 × (-2/5) = 2/5    
    Here, 5/2 is the mutiplicative inverse of 2/5  because 2/5 x 5/2 = 1.    [a/b x c/d = 1]    
        
(vi) -1
    Here, 1/(-1) is the mutiplicative inverse of -1  because -1 x 1/(-1) = 1.    [a/b x c/d = 1] 
    

Question-5 :-  Name the property under multiplication used in each of the following.
(i) -4/5 × 1 = 1 × (-4/5) = -4/5
(ii) -13/17 × (-2/7) = -2/7 × (-13/17)
(iii) -19/29 × 29/(-19) = 1

Solution :-
(i) -4/5 × 1 = 1 × (-4/5) = -4/5
    Here, a x 1 = 1 x a = a in the form.
    So, this is multiplicative identity. 

(ii) -13/17 × (-2/7) = -2/7 × (-13/17)
    Here, a x b = b x a in the form.
    So, this is commutative property. 

(iii) -19/29 × 29/(-19) = 1
    Here, a x 1/a = 1 in the form.
    So, this is multiplicative inverse property.
    

Question-6 :-  Multiply 6/13 by the reciprocal of -7/16.

Solution :-
(i) x = 11/15 
    The additive inverse of x = 11/15 is -x = -11/15
    Then -11/15 + 11/15 = 0
    Now, - (- x) = - (- 11/15) = 11/15

(ii) x = -13/17
    The additive inverse of x = -13/17 is -x = 13/17
    Then -13/17 + 13/17 = 0
    Now, - (- x) = - [- (-13/17)] = -13/17
    

Question-7 :-  Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3.

Solution :-
   Here, a x (b x c) = (a x b) x c in the form.
   So, this is associative property.
    

Question-8 :-  Is 8/9 the multiplicative inverse of -118 .

Solution :-
   If it will be the multiplicative inverse then their product will be 1.
   -118 = -7/8
   8/9 × -7/8 = -7/9 ≠ 1                    [a/b x c/d = 1]
   Hence, 8/9 is not the multiplicative inverse.
    

Question-9 :-  Is 0.3 the multiplicative inverse of 313 ? Why or why not?

Solution :-
   If it will be the multiplicative inverse then their product will be 1.
   313 = 10/3
   also, 0.3 = 3/10
   3/10 × 10/3 = 1                          [a/b x c/d = 1]
   Hence, 0.3 is the multiplicative inverse.
    

Question-10 :-  Write.
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.

Solution :-
(i) 0 is the rational number that does not have a reciprocal.
(ii) 1 and -1 are the rational numbers that are equal to their reciprocals.
(iii) 0 is the rational number that is equal to its negative.
    

Question-11 :-  Fill in the blanks.
(i) Zero has ________ reciprocal.
(ii) The numbers ________ and ________ are their own reciprocals.
(iii) The reciprocal of –5 is ________.
(iv) Reciprocal of 1/x, where x ≠ 0 is ________.
(v) The product of two rational numbers is always a _______.
(vi) The reciprocal of a positive rational number is ________.

Solution :-
(i) Zero has no reciprocal.
(ii) The numbers 1 and -1 are their own reciprocals
(iii) The reciprocal of -5 is -1/5.
(iv) Reciprocal of 1/x, where x ≠ 0 is x.
(v) The product of two rational numbers is always a rational numbers.
(vi) The reciprocal of a positive rational number is positive rational numbers.
    
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