(a)

  (i)  If the system is additive , following relation is true .

  Now if the input is identically zero , that is

  When this is given as input to the given system , we get

  (ii) Now , if the system is homogenous , we proceed as follows :

 (b) Consider a system, which is described by the following relation

   Observe that it has following properties : Neither additive nor homogenous ...

   But if the input is identically zero , sine of the input would also be identically zero .

  (c) NO. It cannot be concluded that if the input is zero for a time interval , the output must also be zero for the
       same interval.

     For example, consider a system which gives an output which is a delayed version of the input :-

                                              y ( t )  =   x ( t  - 1 )

     Now if                              x (t) = 0        3 < t < 4
                                                     = 1        elsewhere

    Then the output will not be zero in the interval (3,4) ...