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Please find below the solution to the asked query

Let us denote the genotype of purple flowered plants as PP and white flowered plants as pp. Then 

(a). ​the genotypes of F₂ progeny will be PP, Pp, Pp and pp. 

This can be expalined by follwoing cross

Parent:                   PP          ×      pp 
                           (purple)​         (white)​
Gametes                  P                   p

F₁:                                  Pp (All purple flowered plants)

Selfing                    Pp       x      Pp
                         
Gametes                 P, p           P, p

F₂ progeny

	P	p
P	PP (Purple)	Pp (Purple)
p	Pp (Purple)	pp (white)

​
Phenotypic Ratio =    3     :         1 
                               (Purple)     (white)

(b).  The different types of plants may be produced by crossing hybrid individuals of F₂ progeny i.e Pp and Pp. This can be explained by punnet square as:

F₂ individuals as Parents               Pp                 x              Pp
                                             (Purpled flowered)        (Purpled flowered)          
Gametes                                           P, p                          P, p

F₃ progeny

	P	p
P	PP (Purple)	Pp (Purple)
p	Pp (Purple)	pp (white)

(c). 
(i). The white flowered plants produced only white flowered F₃ plants because white colour is recessive trait and expresses only in homozygous condition i.e ww. Hence, crossing homozygous parents (ww x ww) produces similar progeny (ww only).

(ii). Some of the purple flowered plants produced only purple flowered F₃ plants because they were homozygous i.e WW. Crossing homozygous parents (WW x WW) produces similar progeny (WW only).

Hope this information will clear your doubts about the topic.
Regards