Concept: The question tests the analytical reasoning associated with Mode as a measure of Central Tendency.

Solution:

Mode - is the number that occurs most frequently 

(Highest frequency of occurrence in the list)

St(1)

Let us consider that L1 and L2 together include all the numbers L has

So, if  L= {1, 1, 3, 17, 17, 17, 20, 21,}, 

L1 = {1, 1, 3, 17, 17, 17, 20}  (mode 17 here)and 

L2 = {3, 17, 17, 17, 20, 21} (mode 17 here)


 Also possible that the numbers of L are not included in L1 and L2  

If L = {1, 1, 3, 17, 17, 17, 17, 17, 17, 17, 20, 21,},

L1 = {1, 1, 3, 17, 17, 17} and  (mode 17 here)

L2 = {17, 17, 20, 21} (mode 17 here)

then the underlined elements are left out.

In both cases 17 is the most repeated number with highest frequency and there is no other possible case, thus the mode of L is 17.
Sufficient  

St (2)

We have no information of the numbers and we have already noticed(st 1) that some 17s can be left out.

Thus not sufficient. Hence option(A).