Show that `(a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)`

Factorise (a-2b)^3+(2b-3c)^3+(3c-a)^3

If b is small compared to 'a' show that (c )/((a-b)^(2)) -(c )/((a+b)^(2))=(4bc)/(a^(3)) approximately.

Show that |{:(a-b,b-c,c-a),(b-c,c-a,a-b),(c-a,a-b,b-c):}|=0

Show that |{:(a,b,c),(b,c,a),(c,a,b):}|^2=|{:(2bc-a^(2),c^(2),b^(2)),(c^(2),2ac-b^(2),a^(2)),(b^(2),a^(2),2ab-c^(2)):}|=(a^(3)+b^(3)+c^(3)-3abc)^(2)

If a +b +c=6 , then the value of (2-a) ^3+(2-b) ^3+(2-c) ^3- 3(2-a) (2-b) (2-c) is

Show that |{:(a,b,c),(b,c,a),(c,a,b):}|=|{:(2bc-a^2,c^2,b^2),(c^2,2ac-b^2,a^2),(b^2,a^2,2ab-c^2):}|=(a^3+b^3+c^3-3abc)^2

Match the following from List - I to List - II {:("List-I","List-II"),((I)|{:(1,1,1),(a,b,c),(bc,ca,ab):}|=,(a)(a-b)(b-c)(c-a)),((II)|{:(a,b,c),(a^(2),b^(2),c^(2)),(a^(3),b^(3),c^(3)):}|=,(b)(a-b)(b-c)(c-a)abc),((III)|{:(1,1,1),(a,b,c),(a^(3),b^(3),c^(3)):}|=,(c)(a-b)(b-c)(c-a)(a+b+c)):}

Show that |(b+c,c+a,a+b),(a+b,b+c,c+a)(a,b,c)|=a^(3)+b^(3)+c^(3)-3abc .

Show that |{:(a,b,c),(a^(2),b^(2),c^(2)),(a^(2),b^(3),c^(3)):}|=abc(a-b)(b-c)(c-a)