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|(1+a,b,c),(a,1+b,c),(a,b,1+c)|=...

`|(1+a,b,c),(a,1+b,c),(a,b,1+c)|=`

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In DeltABC if, |{:(1,b,c),(1,c,a),(1,b,c):}|=0 then sin^(2)A+sin^(2) C is equal to

|(b+c,a,1),(c+a,b,1),(a+b,c,1)|=

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)):}

|(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))|=

Show that |{:(bc,b-c,1),(ca,c+a,1),(ab,a+b,1):}|=(a-b)(b-c)(c-a)

Show that |{:(bc,b+c,1),(ca,c+a,1),(ab,a+b,1):}|=(a-b)(b-c)(c-a)

S.T |(1,a,a^2),(1,b,b^2),(1,c,c^2)| = (a-b)(b-c)(c-a) .

IF Delta_1=|{:(a_1^2+b_1+c_1,a_1a_2+b_2+c_2,a_1a_3+b_3+c_3),(b_1b_2+c_1,b_2^2+c_2,b_2b_3+c_1),(c_3c_1,c_3c_2,c_3^2):}|,Delta_2|{:(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3):}| then find the value of (Delta_1)/(Delta_2)

If a , b, c are real numbers such that a-b=1, b-c=3 , then the number of matrices of the form A[(1,1,1),(a,b,c),(a^(2),b^(2),c^(2))] such that |A|=-12 , is