Evaluate: ` /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]| `

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

If p(x),q(x) and r(x) be polynomials of degree one and a_1,a_2,a_3 be real numbers then |(p(a_1), p(a_2),p(a_3)),(q(a_1), q(a_2),q(a_3)),(r(a_1), r(a_2),r(a_3))|= (A) 0 (B) 1 (C) -1 (D) none of these

If a_1,a_2,a_3,………….a_12 are in A.P. and /_\_1 =|(a_1a_5, a_1,a_2),(a_2a_6,a_2,a_3),(a_3a_7,a_3,a_4)|, |(a_2a_10, a_2,a_3),(a_3a_11,a_3,a_4),(a_4_12,a_4,a_5)| then /_\_1:/_\_2= (A) 1:2 (B) 2:1 (C) 1:1 (D) none of these

If (a_2 a_3)/(a_1 a_4) = (a_2 + a_3)/(a_1 + a_4) = 3 ((a_2 - a_3)/(a_1 - a_4)) then a_1, a_2, a_3, a_4 are in

If Delta=|[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]| and A_1,B_1,C_1 denote the co-factors of a_1, b_1, c_1 respectively, then the value of the determinant |[A_1,B_1,C_1],[A_2,B_2,C_2],[A_3,B_3,C_3]| is-

If (1 + ax + b x^2)^4 = a_0 +a_1 x + a_2 x^2 +...+a_8 x^8 when a,b,a_0,a_1,a_2...,a_8 in R such that a_0+a_1+a_2 != 0 and |(a_0,a_1,a_2),(a_1,a_2,a_3),(a_2,a_0,a_1)| then the value of 5a/b (A) 6 (B) 8 (C) 10 (D) 12

The average of a_1, a_2, a_3, a_4 is 16. Half of the sum of a_2, a_3, a_4 is 23. What is the value of a_1