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]|`

Consider the determinant Delta = |[a_1+b_1x^2,a_1x^2+b_1,c_1],[a_2+b_2x^2,a_2x^2+b_2,c_2],[a_3+b_3x^2,a_3x^2+b_3,c_3]| = 0 , \ w h e r e \ a_i ,b_i , c_i in R \ (i = 1,2,3) \ a n d \ x in R . Statement 1: The value of x satisfying Delta=0 are x=1,-1. Statement 2: If |[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]|=0,t h e n \ Delta=0.

If w is a complex cube root of unity, then value of =|[a_1+b_1w, a_1w^2+b_1,c_1+b_1 w],[ a_2+b_2w, a_2w^2+b_2,c_2+b_2 w],[ a_3+b_3w ,a_3w^2+b_3,c_3+b_3 w] | is a. 0 b. -1 c. 2 d. none of these

If A_1,B_1,C_1, are respectively, the cofactors of the elements a_1, b_1,c_1, of the determinant "Delta"=|[a_1b_1c_1][a_2b_2c_2][a_3b_3c_3]|,"Delta"!=0 , then the value of |[B_2C_2][B_3C_3]| is equal to a. a1^2Δ b. a_1Δ c. a_1^2Δ d. a1 2^2Δ

If x_i=a_i b_i c_i,i=1,2,3 are three-digit positive integer such that each x_1 is a multiple of 19, then for some integers n , prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]| is divisible by 19.

If (b_2-b_1)(b_3-b_1)+(a_2-a_1)(a_3-a_1)=0 , then prove that the circumcenter of the triangle having vertices (a_1,b_1),(a_2,b_2) and (a_3,b_3) is ((a_2+a_3)/2,(b_(2+)b_3)/2)

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat k and vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a and vec b . If the angle between a and b is pi/6, then prove that |[a_1,a_2,a_3],[b_1,b_2,b_3],[c_1,c_2,c_3]|^2=1/4(a_1 ^2+a_2 ^2+a_3 ^2)(b_1 ^2+b_2 ^2+b_3 ^2)

The total number of infections (one-one into mappings) from {a_1,a_2,a_3,a_4} to {b_1,b_2,b_3,b_4,b_5,b_6,b_7} is

If a_1b_1c_1,a_2b_2c_2 and a_3b_3c_3 are three digit even natural numbers and Delta =|[c_1,a_1,b_1],[c_2,a_2,b_2],[c_3,a_3,b_3]| ,t h e n Delta i s divisible by 2 but not necessarily by 4 divisible by 4 but not necessarily by 8 divisible by 8 none of these

If a_r=(cos2rpi+i sin 2rpi)^(19) , then prove that |[a_1,a_2,a_3],[a_4,a_5,a_6],[a_7,a_8,a_9]|=0 .

Find the value of lambda for which |{:(2a_1+b_1 , 2a_2+b_2 , 2a_3+b_3),(2b_1+c_1, 2b_2+c_2 , 2b_3+c_3),(2c_1+a_1,2c_2+a_2, 2c_3+a_3):}|=lambda|{:(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3):}|