If `A_1B_1C_1 , A_2B_2C_2 and A_3B_3C_3` are three digit numbers, each of which is divisible by k, then `Delta = |(A_1,B_1,C_1),(A_2,B_2,C_2),(A_3,B_3,C_3)|` is

If a_1b_1, c_1,a_2b_2c_2a n da_3b_3c_3 are three digit even natural numbers and =|c_1a_1b_1c_2a_2b_2c_3a_3b_3|,t h e ni 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_(i),B_(i),C_(i) are the cofactors of a_(i),b_(i),c_(i) respectively,i=1,2,3 in Delta =|(a_(1),b_(2),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| show that |(A_(1),B_(1),C_(1)),(A_(2),B_(2),C_(2)),(A_(3),B_(3),C_(3))|=Delta^(0)

(a) If A=((4,2),(-1,x)) and such that (A-2I)(A-3I)=0. find the value of x. (b) If A_(i),B_i,C_(i) are the cofactors of a_(i),b_(i),c_(i) , respectively, i=1 to 3 in |A|=|(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(3)),(a_(3),b_(3),c_(3))| , show that |(A_(1),B_(1),C_(1)),(A_(2),B_(2),C_(2)),(A_(3),B_(3),C_(3))|=|A|^(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 ; vec c= c_1hat 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 & vec b . If the angle between vec a and vec b is pi/6 , then |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2=

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 A_(1) ,B_(1),C_(1) ……. are respectively the cofactors of the elements a_(1) ,b_(1),c_(1)…… of the determinant Delta = |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}|, Delta ne 0 then the value of |{:(B_(2),,C_(2)),(B_(3),,C_(3)):}| is equal to

The area of the triangle whose vertices are A(1,-1,2),B(2,1-1)C(3,-1,2) is …….

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)