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.

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)

(b) answer any one question : (i) prove that |[1,a,a^2],[1,b,b^2],[1,c,c^2]| =(a-b)(b-c)(c-a)

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]|

The smallest positive integer n for which ((1+i)/(1-i))^n=1 is (a) 8 (b) 16 (c) 12 (d) None of these

If i=sqrt-1 and n is a positive integer, then i^(n)+i^(n+1)+i^(n+3_ is equal to

Prove without expanding |[a,a^2,bc],[b,b^2,ca],[c,c^2,ab]|=|[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|

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_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot

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.

A function is defined as f:{a_1,a_2,a_3,a_4,a_5,a_6}rarr{b_1,b_2,b_3} then number of functions in which f(a_i)neb_i is divisible by