If the vectors `a=hat(i)+a hat(j)+a^(2) hat(k), b=hat(i)+b hat(j)+b^(2) hat(k)` and `c=hat(i)+chat(j)+c^(2) hat(k)` are three non-coplanar vectors and `|(a,a^(2),1+a^(3)),(b,b^(2),1+b^(3)),(c,c^(2),1+c^(3))|=0`, then the value of abc is

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

D

Explanation

Since, a, b and c are non-coplanar vectoprs therefore `[abc]ne0`
`rArr" "Delta=|{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|ne0rArrDeltane0`
`"Now, "|{:(1,a^2,1+a^3),(1,b,1+b^3),(1,c^2,1+c^3):}|=0`
`rArr" "|{:(1,a^2,1),(1,b,1),(1,c^2,1):}|+|{:(a,a^2,a^3),(b,b^2, b^3),(c,c^2,c^3):}|=0`
`rArr" "Delta(1+abc)=0rArr" "abc=-1[because Delta=ne]`