If ` vec u ,"" vec v , vec w` are noncoplanar vectors and p, q are real numbers, then the equality `[3 vec u ,""p vec v , p vec w]-[p vec v ,"" vec w , q vec u]-[2 vec w ,""q vec v , q vec u]=0` holds for (1) exactly one value of (p, q) (2) exactly two values of (p, q) (3) more than two but not all values of (p, q) (4) all values of (p, q)

Options

A. exactly two values of (p,q)
B. more than two but not all value of (p,q)
C. all values of (p,q)
D. exactly one value of (p,q)

Answer

D

Explanation

Since, `[3upvpw]`
`-[pvwqu]-[2wqvqu]=0`
`therefore3p^(2)[u*(vxxw)]-pq[v*(wxxu)]`
`-2q^(2)[w*(vxxu)]=0`
`implies(3p^(2)-pq+2q^(2))[u*(vxxw)]=0`
but `[uvw]ne0`
`implies(3p^(2)-pq+2q^(2))=0`
`impliesp=q=0`