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If a, b, c are non-coplanar vectors and λ is a real number, then the vectors a + 2b + 3c, λb + 4c and (2λ - 1)c are non coplanar for

1. no value of λ
2. all except one value of λ
3. all except two values of λ
4. all values of λ

Correct answer: all except two values of λ

Solution

The three vectors are non-coplanar iff the determinant of their coefficient matrix [[1,2,3],[0,lambda,4],[0,0,2lambda-1]] = lambda(2lambda-1) is nonzero, which fails only at lambda=0 and lambda=1/2. So they are non-coplanar for all except two values of lambda.

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