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

a¯+2b¯+3c¯,λb¯+4c¯,(2λ−1)c¯ are non coplanar vectors ⇒ 1230λ400(2λ−1)≠0⇒λ(2λ−1)≠0⇒λ≠0,12∴All values except, 2 values of λ

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

a¯+2b¯+3c¯,λb¯+4c¯,(2λ−1)c¯ are non coplanar vectors ⇒ 1230λ400(2λ−1)≠0⇒λ(2λ−1)≠0⇒λ≠0,12∴All values except, 2 values of λ

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

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All values of $λ$

All except three values of $λ$

All except two values of $λ$

No value of $λ$

answer is C.

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Detailed Solution

$\bar{a} + 2 \bar{b} + 3 \bar{c}, λ \bar{b} + 4 \bar{c}, (2 λ - 1) \bar{c} are non coplanar vectors \Rightarrow |\begin{array}{ccc} 1 & 2 & 3 \\ 0 & λ & 4 \\ 0 & 0 & (2 λ - 1) \end{array}| \neq 0$
$\Rightarrow λ (2 λ - 1) \neq 0 \Rightarrow λ \neq 0, \frac{1}{2} ∴$ All values except, 2 values of $λ$