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If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\overline{\mathrm{a}}+2 \overline{\mathrm{b}}+3 \overline{\mathrm{c}}, \lambda \overline{\mathrm{b}}+4 \overline{\mathrm{c}}$ and $(2 \lambda-1) \overline{\mathrm{c}}$ are non-coplanar for
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all except two values of $\lambda$
all except two values of $\lambda$
Condition for given three vectors to be coplanar is $\left|\begin{array}{ccc}1 & 2 & 3 \\ 0 & \lambda & 4 \\ 0 & 0 & 2 \lambda-1\end{array}\right|=0 \Rightarrow \lambda=0,1 / 2$
Hence given vectors will be non coplanar for all real values of $\lambda$ except $0,1 / 2$
Hence given vectors will be non coplanar for all real values of $\lambda$ except $0,1 / 2$
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