Question

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

• A. All except two values of $\lambda$
• B. All except one value of $\lambda$
• C. For all values of $\lambda$
• D. No value of $\lambda$

Answer 1

Answer: (A)

All except two values of $\lambda$

Let they are coplanar
$(2\lambda -1)\overrightarrow{c}=\mu (\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c})+\gamma (\alpha \overrightarrow{B}+4\overrightarrow{c})$
$\mu =o\lambda \gamma =o4\gamma =2\lambda -1$
$4\gamma \lambda =(2\lambda -1)\lambda$
$o=(2\lambda -1)\lambda$
$\lambda =0,\lambda =\frac{1}{2}$
So $\lambda =0,\lambda =\frac{1}{2}$ for which equation are coplanar
So $\lambda ϵR-\{0,\frac{1}{2}\}$