The sum of the distinct real values of

Question:

The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k}, \quad \hat{i}+\mu \hat{j}+\hat{k}$, $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$ are co-planer, is :

  1. 2

  2. 0

  3. -1

  4. 1

Correct Option: , 3

Solution:

$\left|\begin{array}{ccc}\mu & 1 & 1 \\ 1 & \mu & 1 \\ 1 & 1 & \mu\end{array}\right|=0$

$\mu\left(\mu^{2}-1\right)-1(\mu-1)+1(1-\mu)=0$

$\mu^{3}-\mu-\mu+1+1 \mu=0$

$\mu^{3}-3 \mu+2=0$

$\mu^{3}-1-3(\mu-1)=0$

$\mu=1, \mu^{2}+\mu-2=0$

$\mu=1, \mu=-2$

sum of distinct solutions $=-1$

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