Question

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

• A. $2$
• B. $0$
• C. $-1$
• D. $3$

Answer 1

Answer: (C)

$-1$

$\left|\begin{array}{ccc}\mu & 1& 1\\ 1& \mu & 1\\ 1& 1& \mu \end{array}\right|=0$
$\mu ({\mu }^2-1)-1(\mu -1)+1(1-\mu )=0$

${\mu }^3-\mu -\mu +1+1-\mu =0$
${\mu }^3-3\mu +2=0$

${\mu }^3-1-3(\mu -1)=0$
$\mu =1,{\mu }^2+\mu -2=0$

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

sum of distinct solutions $=-1$