HCU MCA 2013 Paper

(a) Let $$p = a i + j + k ,
$$q = i + b j+ k
and $$r = i + j + ck
If these vectors are coplanar, then
$$[p q r] = 0
$\left|\begin{array}{ccc}\mathit{a}& 1& 1\\ 1& \mathit{b}& 1\\ 1& 1& \mathit{c}\end{array}\right|=0$
using operation ${\mathit{R}}_2\to {\mathit{R}}_2+{\mathit{R}}_1,{\mathit{R}}_3\to {\mathit{R}}_3-{\mathit{R}}_1$
$\left|\begin{array}{ccc}\mathit{a}& 1& 1\\ 1-\mathit{a}& \mathit{b}-1& 0\\ 1-\mathit{a}& 0& \mathit{c}-1\end{array}\right|=0$
Expanding along R${}_1$,
$\mathrm{a}\left(\mathit{b}-1\right)\left(\mathit{c}-1\right)-\left(1-\mathit{a}\right)\left(\mathit{c}-1\right)-\left(1-\mathit{a}\right)\left(\mathit{b}-1\right)=0$
$\Rightarrow \left[\left(\mathit{a}-1\right)+1\right]\left(\mathit{b}-1\right)\left(\mathit{c}-1\right)+\left(1-\mathit{a}\right)\left(1-\mathit{c}\right)+$$\left(1-\mathit{a}\right)\left(1-\mathit{b}\right)=0$
$\Rightarrow [\left(\mathit{a}-1\right)\left(\mathit{b}-1\right)\left(\mathit{c}-1\right)+\left(\mathit{b}-1\right)\left(\mathit{c}-1\right)+\left(1-\mathit{a}\right)\left(1-\mathit{c}\right)$$+\left(1-\mathit{a}\right)\left(1-\mathit{b}\right)=0]$
$\Rightarrow -\left(1-\mathit{a}\right)\left(1-\mathit{b}\right)\left(1-\mathit{c}\right)+\left(-\mathit{b}+1\right)\left(-\mathit{c}+1\right)+\left(1-\mathit{a}\right)$$\left(1-\mathit{c}\right)+\left(1-\mathit{a}\right)\left(1-\mathit{b}\right)=0$
On dividing both sides by (1 - a) (1 - b) (1 - c), we get
$-1+\frac{1}{1-\mathit{a}}+\frac{1}{1-\mathit{b}}+\frac{1}{1-\mathit{c}}=0$
$\Rightarrow \frac{1}{1-\mathit{a}}+\frac{1}{1-\mathit{b}}+\frac{1}{1-\mathit{c}}=1$