Mark the tick against the correct answer in the following: The solution set of the equation $\left|\begin{array}{ccc}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{array}\right|=0$ is A. $\{a, 0\}$ B. $\{3 a, 0\}$ C. $\{a, 3 a\}$ D. None of these

Mark the tick against the correct answer in the following: The solution set of the equation $\left|\begin{array}{ccc}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{array}\right|=0$ is
A. $\{a, 0\}$
B. $\{3 a, 0\}$
C. $\{a, 3 a\}$
D. None of these

CBSE | Class 12 | Determinants | Maths | RS Aggarwal

Solution:

Option(B)
To find: Value of $x$
We have, $\left|\begin{array}{ccc}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{array}\right|=0$
Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$
$\Rightarrow\left|\begin{array}{ccc} 2 x & -2 x & 0 \\ a-x & a+x & a-x \\ a-x & a-x & a+x \end{array}\right|=0$
Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$
$\Rightarrow\left|\begin{array}{ccc} 2 x & -2 x & 0 \\ 0 & 2 x & -2 x \\ a-x & a-x & a+x \end{array}\right|=0$
Taking 2 common from $R_{1}$
$\Rightarrow 2\left|\begin{array}{ccc} \mathrm{x} & -\mathrm{x} & 0 \\ 0 & 2 \mathrm{x} & -2 \mathrm{x} \\ \mathrm{a}-\mathrm{x} & \mathrm{a}-\mathrm{x} & \mathrm{a}+\mathrm{x} \end{array}\right|=0$
Taking 2 common from $R_{2}$
$\Rightarrow 2 \times 2\left|\begin{array}{ccc} x & -x & 0 \\ 0 & x & -x \\ a-x & a-x & a+x \end{array}\right|=0$
Applying $R_{3} \rightarrow R_{1}+R_{3}$
$\Rightarrow 4\left|\begin{array}{ccc}x & -x & 0 \\ 0 & x & -x \\ a & a-2 x & a+x\end{array}\right|=0$
Expanding along $R_{1}$
$\Rightarrow 4[x\{(x)(a+x)-(-x)(a-2 x)\}]-(-x)\{(0)(a+x)-(-x)(a)\}]=0$
$\begin{array}{l} \left.\Rightarrow 4\left[x\left\{a x+x^{2}+a x-2 x^{2}\right\}\right]-(-x)\{a x\}\right]=0 \\ \left.\Rightarrow 4\left[x\left\{2 a x-x^{2}\right\}\right]+a x^{2}\right]=0 \\ \Rightarrow 4\left[2 a x^{2}-x^{3}+a x^{2}\right]=0 \\ \Rightarrow-x^{2}+3 a x=0 \\ \Rightarrow-x(x-3 a)=0 \\ \Rightarrow x=0, \text { or } x=3 a \end{array}$