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Explain Solution R .D.Sharma Class 12 Chapter  deteminants Exercise 5.2 Question 52 sub question 3 maths Textbook Solution.

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Answer: x=\frac{2}{3}, \frac{11}{3}, \frac{11}{3}

Hint: Use determinant formula

Given: $$ \left|\begin{array}{ccc} 3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8 \end{array}\right|=0

Solution:

$$ \text { L.H.S }\left|\begin{array}{ccc} 3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8 \end{array}\right|

$$ \begin{aligned} &\text { Apply C }_{1} \rightarrow C_{1}+C_{2}+C_{3} \\ &\left|\begin{array}{ccc} 3 x-2 & 3 & 3 \\ 3 x-2 & 3 x-8 & 3 \\ 3 x-2 & 3 & 3 x-8 \end{array}\right| \end{aligned}

$$ \begin{aligned} &(3 x-2) \text { common from } \mathrm{C}_{1}\\ &=(3 x-2)\left|\begin{array}{ccc} 1 & 3 & 3 \\ 1 & 3 x-8 & 3 \\ 1 & 3 & 3 x-8 \end{array}\right|\\ &\text { Apply } \mathrm{R}_{2} \rightarrow R_{2}-R_{1} \& R_{3} \rightarrow R_{3}-R_{1}\\ &=(3 x-2)\left|\begin{array}{ccc} 1 & 3 & 3 \\ 0 & -3 x-11 & 0 \\ 0 & 0 & -3 x-11 \end{array}\right| \end{aligned}

\begin{aligned} &\text { Expand from } \mathrm{C}_{1}\\ &=(3 x-2)(3 x-11)^{2}=0\\ &3 x-2=0,3 x-11=0\\ &3 x=2,3 x=11\\ &x=\frac{2}{3}, x=\frac{11}{3}, \frac{11}{3}\\ &x=\frac{2}{3}, \frac{11}{3}, \frac{11}{3} \end{aligned}

 

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