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  • Provide Solution For R.D. Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 43 Maths Textbook Solution.

Provide Solution For  R.D. Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 43 Maths Textbook Solution.

Answers (1)

Answer:(x+y+z)(x-z)^{2}

Hint Use determinant formula

Given:\left|\begin{array}{lll} y+z & x & y \\ z+x & z & x \\ x+y & y & z \end{array}\right|=(x+y+z)(x-z)^{2}

Solution:

\begin{aligned} &\text { L.H.S }\left|\begin{array}{ccc} y+z & x & y \\ z+x & z & x \\ x+y & y & z \end{array}\right| \\ &\text { Apply } \mathrm{R}_{1} \rightarrow R_{1}+R_{2}+R_{3} \\ &=\left|\begin{array}{ccc} 2(x+y+z) & x+y+z & x+y+z \\ z+x & z & x \\ x+y & y & z \end{array}\right| \end{aligned}

\begin{aligned} &(x+y+z) \text { common from } \mathrm{R}_{1}\\ &=(x+y+z)\left|\begin{array}{ccc} 2 & 1 & 1 \\ z+x & z & x \\ x+y & y & z \end{array}\right|\\ &\text { Apply } \mathrm{R}_{1} \rightarrow R_{1}-R_{2}\\ &=(x+y+z)\left|\begin{array}{lll} 1 & 1 & 1 \\ x & z & x \\ x & y & z \end{array}\right|\\ &\text { Apply } \mathrm{C}_{2} \rightarrow C_{2}-C_{1} \& \mathrm{C}_{3} \rightarrow C_{3}-C_{1}\\ &=(x+y+z)\left|\begin{array}{ccc} 1 & 0 & 0 \\ x & z-x & 0 \\ x & y-x & z-x \end{array}\right| \end{aligned}

Expanding w.r.t C_{3}

\begin{aligned} &=(x+y+z)\left((z-x)^{2}-0\right) \\ &=(x+y+z)(x-z)^{2} \\ &=R \cdot H \cdot S \end{aligned}

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