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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 44 Maths Textbook Solution.

Provide Solution For  R.D.Sharma Maths Class 12 Chapter  determinants Exercise 5.2 Question 44 Maths Textbook Solution.

Answers (1)

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

Hint Use determinant formula

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

Solution:

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

(a+x+y+z) \text { common from } \mathrm{C}_{1}

\begin{aligned} &=(a+x+y+z)\left|\begin{array}{ccc} 1 & y & z \\ 1 & a+y & z \\ 1 & y & a+z \end{array}\right|\\ &\text { Apply } \mathrm{R}_{1} \rightarrow R_{1}-R_{2} \& R_{2} \rightarrow R_{2}-R_{3}\\ &=(a+x+y+z)\left|\begin{array}{ccc} 0 & -a & 0 \\ 0 & a & -a \\ 1 & y & a+z \end{array}\right|\\ &\text { Expanding w.r.t } \mathrm{C}_{3}\\ &=(a+x+y+z)\left(1\left(a^{2}\right)\right)\\ &=(a+x+y+z) a^{2}\\ &=R \cdot H . S \end{aligned}

 

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