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

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}