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Need Solution for R.D.Sharma Maths Class 12 Chapter determinants  Exercise 5.2 Question 53 Maths Textbook Solution.

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Answer: 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0

Hint Use determinant formula

Given:\left|\begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{array}\right|=0 \text { p.t } \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1=0

Solution:

\begin{aligned} &\text { Divide a to } \mathrm{C}_{1}, \mathrm{~b} \text { to } \mathrm{C}_{2} \text { and } \mathrm{c} \text { to } \mathrm{C}_{3}\\ &(a \times b \times c)\left|\begin{array}{ccc} \frac{1}{a}+1 & \frac{1}{b} & \frac{1}{c} \\ \frac{1}{a} & \frac{1}{b}+1 & \frac{1}{c} \\ \frac{1}{a} & \frac{1}{b} & \frac{1}{c}+1 \end{array}\right|=0 \end{aligned}

\begin{aligned} &\text { Apply } \mathrm{C}_{1} \rightarrow C_{1}+C_{2}+C_{3} \\ &\text { (abc) }\left|\begin{array}{lll} \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1 & \frac{1}{b} & \frac{1}{c} \\ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1 & \frac{1}{b}+1 & \frac{1}{c} \\ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1 & \frac{1}{b} & \frac{1}{c}+1 \end{array}\right|=0 \end{aligned}

\begin{aligned} &\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\right) \text { common from } \mathrm{C}_{1}\\ &(a b c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\right)\left|\begin{array}{lll} 1 & \frac{1}{b} & \frac{1}{c} \\ 1 & \frac{1}{b}+1 & \frac{1}{c} \\ 1 & \frac{1}{b} & \frac{1}{c}+1 \end{array}\right|=0\\ &\text { Apply } \mathrm{R}_{2} \rightarrow R_{2}-R_{1} \& R_{3} \rightarrow R_{3}-R_{1}\\ &a b c\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\right)\left|\begin{array}{lcc} 1 & \frac{1}{b} & \frac{1}{c} \\ 0 & \frac{1}{b}+1-\frac{1}{b} & \frac{1}{c}-\frac{1}{c} \\ 0 & \frac{1}{b}-\frac{1}{b} & \frac{1}{c}+1-\frac{1}{c} \end{array}\right|\\ &a b c\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\right|\left|\begin{array}{ccc} 1 & \frac{1}{b} & \frac{1}{c} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right| \end{aligned}

\begin{aligned} &\text { Expand from } \mathrm{R}_{3}\\ &a b c\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\right|=0\\ &a, b, c \neq 0, \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1=0\\ &a b c \neq 0 \end{aligned}

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