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

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

Answers (1)

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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