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  • Explain Solution R. D. Sharma Class 12 Chapter deteminants Exercise 5.2 Question 47 maths Textbook Solution.

Explain Solution R. D. Sharma Class 12 Chapter  deteminants Exercise 5.2 Question 47 maths Textbook Solution.

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Hint a, b, c \text { are in AP }

Given: \left|\begin{array}{lll} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{array}\right|=0 \text { where } a, b, c \text { are in } \mathrm{AP}

Solution:

\begin{aligned} &\text { L.H.S }\left|\begin{array}{ccc} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{array}\right| \\ &\text { Now } R_{3} \rightarrow-2 R_{2}+R_{1}+R_{3} \\ &=\left|\begin{array}{ccc} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ 0 & 0 & a+c-2 b \end{array}\right| \end{aligned}

As a b_{2} c \text { are in } \mathrm{AP}, 2 \mathrm{~b}=\mathrm{a}+\mathrm{c}

Substituting the value in the denominator,

\begin{aligned} &=\left|\begin{array}{ccc} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ 0 & 0 & a+c-a-c \end{array}\right| \\ &=\left|\begin{array}{ccc} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ 0 & 0 & 0 \end{array}\right| \end{aligned}

Expanding along R_{3}

=0

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