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Please solve RD Sharma class 12 chapter Determinants exercise multiple choise question 25 maths textbook solution

Answers (1)

Answer:

Correct option (d)

Hint:

Simply solve the determinant.

Given:

x, y, z are non-zero value and

        \begin{vmatrix} 1+x &1 &1 \\ 1 &1+y &1 \\ 1 &1 &1+z \end{vmatrix}=0

We have to find

        x^{-1}+y^{-1}+z^{-1}\; \; i.e.\; \; \frac{1}{x}+\frac{1}{y}+\frac{1}{z}

Solution:

Here\; \; \; \begin{vmatrix} 1+x &1 &1 \\ 1 &1+y &1 \\ 1 &1 &1+z \end{vmatrix}=0

Applying C1→C1-C3; C2→C2-C3

        \Rightarrow \begin{vmatrix} x &0 &1 \\ 0 &y &1 \\ -z &-z &1+z \end{vmatrix}=0

Expanding along R1

        \Rightarrow x[y(1+z)+z]+0+(0+zy)=0

        \Rightarrow x(y+yz+z)+zy=0

        \Rightarrow xy+xyz+xz+zy=0

        \Rightarrow xy+xz+zy=-xyz

        \Rightarrow \frac{xy}{xyz}+\frac{xz}{xyz}+\frac{zy}{xyz}=\frac{-xyz}{xyz}

        \Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{z}{z}=-1

        \Rightarrow x^{-1}+y^{-1}+z^{-1}=-1

Hence, the required solution.

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Gurleen Kaur

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