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

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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