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  • Please solve RD Sharma class 12 chapter Determinants exercise multiple choise question 29 maths textbook solution

Please solve RD Sharma class 12 chapter Determinants exercise multiple choise question 29 maths textbook solution

Answers (1)

Answer:

Correct option (b)

Hint:

Solve determinant by applying row and column operation.

Given:

\begin{aligned} &\text { Let } \Delta=\left|\begin{array}{ccc} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right| \end{aligned}

We have to find the value of \Delta

Solution:

\begin{aligned} &\text { Here } \: \: \Delta=\left|\begin{array}{ccc} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right| \end{aligned}

Applying R1 → R1 - R2

        \Rightarrow \Delta =\left|\begin{array}{ccc} -2y & y &y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right|

Applying R3 → R3 - R2

        \Rightarrow \Delta =\left|\begin{array}{ccc} -2y & y &y \\ x+2 y & x & x+y \\ -y & 2 y & -y \end{array}\right|

Taking common y from R1 and R3

        \Rightarrow \Delta =y^{2}\left|\begin{array}{ccc} -2 & 1 &1 \\ x+2 y & x & x+y \\ -1 & 2 & -1 \end{array}\right|

Applying C2 → C2 + 2C1; C3 → C3 -C1

        \Rightarrow \Delta =y^{2}\left|\begin{array}{ccc} -2 & -3 &3 \\ x+2 y & x & x+y \\ -1 & 0 & 0 \end{array}\right|

Expanding along R3, we get

        \begin{aligned} &\Delta=y^{2}[(-1)(3 y-9 x-12 y)] \\ &\Delta=y^{2}[9 x+9 y] \end{aligned}

        \begin{aligned} &\Delta=9y^{2}[ x+ y] \end{aligned}

\begin{aligned} &\text { Hence } \left|\begin{array}{ccc} x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x \end{array}\right| \end{aligned}=9y^{2}[x+y]

Posted by

Gurleen Kaur

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