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Provide solution for RD Sharma maths class 12 chapter Determinants exercise multiple choise question 18

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Correct option (a)


If a,b,c are in A.P \Rightarrow 2b=a+c


If a,b,c are in A.P

        Let \: \: \Delta= \begin{vmatrix} x+2 &x+3 &x+2a \\ x+3 &x+4 &x+2b \\ x+4 &x+5 &x+2c \end{vmatrix}

We have to find the value of given determinant \Delta


        \Delta= \begin{vmatrix} x+2 &x+3 &x+2a \\ x+3 &x+4 &x+2b \\ x+4 &x+5 &x+2c \end{vmatrix}

If a,b,c are in A.P \Rightarrow 2b=a+c

        \Rightarrow \Delta= \begin{vmatrix} x+2 &x+3 &x+2a \\ x+3 &x+4 &x+a+c \\ x+4 &x+5 &x+2c \end{vmatrix}

Applying R1→R1-R2 ; R3→R3-R2

        \Rightarrow \Delta= \begin{vmatrix} -1 &-1 &a-c \\ x+3 &x+4 &x+a+c \\ 1 &1 &c-a \end{vmatrix}

Applying R1→R1+R3

        \Rightarrow \Delta= \begin{vmatrix} 0 &0 &0 \\ x+3 &x+4 &x+a+c \\ 1 &1 &c-a \end{vmatrix}

Here R1 is zero

        \Rightarrow \Delta= 0

Hence \: \: \begin{vmatrix} x+2 &x+3 &x+2a \\ x+3 &x+4 &x+2b \\ x+4 &x+5 &x+2c \end{vmatrix}=0

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Gurleen Kaur

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