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

Correct option (a)

Hint:

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

Given:

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 $\Delta$

Solution:

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

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

$\Rightarrow \Delta= 0$

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