Question

any one solve this please​

Answer 1

$\large\underline{\sf{Solution-}}$

Since, a, b, c are in AP.

$\rm :\implies\:2b = a + c - - - - - (1)$

Now,

Consider,

$\rm :\longmapsto\:\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}2y + 4&5y + 7&8y + a\\3y + 5&6y + 8&9y + b\\4y + 6&7y + 9&10y + c\end{array}\right | \end{gathered}$

$\red{\rm :\longmapsto\:OP \: R_1\to R_1 + R_3}$

$\rm \:=\: \begin{gathered}\sf \left | \begin{array}{ccc}2y + 4 + 4y + 6&5y + 7 + 7y + 9&8y + a + 10y + c\\3y + 5&6y + 8&9y + b\\4y + 6&7y + 9&10y + c\end{array}\right | \end{gathered}$

$\rm \: \: = \: \: \begin{gathered}\sf \left | \begin{array}{ccc}6y + 10&12y + 16&18y + a + c\\3y + 5&6y + 8&9y + b\\4y + 6&7y + 9&10y + c\end{array}\right | \end{gathered}$

$\rm \: \: = \: \: \begin{gathered}\sf \left | \begin{array}{ccc}6y + 10&12y + 16&18y + 2b\\3y + 5&6y + 8&9y + b\\4y + 6&7y + 9&10y + c\end{array}\right | \end{gathered}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: a + c = 2b\bigg \}}$

On taking out 2 common from Row 1, we get

$\rm \: \: = \: 2 \: \begin{gathered}\sf \left | \begin{array}{ccc}3y + 5&6y + 8&9y + b\\3y + 5&6y + 8&9y + b\\4y + 6&7y + 9&10y + c\end{array}\right | \end{gathered}$

$\rm \: = \: \: 2 \times 0$

[ Row 1 and Row 2 are identical, so determinant value is 0]

Hence,

$\rm :\longmapsto\:\bf \: \: \: \: \begin{gathered}\bf \left | \begin{array}{ccc}2y + 4&5y + 7&8y + a\\3y + 5&6y + 8&9y + b\\4y + 6&7y + 9&10y + c\end{array}\right | \end{gathered} = 0$

Additional Information :-

1. The determinant remains unaltered if its rows are changed into columns and the columns into rows.

2. If all the elements of a row (or column) are zero, then the determinant is zero.

3. If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant value is zero.

4. The interchange of any two successive rows (or columns) of the determinant changes its sign.

5. If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.

6. If any row or column is multiplied by any constant 'k', the determinant value is also multiplied by 'k'.

7. The determinant value remains unaltered if rows or columns are added or subtracted.