If a, b, c are in A.P.,find the value of left | vedclass

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(A) Given that $a, b, c$ are in $A.P.$,we have the condition $2b = a + c$,or $a + c - 2b = 0$.Let the determinant be $\Delta = \left|\begin{array}{ccc

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) Given that $a, b, c$ are in $A.P.$,we have the condition $2b = a + c$,or $a + c - 2b = 0$.Let the determinant be $\Delta = \left|\begin{array}{ccc} 2y+4 & 5y+7 & 8y+a \ 3y+5 & 6y+8 & 9y+b \ 4y+6 & 7y+9 & 10y+c \end{array}ight|$.Applying the row operation $R_{1} ightarrow R_{1} + R_{3} - 2R_{2}$:For the first column: $(2y+4) + (4y+6) - 2(3y+5) = 6y + 10 - 6y - 10 = 0$.For the second column: $(5y+7) + (7y+9) - 2(6y+8) = 12y + 16 - 12y - 16 = 0$.For the third column: $(8y+a) + (10y+c) - 2(9y+b) = 18y + a + c - 18y - 2b = a + c - 2b$.Since $a, b, c$ are in $A.P.$,$a + c - 2b = 0$.Thus,the first row becomes $[0, 0, 0]$.Since all elements of the first row are zero,the value of the determinant is $0$.

If $a, b, c$ are in $A.P.$,find the value of $\left|\begin{array}{ccc} 2y+4 & 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{array}\right|$.

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