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Question

If a,b,c are in A.P, find value of ∣ ∣2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c∣ ∣

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Solution

Simplification of given data
Given : a,b,c are in A.P.

ba=cb
bac+b=0
2bac=0 ...(i)

Solve given determinant

=∣ ∣2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c∣ ∣

Multiply & divide by 2

=22∣ ∣2y+45y+78y+a3y+56y+89y+b4y+67y+910y+c∣ ∣

=12∣ ∣ ∣2y+45y+78y+a2(3y+5)2(6y+8)2(9y+b)4y+67y+910y+c∣ ∣ ∣

=12∣ ∣2y+45y+78y+a6y+1012y+1618y+2b4y+67y+910y+c∣ ∣

Applying R2R2R1R3

=12∣ ∣2y+45y+78y+a002bac4y+67y+910y+c∣ ∣

=12∣ ∣2y+45y+78y+a0004y+67y+910y+c∣ ∣ ....[from (i)]

=12×0

=0


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