AP EAMCET · Maths · Determinants

If $a, b, c$ are respectively the $5^{th}, 8^{th}, 13^{th}$ terms of an arithmetic progression, then $|\begin{matrix} a & 5 & 1 \\ b & 8 & 1 \\ c & 13 & 1 \end{matrix}| =$

A $0$
B $1$
C $a b c$
D $520$

Verified Solution

Answer & Solution

Correct Answer

(A) $0$

Step-by-step Solution

Detailed explanation

Since, a,b,c are the 5th ,8th an 13th term of an A.P., respectively, therefore a=A+4D b=A+7D c=A+12D whew, A and D awe the first term and common differince of an A.P.. Now, a51b81c131 Applying R2→R2-R1 and R3→R3-R1 =a51b-a30c-a80 =8b-8a-3c+3a…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If a,b,c are respectively the 5th,8th,13th terms of an arithmetic progression, then a51b81c131=

Step-by-step Solution

If $a, b, c$ are respectively the $5^{th}, 8^{th}, 13^{th}$ terms of an arithmetic progression, then $|\begin{matrix} a & 5 & 1 \\ b & 8 & 1 \\ c & 13 & 1 \end{matrix}| =$