Q47 of 256 Page 7

Show that where a,b,c are in A.P.

Let


Recall that the value of a determinant remains same if we apply the operation Ri→ Ri + kRj or Ci→ Ci + kCj.


Applying R1→ R1 + R3, we get




Given that a, b and c are in an A.P. Using the definition of an arithmetic progression, we have


b – a = c – b


⇒ b + b = c + a


⇒ 2b = c + a


∴ a + c = 2b


By substituting this in the above equation to find Δ, we get




Taking 2 common from R1, we get



Applying R1→ R1 – R2, we get




∴ Δ = 0


Thus, when a, b and c are in A.P.


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