Q. 39

# If a, b and c are all non-zero and , then prove that .(CBSE 2016)

Let

Given that Δ = 0.

We can write the determinant Δ as

Taking a, b and c common from C1, C2 and C3, we get

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

Applying C1 C1 + C2, we get

Applying C1 C1 + C3, we get

Taking common from C1, we get

Applying R2 R2 – R1, we get

Applying R3 R3 – R1, we get

Expanding the determinant along C1, we have

We have Δ = 0.

It is given that a, b and c are all non-zero.

Thus, when and a, b, c are all non-zero.

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