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CBSE Class 12 · 2005 · DeterminantsLong Answer

Show that

Δ = x + 1 x + 2 x + 3 x + 2 x + 3 x + 4 x + a x + b x + c = 0

, where

a, b, c

are in A.P.

Answer: Proved

Solution

Since

a, b, c

are in A.P.,

2 b = a + c

. Apply the row operation

R_{1} \to R_{1} + R_{3} - 2 R_{2}

.
Column

1

:

(x + 1) + (x + 3) - 2 (x + 2) = 0

. Column

2

:

(x + 2) + (x + 4) - 2 (x + 3) = 0

.
Column

3

:

(x + a) + (x + c) - 2 (x + b) = (a + c) - 2 b = 0

.
So the whole first row becomes zero:

Δ = 0 x + 2 x + 3 0 x + 3 x + 4 0 x + b x + c = 0.

A determinant with a zero row is

0

. Proved.

CBSE Class 12 Board · 2005 · Subjective

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