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Question

Choose the correct answer. If a , b , c , are in A.P., then the determinant A. 0 B. 1 C. x D. 2 x

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Solution

The determinant is given as,

Δ=| x+2 x+3 x+2a x+3 x+4 x+2b x+4 x+5 x+2c |

It is given that a, b and c are in A.P.

It can be calculated as,

ba=cb 2bac=0

We have to multiply and divide by 2 in the given determinant.

Δ= 1 2 | x+2 x+3 x+2a 2( x+3 ) 2( x+4 ) 2( x+2b ) x+4 x+5 x+2c | Δ= 1 2 | x+2 x+3 x+2a 2x+6 2x+8 2x+4b x+4 x+5 x+2c |

Apply the row transformation R 2 R 2 R 1 R 3 in the above determinant,

Δ= 1 2 | x+2 x+3 x+2a 2x+6( x+2 )( x+4 ) 2x+8( x+3 )( x+5 ) 2x+4b( x+2a )( x+2c ) x+4 x+5 x+2c | = 1 2 | x+2 x+3 x+2a 0 0 4b2a2c x+4 x+5 x+2c | = 1 2 | x+2 x+3 x+2a 0 0 2( 2bac ) x+4 x+5 x+2c |

By substituting the value of 2bac=0 in the above determinant we get,

Δ= 1 2 | x+2 x+3 x+2a 0 0 0 x+4 x+5 x+2c |

By using the property that if any row or column in a determinant is zero then the value of determinant is zero.

Δ=0

Thus, the correct option is (A).


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