Choose the correct answer. If a , b , c , are in A.P., then the determinant A. 0 B. 1 C. x D. 2 x
The determinant is given as,
Δ = | x + 2 x + 3 x + 2 a x + 3 x + 4 x + 2 b x + 4 x + 5 x + 2 c |
It is given that a , b and c are in A.P.
It can be calculated as,
b − a = c − b 2 b − a − c = 0
We have to multiply and divide by 2 in the given determinant.
Δ = 1 2 | x + 2 x + 3 x + 2 a 2 ( x + 3 ) 2 ( x + 4 ) 2 ( x + 2 b ) x + 4 x + 5 x + 2 c | Δ = 1 2 | x + 2 x + 3 x + 2 a 2 x + 6 2 x + 8 2 x + 4 b x + 4 x + 5 x + 2 c |
Apply the row transformation R 2 → R 2 − R 1 − R 3 in the above determinant,
Δ = 1 2 | x + 2 x + 3 x + 2 a 2 x + 6 − ( x + 2 ) − ( x + 4 ) 2 x + 8 − ( x + 3 ) − ( x + 5 ) 2 x + 4 b − ( x + 2 a ) − ( x + 2 c ) x + 4 x + 5 x + 2 c | = 1 2 | x + 2 x + 3 x + 2 a 0 0 4 b − 2 a − 2 c x + 4 x + 5 x + 2 c | = 1 2 | x + 2 x + 3 x + 2 a 0 0 2 ( 2 b − a − c ) x + 4 x + 5 x + 2 c |
By substituting the value of 2 b − a − c = 0 in the above determinant we get,
Δ = 1 2 | x + 2 x + 3 x + 2 a 0 0 0 x + 4 x + 5 x + 2 c |
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).
The determinant is given as,
It is given that
It can be calculated as,
We have to multiply and divide by
Apply the row transformation
By substituting the value of
By using the property that if any row or column in a determinant is zero then the value of determinant is zero.
Thus, the correct option is (A).
