If the integers a , b , c in order are in A.P., lying between 1 and 9, and a 23 , b 53 , and c 83 are three-digit numbers, then the value of the determinant 2 a 23 a 5 b 53 b 8 c 83 c is:

Question

If the integers a , b , c in order are in A.P., lying between 1 and 9, and a 23 , b 53 , and c 83 are three-digit numbers, then the value of the determinant ​ 2 a 23 a ​ 5 b 53 b ​ 8 c 83 c ​ ​ is:

TestHub · Accepted Answer

We have: ​ 2 a 23 ​ a ​ 5 b 53 ​ b ​ 8 c 83 ​ c ​ ​ = ​ 2 100 a + 20 + 3 a ​ 5 100 b + 50 + 3 b ​ 8 100 c + 80 + 3 c ​ ​ = ​ 2 100 a a ​ 5 100 b b ​ 8 100 c c ​ ​ + ​ 2 20 a ​ 5 50 b ​ 8 80 c ​ ​ + ​ 2 3 a ​ 5 3 b ​ 8 3 c ​ ​ = 100 ​ 2 a a ​ 5 b b ​ 8 c c ​ ​ + 10 ​ 2 2 a ​ 5 5 b ​ 8 8 c ​ ​ + 3 ​ 2 1 a ​ 5 1 b ​ 8 1 c ​ ​ = 0 + 0 + 3 ​ 2 1 a ​ 5 1 b ​ 8 1 c ​ ​ (Applying C 2 ​ → C 2 ​ − C 1 ​ and C 3 ​ → C 3 ​ − C 1 ​ ) = − 3 [ 3 ( c − a ) 6 − ( b − a )] = − 9 [ c − a − 2 b + 2 a ] = − 9 ( a − 2 b + c ) = 0 = − 9 ( a − 2 b + c ) = 0

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