If 1 , log 10 4 x - 2 and log 10 4 x + 18 5 are in arithmetic progression for a real number x then the value of the determinant 2 x - 1 2 x - 1 x 2 1 0 x x 1 0 is equal to:

Step-by-step solution

If three numbers a ,   b ,   c are in arithmetic progression, then 2 b = a + c . Given 1 ,   log 10 4 x - 2 ,   log 10 4 x + 18 5 are in arithmetic progression, then we have 2 log 10 4 x - 2 = 1 + log 10 4 x + 18 5 Using log 10 m n = n log 10 m , we get log 10 4 x - 2 2 = log 10 10 + log 10 4 x + 18 5 Using log 10 m + log 10 n = log 10 m n , we get log 10 4 x - 2 2 = log 10 10 4 x + 18 5 ⇒ 4 x - 2 2 = 10 4 x + 18 5 ⇒ 4 x 2 + 4 - 4 · 4 x = 10 · 4 x + 36 ⇒ 4 x 2 - 14 4