If the roots of the equation 10 x 3 - c x 2 - 54 x - 27 = 0 are in harmonic progression, then the value of c must be equal to

Step-by-step solution

The roots of 10 x 3 - c x 2 - 54 x - 27 = 0 are in H.P. Replacing x by 1 x we will get an equation whose roots will be in A.P. ⇒ 10 x 3 - c x 2 - 54 x - 27 = 0 ⇒ 27 x 3 + 54 x 2 + c x - 10 = 0 Hence, roots of this equation are in A.P. i.e. a - d ,   a ,   a + d a - d + a + a + d = - 54 27 = - 2 ⇒ a = - 2 3 Since a is the root of the given equation ⇒ 27 - 2 3 3 + 54 - 2 3 2 + c - 2 3 - 10 = 0 ⇒ - 8 + 24 - 10 = 2 c 3 ⇒ c = 6 × 3 2 = 9 c = 9