If α and β are the roots of 4 x 2 - 16 x + t = 0 , ∀ t > 0 such that 1 < α < 2 < β < 3 , then the number of integral values of t are

Step-by-step solution

f x = 4 x 2 - 16 x + t Opening upward parabola f 1 > 0 , f 2 < 0 , f 3 > 0 f 1 > 0 ⇒ 4 - 16 + t > 0 ⇒ t > 1 2 f 2 < 0 ⇒ 16 - 32 + t < 0 ⇒ t < 1 6 f 3 > 0 ⇒ 36 - 48 + t > 0 ⇒ t > 1 2 ⇒ 12 < t < 16 ⇒ t = 13, 14, 1 5 Hence, the number of integral values t = 13,14,15 is 3 .