If $\displaystyle \alpha, \beta$ are the roots of the quadratic equation $\displaystyle 4x^2 - 16x + k = 0, k\in R$ such that $\displaystyle 1 < \alpha < 2$ and $\displaystyle 2 < \beta < 3$, then the number of integral values of $k$ is

Let f(x) = 4x^2 - 16x + k. The roots alpha and beta satisfy 1 < alpha < 2 and 2 < beta < 3. This implies f(1) > 0, f(2) < 0, and f(3) > 0. f(1) = 4 - 16 + k = k - 12 > 0 => k > 12. f(2) = 16 - 32 + k = k - 16 < 0 => k < 16. f(3) = 36 - 48 + k = k - 12 > 0 => k > 12. The range is 12 < k < 16. The integers are 13, 14, 15. There are 3 values.