The set of values of k for which roots of the equation `x^(2) - 3x + k = 0` lie in the interval (0, 2), is

Options

A. `(2, oo)`
B. `(0, oo)`
C. `(-oo, 9//4)`
D. `(2, 9//4)`

Answer

D

Explanation

Let `f(x) = x^(2) - 3x + k`. Clearly, it represents a parabola opening upward. If roots of f(x) lie in the interval (0, 2), then we must have
(i) Discriminate `ge` 0, (ii) `f(0) ge 0, f(2) ge 0` (iii) x-coordinate of the vertex lies between 0 and 2.
Now,
(i) Discriminant `ge 0 rArr 9 - 4k ge 0 rArr k le (9)/(4)`
(ii) `f(0) gt 0 and f(2) gt 0 rArr k gt 0 and k-2 gt 0 rArr k gt 2`
(iii) The x-coordinate of the vertex is `(3)/(2)`, which lies between 0 and 2.
So, this condition is true for all values of k.
Hence, required set of values of k is (2, 9/4].