<p>Number of integral value(s) of <i>k</i> for which the equation <i>4x</i><sup>2</sup> − <i>16x</i> + <i>k</i> = 0 has one root lie between 1 and 2 and other root lies between 2 and 3, is</p>
Step-by-Step Solution
Key Concept: For a quadratic to have roots in specified intervals, the function must have opposite signs at the interval endpoints.
<p>Here, <i>f</i>(1) · <i>f</i>(2) < 0</p><p>⟹ (<i>k</i> − 12)(<i>k</i> − 16) < 0</p><p>⟹ 12 < <i>k</i> < 16 ... (i)</p><p>Also, <i>f</i>(2) · <i>f</i>(3) < 0</p><p>⟹ (<i>k</i> − 16)(<i>k</i> − 12) < 0 ⟹ 12 < <i>k</i> < 16 ... (ii)</p><p>From Eqs. (i) and (ii), we get</p><p><i>k</i> ∈ {13, 14, 15}</p><p>∴ Number of integral values of <i>k</i> = 3</p>
Correct Answer: C