Question

All the values of \(k\) such that the quadratic expression \(2kx^2 - (4k+1)x + 2\) is negative for exactly three integral values of \(x\), lie in the interval:

• A. \(\left[\frac{1}{12}, \frac{1}{10}\right)\)
• B. \(\left(\frac{1}{6}, \frac{1}{5}\right)\)
• C. \([-1,2)\)
• D. \([2,6)\)

Hint:

To find where quadratic is negative for a set number of integral points, analyze roots and vertex, then check integer intervals.

Answer 1

The Correct Option is A

Step 1: Condition for negativity of quadratic for integral values
The quadratic is \(2kx^2 - (4k+1)x + 2\). To be negative for exactly three integral values of \(x\), the parabola must be below the \(x\)-axis at exactly three integer points. Step 2: Find roots and analyze
Find roots of quadratic \(2kx^2 - (4k+1)x + 2 = 0\) for \(x\), then find values of \(k\) such that exactly three integral \(x\) satisfy the inequality. Step 3: Interval
By solving and analyzing inequalities, the interval for \(k\) is \(\left[\frac{1}{12}, \frac{1}{10}\right)\).