Question

The number of integral values of \(k\), for which one root of the equation \[2x^2 - 8x + k = 0\] lies in the interval \((1, 2)\) and its other root lies in the interval \((2, 3)\), is:

• A. 2
• B. 0
• C. 1
• D. 2

Hint:

For problems involving roots of quadratic equations in specific intervals, use the condition f(a) · f(b) < 0 to identify intervals of k and solve the resulting inequalities systematically.

Answer 1

The Correct Option is C

The given quadratic equation is:
\[2x^2 - 8x + k = 0.\]
Step 1: Conditions for Roots in the Specified Intervals
For one root to lie in \((1, 2)\), the product of the values of the quadratic function at the endpoints must be negative:
\[f(1) \cdot f(2) < 0.\]
Similarly, for the other root to lie in \((2, 3)\):
\[f(2) \cdot f(3) < 0.\]
Step 2: Compute \(f(x)\) at Specific Points
The quadratic function is:
\[f(x) = 2x^2 - 8x + k.\]
Compute \(f(1)\), \(f(2)\), and \(f(3)\):
\[f(1) = 2(1)^2 - 8(1) + k = 2 - 8 + k = k - 6,\]
\[f(2) = 2(2)^2 - 8(2) + k = 8 - 16 + k = k - 8,\]
\[f(3) = 2(3)^2 - 8(3) + k = 18 - 24 + k = k - 6.\]
Step 3: Apply the Conditions
For \(f(1) \cdot f(2) < 0\):
\[(k - 6)(k - 8) < 0.\]
The roots of this inequality are \(k = 6\) and \(k = 8\). Using the sign change rule for quadratic inequalities:
\[k \in (6, 8).\]
For \(f(2) \cdot f(3) < 0\):
\[(k - 8)(k - 6) < 0.\]
This inequality is also satisfied for:
\[k \in (6, 8).\]
Step 4: Determine Integral Values of \(k\)
The intersection of both conditions is \(k \in (6, 8)\). The only integer in this interval is:
\[k = 7.\]
Conclusion: The number of integral values of \(k\) is \(\mathbf{1}\) (Option 3).