Explanation
Solution
Let the quadratic function be f(x)=2x2−8x+k.
1. Analysis of the Parabola
The leading coefficient is 2 (positive), so the parabola opens upwards. For one root to lie in (1,2) and the other in (2,3), the value of the function must change sign across these intervals.
2. Applying the Conditions
For the roots to be separated by x=2, and bounded by x=1 and x=3, the following conditions must be satisfied:
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f(1) > 0
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f(2) < 0
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f(3) > 0
3. Solving the Inequalities
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Condition 1: f(1) > 0
k - 6 > 0 \implies k > 6 \quad \dots \text{(i)}
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Condition 2: f(2) < 0
k - 8 < 0 \implies k < 8 \quad \dots \text{(ii)}
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Condition 3: f(3) > 0
k - 6 > 0 \implies k > 6 \quad \dots \text{(iii)}
4. Finding the Intersection
Combining conditions (i), (ii), and (iii), we get:
5. Counting Integral Values
The only integer k that satisfies 6 < k < 8 is k=7.
Therefore, there is only one integral value of k.
Correct Option: (4)
