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To solve this problem, let’s analyze the quadratic equation:

\[
(k^2 + 2)x^2 - 5kx + 4 = 0
\]

We need to determine the values of \( k \) for which this equation has exactly one root in the interval \( (0, 1) \).

### Step 1: Conditions for a Root in the Interval \( (0, 1) \)
For a quadratic equation \( ax^2 + bx + c = 0 \) to have exactly one root in an interval, there are a few possibilities:
1. The quadratic must have exactly one root, meaning the discriminant (\( \Delta \)) of the quadratic equation should be zero.
2. Alternatively, it should have two roots, one inside \( (0, 1) \) and the other outside this interval.

In this case, the discriminant \(\Delta\) of the equation must be non-negative for real roots:
\[
\Delta = b^2 - 4ac
\]
where:
- \( a = k^2 + 2 \)
- \( b = -5k \)
- \( c = 4 \)

Thus,
\[
\Delta = (-5k)^2 - 4(k^2 + 2)(4)
\]
\[
= 25k^2 - 16(k^2 + 2)
\]
\[
= 25k^2 - 16k^2 - 32
\]
\[
= 9k^2 - 32
\]

For real roots, \(\Delta \geq 0\):
\[
9k^2 - 32 \geq 0
\]
\[
9k^2 \geq 32
\]
\[
k^2 \geq \frac{32}{9}
\]
\[
k \geq \frac{\sqrt{32}}{3} \text{ or } k \leq -\frac{\sqrt{32}}{3}
\]

Evaluating the options with this result, we find that option **2: \( 3 \leq k \leq 4 \)** satisfies the required range for \( k \).

### Answer
The correct answer is:
**Option 2: \( 3 \leq k \leq 4 \)**

Would you like a deeper explanation on any part of the process, or do you have additional questions?

### Related Questions:
1. How do we determine if a quadratic equation has only one root?
2. What are the conditions for the discriminant of a quadratic equation to yield real roots?
3. Why do we check the discriminant when finding the nature of the roots?
4. How can we verify if both roots lie within a specified interval?
5. Can a quadratic equation have only one real root without the discriminant being zero?

### Tip:
For quadratic equations, analyzing the discriminant is essential to understand the nature of the roots (real, complex, or identical).

If (k^2 + 2)x^2 - 5kx + 4 = 0 has exactly one root in the interval (0, 1), then which of the following is true for 'k' where 'k' is real?
1) 2 < k < 3
2) 3 ≤ k ≤ 4
3) -3 < k ≤ -2
4) 2 ≤ k ≤ 3

Analyze the quadratic equation (k^2 + 2)x^2 - 5kx + 4 = 0 to determine the values of k that result in exactly one root in the interval (0, 1). This solution explores key algebraic concepts like the discriminant of a quadratic equation and conditions for a specific root interval. Suitable for students in Grades 11-12.

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