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The equation given is:

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

and we are told that it has exactly one root in the interval \( (0, 1) \). We are asked to find the correct range for \( k \) from the options provided.

### Step 1: Analyze the Quadratic Form
The quadratic equation is of the form:

\[
Ax^2 + Bx + C = 0
\]

where:
- \( A = k^2 + 2 \),
- \( B = -5k \),
- \( C = 4 \).

For the equation to have exactly one root in the interval \( (0, 1) \), the discriminant condition or analysis of the behavior of the quadratic within that interval must be considered.

### Step 2: Apply Discriminant Condition
The discriminant \( \Delta \) for a quadratic equation \( Ax^2 + Bx + C = 0 \) is:

\[
\Delta = B^2 - 4AC
\]

Substituting values for \( A \), \( B \), and \( C \):

\[
\Delta = (-5k)^2 - 4(k^2 + 2)(4)
\]

Simplifying the discriminant expression:

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

For the quadratic to have real roots, \( \Delta \geq 0 \):

\[
9k^2 - 32 \geq 0
\]
\[
9k^2 \geq 32
\]
\[
k^2 \geq \frac{32}{9}
\]
\[
k \geq \frac{4\sqrt{2}}{3} \approx 2.121 \quad \text{or} \quad k \leq -\frac{4\sqrt{2}}{3}
\]

Thus, the range for \( k \) should lie outside these bounds. However, we also have the condition that the root is in the interval \( (0, 1) \).

### Step 3: Check the Options
Looking at the available options, the correct choice that fits the discriminant analysis and ensures the existence of a root in the interval \( (0, 1) \) is:

- \( 2 \leq k \leq 3 \) (Option 4).

Thus, the correct answer is **Option 4: \( 2 \leq k \leq 3 \)**.

Would you like further clarification or details on any specific step?

### Follow-up Questions:
1. What is the significance of the discriminant in determining the number of roots?
2. How do we determine whether the root lies in a specific interval?
3. What would happen if the discriminant were negative?
4. Can you explain how changes in the coefficient \( k \) affect the shape of the parabola?
5. What would the condition be for having no roots in \( (0, 1) \)?

#### Tip:
Always check the discriminant first to understand the nature of the roots before analyzing specific intervals for quadratic equations.

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?

Learn how to solve the quadratic equation (k^2 + 2)x^2 - 5kx + 4 = 0, with a focus on finding the real value of 'k' that gives exactly one root in the interval (0, 1). Explore the discriminant method and understand the conditions for real roots. Suitable for students in Grades 10-12.

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