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The image contains a mathematical question, but the text is partially unclear. Here's the visible portion of the question:

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**Q:** The value of \(\alpha\) for which the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(\pi/2\) where \(\mathbf{a} = (\alpha^2 - 4)\mathbf{i} + 2\mathbf{j}\) and \(\mathbf{b} = \alpha\mathbf{i} + \mathbf{j} + 2\mathbf{k}\), is:

**Options:**

1. 1/5  
2. 2  
3. 25  
4. 1/2

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### Solution:

To solve this problem, we'll use the fact that the angle between two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(\pi/2\) if and only if their dot product is zero:

\[
\mathbf{a} \cdot \mathbf{b} = 0
\]

Given vectors:

\[
\mathbf{a} = (\alpha^2 - 4)\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{b} = \alpha\mathbf{i} + \mathbf{j} + 2\mathbf{k}
\]

The dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated as follows:

\[
\mathbf{a} \cdot \mathbf{b} = \left(\alpha^2 - 4\right)\alpha + 2(1) + 0(2) = \alpha^3 - 4\alpha + 2
\]

Set the dot product to zero:

\[
\alpha^3 - 4\alpha + 2 = 0
\]

This is a cubic equation in \(\alpha\). We will need to find the value(s) of \(\alpha\) that satisfy this equation. Among the provided options, we can check which one satisfies the equation.

Let's test each option:

1. For \(\alpha = \frac{1}{5}\):

\[
\left(\frac{1}{5}\right)^3 - 4\left(\frac{1}{5}\right) + 2 = \frac{1}{125} - \frac{4}{5} + 2 \neq 0
\]

2. For \(\alpha = 2\):

\[
2^3 - 4(2) + 2 = 8 - 8 + 2 = 2 \neq 0
\]

3. For \(\alpha = 25\):

\[
25^3 - 4(25) + 2 = 15625 - 100 + 2 = 15527 \neq 0
\]

4. For \(\alpha = \frac{1}{2}\):

\[
\left(\frac{1}{2}\right)^3 - 4\left(\frac{1}{2}\right) + 2 = \frac{1}{8} - 2 + 2 = \frac{1}{8} \neq 0
\]

Given these calculations, none of the provided options seems to directly satisfy the equation. However, the expected method involves checking more accurate solutions through a cubic solution process or numeric computation.

Let me know if you would like a further breakdown or more details on this process. 

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**Would you like more details or have any questions?**

**Here are some related questions for further practice:**

1. How do you find the dot product of two vectors in three-dimensional space?
2. What are the general solutions to a cubic equation?
3. How can you determine if two vectors are perpendicular using their components?
4. What methods can be used to approximate the roots of a cubic equation?
5. How does the angle between two vectors affect their dot product?

**Tip:** When solving for the roots of a cubic equation, consider using methods such as synthetic division or numerical approximations if factorization is not straightforward.

Learn how to solve for the value of alpha where the angle between vectors a and b is pi/2. Understand the concept of dot product and cubic equations. Suitable for high school students learning advanced vector algebra and equation solving.

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