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I see you've uploaded an image of a math test with multiple questions. Let's break down and solve the first question mentioned (Question 33):

**Question 33:**  
Find the value of \( k \) if the points \( P(2, 3) \), \( Q(4, k) \), and \( R(6, -3) \) are collinear.

### Solution:
For three points to be collinear, the slope between any two pairs of points must be the same. Let's use the slope formula for two points:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
1. First, let's find the slope between \( P(2, 3) \) and \( Q(4, k) \):
\[
\text{slope}_{PQ} = \frac{k - 3}{4 - 2} = \frac{k - 3}{2}
\]
2. Next, find the slope between \( Q(4, k) \) and \( R(6, -3) \):
\[
\text{slope}_{QR} = \frac{-3 - k}{6 - 4} = \frac{-3 - k}{2}
\]
3. Since the points are collinear, these slopes must be equal:
\[
\frac{k - 3}{2} = \frac{-3 - k}{2}
\]
4. Simplifying this equation:
\[
k - 3 = -3 - k
\]
\[
2k = 0
\]
\[
k = 0
\]

Thus, the value of \( k \) is **0**.

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Would you like a detailed explanation of any other question from the image?

### Related questions:
1. How would you solve for \( k \) if the points \( P(2, 3) \), \( Q(4, k) \), and \( R(6, -3) \) formed a triangle instead?
2. Can the concept of collinearity be used in a 3D coordinate system? How?
3. How does the slope formula change if two points have the same x-coordinate?
4. What is the geometric interpretation of slope equality in collinearity?
5. If a fourth point \( S(x_1, y_1) \) is added, how can we check if it also lies on the same line?

### Tip:
When dealing with collinearity, always check the slopes between pairs of points to ensure they are equal. This guarantees the points lie on a straight line.

Find the value of k if the points P(2, 3); Q(4, k); and R(6, -3) are collinear.

Learn how to solve for k when points P(2, 3), Q(4, k), and R(6, -3) are collinear. This involves using the slope formula and ensuring the slopes between point pairs are equal. Suitable for students in Grades 9-10.

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