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.

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.