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Let's go through each question step-by-step:

### **Question 11**

#### (a) 
**(i) Illustrate the information in a diagram.**

We are dealing with two buildings where one is 320 m tall, and the angles of depression from the top of the taller building (Point X) to the top (Point Y) and bottom (Point Z) of the shorter building are given as 29° and 41°, respectively.

**Diagram Explanation:**
- Label the tall building's top as X (320 m).
- Point Y is the top of the shorter building.
- Point Z is the base of the shorter building.
- The horizontal line from the base of the tall building is parallel to the line connecting Y and Z.

**(ii) Calculate the height of the other building.**

We need to use trigonometry to find the height of the shorter building. Let \( d \) be the horizontal distance between the two buildings and \( h \) be the height of the shorter building.

Using the tangent function:
1. For the top of the shorter building: 
   \[
   \tan(29^\circ) = \frac{320 - h}{d}
   \]
   
2. For the bottom of the shorter building:
   \[
   \tan(41^\circ) = \frac{320}{d}
   \]

By solving these two equations, we can find the value of \( h \).

#### (b)
To find the initial speed:
Given:
- The time reduction is 30 minutes (0.5 hours).
- The speed increased by 20 km/h.

We use the formula:
\[
\text{Initial time} = \frac{120}{v}, \quad \text{New time} = \frac{120}{v + 20}
\]

Given the time difference:
\[
\frac{120}{v} - \frac{120}{v + 20} = 0.5
\]

### **Question 12**

#### (a)
**(i) Draw a Venn diagram.**
There are two sets: 
- Set \( S \) (Most businessmen are rich).
- Set \( T \) (No salary worker is rich).

**(ii) Determine the validity of statements:**
1. \( \alpha \) Ali is a businessman and is therefore rich.
2. \( \beta \) Ahmed is not rich because he is a salary worker.

#### (b)
Using the frequency distribution table, find:
**(i) Mean and (ii) Variance.**

### **Question 13**

#### (a)
**(i) Plotting points and drawing triangles.**
1. Plot points M(3, 1), N(1, 1), and P(3, 5).
2. Find the coordinates after reflections and transformations as described.

#### (b)
Find the equation of the line joining the points M and \( M_2 \).

Would you like a detailed solution for any of the specific parts of these problems, or a step-by-step explanation of the calculations? 

### Follow-up Questions
1. Would you like help illustrating the diagram for Question 11(a)?
2. Do you need assistance solving the trigonometric equations in Question 11(a)(ii)?
3. Should we calculate the initial speed in Question 11(b)?
4. Would you like a detailed Venn diagram and statement evaluation for Question 12(a)?
5. Shall we proceed with finding the mean and variance for Question 12(b)?

### Tip
When dealing with trigonometry and geometric problems, always try to sketch a diagram; it simplifies the visualization and solution process.

The original question includes trigonometry, speed calculation, logic (Venn diagrams), statistics (mean, variance), and geometry (coordinate plane transformations).

This problem set covers key concepts including trigonometry with angle of depression, speed and time calculation, logical reasoning with Venn diagrams, statistics (mean and variance), and geometric transformations on the coordinate plane. Suitable for high school students in Grades 10-12.

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