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Certainly! I'll help solve the questions from the provided test. Here’s the step-by-step solution for each problem:

### Question 1
Given \( \triangle ABC \) with \( \sin B \) value:
1. \( \sin B \) = \(\frac{\sqrt{3}}{2}\) corresponds to an angle \( B \) of 60°.

So the answer is **A**.

### Question 2
Find the coordinates of each vertex of the square:
2. The coordinates that form a square with equal lengths are (55, 11), (55, 5), (59, 9), (55, 11).

So the answer is **B**.

### Question 3
Evaluate the nature of \( \left( \sqrt{5} - \sqrt{2} \right)^2 \):
3. The expression \( \left( \sqrt{5} - \sqrt{2} \right)^2 \) will give a real and rational number.

So the answer is **A**.

### Question 4
Find the area of a square inside a semicircle of radius 3 cm:
4. The side length \( s \) of the square is 3 cm, so the area is \( s^2 = 9 \, \text{cm}^2 \).

So the answer is **D**.

### Question 5
Evaluate the correct statements about the geometric figure:
5. Correct statements:
   - **EH** is parallel to **CDFG**.
   - The length **BC** is 26 units.

The answers are **(2)** and **(3)**.

### Question 6
For the cylinder with a given height and diameter:
6. Volume \( V = \pi r^2 h \). Here, \( r = 3 \, \text{cm} \), \( h = 4 \, \text{cm} \), so \( V = \pi \times 3^2 \times 4 = 36\pi \, \text{cm}^3 \).

So the answer is \( 36\pi \, \text{cm}^3 \).

### Question 7
Solve the equation \( g(x) = 2x - 3 \) and \( f(x) = -2x \):
7. We find where the two functions intersect:
   - \( g(x) = 2x - 3 \)
   - \( f(x) = -2x \)

Setting them equal: \( 2x - 3 = -2x \implies 4x = 3 \implies x = \frac{3}{4} \).

So the solution is \( x = \frac{3}{4} \).

### Question 8
Solve the inequalities:
8. For \( 3x + 2 < 2x \):
   - \( 3x + 2 < 2x \implies x < -2 \).

The interval is \( (-\infty, -2) \).

### Question 9
Represent the solutions on the number line:
9. \( \frac{3x - 2}{4} \leq 1 \):
   - Multiply both sides by 4: \( 3x - 2 \leq 4 \).
   - Solve for \( x \): \( 3x \leq 6 \implies x \leq 2 \).

The interval is \( (-\infty, 2] \).

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If you have any questions or need further details on any problem, please let me know!

Here are five questions you might want to ask next:
1. Can you explain how to solve quadratic equations?
2. What are the properties of geometric shapes?
3. How do you find the area and perimeter of a circle?
4. Can you explain the rules of exponents?
5. How do you solve systems of linear equations?

**Tip:** When solving algebraic equations, always check your solutions by substituting them back into the original equations to ensure they satisfy all conditions.

Explore detailed step-by-step solutions for geometry, trigonometry, and algebra problems from a given test. Includes explanations on the Pythagorean theorem, area calculations for shapes, and volumes of cylinders. Suitable for high school students.

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