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.