https://r2.erweima.ai/mathbot/image/1718690496015q6zncxc0.pdf

Based on the provided document, here are solutions to some of the problems extracted:

### Problem 1:
Find the solution to the equation \( x^2 - 7x + 12 = 0 \).

**Solution:**
To solve the quadratic equation \( x^2 - 7x + 12 = 0 \), we can factorize it.

\[ x^2 - 7x + 12 = (x - 3)(x - 4) = 0 \]

So, the solutions are:
\[ x = 3 \]
\[ x = 4 \]

### Problem 2:
Find the cosine of angle \( \angle ACB \) for a triangle with sides \( AC = 3 \), \( BC = 5 \), and \( AB = 7 \).

**Solution:**
Using the cosine rule:
\[ \cos(\angle ACB) = \frac{AC^2 + BC^2 - AB^2}{2 \cdot AC \cdot BC} \]

Substitute the given values:
\[ \cos(\angle ACB) = \frac{3^2 + 5^2 - 7^2}{2 \cdot 3 \cdot 5} \]
\[ \cos(\angle ACB) = \frac{9 + 25 - 49}{30} \]
\[ \cos(\angle ACB) = \frac{-15}{30} \]
\[ \cos(\angle ACB) = -0.5 \]

### Problem 3:
Find the volume of a pyramid with a base side length of 2 cm and a height of \( 4\sqrt{6} \) cm.

**Solution:**
The volume \( V \) of a pyramid is given by:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

The base area for a square base with side length \( s \) is:
\[ \text{Base Area} = s^2 \]
For \( s = 2 \) cm:
\[ \text{Base Area} = 2^2 = 4 \text{ cm}^2 \]

Given height \( h = 4\sqrt{6} \) cm:
\[ V = \frac{1}{3} \times 4 \times 4\sqrt{6} \]
\[ V = \frac{16\sqrt{6}}{3} \text{ cm}^3 \]

### Problem 4:
Calculate the probability of selecting a student whose shoe size is greater than the average shoe size.

**Solution:**
Without specific numerical data from the document, we assume a normal distribution of shoe sizes. If the shoe sizes are normally distributed, the probability of selecting a student with a shoe size greater than the average is 0.5 (50%) because the normal distribution is symmetric about the mean.

### Problem 5:
Determine the solutions to the inequality \( x^2 + 3x - 4 < 0 \).

**Solution:**
First, find the roots of the corresponding equation \( x^2 + 3x - 4 = 0 \) by factorizing.

\[ x^2 + 3x - 4 = (x + 4)(x - 1) = 0 \]

The roots are:
\[ x = -4 \]
\[ x = 1 \]

The inequality \( x^2 + 3x - 4 < 0 \) holds between the roots:
\[ -4 < x < 1 \]

### Any questions or further details required?

Feel free to ask any of the following questions for more practice or clarity:
1. How do you solve quadratic equations by factorization?
2. What is the cosine rule and when is it used?
3. How do you calculate the volume of a pyramid?
4. How do you determine the probability of an event in a normal distribution?
5. How do you solve polynomial inequalities?

**Tip:** Always verify the solutions to quadratic equations by substituting them back into the original equation to ensure they satisfy the equation.

Explore solutions to various mathematical problems including quadratic equations, trigonometric calculations using the cosine rule, volume calculation of a pyramid, and probability in normal distribution. Detailed step-by-step solutions provided for Grades 9-12 students.

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