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.