This is a Grade 12 mathematics test covering various topics, divided into three main questions with a total of 50 marks. Here’s a breakdown of each question:

Question 1: Functions (11 marks)

• 1.1: Determine the values of $a$ and $b$ for the functions $f(x) = bx^2$ and $g(x) = ax$.
• 1.2: Suggest a restriction for $f(x)$ so its inverse will be a function.
• 1.3: Find the equation for the inverse of $f$.
• 1.4: Write the inverse of $g$ as $g^{-1}(x) = \ldots$.
• 1.5: Describe the transformation from $g(x)$ to $h(x) = g(x - 2) + 3$ and provide the equation for $h(x)$.

Question 2: Trigonometry (29 marks)

• 2.1: Complete trigonometric expressions (cosine and sine).
• 2.2: Express trigonometric terms in terms of $t$, given $\cos(48^\circ) = t$.
• 2.3: Prove a trigonometric identity.
• 2.4: Calculate a trigonometric expression without using a calculator.
• 2.5: Find the general solution for $x$ in a trigonometric equation.
• 2.6: Given $\sin A = \frac{12}{13}$ and $A$ is within a certain range, determine $\cos(2A)$ using a diagram.

Question 3: Geometry and Tangents (10 marks)

• 3.1: Prove that $\triangle PTS \sim \triangle SPQ$ with reasons.
• 3.2: Show that $PTS \cdot SP = \ldots$.
• 3.3: Prove that $RTR \cdot QR = PR \cdot RS$.

If you need detailed solutions or explanations for any specific part, please let me know!

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Further Questions for Practice:

1. How would you find the inverse of a quadratic function?
2. What transformations occur when modifying the input and output of a function, such as $g(x - 2) + 3$?
3. What is the general solution approach for trigonometric equations?
4. How do you prove similarity between triangles in geometry?
5. How can you calculate angles without a calculator using trigonometric identities?

Tip: When working with trigonometric identities, it helps to remember key identities, like $\sin^2 x + \cos^2 x = 1$, to simplify complex expressions.