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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.

This is a Grade 12 mathematics test covering various topics, divided into three main questions with a total of 50 marks.

Explore a Grade 12 mathematics test with questions on functions, trigonometric identities, and geometric proofs. This test covers inverse functions, transformations, general solutions for trigonometric equations, and similarity in triangles. Total marks: 50.

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