# Mathematics Grade 9 Exam
## Instructions:
1. **Time Allowed:** 2 hours
2. **Total Marks:** 100
3. **Answer all questions.**
---
### Section A: Multiple Choice Questions (20 marks)
**Instructions:** Choose the correct answer for each question. Each question carries 2 marks.
1. **Which of the following is a linear equation?**
- A) \( y = x^2 + 3 \)
- B) \( y = 2x + 5 \)
- C) \( y = \frac{1}{x} \)
- D) \( y = \sqrt{x} \)
2. **Solve for \( x \) in the equation \( 3x - 7 = 8 \).**
- A) \( x = 5 \)
- B) \( x = 3 \)
- C) \( x = 2 \)
- D) \( x = 1 \)
3. **What is the slope of the line \( 2y = 4x + 6 \)?**
- A) 2
- B) 4
- C) 3
- D) 1
4. **Simplify \( \frac{3x^2 + 6x}{3x} \).**
- A) \( x + 2 \)
- B) \( x^2 + 2 \)
- C) \( x + 6 \)
- D) \( x^2 + 6 \)
5. **Which of the following is a quadratic equation?**
- A) \( y = 3x + 2 \)
- B) \( y = x^2 + 4x - 5 \)
- C) \( y = \frac{2}{x} \)
- D) \( y = \sqrt{x} \)
---
### Section B: Short Answer Questions (40 marks)
**Instructions:** Answer each question in detail. Each question carries 8 marks.
6. **Solve the equation \( 2x^2 - 5x - 3 = 0 \) using the quadratic formula.**
7. **Find the equation of the line that passes through the points (2, 3) and (4, 7).**
8. **Simplify the expression \( \frac{4x^2 - 9}{2x + 3} \).**
9. **Determine the value of \( k \) for which the system of equations \( 2x + 3y = 6 \) and \( kx + 9y = 18 \) has infinitely many solutions.**
10. **Factorize the expression \( x^2 - 7x + 12 \).**
---
### Section C: Essay Questions (40 marks)
**Instructions:** Write a detailed answer for each question. Each question carries 20 marks.
11. **Discuss the importance of understanding linear equations in real-world applications. Provide examples from different fields such as economics, physics, and engineering.**
12. **Explain the process of solving a quadratic equation by completing the square. Provide a step-by-step example to illustrate your explanation.**
---
## Answer Key
### Section A: Multiple Choice Questions
1. **B**
2. **A**
3. **A**
4. **A**
5. **B**
### Section B: Short Answer Questions
6. **Solution:**
\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2} = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm \sqrt{49}}{4} = \frac{5 \pm 7}{4}
\]
\[
x = \frac{12}{4} = 3 \quad \text{or} \quad x = \frac{-2}{4} = -\frac{1}{2}
\]
7. **Solution:**
\[
\text{Slope} = \frac{7 - 3}{4 - 2} = 2
\]
\[
\text{Equation: } y - 3 = 2(x - 2) \Rightarrow y = 2x - 1
\]
8. **Solution:**
\[
\frac{4x^2 - 9}{2x + 3} = \frac{(2x - 3)(2x + 3)}{2x + 3} = 2x - 3
\]
9. **Solution:**
\[
\text{For infinitely many solutions, } \frac{k}{2} = \frac{9}{3} \Rightarrow k = 6
\]
10. **Solution:**
\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]
### Section C: Essay Questions
11. **Model Answer:**
Linear equations are fundamental in various real-world applications. In economics, linear equations model supply and demand curves, helping to predict market behavior. In physics, they describe motion and force relationships. In engineering, they are used in circuit analysis and structural design. Understanding these equations allows for accurate predictions and efficient problem-solving in these fields.
12. **Model Answer:**
Completing the square involves transforming a quadratic equation into a perfect square trinomial. For example, solve \( x^2 + 6x + 5 = 0 \):
\[
x^2 + 6x + 9 - 9 + 5 = 0 \Rightarrow (x + 3)^2 - 4 = 0 \Rightarrow (x + 3)^2 = 4 \Rightarrow x + 3 = \pm 2
\]
\[
x = -1 \quad \text{or} \quad x = -5
\]
Notis
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