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Here are the solutions for each of the multiple-choice questions (MCQs) in the image:

1. **If \( \frac{3^2}{3^4} = 3^x \), then \( x \) is equal to:**
   - Simplifying the expression: \( \frac{3^2}{3^4} = 3^{2-4} = 3^{-2} \).
   - So, \( x = -2 \).
   - **Answer: (B) -2**

2. **Ratio has ____ unit:**
   - A ratio compares quantities of the same unit.
   - **Answer: (C) Same**

3. **A Triangle having two sides congruent is called:**
   - A triangle with two equal sides is called an isosceles triangle.
   - **Answer: (A) Isosceles**

4. **The logarithm of unity to any base is:**
   - The logarithm of 1 to any base is 0: \( \log_b(1) = 0 \).
   - **Answer: (B) 0**

5. **\( a^2 - ab = ? \):**
   - This is a simplification based on the choices.
   - \( a^2 - ab = a(a-b) \).
   - **Answer: (D) \( a(a-b) \)**

6. **Find \( x \) in \( (2x + 5)^2 = 25 \):**
   - \( (2x + 5) = \pm 5 \), so \( 2x + 5 = 5 \) or \( 2x + 5 = -5 \).
   - Solving gives \( x = 0 \) or \( x = -5 \).
   - **Answer: (A) 0 or -5**

7. **If \( a = 4 \), what is \( x^2 = a \) a complete square:**
   - This looks like an incomplete question. Based on the options, when \( a = 4 \), \( x = 2 \) makes \( x^2 = 4 \), a perfect square.
   - **Answer: (B) 2**

8. **Simplify: \( \frac{\frac{3}{4} - \frac{1}{2}}{\frac{1}{3} + \frac{1}{6}} \):**
   - \( \frac{3/4 - 1/2}{1/3 + 1/6} = \frac{1/4}{1/2} = \frac{1/4 \times 2/1}{1} = \frac{1}{2} \).
   - **Answer: (A) \(\frac{1}{2}\)**

9. **\( x \) is a solution of the inequality \( 2x + 5 > 9 \):**
   - Solve the inequality: \( 2x + 5 > 9 \) gives \( x > 2 \).
   - **Answer: (A) \( x > 2 \)**

10. **The distance between points \( (0, 0) \) and \( (1,1) \) is:**
    - Using the distance formula: \( \sqrt{(1-0)^2 + (1-0)^2} = \sqrt{1 + 1} = \sqrt{2} \).
    - **Answer: (A) \( \sqrt{2} \)**

11. **The median used for different purposes between two triangles is:**
    - The Centroid Theorem is used.
    - **Answer: (A) Centroid**

12. **Measure of straight \( \angle B \):**
    - This question lacks sufficient information, but based on options:
    - **Answer: (A) 90°**

13. **The perpendicular bisector of a chord:**
    - A perpendicular bisector of a chord in a circle passes through the center.
    - **Answer: (A) Bisector**

14. **The area enclosed by a triangle having vertices at (0, 0), (1, 0), and (0, 1) is:**
    - The area of the triangle is \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \).
    - **Answer: (B) \( \frac{1}{2} \)**

15. **\( 3x^2 = 36 \), then \( x = \):**
    - Solve: \( 3x^2 = 36 \) gives \( x^2 = 12 \), so \( x = \pm \sqrt{12} \).
    - **Answer: (B) \( 6 \)**

If you need any clarification on these solutions or have further questions, feel free to ask!

**Related Questions:**
1. What is the general formula for the distance between two points?
2. How do you determine if a number is a perfect square?
3. What is the relationship between logarithms and exponents?
4. How do you solve quadratic equations by factoring?
5. What are the properties of the centroid in a triangle?

**Tip:** When dealing with inequalities, always check if the inequality sign changes when multiplying or dividing by a negative number.

Explore solutions to multiple-choice questions (MCQs) covering exponents, ratios, triangles, logarithms, inequalities, and quadratic equations. Understand key mathematical concepts and formulas required for each question. Suitable for Grades 7-9.

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