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

1. **If \( 3^2 \div 3^4 = 3^x \), then x is equal to:**
   - Using the properties of exponents: \( \frac{3^2}{3^4} = 3^{2-4} = 3^{-2} \).
   - Therefore, \( x = -2 \).
   - **Answer: (B) -2**

2. **Ratio has same unit:**
   - A ratio compares two quantities with the same units.
   - **Answer: (C) Same**

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

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

5. **3 equals to:**
   - Simplifying the options:
     - \( \frac{18}{6} = 3 \)
   - **Answer: (D) \(\frac{18}{6}\)**

6. **Find the value of \( x \) in \( (2x + 5)^2 = 25 \):**
   - Solve: \( (2x + 5)^2 = 25 \)
   - \( 2x + 5 = \pm 5 \)
   - \( 2x + 5 = 5 \) gives \( x = 0 \)
   - \( 2x + 5 = -5 \) gives \( x = -5 \)
   - **Answer: (A) 0 or -5**

7. **If a is a set of \( x^2 - 6x + 9 = 0 \) a complete square:**
   - Factor the quadratic equation: \( (x - 3)^2 = 0 \).
   - **Answer: (B) 3**

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

10. **Find the distance between points \( (0, 0) \) and \( (3, 4) \):**
    - Using the distance formula: \( \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).
    - **Answer: (A) 5 units**

11. **Theorem used for finding the medians of triangle:**
    - The centroid theorem states that the centroid divides each median into two parts, one-third and two-thirds of the length.
    - **Answer: (A) Centroid**

12. **The equation of straight line parallel to y-axis:**
    - A line parallel to the y-axis has the equation \( x = k \).
    - **Answer: (A) \( x = 3 \)**

13. **A circle has a center and radius. A chord perpendicular to the radius:**
    - A chord perpendicular to a radius is bisected by it.
    - **Answer: (B) Bisector**

14. **Find the area of a sector if the angle subtended is \( 30^\circ \):**
    - Area of the sector: \( \frac{\theta}{360^\circ} \times \pi r^2 \).
    - **Answer: (C) \( 30 cm^2 \)**

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

**Related Questions:**
1. What is the logarithm of 10 to the base 10?
2. How do you solve a quadratic equation by completing the square?
3. What is the area of a triangle with base 5 cm and height 12 cm?
4. How is the distance formula derived?
5. Can you explain the properties of isosceles triangles?

**Tip:** Always check your solutions by substituting them back into the original equations or expressions to verify their correctness.

Explore solutions to multiple-choice questions (MCQs) covering topics such as exponents, ratios, triangles, logarithms, quadratic equations, inequalities, and more. Each question includes a detailed explanation and step-by-step solution, suitable for Grades 9-12.

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