https://file.aiquickdraw.com/mathbot/file/17259593696142yd2nz3m.jpg

The image you uploaded is a mathematics exam paper for SSC-I (Science Group). It contains several questions from Sections B and C. Here is an overview of the questions present in the image:

### Section B (Marks: 36)
You are asked to attempt any nine parts, with each part carrying equal marks. The parts are:

1. Simplify the given expression:  
   \[
   \frac{4(3)^x}{3^{x+1} - 3^x}
   \]

2. Find the values of \(x\) and \(y\) given:  
   \[
   (3 - 2)(x + y) = 2(x - y) + 2 - 1
   \]

3. Solve the logarithmic equation:  
   \[
   \log_{10} (0.001) = 3
   \]

4. If \(x + y = 1, x - y = 3\), then find the values of:  
   - \(4xy\)  
   - \(2(x^2 + y^2)\)  
   - \(8xy(x^2 + y^2)\)

5. If the area of rectangles is \(48x^2 - 81\), find the sides using factorization.  
   - (a) Sides of rectangle  
   - (b) Perimeter of the rectangle  

6. Factorize the quadratic expression:  
   \[
   2x^4 + x^3 - 2x - 1
   \]

7. Find the square root by factorization:  
   \[
   (x^4 + x^3) - \left(5x^2 + \frac{1}{4}\right)
   \]

8. Find the solution set for the equation:  
   \[
   \frac{2}{x - 4} = \frac{1}{x + 2} + \frac{1}{x - 4}
   \]

9. Find the values of \(m\) and \(c\) by comparing with the equation \(2x + 3y = 10\).

10. Verify whether the points \(O(0,0), A(\sqrt{5},1), B(\sqrt{5}, -1)\) are the vertices of an equilateral triangle.

11. Prove that the right bisectors of the sides of a triangle are concurrent.

12. In the figure, \(PL\) is perpendicular to line \(AB\) and \(\angle NLX = \angle LMX\). Prove that \(mPX > mPM\).

13. In triangle \(\triangle ABC\), shown in figure, line \(CD\) bisects \(\angle C\). If \(\angle ACD = 3\), \(\angle CBD = 6\), and \(\angle DBC = 7\), then find:  
    - (a) \(m_{\angle AD}\)  
    - (b) \(m_{\angle DB}\)

14. A student travels to school by the route shown in the figure. Find the direct distance from the house to the school.

### Section C (Marks: 24)
Attempt any three questions, each carrying equal marks.

1. **Q3:** The length of a rectangular playground is twice its width. The perimeter of the ground is 30m. Find its dimensions using Cramer's rule.
   
2. **Q4:** In a correspondence of two triangles, if three sides of one triangle are congruent to the corresponding three sides of the other, then prove that two triangles are congruent.

3. **Q5:** In a right-angled triangle, prove that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

4. **Q6:** Prove that the medians of a triangle are concurrent, and their point of concurrency is the point of trisection of each median.

5. **Q7:** Construct a triangle \(\triangle PQR\) with given sides, draw its altitude, and find the area of the triangle.

### How would you like to proceed?
- Would you like me to solve a specific question or provide detailed solutions for multiple questions?
- Please specify which question(s) you are interested in.

### Follow-up Questions
1. Would you like a detailed solution for Question 2 part (i) involving the simplification of an algebraic fraction?
2. Should I solve the quadratic factorization in Question 2 part (vi)?
3. Do you need help with the logarithmic equation in Question 2 part (iii)?
4. Are you interested in proving geometric properties from Section C, such as the concurrency of medians?
5. Would you like a step-by-step construction guide for Question 7?

### Tip
When solving geometry problems, always start by drawing a clear diagram to visualize the given information and relationships.

The uploaded image contains several math problems from an SSC-I (Science Group) mathematics exam paper. It includes algebraic expressions, factorization, solving equations, logarithmic equations, geometry problems, and theorems.

This SSC-I mathematics exam paper covers various algebraic problems, quadratic equations, logarithmic equations, factorization, geometry, and important theorems like triangle congruence and Pythagoras' theorem. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation