https://file.aiquickdraw.com/mathbot/file/17267031810391v4pq45y.jpg

The image you provided seems to be a question paper or exercise sheet with mathematical problems written in Bengali. Here's a transcription and interpretation of the problems I can read:

1. **Set Theory Question:**
   Given sets:
   \[
   P = \{a, b\}, \quad Q = \{1, 2, 3\}, \quad R = \{2, 3\}
   \]
   You are asked to find the following:
   \[
   (P \times Q) \cap (P \times R) \quad \text{(P cross Q intersect P cross R)}
   \]
   This requires calculating the Cartesian product of sets and finding the intersection.

2. **Arithmetic Progression Question:**
   Find the sum of the sequence:
   \[
   25, 125, -625, \dots
   \]
   This appears to be a geometric series where you are asked to either find the sum or a particular term of the series.

3. **Logarithmic Calculation:**
   You are asked to solve or simplify:
   \[
   x \log_b y
   \]

4. **Logarithmic Property Problem:**
   Use the property:
   \[
   \log_a b \times \log_b c
   \]
   to either prove or simplify the expression.

5. **Geometrical Progression and Triangle Problem:**
   There is also a question involving a geometric progression and triangles, but part of the image is obscured.

Would you like a solution for one of these questions or explanations in detail?

Here are five related questions:
1. How is the Cartesian product of two sets calculated?
2. What is the general formula for the sum of a geometric series?
3. How can you simplify expressions using logarithmic properties?
4. What is the relation between geometric progression and sums?
5. How are arithmetic and geometric progressions different in terms of their properties?

**Tip:** For logarithms, understanding their properties like the product, quotient, and power rules helps simplify many complex expressions.

Given sets P = {a,b}, Q = {1,2,3}, and R = {2,3}, find (P × Q) ∩ (P × R). Also, solve arithmetic and geometric progressions and logarithmic problems from the provided set of questions.

This page provides a detailed solution to problems related to set theory, arithmetic and geometric progressions, and logarithmic calculations for Grade 10-12 students. It covers key formulas and theorems, including Cartesian products, geometric series sums, and logarithmic identities.

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