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The image contains a series of linear programming problems and optimization tasks. Let's go through each question one by one and provide detailed solutions:

### Question 3:
**Problem Statement:**
Two grades of paper, \(x\) and \(y\), are produced on a paper machine. Due to raw material restrictions, not more than 400 tonnes of grade \(x\) and 300 tonnes of grade \(y\) can be produced in a week. There are 160 production hours available per week. It requires 0.2 hours to produce one tonne of product \(x\) and 0.4 hours for product \(y\). The profit is ₹200 per tonne of \(x\) and ₹500 per tonne of \(y\). Find the optimum product mix using the graphical solution method.

**Mathematical Formulation:**
Objective: Maximize profit, \( Z = 200x + 500y \)

**Constraints:**
1. \( x \leq 400 \) (Maximum production limit of grade \(x\))
2. \( y \leq 300 \) (Maximum production limit of grade \(y\))
3. \( 0.2x + 0.4y \leq 160 \) (Production hours constraint)
4. \( x \geq 0, y \geq 0 \) (Non-negativity constraint)

**Solution:**
To solve this using a graphical method:
1. Plot the constraints on a graph.
2. Identify the feasible region.
3. Calculate the profit at each vertex of the feasible region.
4. The optimal solution occurs at the vertex where the profit is maximized.

From the given answer:
Optimal solution: \( x = 200 \) tonnes, \( y = 300 \) tonnes, \( Z = ₹190000 \).

### Question 4:
**Use the graphical method to solve the following problems:**

#### (i) (a) Minimize \( Z = 2x_1 + x_2 \)
**Constraints:**
1. \( 2x_1 + 4x_2 \geq 8 \)
2. \( x_1 - 3x_2 \leq 6 \)
3. \( x_1, x_2 \geq 0 \)

**(b) Maximize \( Z = 2x_1 + 3x_2 \)** with the same constraints as above.

**Solution for (i):**
Plot the constraints and find the feasible region. Check each vertex of the feasible region to find the minimum and maximum values of \(Z\).

Given answer:
- \( x_1 = 0, x_2 = 2 \) (for both min and max solutions)
- Min: \( Z = 2 \), Max: \( Z = 6 \) (if unbounded or specific to solution details).

#### (ii) Maximize \( Z = 3x_1 + 4x_2 \)
**Constraints:**
1. \( x_1 - x_2 \geq 0 \)
2. \( 2.5x_1 - x_2 \leq 3 \)
3. \( x_1, x_2 \geq 0 \)

**Solution for (ii):**
Graph the constraints and find the feasible region. From the provided answer:
- No solution.

#### (iii) Maximize \( Z = 8x_1 + x_2 \)
**Constraints:**
1. \( 2x_1 + x_2 \leq 6 \)
2. \( 3x_1 + x_2 \leq 8 \)
3. \( x_1 + 6x_2 \leq 8 \)
4. \( x_1, x_2 \geq 0 \)

**Solution for (iii):**
Check all intersection points of these inequalities. From the provided answer:
- Multiple optimal solutions.

### Question 5:
**Express the following LPP in standard form:**
Maximize \( Z = 3x_1 + 2x_2 + 5x_3 \)

**Subject to:**
1. \( 2x_1 + 3x_2 - 2x_3 \leq 40 \)
2. \( 4x_1 - 2x_2 + x_3 \leq 94 \)
3. \( x_1 - 5x_2 - 6x_3 \geq 2 \)
4. \( x_1 \geq 0 \)

**Standard Form:**
To express in standard form, convert all inequalities to equalities using slack or surplus variables:
1. \( 2x_1 + 3x_2 - 2x_3 + s_1 = 40 \)
2. \( 4x_1 - 2x_2 + x_3 + s_2 = 94 \)
3. \( x_1 - 5x_2 - 6x_3 - s_3 = 2 \)
4. \( x_1, x_2, x_3, s_1, s_2, s_3 \geq 0 \)

Would you like more details on any of these solutions or have any specific questions?

Here are 5 related questions to explore further:
1. How can we solve linear programming problems using the Simplex method?
2. What are the differences between graphical and algebraic methods for solving LP problems?
3. How do we handle unbounded solutions in linear programming?
4. What are the steps to formulate a linear programming problem from a real-world scenario?
5. How do slack, surplus, and artificial variables

Solve the linear programming problems involving two grades of paper and optimization using the graphical method.

Solve linear programming problems for paper production optimization with constraints on raw materials and time using the graphical method. This solution explains step-by-step the process of finding the optimum mix of two grades of paper for maximum profit.

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