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I can help you with solving the questions in the image you've uploaded! Let's break them down one by one.

### Section A:

1. **Question 1 (i)**: In what ratio is the line joining the points (4,2) and (3,5) divided by X-axis? Also, find the co-ordinates of point of intersection.
   - **Solution**: The X-axis divides the line joining two points where the y-coordinate becomes zero. You can use the section formula and find the point of intersection.

2. **Question 1 (ii)**: A straight line passes through the point (3,2). It meets the Y-axis at point B. If \( \frac{PA}{PB} = \frac{2}{3} \), find the equation of the line that passes through point P and is perpendicular to line AB.
   - **Solution**: This involves the point-slope form and condition of perpendicularity to solve for the equation of the line.

3. **Question 1 (iii)**: Draw a circle circumscribing a regular hexagon with side 5 cm.
   - **Solution**: For a regular hexagon, the circumcircle radius is equal to the side length. So, the radius of the circle is 5 cm.

### Section B:

1. **Question 4 (i)**: What length of a solid cylinder, 2 cm in diameter, must be taken to recast into a hollow cylinder of external diameter 20 cm, 0.25 cm thick, and 15 cm long?
   - **Solution**: You will use the volume formula of the cylinder here. Equate the volume of the solid cylinder to the volume of the hollow cylinder and solve for the required length.

2. **Question 4 (ii)**: Evaluate \( 3 \cos 80^\circ + \csc 10^\circ + 2 \sin 59^\circ \sec 31^\circ \).
   - **Solution**: Plug the values of the trigonometric functions and simplify to find the result.

### Question 5:

You have a table where you need to evaluate some mathematical expressions for different values of \(x\) and \(f\). You need to solve for these using the given data.

Let me know which specific problem you'd like detailed help with, and I can provide step-by-step solutions!

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Would you like a detailed solution to any specific question? Here are 5 related questions to further explore:

1. How do you find the equation of a line perpendicular to another?
2. What is the section formula in coordinate geometry?
3. How do you calculate the volume of a hollow cylinder?
4. How can you simplify trigonometric expressions using identities?
5. What is the mean of a frequency distribution?

**Tip**: For geometry questions, always visualize the shapes, and for algebraic questions, break the problem into smaller parts.

In what ratio is the line joining the points (4,2) and (3,5) divided by X-axis? Also, find the co-ordinates of point of intersection. A straight line passes through the point (3,2). It meets the Y-axis at point B. If PA/PB = 2/3, find the equation of the line that passes through point P and is perpendicular to line AB. Draw a circle circumscribing a regular hexagon with side 5 cm. What length of a solid cylinder, 2 cm in diameter, must be taken to recast into a hollow cylinder of external diameter 20 cm, 0.25 cm thick, and 15 cm long? Evaluate 3cos(80°) + csc(10°) + 2sin(59°) sec(31°).

Explore key mathematical concepts like coordinate geometry, trigonometric identities, and the volume of cylinders. Solve problems involving line divisions by the X-axis, perpendicular lines, and circumcircle radius for regular hexagons. Learn how to calculate the volume of a hollow cylinder and simplify trigonometric expressions like 3cos(80°) + csc(10°) + 2sin(59°) sec(31°). Suitable for students in Grades 9-12.

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