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Let's break down the questions and provide answers step by step:

### Question 1:
**A candle is placed in front of a flat mirror at 20 cm from the mirror.**

- **(a) Construct a ray diagram for this arrangement.**
    - In a flat mirror, the rays from the object reflect symmetrically, following the law of reflection (angle of incidence = angle of reflection). The image is formed at the same distance behind the mirror as the object is in front of it (in this case, 20 cm behind the mirror). The image will be virtual, upright, and of the same size as the object.

- **(b) Write the properties for the image.**
    - The properties of the image in a flat mirror are:
        1. The image is virtual.
        2. It is the same size as the object (not magnified).
        3. The image is upright.
        4. It is laterally inverted (left-right inversion).
        5. The image is at the same distance behind the mirror as the object is in front (20 cm in this case).

### Question 2:
**A candle is placed on the principal axis of a concave mirror at a distance of 20 cm from the mirror. The image is inverted and formed in front of the mirror, and it is magnified 3 times.**

- **(a) Without using the mathematical solution, is the image real or virtual?**
    - The image is real. Since the image is inverted and magnified, and formed in front of the concave mirror, this indicates that the image is real. Real images in concave mirrors are always inverted and formed on the same side as the object.

- **(b) Find the image distance.**
    - Since the image is magnified 3 times, we can use the magnification formula:
      \[
      \text{Magnification (m)} = \frac{-\text{Image distance} (v)}{\text{Object distance} (u)}
      \]
      Given that \( m = 3 \) and \( u = -20 \) cm (since object distance is negative for real objects in mirror convention):
      \[
      3 = \frac{-v}{-20} \implies v = -60 \text{ cm}.
      \]
      The image distance is \( v = -60 \) cm, meaning the image is 60 cm in front of the mirror.

- **(c) Find the focal length.**
    - Using the mirror equation:
      \[
      \frac{1}{f} = \frac{1}{v} + \frac{1}{u}
      \]
      Substituting the known values:
      \[
      \frac{1}{f} = \frac{1}{-60} + \frac{1}{-20}
      \]
      \[
      \frac{1}{f} = \frac{-1}{60} - \frac{1}{20} = \frac{-1 - 3}{60} = \frac{-4}{60} = \frac{-1}{15}
      \]
      Therefore, the focal length \( f = -15 \) cm.

- **(d) Construct a ray diagram for this arrangement.**
    - To construct the ray diagram:
        1. Draw the concave mirror.
        2. Place the object (candle) on the principal axis at 20 cm from the mirror.
        3. One ray should be parallel to the principal axis, reflecting through the focal point.
        4. Another ray should go through the center of curvature (30 cm from the mirror), reflecting back on itself.
        5. The rays will intersect in front of the mirror at 60 cm from the mirror, forming a real, inverted, and magnified image.

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Do you want further details on any of these points, or need clarifications?

Here are five related questions to expand on this:

1. How does the distance of the object affect the nature of the image in concave and convex mirrors?
2. What is the difference between real and virtual images in terms of light ray convergence?
3. How would the properties of the image change if the mirror was convex instead of concave?
4. Can an object placed at the focal point of a concave mirror form an image? Why or why not?
5. What happens to the size and position of the image as the object is moved closer to a concave mirror?

**Tip:** When dealing with mirrors, always remember the sign conventions for object and image distances to avoid errors in calculations.

The original question submitted by the user as text.

Explore the fundamentals of image formation in flat and concave mirrors, featuring detailed ray diagram construction and calculations of image properties and focal length. This lesson is suited for high school students studying optics.

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