To solve the problem step by step, we will analyze the situation involving the convex lens and the concave mirror.
### Step 1: Identify the given values
- Object distance from the convex lens, \( u_1 = -40 \, \text{cm} \) (negative as per sign convention)
- Focal length of the convex lens, \( f_1 = +20 \, \text{cm} \) (positive for convex lens)
- Focal length of the concave mirror, \( f_2 = -10 \, \text{cm} \) (negative for concave mirror)
- Distance from the lens to the mirror, which we will calculate.
### Step 2: Calculate the image distance from the convex lens
Using the lens formula:
\[
\frac{1}{f} = \frac{1}{v} - \frac{1}{u}
\]
Rearranging gives:
\[
\frac{1}{v_1} = \frac{1}{f_1} + \frac{1}{u_1}
\]
Substituting the known values:
\[
\frac{1}{v_1} = \frac{1}{20} + \frac{1}{-40}
\]
Calculating the right-hand side:
\[
\frac{1}{v_1} = \frac{1}{20} - \frac{1}{40} = \frac{2 - 1}{40} = \frac{1}{40}
\]
Thus,
\[
v_1 = 40 \, \text{cm}
\]
This image is formed on the opposite side of the lens.
### Step 3: Determine the object distance for the concave mirror
The distance from the lens to the mirror is \( 60 \, \text{cm} \) (since the total distance from the object to the mirror is \( 100 \, \text{cm} \) and the object is \( 40 \, \text{cm} \) from the lens). Therefore, the object distance for the concave mirror is:
\[
u_2 = - (60 - 40) = -20 \, \text{cm}
\]
(negative because the object is on the same side as the incoming light).
### Step 4: Calculate the image distance from the concave mirror
Using the mirror formula:
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\]
Rearranging gives:
\[
\frac{1}{v_2} = \frac{1}{f_2} - \frac{1}{u_2}
\]
Substituting the known values:
\[
\frac{1}{v_2} = \frac{1}{-10} - \frac{1}{-20}
\]
Calculating the right-hand side:
\[
\frac{1}{v_2} = -\frac{1}{10} + \frac{1}{20} = -\frac{2}{20} + \frac{1}{20} = -\frac{1}{20}
\]
Thus,
\[
v_2 = -20 \, \text{cm}
\]
This means the image formed by the concave mirror is \( 20 \, \text{cm} \) in front of the mirror (on the same side as the object).
### Step 5: Determine the object distance for the convex lens again
The distance from the lens to the image formed by the mirror is \( 60 \, \text{cm} - 20 \, \text{cm} = 40 \, \text{cm} \). Therefore, the object distance for the convex lens is:
\[
u_3 = -40 \, \text{cm}
\]
### Step 6: Calculate the final image distance from the convex lens
Using the lens formula again:
\[
\frac{1}{v_3} = \frac{1}{f_1} + \frac{1}{u_3}
\]
Substituting the known values:
\[
\frac{1}{v_3} = \frac{1}{20} + \frac{1}{-40}
\]
Calculating the right-hand side:
\[
\frac{1}{v_3} = \frac{1}{20} - \frac{1}{40} = \frac{2 - 1}{40} = \frac{1}{40}
\]
Thus,
\[
v_3 = 40 \, \text{cm}
\]
### Conclusion
The final image formed after refraction from the lens, reflection from the mirror, and again refraction from the lens is located at:
- **Final image distance from the lens = 40 cm (on the opposite side of the lens)**
- **Nature of the image: inverted and of the same size as the object.**
