Final answer:
To find the focal length of the convex lens, we utilize the lens formula and the magnification equation. However, without exact object and image distances, we can't solve this problem numerically. We know that moving the lens through its focal point changes the nature of the image, which helps to understand how magnification is related to lens position. We cannot find a numerical answer with definite options A to D.
Step-by-step explanation:
The question involves finding the focal length of a convex lens where an object is placed between the lens and the screen resulting in two images with different sizes.
Given that the image size changes by a factor of 4 when the lens is moved, we know that there are two different image distances corresponding to two positions of the lens. One position creates a smaller image that is real and inverted, and another creates an image that is 4 times larger, likely to be virtual and erect if obtained by diverging rays due to the lens being within the focal length of the object.
Without concrete distances for object and image positions, we can't solve this numerically, but we understand that the lens is being moved through its focal point because that's when the nature of the image transitions from real and inverted to virtual and erect. Considering the information given in the examples provided on lenses, we can deduce that the image forming 4 times larger than the object implies that the lens must have moved closer to the object than its focal length. This is supported by the concept that a convex lens used as a magnifier (where the image is larger and virtual) must be within one focal length of the object.
However, without the specific distances for lens positions or an exact object distance (other than it being within the focal length for the larger image), we cannot find a numerical answer with definite options A to D.