Firstly, let's break down the given information to find out what we need to solve this problem:

• Distance from light source to screen (x) = 1.5 m
• Distance from new screen position to light source (x') = 3 m
• Image size from first position (I) = 18 mm
• Image size from second position (I') = 96 mm
• Focal length of lens (f)
• Size of light source (S)

Using the thin lens formula, we can solve for the focal length:

1/f = 1/x + 1/x'

1/f = 1/1.5 + 1/3

1/f = 3/6

f = 6/3 = 2 m

Now, let's calculate the magnification (M) of the lens in both positions:

M = -x'/x = -3/1.5 = -2

The magnification (M) can also be calculated using the image and object distances:

M = I'/I = 96/18 = 5.33

Since we know that the magnification (M) is equal to -2, we can set up the following equation:

I'/I = -2 = -x'/x

-2 = -3/1.5

x' = -3/2 m

To find the size of the light source (S), we can use the equation:

M = -x'/x = S'/S

-2 = -3/2S

S = 1.5/2 = 0.75 m

Therefore, the focal length of the lens is 2 m and the size of the light source is 0.75 m.