1. a biconveex lens of focal length 15 cm is lcaed inn front of plane mirror 10 cm away from the lens . what is the nature and the postion of final image draw the ray diagram also???

Dear Student ,
Here in this case from the following figure the lens has a focal length of 15 cm and let the object be at a distance 2f from the lens .Therefore a real image I₁ will be formed at a distance 2f(= 30 cm) on the other side of the lens if the mirror is absent .sinc the image I₁ is at a distance 20 cm from the plane mirror and a real image I₂ would be formed ( by the reflection of the rays at the plane mirror ) at a distance 20 cm from the mirror .But the returning rays are further refracted by the lens and the final image is formed at I₃ .The image I₂ serves as the object for the lens for the formation of the final image I₃ .

As clear from the figure the object distance u₃ for the formation of final image is 10 cm .Therefore using lens formula and cartesian sign convention we can write that ,
$$\begin{gathered} \frac{1}{v}-\frac{1}{u}=\frac{1}{f} \\ \Rightarrow \frac{1}{v_3}-\frac{1}{10}=\frac{1}{15} \\ \Rightarrow \frac{1}{v_3}=\frac{1}{10}+\frac{1}{15} \\ \Rightarrow v_3= 6 cm \\ therefore the dis\tan ce of the final image from the mirror = 10+6 =16 cm . \\ Also the image formed will be real . \end{gathered}$$
Regards