Let us first consider the situation when there is no liquid between lens and plane mirror and the image is formed at 30 cm i.e., at the position of object.
As the image is formed on the object position itself, the object must be placed at focus of Biconvex lens.
fo = 30 cm
Radius of curvature of convex lens can be calculated

1

f

= (

a

‌

µg−1) (

1

R1

−

1

R2

)
or

1

30

= (

3

2

−1)(

1

R

−

1

−R

)
or

1

30

=

1

2

(

2

R

) ⇒ R = 30 cm
Now a liquid is filled between lens and plane mirror and the image is formed at position of object at 45 cm.
The image is formed on the position of object itself, the object must be placed at focus of equivalent lens of Biconvex of glass and Plano convex lens of liquid

1

feq

=

1

f1

+

1

f2

... (i)
Equivalent total length feq = 45 cm
Focal length of Biconvex lens f1 = 30 cm
Focal length of plano convex lens

1

f2

= [µ - 1] [

1

−R

−

1

∞

]
or

1

f2

= [µ - 1] [−

1

30

] , f2 =

−30

µ−1

Now equation (i),

1

feq

=

1

f1

+

1

f2

⇒

1

45

=

1

30

−(

µ−1

30

)

µ−1

30

=

1

90

or µ - 1 =

1

3

, µ =

1

3

+1 =

4

3