$A$ thin lens of refractive index $1.5$ has an optical power of $-5 \ D$ in air. Its optical power in a liquid medium with a refractive index of $1.6$ is $D$.

$A$ magnifying glass is to be used at a fixed object distance of $1 \text{ inch}$. If it is to produce an erect image magnified $5$ times,its focal length should be ........ $\text{inch}$.

$A$ double convex lens has two surfaces of equal radii $15 \, cm$ and refractive index $\mu = 1.5$. Its focal length is equal to ..... $cm$.

The radius of curvature of the curved face of a plano-convex lens is $12 \,cm$ and its refractive index is $1.5$. Then,the focal length of the lens is (in $\,cm$)

$A$ thin convex lens is made of two materials with refractive indices $n_1$ and $n_2$,as shown in the figure. The radii of curvature of the left and right spherical surfaces are equal. $f$ is the focal length of the lens when $n_1 = n_2 = n$. The focal length is $f + \Delta f$ when $n_1 = n$ and $n_2 = n + \Delta n$. Assuming $\Delta n \ll (n - 1)$ and $1 < n < 2$,which of the following statement$(s)$ is/are correct?
$(1)$ The relation between $\frac{\Delta f}{f}$ and $\frac{\Delta n}{n}$ remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.
$(2)$ $\left|\frac{\Delta f}{f}\right| < \left|\frac{\Delta n}{n}\right|$
$(3)$ For $n = 1.5, \Delta n = 10^{-3}$ and $f = 20 \text{ cm}$,the value of $|\Delta f|$ will be $0.04 \text{ cm}$.
$(4)$ If $\frac{\Delta n}{n} < 0$ then $\frac{\Delta f}{f} > 0$.

$A$ student measures the focal length of a convex lens by obtaining an image of an object pin at a distance '$v$' from the lens,with the object placed at a distance '$u$' from the lens. What will the graph between '$u$' and '$v$' look like?