Suppose you have three concave mirrors $A$, $B$ and $C$ of focal lengths $10 \text{ cm}$, $15 \text{ cm}$ and $20 \text{ cm}$ respectively. For each concave mirror, you perform an experiment of image formation for three object distances of $10 \text{ cm}$, $20 \text{ cm}$ and $30 \text{ cm}$. Giving reasons, answer the following:
$(a)$ For the three object distances, identify the mirror$(s)$ which will form an image of magnification $m = -1$.
$(b)$ Out of the three mirrors, identify the mirror which would be preferred for shaving or makeup purposes.
$(c)$ For mirror $B$, draw the ray diagrams for image formation for object distances $10 \text{ cm}$ and $20 \text{ cm}$.

(N/A) Given: Focal lengths $f_A = 10 \text{ cm}$, $f_B = 15 \text{ cm}$, $f_C = 20 \text{ cm}$. Object distances $u = 10 \text{ cm}, 20 \text{ cm}, 30 \text{ cm}$.
$(a)$ Magnification $m = -1$ occurs when the object is placed at the center of curvature, i.e., $u = 2f = R$.
- For mirror $A$ $(f=10 \text{ cm})$, $2f = 20 \text{ cm}$. Thus, at $u = 20 \text{ cm}$, $m = -1$.
- For mirror $B$ $(f=15 \text{ cm})$, $2f = 30 \text{ cm}$. Thus, at $u = 30 \text{ cm}$, $m = -1$.
- Mirror $C$ $(f=20 \text{ cm})$ has $2f = 40 \text{ cm}$, which is not among the given object distances.
$(b)$ For shaving or makeup, a concave mirror is used to obtain an erect and magnified image. This happens when the object is placed between the pole and the focus $(u < f)$.
- Mirror $A$: $u=10 \text{ cm}$ is at the focus $(u=f)$, image at infinity.
- Mirror $B$: $u=10 \text{ cm}$ $(u < 15 \text{ cm})$, so it forms an erect and magnified image.
- Mirror $C$: $u=10 \text{ cm}$ and $u=20 \text{ cm}$ ($u < 20 \text{ cm}$ and $u=f$ respectively). $u=10 \text{ cm}$ forms an erect and magnified image.
Mirror $B$ or $C$ can be used, but $C$ provides a larger field of view and magnification range for $u < 20 \text{ cm}$.
$(c)$ Refer to the provided ray diagram.