An image formed by a concave mirror of focal length $6 \, cm$ is $3$ times the size of the object. The distance of the object from the mirror is......$cm$.

$A$ thin cylindrical rod of length $10\,cm$ is placed horizontally on the principal axis of a concave mirror of focal length $20\,cm$. The rod is placed in such a way that the midpoint of the rod is at $40\,cm$ from the pole of the mirror. The length of the image formed by the mirror will be $\frac{x}{3}\,cm$. The value of $x$ is $............$.

In an experiment to determine the focal length $(f)$ of a concave mirror by the parallax method,a student places the object pin $\text{A}$ on the principal axis at a distance $x$ from the pole $\text{P}$. The student looks at the pin and its inverted image from a distance,keeping his/her eye in line with $\text{PA}$. When the student shifts his/her eye towards the left,the image appears to the right of the pin. Then:

Assertion $(A):$ The relation among $u, v$ and $f$ for a spherical mirror is valid only for mirrors whose sizes are very small compared to their radii of curvature.
Reason $(R):$ The laws of reflection are strictly valid for plane surfaces but not for large spherical surfaces.

$A$ $4.5 \;cm$ needle is placed $12 \;cm$ away from a convex mirror of focal length $15 \;cm$. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.

$A$ concave mirror has a focal length $20\,cm$. The distance between the two positions of the object for which the image size is double of the object size is......$cm$