An object of length $6 \ cm$ is placed at a distance of $4f$ on the principal axis of a concave mirror of focal length $f$. The length of the image will be ......$cm$.

Use the mirror equation to deduce that:
$(a)$ an object placed between $f$ and $2f$ of a concave mirror produces a real image beyond $2f$.
$(b)$ a convex mirror always produces a virtual image independent of the location of the object.
$(c)$ the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.
$(d)$ an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.

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:

The sun (diameter $D$) subtends an angle $\theta$ at the pole of a concave mirror of focal length $f$. What will be the diameter of the image formed by the mirror?

$A$ rod of length $10\, cm$ lies along the principal axis of a concave mirror of focal length $10\, cm$ in such a way that its end closer to the pole is $20\, cm$ away from the mirror. The length of the image is ......$cm$.

$A$ $4.5 \;cm$ 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.