If we want to obtain a virtual and magnified image of an object by using a concave mirror of focal length $18\text{ cm}$, where should the object be placed? Use mirror formula to determine the object distance for an image of magnification $+2$ produced by this mirror to justify your answer.

To obtain a virtual and magnified image of an object by using a concave mirror of focal length $18\text{ cm}$, the object should be placed between the focus and the pole of the mirror. Using the mirror formula, we can determine the object distance for an image of magnification $+2$ produced by this mirror as follows:

$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$

where $v$ is the image distance, $u$ is the object distance, and $f$ is the focal length of the mirror.

Given that the magnification is $+2$, we can write:

$$m = \frac{v}{u} = +2$$

Solving for $v$, we get:

$$v = 2u$$

Substituting this value of $v$ in the mirror formula, we get:

$$\frac{1}{2u} + \frac{1}{u} = \frac{1}{18}$$

Simplifying, we get:

$$\frac{3}{2u} = \frac{1}{18}$$

Solving for $u$, we get:

$$u = 27\text{ cm}$$

Therefore, the object should be placed at a distance of $27\text{ cm}$ from the mirror to obtain a virtual and magnified image.