Note: I assume the mirror is concave, so that its focal length is positive (it is not specified in the text)
1a) We can use the mirror equation to find the distance of the image from the mirror:
[tex] \frac{1}{f}= \frac{1}{p}+ \frac{1}{q} [/tex]
where
[tex]f=5 cm[/tex] is the focal length
[tex]p=10 cm[/tex] is the distance of the object from the mirror
[tex]q[/tex] is the distance of the image from the mirror.
Rearranging the equation, we find
[tex] \frac{1}{q}= \frac{1}{f}- \frac{1}{p}= \frac{1}{5}- \frac{1}{10}= \frac{1}{10 cm} [/tex]
so, the distance of the image from the mirror is [tex]q=10 cm[/tex].
1b) The image height is given by the magnification equation:
[tex] \frac{h_i}{h_o}=- \frac{p}{q} [/tex]
where [tex]h_i[/tex] is the heigth of the image and [tex]h_o=1 cm[/tex] is the height of the object. By rearranging the equation and using p and q, we find
[tex]h_i=-h_o \frac{p}{q}=-(1 cm) \frac{10 cm}{10 cm}=-1 cm [/tex]
and the negative sign means the image is inverted.
2) As before, we can find the distance of the image from the mirror by using the mirror equation:
[tex]\frac{1}{f}= \frac{1}{p}+ \frac{1}{q}[/tex]
Rearranging it, we find
[tex]\frac{1}{q}= \frac{1}{f}- \frac{1}{p}= \frac{1}{2}- \frac{1}{10}= \frac{4}{10 cm} [/tex]
so, the distance of the image from the mirror is
[tex]q= \frac{10}{4}cm= 2.5 cm [/tex]
3) As before, we find the distance of the image from the mirror by using the mirror equation:
[tex]\frac{1}{f}= \frac{1}{p}+ \frac{1}{q}[/tex]
Rearranging it, we find
[tex]\frac{1}{q}= \frac{1}{f}- \frac{1}{p}= \frac{1}{2}- \frac{1}{10}= \frac{4}{10 cm} [/tex]
so, the distance of the image from the mirror is
[tex]q= \frac{10}{4}cm= 2.5 cm [/tex]
And now we can use the magnification equation to find the image height:
[tex]\frac{h_i}{h_o}=- \frac{p}{q}[/tex]
Rearranging it, we find
[tex]h_i=-h_o \frac{p}{q}=-(3cm) \frac{10 cm}{2.5 cm}=-12 cm[/tex]
and the negative sign means the image is inverted.
