A man views a small object located at his near point. Without...

The angular magnification (or magnifying power) is defined as the ratio of the angle subtended by the image at the eye ($\theta_i$) to the angle subtended by the object at the unaided eye when placed at the near point ($\theta_o$).

When the object is viewed with the unaided eye at the near point $D$, the angle subtended is:

$\theta_o = \dfrac{h}{D}$

When the simple microscope (convex lens) is placed in front of the eye without altering the object's position, the object distance from the lens remains $u = D$.

The angle subtended by the image at the eye (assuming the lens is very close to the eye) is:

$\theta_i = \dfrac{h'}{v}$

Using the transverse magnification formula for a lens, $\dfrac{h'}{h} = \dfrac{v}{u}$, we can rewrite this as:

$\dfrac{h'}{v} = \dfrac{h}{u}$

Since the object is still at the near point, $u = D$. Therefore, the angle subtended by the image is:

$\theta_i = \dfrac{h}{D}$

The angular magnification achieved in this specific position is:

$M = \dfrac{\theta_i}{\theta_o} = \dfrac{h/D}{h/D} = 1$

The given magnifying power of $5 \text{ X}$ is the maximum magnifying power (when the object is moved closer to the lens such that the image forms at the near point), which does not apply here since the object's position was not altered.