Institution: Bank Asia Ltd Exam: Bank Asia Ltd - Management Trainees - 17.06.2011 ||
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To solve the problem, we start by understanding the geometric relationships between the circle and the inscribed square. The square is inscribed in a circle of diameter \( d \). The radius \( r \) of the circle is therefore \( r = \frac{d}{2} \).

The area \( A_{\text{circle}} \) of the circle is calculated using the formula:
\[
A_{\text{circle}} = \pi r^2 = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}.
\]

Next, we find the area \( A_{\text{square}} \) of the inscribed square. The diagonal of the square equals the diameter of the circle \( d \). If \( s \) is the length of a side of the square, the relationship between the side and the diagonal is given by:
\[
s\sqrt{2} = d \implies s = \frac{d}{\sqrt{2}}.
\]
The area of the square is:
\[
A_{\text{square}} = s^2 = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}.
\]

Now we compute the area of the shaded regions, which is the area of the circle minus the area of the square:
\[
A_{\text{shaded}} = A_{\text{circle}} - A_{\text{square}} = \frac{\pi d^2}{4} - \frac{d^2}{2}.
\]
To combine these, we express \( \frac{d^2}{2} \) with a common denominator:
\[
\frac{d^2}{2} = \frac{2d^2}{4}.
\]
Thus:
\[
A_{\text{shaded}} = \frac{\pi d^2}{4} - \frac{2d^2}{4} = \frac{(\pi - 2)d^2}{4}.
\]
Finally, the expression can be rearranged to match the given options:
\[
A_{\text{shaded}} = d^2 \left( \frac{\pi}{4} - \frac{1}{2} \right),
\]
which corresponds to option A: \( d^2 \left( \frac{\pi}{4} - \frac{1}{2} \right) \). Thus, option A is the correct choice.