Institution: SONALI BANK LTD Exam: Sonali Bank Ltd - Sub-Assistant Engineer (Mechanical) - 30.09.2016 ||
Topic: Mathematics (সাধারণ গণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To understand why the correct answer is \(2P \cos(\theta/2)\), we need to analyze the situation involving two equal forces acting at an angle \(\theta\).

When two forces of equal magnitude \(P\) act at an angle \(\theta\) with respect to each other, the resultant force can be found using the law of cosines or the parallelogram law. According to the parallelogram law, if two forces are represented as two adjacent sides of a parallelogram, the magnitude of the resultant is given by:

\[
R = \sqrt{P^2 + P^2 + 2P \cdot P \cdot \cos(\theta)} = \sqrt{2P^2(1 + \cos(\theta))} = P\sqrt{2(1 + \cos(\theta))}
\]

However, we can also utilize the trigonometric relationship. The angle at which the resultant is directed can be divided into two components, each making an angle \(\theta/2\) with the resultant. Hence, the vertical component contributed by each force, when the angle is \(\theta\), can be calculated using the cosine for the component along the resultant.

Specifically, for the resultant vector's length in the direction of force, we can represent it as:

\[
R = 2P \cos(\theta/2)
\]

This arises because, for each force, when resolved along a line bisecting the angle \(\theta\), the vertical components sum up to yield \(2P \cos(\theta/2)\). Thus, the resultant force acting along this direction is indeed:

\[
\boxed{2P \cos(\theta/2)}
\]

This equation highlights why \(2P \cos(\theta/2)\) represents the effective sum of the two forces when acting at the specified angle.