2025 JPJC P2 Q1

To prepare for the General Election, a political party studies the effectiveness of spreading information on Instantglam, a social media platform, by uploading a particular video. The number of Instantglam users who have viewed the video increases at a rate that is directly proportional to the number of Instantglam users who have yet to view the video. The number of Instantglam users who have viewed the video \(t\) hours after the video is uploaded is \(x\). The total number of Instantglam users is \(P\), assumed to be fixed.

1. Write down a differential equation and show that \(x = P(1 - e^{-kt})\), where \(k\) is a positive constant.[4]

It is known that \(12\) hours after the video is uploaded, half the total number of Instantglam users have viewed the video.

2. Estimate to the nearest hour, the time needed for \(80\%\) of the total number of Instantglam users to have viewed the video.[3]
3. Sketch the graph of \(x\) against \(t\).[2]