IB MATH AA HL, Paper 2, May, 2022, TZ2, Solved Past Paper

The scenario presents a circle centered at (O) with a radius stretching out 5 meters. On this circle's boundary, imagine two distinct points, (A) and (B), which are endpoints of a curve along the circle's edge.

These points are connected to the circle's center (O) forming lines that meet at the center to create an angle. This angle, referred to as the central angle, measures 1.9 radians.

We have a function, let's call it (g), for which we know the rate of change at any point (x). This rate is described by the equation (g'(x) = 3x² + 5e^x), with $x$ being any real number. The function (g) also has a specific feature: its curve goes through a particular spot on the graph, namely the point (0, 4).

What we're looking to find out is the original function (g(x)), knowing these details. Keep in mind that finding (g(x)) will involve integration, since we're given its derivative (g'(x)), and that we'll need to use the point (0, 4) to solve for the constant of integration.

Can you determine what (g(x)) is?

We're looking at two separate scenarios, Event A and Event B, which do not influence each other in any way. The probability of Event A happening is exactly three times the probability of Event B occurring.

Now, if we consider the combined situation where either Event A or Event B or both could happen, this combined probability is 0.68. With this information, we want to figure out the probability of just Event B happening on its own.