6.2 Extended Questions (Paper 2, HL)

A boat is moving such that its position vector when viewed from above at time t  seconds can be modelled by

with respect to a rectangular coordinate system from a point O, where the non-zero constants   and  can be determined. All distances are given in metres. 

The boat leaves its mooring point at time  seconds and 5 minutes later is at the point with coordinates . 

Find

the values of  and , 

Find the velocity vector of the boat at time t seconds.

After setting off, the boat reaches a point P where it is moving parallel to the -axis.

Find OP.

Find the time that the boat returns to its mooring point and the acceleration of the boat at this moment.

On a particular island, a particular species of bird was initially recorded as having a population of 80 at the start of a programme of observations.  Over time, the scientists conducting the programme determined that the growth rate of the bird population could be modelled by the following differential equation

where  is the size of the bird population, and  is the length of time in years since the start of the programme. 

Find the population of the bird species two years after the start of the programme.

When the population of the bird species reaches 2000, a new reptile species is introduced to the island in order to control the bird population. Initially 280 reptiles are introduced to the island. Based on their research the scientists believe that the interaction between the two species after the introduction of the reptiles can be modelled by the system of coupled differential equations

Where  and  represent the size of the bird and reptile populations respectively.

Using the Euler method with a step size of 0.5, find an estimate for

Explain how the approximation in part (b) could be improved.

Show that the origin is an equilibrium point for the system, and determine the coordinates of the other equilibrium point.

In a game, enemies appear independently and randomly at an average rate of 2.5 enemies every minute. 

Find the probability that exactly 3 enemies will appear during one particular minute.

Find the probability that exactly 10 enemies will appear in a five-minute period.

Find the probability that at least 3 enemies will appear in a 90-second period.

The probability that at least one enemy appears in seconds is 0.999. Find the value of  correct to 3 significant figures.

A 10-minute interval is divided into ten 1-minute periods (first minute, second minute, third minute, etc.). Find the probability that there will be exactly two of those 1-minute periods in which no enemies appear.

On the next level of the game, there is a boss enemy and a number of additional henchmen to fight against. 

The number of times that the boss enemy appears in a one-minute period can be modelled by a Poisson distribution with a mean of 1.1. 

The number of times that an individual henchman appears in a one-minute period can be modelled by a Poisson distribution with a mean of 0.6. 

It may be assumed that the boss enemy and the henchmen each appear randomly and independently of one another. 

Each time that the boss enemy or any particular henchman appears, it is counted as one ‘enemy appearance’. 

Determine the least number of henchmen required in order that the probability of 40 or more ‘enemy appearances’ occurring in a 3-minute period is greater than 0.38. You may assume that neither the boss enemy nor any of the henchmen are able to be totally eliminated from the game during this 3-minute period.

James throws a throws ball to his friend Mia. The height, , in metres, of the ball above the ground is modelled by the function

where  is the time, in seconds, from the moment that James releases the ball.

Write down the height of the ball when James releases it.

After 4 seconds the ball is at a height of metres above the ground.

Find the value of .

Find

Find the maximum height reached by the ball and write down the corresponding time .

James then drives a remote-controlled car in a straight horizontal line from a starting position right in front of his feet.  The velocity of the remote-controlled car in  is given by the equation

Find an expression for the horizontal displacement of the remote-controlled car from its starting position at time seconds.

Find the total horizontal distance that the remote-controlled car has travelled in the first 5 seconds.

Consider the following system of differential equations:

Find the eigenvalues and corresponding eigenvectors of the matrix  .

Hence write down the general solution of the system.

When  ,  and .

Use the given initial condition to determine the exact solution of the system.

Hence sketch the solution trajectory of the system for .