Practice Paper 2 (DP IB Maths: AI HL)

The table below shows the distribution of the number of baskets scored by 150 netball players during a weekly game.

Calculate

Find the median number of baskets scored.

Find the interquartile range.

Determine if a player who scored 8 baskets would be considered an outlier.

Two players are randomly chosen.

Given that the first player scored 2 or less baskets, find the probability that both players scored exactly 1 basket.

The number of hours each player trains each week is normally distributed with a mean of 5 hours and standard deviation of 0.8 hours.

Chun-hee is creating some packaging in the shape of a square based pyramid where the base has length  cm and the perpendicular height of the pyramid is  cm. Chun-hee wants to keep the distance from the apex of the pyramid to the midpoint of the base edge fixed at 7 cm.

Write down an equation for the volume, , of the packaging in terms of  and .

Show that  can be expressed by .

Find .

Find the value of  for which the volume of the pyramid is maximised.

Find the value of  when the volume of the pyramid is maximised.

Chun-hee decides to make the packaging using the dimensions required to maximise the volume. The material for the packaging costs 4 KRW / cm².

Calculate the number of units that Chun-hee can make given that she has 90, 000 KRW.

Chun-hee takes out a 3 year loan for 90,000 KRW at a nominal annual interest rate of 2.3% compounded monthly. Repayments are made at the end of each month.

Find the value of the repayments that Chun-hee must make to pay off the loan.

In the town of Manh, all the residents belong to either one or the other of the town’s two fitness clubs – Giang’s House of Fitness (G) or Thu’s Wonder Gym (T). Each year 30% of the members of switch to  and 25% of the members of  switch to . Any other losses or gains of members by the two fitness clubs may be ignored.

Write down a transition matrix  representing the movement of members between the two clubs in a particular year.

Find the eigenvalues and corresponding eigenvectors of .

Hence write down matrices  and  such that .

Initially there are 2500 members of  and 800 members of .

Using the matrix power formula, show that the numbers of members of  and  after  years will be  and , respectively.

Hence write down the number of customers that each of the fitness clubs can expect to have in the long term.

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.

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

Find the value of .

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.