Practice Paper 1 (DP IB Maths: AI HL)

A function is defined by ,  for  

Find the range of .

Find an expression for the inverse function . The domain is not required.

Write down the range of .

It is claimed that women from Japan are taller on average than women from India. The heights, in cm, of 11 women from each country have been collated in the table below.

A -test is to be performed at the 5% significance level.
State the null and alternative hypotheses.

Find the -value for this test.

State whether or not the initial claim is justified. Give a reason for your answer.

Adah would like to estimate the height of a tree located at point  on the edge of a  riverbank, with the top of the tree at point . However, due to a raging river, she is unable to reach the base of the tree. From point  she measures an angle of elevation of 20° to the top of the tree, and then from point  (which is on the edge of Adah's bank of the river) she measures an angle of elevation of 35° to the top of the tree. Between the points  and she measures a horizontal distance of 12 m. Points  and  all lie on a single horizontal line, and point  is vertically above point . The diagram below shows this  information.

Calculate the length of .

Calculate the height of the tree.

Adah borrows a boat and crosses the river at a rate of 50 metres per 15 minutes.

Assuming that she crosses in a straight line directly from point  to point , find out how long it takes her to cross the river.

A new car costs $20 000 and its value depreciates to $14 792 after 2 years.

Calculate

Find the number of years and months it will take for the car's value to be approximately $4000.

The function  intercepts the -axis at —16, has an -intercept when  and can be obtained by an appropriate translation of the graph 

Write down .

Find the other -intercept of .

Write down the coordinates of the vertex of .

Ashley and Emma are attempting to swim a total of 2000 m each by completing laps of a 25 m pool. Ashley swims her first lap in 17 s and takes 0.2 s longer each lap after that.
Emma swims her first lap in 16.5 s and takes 1.01 s times longer each lap after that.

Consider the lines   and defined by the equations:

Given that   and  are identical, find the value of  and .

A carpet salesman in interested how his sales are distributed and records his sales results over a period of six months. The data is shown in the table.

A chi-squared goodness of fit test is to be performed on the data at the 5% significance level to find out whether the data fits a uniform distribution.

The critical value for the test is 11.070 and the hypotheses are

                     H₀: The data satisfies the model.
                     H₁: The data does not satisfy the model.

Find an estimate of how many carpets the salesman expects to sell each month.

Write down the number of degrees of freedom for this test.

State the conclusion of the test. Give a reason for your answer.

Let be a normally distributed random variable that represents the distance travelled in metres by a slug in one day. The distance covered by a random sample of 21 slugs on a randomly selected day can be summarized as follows

Find an unbiased estimate of the mean, , of .

Use the formula to find an unbiased estimate of the variance of .

Find a 95% confidence interval for μ.

Justin believes that the average slug travels 15 m per day.

State whether or not Justin’s statement is valid. Give a reason for your answer.

The graph below shows 7 towns and the train tracks that the connect them, with the vertices representing the towns and the weighting of each edge indicating the time taken in minutes to walk along the section of track.

State the degree of each vertex.

Explain why  does not contain an Eulerian circuit.

The railway company in charge of maintaining the track wishes to inspect all sections of the track for defects after a storm event.

Find the minimum time it would take for a railway worker who was walking to inspect all of the track.

It is given that that  and  

Find the value of for  

Find the least value of such that 

The diagram below shows the plan of a school building.

Construct a graph to represent this information, using vertices to represent rooms and edges to represent the connecting doors.

For a prank, a student releases a monkey into the English classroom at 8:30 pm on Thursday.  The monkey continues to wander at random through the school building all night until the cleaner arrives at 6 am on Friday morning. 

Stating clearly any assumptions you have made, find an approximation for the total length of time the monkey is likely to have spent in the Maths classroom before the cleaner arrives.

Points in a plane are subjected to a transformation  where  is defined by:

Given that a point  is mapped to , find  and , where .

Given that comprises two individual transformations describe in full the composite transformation .

The number of spam emails that Arturo receives per day is modelled by a Poisson distribution with a mean of 25 spam emails per day.

After changing the settings on his spam filter, Arturo decides to test whether the new settings have reduced the number of spam emails he receives. To do this he records the number of spam emails he receives over a period of one week. He decides to use a 5% level of significance for his test.

State the null and alternative hypotheses for the test.

During the 1-week period, Arturo receives 149 spam emails.

A particle  moves in a straight line, such that its displacement  at time  is defined by the differential equation . At time

By using Euler’s method with a step length of 0.2, find an approximate value for when

Solve the differential equation to find the actual value of  when

A manufacturing process takes place inside a sealed chamber and produces a pollutant that decays over time.  After the process is completed, at time t = 0 seconds, the amount of pollutant, P  ppm (parts per million) in the chamber is modelled by

The half-life of the pollutant is .

Find the value of .