Practice Paper 1 (DP IB Maths: AA SL)

Given that  and that the graph of  passes through the point (0, —1), find an expression for  in terms of . 

The functions and are defined such that and .
Show that 

Given that , find the value of .

Thus prove, given , , that the difference between an odd natural number greater than 1 and its cube is always even.

The following diagram shows triangle , with  and .

Given that ,  find the possible values of .

Given that is obtuse, find the precise value of .

Find the length of AC.

Show that  

Hence or otherwise solve for

Let 

Find 

The graph of  has horizontal tangents at the points where  and , .

Find the value of  and the value of .

The tangent to the graph of  at  and the normal to the graph of  at  intersect At the point .

Find the value of  and the value of .

Let  where , 

Show that .

The graph of  has exactly one maximum point .

Find the -coordinate of .

The second derivative of  is given by . The graph of  has exactly one point of inflexion B.

Show that the -coordinate of  is .

The region  is enclosed by the graph of  , the -axis, and the vertical lines through the maximum point and the point of inflexion .

Calculate the area of  in terms of  and show that the value of the area is independent of .

A school surveyed 80 of its final year students to find out how much time they spent reading the news on a given day. The results of the survey are shown in the following cumulative frequency diagram.

Find the median number of minutes spent reading the news.

Find the number of students whose reading time is within 2.5 minutes of the median.

Only 15% of students spent more than  minutes reading.

Find the value of .

The results of the survey can also be displayed on the following box-and-whisker diagram.

Write down the value of .

Determine whether someone who spends 30 minutes reading the news would be an outlier.