Practice Paper 1 (DP IB Maths: AA HL)

Let  and  be events such that  and . 

Find .

The functions and  are defined such that and  .

Show that .            

Given that , find the value of .                                           

The following diagram shows the graph of . The graph has a horizontal asymptote at  .  The graph crosses the -axis at   and ,  and the -axis at 

On the following set of axes, sketch the graph of , clearly showing any asymptotes with their equations along with the coordinates of any local maxima or minima.

Given that  and that the graph of  passes through the point , find an expression for  in terms of .

The plane has the Cartesian equation .

The line  has the vector equation .  The acute angle between the line  and the plane  is . 

Find the possible values of .        

Show that

Hence or otherwise  for 

The function is defined by  The graph of  is shown in the following diagram.

Find the largest value of  such that  has an inverse function.

For this value of , find an expression for , stating its domain.

A continuous random variable  has the probability density function  given by

Find    

Express  in the form  , where  and .

Let the roots of the equation  be  and .

Find  and expressing your answers in the form , where and .                 

On an Argand diagram and  are represented by the points  and  respectively.

Find the area of triangle .

By considering the sum of the roots and , show that

The function  is defined by  .

Find the first two derivatives of and hence find the Maclaurin series for  up to and including the  term.

Show that the coefficient of  in the Maclaurin series for  is zero.

Using the Maclaurin series for and , find the Maclaurin series for  up to and including the  term.

Hence, or otherwise, find 

Let  where .

Show that   . 

The graph of has exactly one maximum point A.

Find the -coordinate of A.         

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

Show that the -coordinate of B is . 

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

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