Practice Paper 1 (DP IB Maths: AA HL)

Show that the equation can be written in the form , where  and  are integers to be found.

Hence, or otherwise, solve the equation  for 

In the expansion of  where  , the coefficient of the term in is 320.

Find the possible values of .

The diagram below shows part of the graph of , where  is the function defined by

Points  and  are the three places where the graph intercepts the -axis.

Find 

Show that the coordinates of point are (-2, 0).

Find the equation of the tangent to the curve at point .

The points A, B, C and D form the vertices of a parallelogram with position vectors and  respectively.

Show that the area of the parallelogram is 

The following triangle shows triangle ABC, with AB = 3, BC =  and AC = 7.

Given that  , find the area of the triangle. Give your answer in the form where .

 and are non-real roots of the equation ,  where  is a constant. 

Find and , in terms of .

Given that , show that .

Two lines,  and  , are parallel and their vector equations are given below:

          (i)     State the values of  and .

(ii)   Show that and are not collinear.

Use the substitution   to find 

Consider the function  defined by  , for
The following diagram shows the graph of 

The graph of f touches the x-axis at point A as shown. Point B is a local minimum of .
The shaded region is the area between the graph of  and the -axis, between the
points A and B.

 Find the -coordinates of A and B.

Show that the area of the shaded region is 3 units² .

The right cone in the diagram below has a curved surface area of twice the shaded area in
the previous part of the question.
The cone has a slant height of , base radius , and height .

Find the value of .

Hence find the volume of the cone.

A particle is moving in a vertical line and its acceleration, in , at time t seconds,  is given by  where is the velocity in meters per second and

The particle starts at a fixed origin O with initial velocity .

By solving a suitable differential equation, show that the particle’s velocity at time t is given by

The particle moves down in the negative direction, until its displacement relative to the origin reaches a minimum. Then the particle changes direction and starts moving up, in a positive direction. 

(i)

If the initial velocity of the particle is , find the time at which the minimum displacement of the particle from the origin occurs, giving your answer in exact form.

(ii)

If T is the time in seconds when the displacement reaches its smallest value, show that .

Let  represent the particle’s velocity k seconds before the minimum displacement and  the particle’s velocity k seconds after the minimum displacement. 

(i)     Show that

(ii)    Given that show that

The diagram below shows the graph of  The graph has rotational symmetry of order 2 about the origin.

A different function, g, is described by g

(i)

Describe the sequence of transformations that transforms to g.

(ii)

Sketch the graph of  g on the axes above.

(iii)

Using your answers to parts (i) and (ii) to help you, describe the relationship between and  .

Using the results from parts (a) and (b), evaluate leaving your answer in exact form.