Practice Paper 2 (DP IB Maths: AA HL)

A scientist is studying the movement of snails and has observed that the distribution of their speeds, S, follows a normal distribution with a mean of 48 m/h and a standard deviation of 1.5 m/h.

Sketch a diagram to represent this information.

Find the probability that a randomly selected snail has a speed of less than 46.5 m/h.

From a sample of 80 snails, calculate the expected number of snails that would have a speed of less than 46.5 m/h. Give your answer to the nearest integer.

A hamster runs in its exercise wheel, rotating the wheel at a constant speed. The wheel has a diameter of 14 centimetres and the top of the wheel is positioned at a height of  centimetres above the floor of the cage.

A point at the top of the wheel is marked before the hamster starts to run, turning the wheel clockwise. The hamster takes 4 seconds to turn the wheel one complete revolution.

After  seconds, the height of the mark on the wheel above the floor of the cage is given by

After 26 seconds, the mark is 3 cm above the cage floor. Find .

Find the value of .

A particle moves along a straight line with a velocity, , given by where  is measured in seconds such that .

Find the acceleration of the particle at time .

State the time when the particle comes to rest.

Find the total distance travelled by the particle.

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

Find .

A plane lies parallel to the line with equation  and contains the points  and  with coordinates  and respectively. 

Find the vector .

By appropriate use of the vector product, find the normal to the plane.

Hence find the Cartesian equation of the plane.

A continuous random variable  has the probability density function given by

Find the value of .

Giving your answers to three significant figures, find

(i)

the mean of ,

(ii)

the mode of .

It is given that that  and  

Find the value of  for   

Find the least value of such that 

In a robotics research facility two robots A and B are moving along parallel tracks [PQ] and [RS].  Robot A begins at point P and moves towards point Q, and robot B begins at point R and moves towards point S. [PR] is perpendicular to both [PQ] and [RS], and the distance from P to R is 20 metres. 

Both robots start moving at the same moment, and the distances travelled by robot A and robot B after  seconds are  metres and  metres, respectively.  The angle  is the radian measure of angle , where points A and B indicate the positions of robots A and B respectively at any time .  This information is shown on the following diagram.

Show that

At time , the following conditions are true: 

           Robot B has travelled 4 metres further than robot A.

The speed of Robot A is only two thirds of the speed of robot B.

          The rate of change of the angle is -0.05 radians per second. 

Find the speed of robot A at time .

The diagram below shows a part of the graph of the function

Point A is the point of intersection between the graph and the -axis. Write down the coordinates of point A.

Find .

Using the graph, explain why the equation  must have exactly two distinct real solutions.

Point  is the point on the graph with -coordinate .

Find the gradient of the tangent line to the graph at point .

Points  and  are the points on the graph at which the tangent lines are perpendicular to the tangent line at point .

By first determining the gradient of the tangents at points  and , find the -coordinates of points  and .

Given that point  lies between points  and  on the graph, find the equation of the tangent line to the graph at point .

The function  is defined by ,  for  

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

Find an expression for .

The graph of  has exactly one point of inflection. 

Find the -coordinate of the point of inflection.

Sketch the graph of for showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.

The function g is defined by  for . 

Find the equations of all the asymptotes on the graph of  .

By considering the graph of  or otherwise, solve  for .

The derivative of the function  is given by  where  is a real constant. 

By finding appropriate constants  and  in terms of , show that the expression for can be written in the form  where

Hence find an expression for .

Consider a population of lizards, , which has an initial size of 800. The rate of change of the population can be modelled by the differential equation ,  where  is the time measured in years,  and  is the maximum sustainable population. 

By solving the differential equation, show that

At  the lizard population has reduced in size to three fourths of its original value. 

Find the value of , giving your answer correct to four significant figures.

Find the value of  when the population is decreasing at a rate of 16 lizards per year.