Practice Paper 1 (DP IB Maths: AA SL)

Prove that the square of an odd number is always odd.

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 .

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

The following triangle shows triangle with = ,  =  and  = .

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

The following cumulative frequency curve shows the number of hours, , students took to complete their online driving course. The data is taken from 80 students, randomly selected from a large sample over a  month period.

Find the median number of hours spent completing the online driving course.

Find the number of students whose online course time was within  hour of the
median.

Calculate the interquartile range.

The same information is represented by the following table.

Hours,
Frequency

Find the value of  and the value of .

It is known that of students take longer than  hours to complete the online driving course.

Find the value of .

It is known that over a 12 month period, 4000 students in total sat the online driving course.

Estimate the number of students over a 12 month period who took less than 3 hours to complete the course.

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

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

Find the -coordinates of  and .

Show that the area of the shaded region is  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 3, base radius , and height .

Find the value of .

Hence find the volume of the cone.

Particle  travels in a straight line such that its displacement,  meters, from a fixed origin after  seconds is given by for , as shown in the following diagram.

Particle  starts at the origin and passes through the origin again when .

Find the value of .

Particle  changes direction when 

Find the value of .

Find the displacement of particle from the origin when .

Find the distance of particlefrom the origin when .

The total distance travelled by particle  is given by .

Find the value of .

A second particle, particle , travels along the same straight line such that its velocity is given by for .
When  , the distance travelled by particle  is .

Find the value of .