Question 1
SLPaper 1The diameter of a spherical planet is km.
Write down the radius of the planet.
The volume of the planet can be expressed in the form where and .
Find the value of and the value of .
Question 2
SLPaper 1Solve the equation . Give your answer in the form where .
Question 3
SLPaper 1In the Canadian city of Ottawa:
97% of the population speak English, 38% of the population speak French, 36% of the population speak both English and French.
The total population of Ottawa is .
Calculate the percentage of the population of Ottawa that speak English but not French.
Calculate the number of people in Ottawa that speak both English and French.
Write down your answer to part (b) in the form where and .
Question 4
SLPaper 1Consider the binomial expansion where and .
Show that .
The third term in the expansion is the mean of the second term and the fourth term in the expansion.
Find the possible values of .
Question 5
HLPaper 2Consider the identity , where .
Find the value of and the value of .
Hence, expand in ascending powers of , up to and including the term in .
Give a reason why the series expansion found in part (b) is not valid for .
Question 6
HLPaper 1Consider the following system of equations where .
Find the solution of the system of equations when .
Question 7
SLPaper 1Consider .
Expand and simplify in ascending powers of .
By using a suitable substitution for , show that
.
Show that where is a positive real constant.
Question 8
HLPaper 1The first term in an arithmetic sequence is and the fifth term is .
Find the common difference of the sequence, expressing your answer in the form , where .
Question 9
HLPaper 3This question will explore connections between complex numbers and regular polygons. The diagram below shows a sector of a circle of radius 1, with the angle subtended at the centre being , . A perpendicular is drawn from point to intersect the -axis at . The tangent to the circle at intersects the -axis at .
By considering the area of two triangles and the area of the sector show that
.
Hence show that .
Let , , , . Working in modulus/argument form find the solutions to this equation.
Represent these solutions on an Argand diagram. Let their positions be denoted by placed in order in an anticlockwise direction round the circle, starting on the positive -axis. Show the positions of , and .
Show that the length of the line segment is .
Hence, write down the total length of the perimeter of the regular -sided polygon .
Using part (b) find the limit of this perimeter as .
Find the total area of this sided polygon.
Question 10
HLPaper 1Consider the series , where , and , .
Consider the case where the series is geometric.
Show that .
Hence or otherwise, show that the series is convergent.
Given that and , find the value of .
Now consider the case where the series is arithmetic with common difference .
Show that .
Write down in the form , where .
The sum of the first terms of the series is .
Find the value of .