Algebra
workbook
~ categorized past IB Paper 1 and Paper 2 examination questions ~
IB DP Mathematics Higher Level
Topic 1
This workbook contains past Paper 1 and Paper 2 IB examination questions categorized
according to major concepts in this topic.
Contents
• 1.1 Sequences and Series
Paper 1 Questions
30 questions; 156 marks
Paper 2 Questions
34 questions; 232 marks
• 1.2 Exponents and Logarithms
Paper 1 Questions
17 questions; 91 marks
Paper 2 Questions
6 questions; 34 marks
• 1.3 Binomial Theorem
Paper 1 Questions
15 questions 79 marks
Paper 2 Questions
14 questions; 72 marks
• 1.4 Counting Principles
Paper 1 Questions
5 questions 21 marks
Paper 2 Questions
4 questions; 24 marks
• 1.5 Mathematical Induction
Paper 1 Questions
10 questions 101 marks
Paper 2 Questions
1 question; 11 marks
• 1.6 Complex Numbers
Paper 1 Questions
21 questions 137 marks
Paper 2 Questions
7 questions; 37 marks
page 1
page 15
page 36
page 44
page 47
page 53
page 58
page 60
page 62
page 68
page 69
page 80
• 1.7 De Moivre’s Theorem
Paper 1 Questions
10 questions 102 marks
Paper 2 Questions
3 questions; 26 marks
• 1.8 Conjugate Roots
Paper 1 Questions
9 questions 64 marks
page 83
page 89
page 91
The use of GDC is not permitted for Paper 1 but is required for Paper 2 questions.
1.1 Sequences and Series
1.
Paper 1
An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series.
(Total 4 marks)
2.
In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term.
(Total 4 marks)
3.
Find the sum of the infinite geometric series 2
3
4
9
8
27
16 ...
81
(Total 4 marks)
1
4.
Each day a runner trains for a 10 km race. On the first day she runs 1000 m, and then increases the distance by 250
m on each subsequent day.
(a)
On which day does she run a distance of 10 km in training?
(b)
What is the total distance she will have run in training by the end of that day? Give your answer exactly.
(Total 4 marks)
5.
In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11 998.
(a)
Find the common difference d.
(b)
Find the value of n.
(Total 6 marks)
2
6.
The following table shows four series of numbers. One of these series is geometric, one of the series is arithmetic
and the other two are neither geometric nor arithmetic.
(a)
Complete the table by stating the type of series that is shown.
Series
(b)
Type of series
(i)
1 11 111 1111 11111 …
(ii)
1
(iii)
9
27
3
…
4 16 64
0.9 0.875 0.85 0.825 0.8 …
(iv)
1
2
2
3
3
4
4
5
5
6
The geometric series can be summed to infinity. Find this sum.
(Total 6 marks)
7.
Let Sn be the sum of the first n terms of an arithmetic sequence, whose first three terms are u1, u2 and u3. It is
known that S1 = 7, and S2 = 18.
(a)
Write down u1.
(b)
Calculate the common difference of the sequence.
(c)
Calculate u4.
(Total 6 marks)
3
8.
The first term of an infinite geometric sequence is 18, while the third term is 8. There are two possible sequences.
Find the sum of each sequence.
(Total 6 marks)
9.
Consider the arithmetic sequence 2, 5, 8, 11, .....
(a)
Find u101.
(3)
(b)
Find the value of n so that un = 152.
(3)
(Total 6 marks)
4
10.
The second term of an arithmetic sequence is 7. The sum of the first four terms of the arithmetic sequence is 12.
Find the first term, a, and the common difference, d, of the sequence.
(Total 4 marks)
11.
6 . If each term of this
13
sequence is positive, and the product of the first term and the third term is 32, find the sum of the first 100 terms
of this sequence.
The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is
(Total 7 marks)
5
12.
An arithmetic sequence has 5 and 13 as its first two terms respectively.
(a)
Write down, in terms of n, an expression for the nth term, an.
(b)
Find the number of terms of the sequence which are less than 400.
(Total 4 marks)
13.
The sum of the first n terms of an arithmetic sequence is Sn = 3n2 – 2n. Find the nth term un.
(Total 3 marks)
6
14.
Find the sum of the positive terms of the arithmetic sequence 85, 78, 71, ....
(Total 3 marks)
15.
Find the sum to infinity of the geometric series – 12 8 –
16
....
3
(Total 3 marks)
16.
The nth term, un, of a geometric sequence is given by un = 3(4)n+1, n
+
.
(a)
Find the common ratio r.
(b)
Hence, or otherwise, find Sn, the sum of the first n terms of this sequence.
(Total 3 marks)
7
17.
Consider the infinite geometric series 1
2x
3
2x
3
(a)
For what values of x does the series converge?
(b)
Find the sum of the series if x = 1.2.
2
2x
3
3
.....
(Total 3 marks)
18.
A sequence {un} is defined by u0 = 1, u1 = 2, un+1 = 3un – 2un–1 where n
(a)
+
.
Find u2, u3, u4.
(3)
(b)
(i)
Express un in terms of n.
(ii)
Verify that your answer to part (b)(i) satisfies the equation un+1 = 3un – 2un – 1.
(3)
(Total 6 marks)
8
19.
A geometric sequence has all positive terms. The sum of the first two terms is 15 and the sum to infinity is 27.
Find the value of
(a)
the common ratio;
(b)
the first term.
(Total 6 marks)
20.
The sum of the first n terms of a series is given by Sn = 2n2 – n, where n
+
.
(a)
Find the first three terms of the series.
(b)
Find an expression for the nth term of the series, giving your answer in terms of n.
(Total 6 marks)
9
21.
The sum of the first n terms of an arithmetic sequence {un} is given by the formula Sn = 4n2 – 2n. Three terms of
this sequence, u2, um and u32, are consecutive terms in a geometric sequence. Find m.
(Total 6 marks)
22.
In an arithmetic sequence the second term is 7 and the sum of the first five terms is 50. Find the common
difference of this arithmetic sequence.
(Total 6 marks)
10
23.
The first and fourth terms of a geometric series are 18 and –
(a)
2
respectively. Find
3
the sum of the first n terms of the series;
(4)
(b)
the sum to infinity of the series.
(2)
(Total 6 marks)
24.
A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence. The angle of the largest sector
is twice the angle of the smallest sector. Find the size of the angle of the smallest sector.
(Total 5 marks)
11
25.
The common ratio of the terms in a geometric series is 2x.
(a)
State the set of values of x for which the sum to infinity of the series exists.
(2)
(b)
If the first term of the series is 35, find the value of x for which the sum to infinity is 40.
(4)
(Total 6 marks)
26.
A geometric sequence u1, u2, u3, ... has u1 = 27 and a sum to infinity of
(a)
81
.
2
Find the common ratio of the geometric sequence.
(2)
An arithmetic sequence v1, v2, v3, ... is such that v2 = u2 and v4 = u4.
N
(b)
vn
Find the greatest value of N such that
0.
n 1
(5)
(Total 7 marks)
12
27.
An arithmetic sequence has first term a and common difference d, d ≠ 0. The 3rd, 4th and 7th terms of the
arithmetic sequence are the first three terms of a geometric sequence.
(a)
Show that a =
3
d.
2
(3)
(b)
Show that the 4th term of the geometric sequence is the 16th term of the arithmetic sequence.
(5)
(Total 8 marks)
28.
The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty terms of the arithmetic
sequence is 16. Find the value of the 15th term of the sequence.
(Total 6 marks)
13
29.
The sum, Sn, of the first n terms of a geometric sequence, whose nth term is un, is given by Sn =
7n
an
7n
, where
a > 0.
(a)
Find an expression for un.
(2)
(b)
Find the first term and common ratio of the sequence.
(4)
(c)
Consider the sum to infinity of the sequence.
(i)
Determine the values of a such that the sum to infinity exists.
(ii)
Find the sum to infinity when it exists.
(2)
(Total 8 marks)
30.
An 81 metre rope is cut into n pieces of increasing lengths that form an arithmetic sequence with a common
difference of d metres. Given that the lengths of the shortest and longest pieces are 1.5 metres and 7.5 metres
respectively, find the values of n and d.
(Total 4 marks)
14
1.1 Sequences and Series
1.
Paper 2
Find the sum of the arithmetic series 17 + 27 + 37 +...+ 417.
(Total 4 marks)
2.
$1000 is invested at the beginning of each year for 10 years. The rate of interest is fixed at 7.5% per annum.
Interest is compounded annually. Calculate, giving your answers to the nearest dollar
(a)
how much the first $1000 is worth at the end of the ten years;
(b)
the total value of the investments at the end of the ten years.
(Total 4 marks)
15
3.
The first three terms of an arithmetic sequence are 7, 9.5, 12.
(a)
What is the 41st term of the sequence?
(b)
What is the sum of the first 101 terms of the sequence?
(Total 4 marks)
4.
Gwendolyn added the multiples of 3, from 3 to 3750 and found that 3 + 6 + 9 + … + 3750 = s. Calculate s.
(Total 6 marks)
16
5.
Arturo goes swimming every week. He swims 200 metres in the first week. Each week he swims 30 metres more
than the previous week. He continues for one year (52 weeks).
(a)
How far does Arturo swim in the final week?
(b)
How far does he swim altogether?
(Total 6 marks)
6.
A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive
row of seats has two more seats in it than the previous row.
(a)
Calculate the number of seats in the 20th row.
(b)
Calculate the total number of seats.
(Total 6 marks)
17
7.
A sum of $5000 is invested at a compound interest rate of 6.3% per annum.
(a)
Write down an expression for the value of the investment after n full years.
(b)
What will be the value of the investment at the end of five years?
(c)
The value of the investment will exceed $10000 after n full years,
(i)
Write down an inequality to represent this information.
(ii)
Calculate the minimum value of n.
(Total 6 marks)
8.
Consider the infinite geometric series 405 + 270 + 180 +....
(a)
For this series, find the common ratio, giving your answer as a fraction in its simplest form.
(b)
Find the fifteenth term of this series.
(c)
Find the exact value of the sum of the infinite series.
(Total 6 marks)
18
9.
Consider the infinite geometric sequence 25, 5, 1, 0.2, … .
(a)
Find the common ratio.
(b)
Find
(c)
(i)
the 10th term;
(ii)
an expression for the nth term.
Find the sum of the infinite sequence.
(Total 6 marks)
10.
The first four terms of a sequence are 18, 54, 162, 486.
(a)
Use all four terms to show that this is a geometric sequence.
(2)
(b)
(i)
Find an expression for the nth term of this geometric sequence.
(ii)
If the nth term of the sequence is 1062 882, find the value of n.
(4)
(Total 6 marks)
19
11.
The Acme insurance company sells two savings plans, Plan A and Plan B. For Plan A, an investor starts
with an initial deposit of $1000 and increases this by $80 each month, so that in the second month,
the deposit is $1080, the next month it is $1160 and so on. For Plan B, the investor again starts with
$1000 and each month deposits 6% more than the previous month.
(a)
Write down the amount of money invested under Plan B in the second and third months.
(2)
Give your answers to parts (b) and (c) correct to the nearest dollar.
(b)
Find the amount of the 12th deposit for each Plan.
(4)
(c)
Find the total amount of money invested during the first 12 months
(i)
under Plan A;
(2)
(ii)
under Plan B.
(2)
(Total 10 marks)
20
12.
Portable telephones are first sold in the country Cellmania in 1990. During 1990, the number of units sold is 160.
In 1991, the number of units sold is 240 and in 1992, the number of units sold is 360. In 1993 it was noticed that
the annual sales formed a geometric sequence with first term 160, the 2nd and 3rd terms being 240 and 360
respectively.
(a)
What is the common ratio of this sequence?
(1)
Assume that this trend in sales continues.
(b)
How many units will be sold during 2002?
(3)
(c)
In what year does the number of units sold first exceed 5000?
(4)
Between 1990 and 1992, the total number of units sold is 760.
(d)
What is the total number of units sold between 1990 and 2002?
(2)
During this period, the total population of Cellmania remains approximately 80 000.
(e)
Use this information to suggest a reason why the geometric growth in sales would not continue.
(1)
(Total 11 marks)
21
13.
Ashley and Billie are swimmers training for a competition.
(a)
Ashley trains for 12 hours in the first week. She decides to increase the amount of time she spends training
by 2 hours each week. Find the total number of hours she spends training during the first 15 weeks.
(3)
(b)
Billie also trains for 12 hours in the first week. She decides to train for 10% longer each week than the
previous week.
(i)
Show that in the third week she trains for 14.52 hours.
(ii)
Find the total number of hours she spends training during the first 15 weeks.
(4)
(c)
In which week will the time Billie spends training first exceed 50 hours?
(4)
(Total 11 marks)
22
14.
The diagram shows a square ABCD of side 4 cm.
The midpoints P, Q, R, S of the sides are joined to form a second square.
(a)
(i)
Show that PQ = 2 2 cm.
(ii)
Find the area of PQRS.
Q
A
B
P
R
(3)
D
S
C
Q
B
The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a third square as shown.
(b)
(i)
Write down the area of the third square, WXYZ.
(ii)
Show that the areas of ABCD, PQRS, and WXYZ
form a geometric sequence.
Find the common ratio of this sequence.
A
X
W
P
R
(3)
Y
Z
D
S
C
The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely.
(c)
(i)
Find the area of the 11th square.
(ii)
Calculate the sum of the areas of all the squares.
(4)
(Total 10 marks)
23
15.
The diagrams below show the first four squares in a sequence of squares which are subdivided in half.
1
The area of the shaded square A is .
4
(a)
(i)
Find the area of square B and of square C.
(ii)
Show that the areas of squares A, B and C are in geometric progression.
(iii)
Write down the common ratio of the progression.
(5)
(b)
(i)
Find the total area shaded in diagram 2.
(ii)
Find the total area shaded in the 8th diagram of this sequence.
Give your answer correct to six significant figures.
(4)
(c)
The dividing and shading process illustrated is continued indefinitely. Find the total area shaded.
(2)
(Total 11 marks)
A
A
B
Diagram 1
Diagram 2
A
A
B
B
C
Diagram 3
C
Diagram 4
24
16.
A company offers its employees a choice of two salary schemes A and B over a period of 10 years.
Scheme A offers a starting salary of $11000 in the first year and then an annual increase of $400 per year.
(a)
(i)
Write down the salary paid in the second year and in the third year.
(ii)
Calculate the total (amount of) salary paid over ten years.
(3)
Scheme B offers a starting salary of $10000 dollars in the first year and then an annual increase of 7% of the
previous year’s salary.
(b)
(i)
Write down the salary paid in the second year and in the third year.
(ii)
Calculate the salary paid in the tenth year.
(4)
(c)
Arturo works for n complete years under scheme A. Bill works for n complete years under scheme B.
Find the minimum number of years so that the total earned by Bill exceeds the total earned by Arturo.
(4)
(Total 11 marks)
25
17.
(a)
Consider the geometric sequence −3, 6, −12, 24, ….
(i)
Write down the common ratio.
(ii)
Find the 15th term.
Consider the sequence x − 3, x +1, 2x + 8, ….
(3)
(b)
When x = 5, the sequence is geometric.
(i)
Write down the first three terms.
(ii)
Find the common ratio.
(2)
(c)
Find the other value of x for which the sequence is geometric.
(4)
(d)
For this value of x, find
(i)
the common ratio;
(ii)
the sum of the infinite sequence.
(3)
(Total 12 marks)
26
18.
Clara organizes cans in triangular piles, where each row has one less can than the row below. For example, the
pile of 15 cans shown has 5 cans in the bottom row and 4 cans in the row above it.
(a)
A pile has 20 cans in the bottom row. Show that the pile contains 210 cans.
(4)
(b)
There are 3240 cans in a pile. How many cans are in the bottom row?
(4)
(c)
(i)
There are S cans and they are organized in a triangular pile with n cans in the bottom row. Show that
n2 + n − 2S = 0.
(ii)
Clara has 2100 cans. Explain why she cannot organize them in a triangular pile.
(6)
(Total 14 marks)
27
19.
Consider the infinite geometric sequence 3000, – 1800, 1080, – 648, … .
(a)
Find the common ratio.
(2)
(b)
Find the 10th term.
(2)
(c)
Find the exact sum of the infinite sequence.
(2)
(Total 6 marks)
20.
The sum of an infinite geometric sequence is 13 12 , and the sum of the first three terms is 13. Find the first term.
(Total 3 marks)
28
21.
Consider the arithmetic series 2 + 5 + 8 +....
(a)
Find an expression for Sn, the sum of the first n terms.
(b)
Find the value of n for which Sn = 1365.
(Total 6 marks)
22.
The first four terms of an arithmetic sequence are 2, a – b, 2a +b + 7, and a – 3b, where a and b are constants.
Find a and b.
(Total 6 marks)
29
23.
The three terms a, 1, b are in arithmetic progression. The three terms 1, a, b are in geometric progression.
Find the value of a and of b given that a b.
(Total 6 marks)
24.
The diagram shows a sector AOB of a circle of radius 1 and centre O, where AÔB = . The lines (AB1), (A1B2),
(A2B3) are perpendicular to OB. A1B1, A2B2 are all arcs of circles with centre O. Calculate the sum to infinity of
the arc lengths AB + A1B1 + A2B2 + A3B3 + …
(Total 6 marks)
A
A1
A2
O
B3
B2
B1
30
B
25.
A sum of $5 000 is invested at a compound interest rate of 6.3% per annum.
(a)
Write down an expression for the value of the investment after n full years.
(b)
What will be the value of the investment at the end of five years?
(c)
The value of the investment will exceed $10 000 after n full years.
(i)
Write an inequality to represent this information.
(ii)
Calculate the minimum value of n.
(Total 6 marks)
26.
The sum to infinity of a geometric series is 32. The sum of the first four terms is 30 and all the terms are positive.
Find the difference between the sum to infinity and the sum of the first eight terms.
(Total 6 marks)
31
27.
A sum of $100 is invested.
(a)
If the interest is compounded annually at a rate of 5
20 years.
(b)
If the interest is compounded monthly at a rate of
per year, find the total value V of the investment after
5
12
per month, find the minimum number of months
for the value of the investment to exceed V.
(Total 6 marks)
28.
k
2 4 3x .
An infinite geometric series is given by
k 1
(a)
Find the values of x for which the series has a finite sum.
(b)
When x = 1.2, find the minimum number of terms needed to give a sum which is greater than 1.328.
(Total 6 marks)
32
29.
Consider the arithmetic series −6 +1 +8 +15 +.... Find the least number of terms so that the sum of the series is
greater than 10 000.
(Total 6 marks)
30.
Consider the arithmetic sequence 8, 26, 44, ....
(a)
Find an expression for the nth term.
(1)
(b)
Write down the sum of the first n terms using sigma notation.
(1)
(c)
Calculate the sum of the first 15 terms.
(2)
(Total 4 marks)
33
31.
In the arithmetic series with nth term un, it is given that u4 = 7 and u9 = 22.
Find the minimum value of n so that u1 + u2 + u3 + ... + un > 10 000.
(Total 5 marks)
32.
Find the sum of all three-digit natural numbers that are not exactly divisible by 3.
(Total 5 marks)
34
33.
A geometric sequence has a first term of 2 and a common ratio of 1.05. Find the value of the smallest term that is
greater than 500.
(Total 5 marks)
34.
A sum of $ 5000 is invested at a compound interest rate of 6.3 % per annum.
(a)
Write down an expression for the value of the investment after n full years.
(1)
(b)
What will be the value of the investment at the end of five years?
(1)
(c)
The value of the investment will exceed $10 000 after n full years.
(i)
Write an inequality to represent this information.
(ii)
Calculate the minimum value of n.
(4)
(Total 6 marks)
35
1.2 Exponents and Logarithms
1.
Solve the equation 9x–1 =
1
3
Paper 1
2x
.
(Total 4 marks)
2.
Solve the equation 43x–1 = 1.5625 × 10–2.
(Total 4 marks)
3.
If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for
(a)
log2 5;
(b)
loga 20.
(Total 4 marks)
36
4.
Let log10P = x , log10Q = y and log10R = z. Express log10
P
QR 3
2
in terms of x , y and z.
(Total 4 marks)
5.
Given that log5 x = y, express each of the following in terms of y.
(a)
log5 x2
(b)
log5
(c)
log25 x
1
x
(Total 6 marks)
37
6.
Let p = log10 x, q = log10 y and r = log10 z. Write the expression log10
x
y
2
z
in terms of p, q and r.
(Total 6 marks)
7.
Let a = log x, b = log y, and c = log z. Write log
x2 y
z3
in terms of a, b and c.
(Total 6 marks)
38
8.
(a)
(b)
Let logc 3 = p and logc 5 = q. Find an expression in terms of p and q for
(i)
log c 15;
(ii)
log c 25.
Find the value of d if log d 6 =
1
.
2
(Total 6 marks)
9.
Let ln a = p, ln b = q. Write the following expressions in terms of p and q.
(a)
ln a3b
(b)
ln
a
b
(Total 6 marks)
39
10.
Given that p = loga 5, q = loga 2, express the following in terms of p and/or q.
(a)
loga 10
(b)
loga 8
(c)
loga 2.5
(Total 6 marks)
11.
Find the exact value of x satisfying the equation (3x)(42x+1) = 6x+2.
ln a
Give your answer in the form
where a, b
.
ln b
(Total 6 marks)
40
12.
Solve log16
3
100 – x 2
1
.
2
(Total 6 marks)
13.
Solve 2(5x+1) = 1 +
3
, giving the answer in the form a + log5 b, where a, b
5x
.
(Total 6 marks)
41
14.
Find an expression for the sum of the first 35 terms of the series ln x2 + ln
answer in the form ln
xm
, where m, n
yn
x2
y
ln
x2
y2
ln
x2
y3
giving your
ℕ.
(Total 5 marks)
15.
Given that 4 ln 2 – 3ln 4 = –ln k, find the value of k.
(Total 5 marks)
42
16.
Solve the equation log3(x + 17) – 2 = log3 2x.
(Total 5 marks)
17.
Solve the equation 22x+2 – 10 × 2x + 4 = 0, x
.
(Total 6 marks)
43
1.2 Exponents and Logarithms
1.
Solve the equation log9 81 + log9
Paper 2
1
+ log9 3 = log9 x.
9
(Total 4 marks)
2.
Solve the equation log27 x = 1 – log27 (x – 0.4).
(Total 6 marks)
44
3.
Find the exact solution of the equation 92x = 27(1–x).
(Total 6 marks)
4.
(a)
Given that log3 x – log3 (x – 5) = log3 A, express A in terms of x.
(b)
Hence or otherwise, solve the equation log3 x – log3 (x – 5) = 1.
(Total 6 marks)
45
5.
(a)
Expand
1
e
e
4
in terms of e.
(4)
(b)
Express
e
1
e
4
+
e
1
e
4
as the sum of three terms.
(2)
(Total 6 marks)
6.
Solve the following system of equations.
logx+1 y = 2
1
logy+1 x =
4
(Total 6 marks)
46
1.3 Binomial Theorem
1.
Paper 1
Use the binomial theorem to complete this expansion.
(3x + 2y)4 = 81x4 + 216x3 y +...
(Total 4 marks)
2.
4
x
1
Consider the binomial expansion (1 x) 4 1
4 2
x
2
4 3
x
3
x4 .
4
1
4
2
(a)
By substituting x = 1 into both sides, or otherwise, evaluate
(b)
Evaluate
9
9
9
9
9
9
1
2
3
4
5
6
9
7
4
.
3
9
.
8
(Total 4 marks)
47
3.
Find the coefficient of x3 in the expansion of (2 – x)5.
(Total 6 marks)
4.
Complete the following expansion:
(2 + ax)4 = 16 + 32ax + …
(Total 6 marks)
5.
Consider the expansion of (x2 – 2)5.
(a)
Write down the number of terms in this expansion.
(b)
The first four terms of the expansion in descending powers of x are x10 – 10x8 + 40x6 + Ax4 + ...
Find the value of A.
(Total 6 marks)
48
6.
Given that 3
(a)
p;
(b)
q.
7
3
= p + q 7 where p and q are integers, find
(Total 6 marks)
7.
One of the terms of the expansion of (x + 2y)10 is ax8 y2. Find the value of a.
(Total 6 marks)
8.
Given that (1 + x)5 (1 + ax)6
1 + bx + 10x2 + ............... + a6 x11, find the values of a, b
*.
(Total 4 marks)
49
8
9.
1
Find the coefficient of x in the binomial expansion of 1 – x .
2
3
(Total 6 marks)
10.
The coefficient of x in the expansion of
x
1
ax 2
7
is 7 . Find the possible values of a.
3
(Total 3 marks)
11.
Express
3 2
3
in the form a 3 b , where a, b
.
(Total 6 marks)
50
12.
(a)
Write down the quadratic expression 2x2 + x – 3 as the product of two linear factors.
(1)
(b)
Hence, or otherwise, find the coefficient of x in the expansion of (2x2 + x – 3)8.
(4)
(Total 5 marks)
13.
Expand and simplify
x
2
2
x
4
.
(Total 4 marks)
51
14.
The diagram below shows a solid with volume V, obtained from a cube with edge a > 1 when a smaller cube
1
1
with edge
is removed. Let x = a
.
a
a
(a)
Find V in terms of x.
(4)
(b)
Hence or otherwise, show that the only value of a for which V = 4x is a =
1
5
2
.
(4)
(Total 8 marks)
diagram not to scale
15.
Determine the first three terms in the expansion of (1− 2x)5 (1+ x)7 in ascending powers of x.
(Total 5 marks)
52
1.3 Binomial Theorem
1.
Paper 2
Find the coefficient of x5 in the expansion of (3x – 2)8.
(Total 4 marks)
2.
Find the coefficient of a3b4 in the expansion of (5a + b)7.
(Total 4 marks)
3.
Find the coefficient of a5b7 in the expansion of (a + b)12.
(Total 4 marks)
53
4.
Determine the constant term in the expansion of
2
x– 2
x
9
.
(Total 4 marks)
5.
Consider the expansion of
1
3x –
x
2
9
.
(a)
How many terms are there in this expansion?
(b)
Find the constant term in this expansion.
(Total 6 marks)
6.
Find the term containing x10 in the expansion of (5 + 2x2)7.
(Total 6 marks)
54
7.
When the expression (2 + ax)10 is expanded, the coefficient of the term in x3 is 414 720. Find the value of a.
(Total 6 marks)
8.
Find the term containing x3 in the expansion of (2 – 3x)8.
(Total 6 marks)
9.
Consider the expansion of the expression (x3 − 3x)6.
(a)
Write down the number of terms in this expansion.
(b)
Find the term in x12.
(Total 6 marks)
55
8
10.
3
Find the term in x in the expansion of
2
x 3 .
3
(Total 5 marks)
11.
Find the coefficient of x7 in the expansion of (2 + 3x)10, giving your answer as a whole number.
(Total 3 marks)
12.
(a)
Find the expansion of (2 + x)5, giving your answer in ascending powers of x.
(b)
By letting x = 0.01 or otherwise, find the exact value of 2.015.
(Total 6 marks)
56
13.
(a)
Simplify the difference of binomial coefficients
n
3
2n
, where n ≥ 3.
2
(4)
(b)
Hence, solve the inequality
n
3
2n
2
> 32n, where n ≥ 3.
(2)
(Total 6 marks)
14.
When 1
(a)
x
2
n
,n
, is expanded in ascending powers of x, the coefficient of x3 is 70.
Find the value of n.
(5)
(b)
Hence, find the coefficient of x2.
(1)
(Total 6 marks)
57
1.4 Counting Principles
1.
Paper 1
Mr Blue, Mr Black, Mr Green, Mrs White, Mrs Yellow and Mrs Red sit around a circular table for a meeting. Mr
Black and Mrs White must not sit together. Calculate the number of different ways these six people can sit at the
table without Mr Black and Mrs White sitting together.
(Total 3 marks)
2.
In how many ways can six different coins be divided between two students so that each student receives at least
one coin?
(Total 3 marks)
3.
How many four-digit numbers are there which contain at least one digit 3?
(Total 3 marks)
58
4.
A committee of four children is chosen from eight children. The two oldest children cannot both be chosen. Find
the number of ways the committee may be chosen.
(Total 6 marks)
5.
There are 25 disks in a bag. Some of them are black and the rest are white. Two are simultaneously selected at
random. Given that the probability of selecting two disks of the same colour is equal to the probability of
selecting two disks of different colour, how many black disks are there in the bag?
(Total 6 marks)
59
1.4 Counting Principles
1.
Paper 2
There are 10 seats in a row in a waiting room. There are six people in the room.
(a)
In how many different ways can they be seated?
(b)
In the group of six people, there are three sisters who must sit next to each other.
In how many different ways can the group be seated?
(Total 6 marks)
2.
Twelve people travel in three cars, with four people in each car. Each car is driven by its owner. Find the number
of ways in which the remaining nine people may be allocated to the cars. (The arrangement of people within a
particular car is not relevant).
(Total 6 marks)
60
3.
Six people are to sit at a circular table. Two of the people are not to sit immediately beside each other. Find
the number of ways that the six people can be seated.
(Total 5 marks)
4.
Three Mathematics books, five English books, four Science books and a dictionary are to be placed on a
student’s shelf so that the books of each subject remain together.
(a)
In how many different ways can the books be arranged?
(4)
(b)
In how many of these will the dictionary be next to the Mathematics books?
(3)
(Total 7 marks)
61
1.5 Mathematical Induction
1.
Paper 1
Using mathematical induction, prove that the number 22n – 3n – 1 is divisible by 9, for n = 1, 2, ..... .
(Total 7 marks)
2.
Use mathematical induction to prove that 5n + 9n + 2 is divisible by 4, for n
+
.
(Total 9 marks)
62
n
3.
(a)
Use mathematical induction to prove that
r 1
1
(2 r 1) (2 r 1)
n
,n
2n 1
+
.
(6)
(b)
Hence show that the sum of the first (n + 1) terms of the series
1
1
1
1
(n 1)
+
+
+
+ ... is
.
(2 n 3)
3
15
35
63
(3)
(Total 9 marks)
4.
Prove by induction that 12n + 2(5n−1) is a multiple of 7 for n
+
.
(Total 10 marks)
63
5.
+
Prove by mathematical induction that, for n
1+ 2
1
2
3
1
2
2
4
1
2
,
3
... n
1
2
n 1
4
n 2
2n
1
.
(Total 8 marks)
n
6.
r (r! )
Prove by mathematical induction
(n 1)! 1, n
+
.
r 1
(Total 8 marks)
64
7.
(a)
Consider the following sequence of equations.
1
(1 × 2 × 3),
3
1
1 × 2 + 2 × 3 = (2 × 3 × 4),
3
1
1 × 2 + 2 × 3 + 3 × 4 = (3 × 4 × 5),
3
.... .
1×2=
(i)
Formulate a conjecture for the nth equation in the sequence.
(ii)
Verify your conjecture for n = 4.
(2)
(b)
+
A sequence of numbers has the nth term given by un = 2n + 3, n
. Bill conjectures that all members of
the sequence are prime numbers. Show that Bill’s conjecture is false.
(2)
(c)
Use mathematical induction to prove that 5 × 7n + 1 is divisible by 6 for all n
+
.
(6)
(Total 10 marks)
65
8.
(a)
The sum of the first six terms of an arithmetic series is 81. The sum of its first eleven terms is 231.
Find the first term and the common difference.
(6)
(b)
The sum of the first two terms of a geometric series is 1 and the sum of its first four terms is 5.
If all of its terms are positive, find the first term and the common ratio.
(5)
(c)
The rth term of a new series is defined as the product of the rth term of the arithmetic series and the rth term
of the geometric series above. Show that the rth term of this new series is (r + 1)2r–1.
(3)
(d)
Using mathematical induction, prove that
n
(r 1)2 r
1
n2 n , n
+
.
r 1
(7)
(Total 21 marks)
66
9.
(a)
Find the sum of the infinite geometric sequence 27, −9, 3, −1, ... .
(3)
(b)
Use mathematical induction to prove that for n
+
, a + ar + ar2 + ... + arn–1 =
a 1 rn
.
1 r
(7)
(Total 10 marks)
10.
Use mathematical induction to prove that 5n + 9n + 2 is divisible by 4, for n
+
.
(Total 9 marks)
67
1.5 Mathematical Induction
1.
(a)
Paper 2
Use mathematical induction to prove that
(1)(1!) + (2)(2!) + (3)(3!) + ...+ (n)(n!) = (n +1)!−1 where n
+
.
(8)
(b)
Find the minimum number of terms of the series for the sum to exceed 109.
(3)
(Total 11 marks)
68
1.6 Complex Numbers
1.
Paper 1
Let z = x + yi. Find the values of x and y if (1 – i)z = 1 – 3i.
(Total 4 marks)
2.
(a)
Evaluate (1 + i)2, where i =
1.
(2)
(b)
Prove, by mathematical induction, that (1 + i)4n = (–4)n, where n
*.
(6)
(c)
Hence or otherwise, find (1 + i)32.
(2)
(Total 10 marks)
69
3.
Let z1 =
(a)
6
i 2
2
, and z2 = 1 – i.
Write z1 and z2 in the form r(cos θ + i sin θ), where r > 0 and –
π
2
θ
π
.
2
(6)
(b)
Show that
z1
= cos
+ i sin
.
12
12
z2
(2)
(c)
Find the value of
z1
in the form a + bi, where a and b are to be determined exactly in radical (surd) form.
z2
Hence or otherwise find the exact values of cos
12
and sin
12
.
(4)
(Total 12 marks)
70
4.
If z is a complex number and |z + 16| = 4 |z + l|, find the value of | z|.
(Total 3 marks)
5.
Consider the equation 2(p + iq) = q – ip – 2 (1 – i), where p and q are both real numbers. Find p and q.
(Total 6 marks)
71
6.
Let the complex number z be given by z = 1 +
i
i– 3
. Express z in the form a +bi, giving the exact values of the
real constants a, b.
(Total 6 marks)
7.
Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z = 60°.
(Total 6 marks)
72
8.
The two complex numbers z1 =
a
1 i
Calculate the value of a and of b.
and z2 =
b
where a, b
1 2i
, are such that z1 + z2 = 3.
(Total 6 marks)
9.
Let z1 and z2 be complex numbers. Solve the simultaneous equations
2z1 + z2 = 7, z1 + iz2 = 4 + 4i
Give your answers in the form z = a + bi, where a, b
.
(Total 6 marks)
73
10.
Find, in its simplest form, the argument of (sin + i (1− cos ))2 where
is an acute angle.
(Total 7 marks)
11.
The complex number z is defined by
z=4
cos
2π
2π
i sin
3
3
4 3 cos
π
π
i sin .
6
6
(a)
Express z in the form rei , where r and
(b)
Find the cube roots of z, expressing in the form rei , where r and
have exact values.
have exact values.
(Total 6 marks)
74
12.
Let z1 = r
cos
π
π
i sin
4
4
and z2 = 1 +
(a)
Write z2 in modulus-argument form.
(b)
Find the value of r if z1 z 2
3
3 i.
= 2.
(Total 6 marks)
13.
Given that
z
z 2
= 2 – i, z
, find z in the form a + ib.
(Total 4 marks)
75
14.
Given that z = cosθ + i sin θ show that
(a)
n
Im z
1
z
n
0, n
+
;
(2)
(b)
Re
z 1
z 1
= 0, z ≠ –1.
(5)
(Total 7 marks)
15.
Consider the complex numbers z = 1 + 2i and w = 2 +ai, where a
(a)
. Find a when
│w│ = 2│z│;
(3)
(b)
Re (zw) = 2 Im(zw).
(3)
(Total 6 marks)
76
16.
Consider w =
z
z
2
1
where z = x + iy, y
0 and z2 + 1
0.
Given that Im w = 0, show that z = 1.
(Total 7 marks)
17.
Given that (a + bi)2 = 3 + 4i obtain a pair of simultaneous equations involving a and b.
Hence find the two square roots of 3 + 4i.
(Total 7 marks)
77
18.
Given that │z│ =
10 , solve the equation 5z +
10
= 6 – 18i, where z* is the conjugate of z.
z*
(Total 7 marks)
19.
Solve the simultaneous equations
iz1 + 2z2 = 3
z1 + (1 – i)z2 = 4
giving z1 and z2 in the form x + iy, where x and y are real.
(Total 9 marks)
78
20.
Find b where
2 bi
1 bi
7 9
i.
10 10
(Total 6 marks)
21.
Given that z = (b + i)2, where b is real and positive, find the value of b when arg z = 60°.
(Total 6 marks)
79
1.6 Complex Numbers
1.
Paper 2
Find the values of a and b, where a and b are real, given that (a + bi)(2 – i) = 5 – i.
(Total 3 marks)
2.
Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z = 60°.
(Total 3 marks)
3.
–1 .
The complex number z satisfies i(z + 2) = 1 – 2z, where i
Write z in the form z = a + bi, where a and b are real numbers.
(Total 3 marks)
80
4.
A complex number z is such that
(a)
z
z
Show that the imaginary part of z is
3i .
3
.
2
(2)
(b)
Let z1 and z2 be the two possible values of z, such that z
3.
(i)
Sketch a diagram to show the points which represent z1 and z2 in the complex plane, where z1
is in the first quadrant.
(ii)
Show that arg z1 =
(iii)
Find arg z2.
π
.
6
(4)
(c)
Given that arg
z1k z 2
2i
= π, find a value of k.
(4)
(Total 10 marks)
81
5.
The complex number z satisfies the equation
z=
2
+ 1 – 4i. Express z in the form x + iy where x, y
1– i
.
(Total 6 marks)
6.
Given that (a + i)(2 – bi) = 7 – i, find the value of a and of b, where a, b
.
(Total 6 marks)
7.
Given that | z | = 2 5 , find the complex number z that satisfies the equation 25
z
15
z*
1 8i.
(Total 6 marks)
82
1.7 De Moivre’s Theorem
1.
Paper 1
Let x and y be real numbers, and
(a)
1+
2
+
be one of the complex solutions of the equation z3 = 1. Evaluate:
;
(2)
(b)
( x+
2
y)(
2
x+
y).
(4)
(Total 6 marks)
2.
(a)
Express z5 – 1 as a product of two factors, one of which is linear.
(2)
(b)
Find the zeros of z5 – 1, giving your answers in the form r(cos θ + i sin θ) where r > 0 and –π < θ π.
(3)
(c)
Express z4 + z3 + z2 + z + 1 as a product of two real quadratic factors.
(5)
(Total 10 marks)
83
3.
(a)
Prove, using mathematical induction, that for a positive integer n,
(cos + i sin )n = cos n + i sin n
where i2 = –1.
(5)
(b)
The complex number z is defined by z = cos + i sin .
1
= cos (– ) + i sin (– ).
z
(i)
Show that
(ii)
Deduce that zn + z–n = 2 cos nθ.
(5)
(c)
(i)
Find the binomial expansion of (z + z–l)5.
(ii)
Hence show that cos5 =
1
(a cos 5 + b cos 3 + c cos ),
16
where a, b, c are positive integers to be found.
(5)
(Total 15 marks)
84
4.
(a)
Solve the equation z3 = –2 + 2i, giving your answers in modulus–argument form.
(6)
(b)
Hence show that one of the solutions is 1 + i when written in Cartesian form.
(1)
(Total 7 marks)
5.
Find the values of n such that (1 +
3 i)n is a real number.
(Total 5 marks)
85
6.
(a)
Let z = x + iy be any non-zero complex number.
1
in the form u + iv.
z
(i)
Express
(ii)
If z
(iii)
Show that if x2 + y2 = 1 then │k│ ≤ 2.
1
z
k, k
, show that either y = 0 or x2 + y2 = 1.
(8)
(b)
Let w = cos θ + i sin θ.
(i)
Show that wn + w–n = 2cos nθ, n
(ii)
Solve the equation 3w2 – w + 2 – w–1 + 3w–2 = 0, giving the roots in the
form x + iy.
.
(14)
(Total 22 marks)
86
7.
1
Express
1 i 3
3
in the form
a
where a, b
b
.
(Total 5 marks)
8.
Let w = cos
(a)
2
5
i sin
2
.
5
Show that w is a root of the equation z5 − 1 = 0.
(3)
(b)
Show that (w − 1) (w4 + w3 + w2 + w + 1) = w5 − 1 and deduce that
w4 + w3 + w2 + w + 1 = 0.
(3)
(c)
Hence show that cos
2
5
cos
4
5
1
.
2
(6)
(Total 12 marks)
87
9.
(a)
Use de Moivre’s theorem to find the roots of the equation z4 = 1 – i.
(6)
(b)
Draw these roots on an Argand diagram.
(2)
(c)
If z1 is the root in the first quadrant and z2 is the root in the second quadrant, find
z2
in the form a + ib.
z1
(4)
(Total 12 marks)
10.
Find the three cube roots of the complex number 8i. Give your answers in the form x + iy.
(Total 8 marks)
88
1.7 De Moivre’s Theorem
1.
(a)
Express the complex number 8i in polar form.
(b)
The cube root of 8i which lies in the first quadrant is denoted by z. Express z
(i)
in polar form;
(ii)
in cartesian form.
Paper 2
(Total 6 marks)
2.
Given that z
, solve the equation z3 – 8i = 0, giving your answers in the form z = r (cos + i sin ).
(Total 6 marks)
89
3.
z1 = 1 i 3
(a)
m
n
and z2 = 1 i .
Find the modulus and argument of z1 and z2 in terms of m and n, respectively.
(6)
(b)
Hence, find the smallest positive integers m and n such that z1 = z2.
(8)
(Total 14 marks)
90
1.8 Conjugate Roots
1.
Paper 1
(z + 2i) is a factor of 2z3–3z2 + 8z – 12. Find the other two factors.
(Total 3 marks)
2.
The polynomial P(z) = z3 + mz2 + nz −8 is divisible by (z +1+ i), where z
Find the value of m and of n.
and m, n
.
(Total 6 marks)
91
3.
(a)
Express the complex number 1+ i in the form
ae
i
π
b
, where a, b
+
.
(2)
(b)
Using the result from (a), show that
1 i
2
n
, where n
, has only eight distinct values.
(5)
(c)
Hence solve the equation z8 −1 = 0.
(2)
(Total 9 marks)
92
4.
Given that z1 = 2 and z2 = 1 + i 3 are roots of the cubic equation z3 + bz2 + cz + d = 0 where b, c, d
(a)
,
write down the third root, z3, of the equation;
(1)
(b)
find the values of b, c and d;
(4)
(c)
write z2 and z3 in the form reiθ.
(3)
(Total 8 marks)
5.
Consider the equation z3 + az2 + bz + c = 0, where a, b, c
. The points in the Argand diagram representing
the three roots of the equation form the vertices of a triangle whose area is 9. Given that one root is –1 + 3i, find
(a)
the other two roots;
(4)
(b)
a, b and c.
(3)
(Total 7 marks)
93
6.
(a)
Expand and simplify (x – 1)(x4 + x3 + x2 + x + 1).
(2)
(b)
Given that b is a root of the equation z5 –1 = 0 which does not lie on the real axis in the Argand diagram,
show that 1 + b + b2 + b3 + b4 = 0.
(3)
(c)
If u = b + b4 and v = b2 + b3 show that
(i)
u + v = uv = –1;
(ii)
u–v=
5 , given that u – v > 0.
(8)
(Total 13 marks)
7.
Given that 2 + i is a root of the equation x3 – 6x2 + 13x – 10 = 0 find the other two roots.
(Total 5 marks)
94
8.
(a)
Show that the complex number i is a root of the equation x4 – 5x3 + 7x2 – 5x + 6 = 0.
(2)
(b)
Find the other roots of this equation.
(4)
(Total 6 marks)
9.
Consider the polynomial p(x) = x4 + ax3 + bx2 + cx + d, where a, b, c, d
.
Given that 1 + i and 1 – 2i are zeros of p(x), find the values of a, b, c and d.
(Total 7 marks)
95