IB · Mathematics: Analysis & Approaches SL & HL · Topic

Number & Algebra: IB Mathematics: Analysis & Approaches questions and answers

10 free IB Mathematics: Analysis & Approaches Number & Algebra questions written in the style of IB Mathematics: Analysis & Approaches papers, each with its answer and mark scheme. 2 of them are A* challenge questions, built to catch the mistakes strong students still make.

This topic covers Sequences & series, Financial applications, Exponents & logarithms, Binomial theorem, Proof, Counting principles, Partial fractions, Complex numbers, Induction, contradiction & counterexample, Systems of linear equations. Work through the questions below, then generate a full Number & Algebra paper with a fresh set every time.

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Number & algebra sub-topics

Sequences & seriesFinancial applicationsExponents & logarithmsBinomial theoremProofCounting principlesPartial fractionsComplex numbersInduction, contradiction & counterexampleSystems of linear equations

Number & algebra practice questions

Question 1Structured · 6 marksA* challenge

A committee of 5 people is chosen from 7 men and 6 women.

(a)

Find the number of committees that contain at least two women.

[3]
(b)

Two particular people refuse to serve on the committee together. Find the number of possible committees.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1056
total 13C5 = 1287 (A1)
subtract committees with 0 or 1 woman: 21 + 210 (M1)
1056 A1
3
(b)1122
committees containing both: 11C3 = 165 (A1)
1287 − 165 (M1)
1122 A1
3
Question 2Structured · 7 marksA* challenge

Account A pays 5.5% per year compounded annually. Account B pays 5.1% per year compounded monthly. Mei invests $20000 for 8 years.

(a)

Determine which account gives the larger final amount, and by how much, to the nearest dollar.

[4]
(b)

Find the minimum nominal annual rate, compounded monthly, needed for an investment to double in 10 years. Give your answer to two decimal places.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)A, by $644
A: 20000(1 + 0.055)8 = 30693.73 (M1)(A1)
B: 20000(1 + 5.11200)96 = 30050.15 (A1)
A by $644 A1
4
(b)6.95%
(1 + r1200)120 = 2 (M1)
attempt to solve, e.g. TVM solver (M1)
r = 6.95% A1
3
Question 3Structured · 7 marks
(a)

Express 4x − 8(x − 1)(x − 3) in partial fractions.

[3]
(b)

Hence find ∫584x − 8(x − 1)(x − 3)dx, giving your answer in the form ln k, where k ∈ ℚ.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2x − 1 + 2x − 3
4x − 8 ≡ A(x − 3) + B(x − 1) M1
A = 2 A1
B = 2 A1
3
(b)ln 122564
[2ln|x − 1| + 2ln|x − 3|]58 A1
substituting limits (M1)
use of log laws to combine M1
ln 122564 A1
4
Question 4Structured · 5 marks

Prove that the sum of the squares of any two consecutive odd integers is 2 more than a multiple of 8.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
proof
let the integers be 2n + 1 and 2n + 3, n ∈ ℤ M1
(2n + 1)2 + (2n + 3)2 = 4n2 + 4n + 1 + 4n2 + 12n + 9 A1
= 8n2 + 16n + 10 A1
= 8(n2 + 2n + 1) + 2 A1
which is 2 more than a multiple of 8 R1
5
Question 5Structured · 6 marks

Consider the 8 letters of the word CALCULUS.

(a)

Find the number of different arrangements of these 8 letters.

[2]
(b)

Find the number of these arrangements in which all the vowels are together.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)5040
8!2! 2! 2! (M1)
5040 A1
2
(b)540
treat the vowels as a single block (M1)
arrangements of the block and 5 consonants: 180 (A1)
arrangements within the block: 3 (A1)
540 A1
4
Question 6Structured · 7 marks

Prove by mathematical induction that 7n − 3n is divisible by 4 for all n ∈ ℤ+.

[7]
Show answers and mark scheme
PartAnswer and marking pointsMarks
proof
n = 1: 7 − 3 = 4 A1
assume 7k − 3k = 4M, M ∈ ℤ M1
7k+1 − 3k+1 = 7·7k − 3·3k M1
= 7(4M + 3k) − 3·3k (A1)
= 28M + 4·3k A1
= 4(7M + 3k), divisible by 4 A1
true for n = 1 and P(k) ⇒ P(k + 1), so true for all n ∈ ℤ+ R1
7
Question 7Structured · 5 marks
(a)

Find the first three terms in the expansion of (4 + 3x)−1/2 in ascending powers of x.

[4]
(b)

State the set of values of x for which the expansion is valid.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)12 − 316x + 27256x2
4−1/2(1 + 34x)−1/2 (M1)
use of (1 + u)n = 1 + nu + n(n − 1)2!u2 + … (M1)
12 − 316x A1
27256x2 A1
4
(b)|x| < 43
|x| < 43 A1
1
Question 8Structured · 6 marks

The second term of a geometric sequence is −54 and the fifth term is 2.

(a)

Find the common ratio.

[3]
(b)

Find the sum to infinity of the sequence.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)r = −13
u1r = −54, u1r4 = 2 (M1)
r3 = −127 (A1)
r = −13 A1
3
(b)2432
u1 = 162 (A1)
S∞ = 1621 − (−1/3) (M1)
2432 A1
3
Question 9Structured · 14 marks

Offer A: a starting salary of $32000 with an increase of $1200 each year.
Offer B: a starting salary of $32000 with an increase of 4% each year.

(a)

Find the salary in year 10 for each offer.

[4]
(b)

Find the total amount earned over 25 years under each offer.

[4]
(c)

Find the first year in which the salary under Offer B is greater than the salary under Offer A.

[3]
(d)

Find the minimum number of complete years after which the total earned under Offer B exceeds the total earned under Offer A.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)A: $42800; B: $45546
A: 32000 + 9(1200) (M1)
$42800 A1
B: 32000(1.04)9 (M1)
$45546 A1
4
(b)A: $1160000; B: $1332669
S25 = 252(2(32000) + 24(1200)) (M1)
$1160000 A1
S25 = 32000(1.0425 − 1)1.04 − 1 (M1)
$1332669 A1
4
(c)year 2
32000(1.04)n−1 > 32000 + (n − 1)(1200) (M1)
attempt to solve, e.g. table or graph (M1)
year 2 A1
3
(d)2 years
equating or comparing the sums (M1)
attempt to solve (M1)
2 A1
3
Question 10Structured · 5 marks

Solve the equation 9x − 2(3x) − 3 = 0, giving your answers in exact form.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
x = 1 only
let u = 3x: u2 − 2u − 3 = 0 (M1)
(u − 3)(u + 1) = 0 A1
3x = 3 (A1)
3x = −1 is impossible since 3x > 0 for all x R1
x = 1 A1
5

Other IB Mathematics: Analysis & Approaches topics

FunctionsGeometry & trigonometryStatistics & probabilityCalculus

Frequently asked questions

What is in IB Mathematics: Analysis & Approaches Number & Algebra?

In the SL & HL syllabus, Number & Algebra covers Sequences & series, Financial applications, Exponents & logarithms, Binomial theorem, Proof, Counting principles, Partial fractions, Complex numbers, Induction, contradiction & counterexample, Systems of linear equations.

Where can I get Number & Algebra questions with mark schemes?

This page has 10 Number & Algebra questions with answers. For more, the free generator on mentorriyaz.in builds a new Number & Algebra paper with a full mark scheme each time you click.

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in