IB · Mathematics: Analysis & Approaches · SL & HL

IB Mathematics: Analysis & Approaches (SL & HL) topical questions with answers

Every IB Mathematics: Analysis & Approaches topic, with free past-paper-style questions and full mark schemes. Pick a chapter to practise its questions, or generate a complete SL & HL-format paper in seconds, from easy warm-ups to A*-level "brutal" papers.

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IB Mathematics: Analysis & Approaches questions by topic

Number & algebra

Sequences & series · Financial applications · Exponents & logarithms · Binomial theorem · Proof · Counting principles · Partial fractions · Complex numbers · Induction, contradiction & counterexample · Systems of linear equations

Functions

Functions, composites & inverses · Quadratics & the discriminant · Rational functions · Exponential & log functions · Transformations of graphs · Polynomials & roots · Modulus, odd/even & inequalities

Geometry & trigonometry

3D geometry · Sine & cosine rules · Radians, arcs & sectors · Trig identities & exact values · Trig equations & graphs · Compound angles & reciprocal trig · Vectors, lines & planes

Statistics & probability

Descriptive statistics · Correlation & regression · Probability · Discrete random variables · Binomial distribution · Normal distribution · Bayes' theorem · Continuous random variables

Calculus

Differentiation · Tangents & normals · Stationary points & optimisation · Integration · Areas · Kinematics · Implicit, related rates & first principles · Integration techniques · Volumes of revolution · Differential equations · Maclaurin series & limits

Practice papers in the real SL & HL format

Paper 1 · SL (no calculator) · 2029 syllabusPaper 2 · SL (calculator) · 2029 syllabusPaper 1 · HL (no calculator) · 2029 syllabusPaper 2 · HL (calculator) · 2029 syllabusPaper 3 · HL problem solving · 2029 syllabusPaper 1 · SL (no calculator) · exams to Nov 2028Paper 2 · SL (calculator) · exams to Nov 2028Paper 1 · HL (no calculator) · exams to Nov 2028Paper 2 · HL (calculator) · exams to Nov 2028Paper 3 · HL problem solving · exams to Nov 2028

Paper 1 · SL (no calculator) · 2029 syllabus (2029 syllabus · SL · 75 marks · 1h 30m · Section A short + Section B long); Paper 2 · SL (calculator) · 2029 syllabus (2029 syllabus · SL · 75 marks · 1h 30m · Section A short + Section B long); Paper 1 · HL (no calculator) · 2029 syllabus (2029 syllabus · HL · 100 marks · 2h · Section A short + Section B long); Paper 2 · HL (calculator) · 2029 syllabus (2029 syllabus · HL · 100 marks · 2h · Section A short + Section B long); Paper 3 · HL problem solving · 2029 syllabus (2029 syllabus · HL · 50 marks · 1h · two extended investigations); Paper 1 · SL (no calculator) · exams to Nov 2028 (SL · 80 marks · 1h 30m · Section A short + Section B long); Paper 2 · SL (calculator) · exams to Nov 2028 (SL · 80 marks · 1h 30m · Section A short + Section B long); Paper 1 · HL (no calculator) · exams to Nov 2028 (HL · 110 marks · 2h · Section A short + Section B long); Paper 2 · HL (calculator) · exams to Nov 2028 (HL · 110 marks · 2h · Section A short + Section B long); Paper 3 · HL problem solving · exams to Nov 2028 (HL · 55 marks · 1h 15m · two extended investigations).

Sample IB Mathematics: Analysis & Approaches questions

A mixed set from across the syllabus. Questions marked A* challenge come from the hardest bank.

Question 1Structured · 6 marksA* challenge

A window is in the shape of a rectangle of width x m with a semicircle on top. The perimeter of the window is 8 m.

(a)

Show that the area of the window is A = 4x − x22 − πx28.

[3]
(b)

Find the width that maximizes the area, and the maximum area.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
x + 2h + πx2 = 8 A1
A = xh + πx28 with h substituted M1
simplifying AG A1
3
(b)x = 2.24 m, A = 4.48 m2
attempt to find the maximum, e.g. A′(x) = 0 or graph (M1)
x = 2.24 A1
A = 4.48 m2 A1
3
Question 2Structured · 7 marks

The table shows the time x hours spent revising and the score y obtained by a sample of students.

x711152022273235
y182328.228.735.436.242.349.5
Table 2.1
(a)

Find the value of Pearson’s product-moment correlation coefficient, r.

[2]
(b)

Find the equation of the regression line of y on x.

[2]
(c)

Use your regression line to estimate the score of a student who revises for 21 hours.

[2]
(d)

Explain why it would not be appropriate to use the line to estimate the score of a student who revises for 59 hours.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.979
use of GDC (M1)
r = 0.979 A1
2
(b)y = 1.02x + 11.2
use of GDC (M1)
y = 1.02x + 11.2 A1
2
(c)32.5
1.02(21) + 11.2 (M1)
32.5 A1
2
(d)extrapolation
59 is outside the range of the data (extrapolation) R1
1
Question 3Structured · 6 marks

The curve y = x2 − 9x + 13 and the line y = −2x + 3 intersect at two points.

(a)

Find the x-coordinates of the points of intersection.

[2]
(b)

Find the exact area of the region enclosed by the curve and the line.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)x = 2, x = 5
x2 − 7x + 10 = 0 (M1)
x = 2, x = 5 A1
2
(b)92
∫25((−2x + 3) − (x2 − 9x + 13))dx M1
∫25(−x2 + 7x − 10)dx A1
correct integration and substitution (A1)
92 A1
4
Question 4Structured · 14 marks

The masses of apples from an orchard are normally distributed with mean 150 g and standard deviation 12 g. An apple is classified as large if its mass is greater than 175 g.

(a)

Find the probability that a randomly chosen apple is large.

[2]
(b)

Find the probability that an apple has a mass between 135 g and 175 g.

[2]
(c)

The lightest 15% of apples are used for juice. Find the maximum mass of an apple that is used for juice.

[3]
(d)

A bag contains 8 apples chosen at random. Find the probability that at least 2 of them are large.

[4]
(e)

Given that an apple has a mass greater than 150 g, find the probability that it is large.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.0186
normal CDF (M1)
0.0186 A1
2
(b)0.876
normal CDF (M1)
0.876 A1
2
(c)138 g
P(X < k) = 0.15 (M1)
inverse normal (M1)
138 g A1
3
(d)0.00900
Y ~ B(8, 0.0186) (M1)(A1)
1 − P(Y ≤ 1) (M1)
0.00900 A1
4
(e)0.0372
P(X > 175)P(X > 150) (M1)
0.01860.5 (A1)
0.0372 A1
3
Question 5Structured · 14 marks

Offer A: a starting salary of $35000 with an increase of $1200 each year.
Offer B: a starting salary of $35000 with an increase of 3.5% each year.

(a)

Find the salary in year 5 for each offer.

[4]
(b)

Find the total amount earned over 20 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: $39800; B: $40163
A: 35000 + 4(1200) (M1)
$39800 A1
B: 35000(1.035)4 (M1)
$40163 A1
4
(b)A: $928000; B: $989789
S20 = 202(2(35000) + 19(1200)) (M1)
$928000 A1
S20 = 35000(1.03520 − 1)1.035 − 1 (M1)
$989789 A1
4
(c)year 2
35000(1.035)n−1 > 35000 + (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 6Structured · 5 marks

Consider the system of equations
x + 2y − z = 3
−x + y = −3
4x + 2y + kz = c
where k, c ∈ ℝ.

(a)

Find the value of k for which the system does not have a unique solution.

[3]
(b)

For this value of k, find the value of c for which the system has infinitely many solutions.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)k = −2
attempt to eliminate a variable, or set the determinant equal to 0 M1
row 3 = 2×row 1 − 2×row 2 for x and y (A1)
k = −2 A1
3
(b)c = 12
c = 2(3) − 2(−3) (M1)
c = 12 A1
2

Frequently asked questions

Are these real SL & HL past paper questions?

No. They are original questions written to match the IB Mathematics: Analysis & Approaches SL & HL syllabus and the style of real IB papers, so you can practise beyond the past papers you have already done.

Do the Mathematics: Analysis & Approaches questions have answers?

Yes. Every question on these pages has its answer, and every generated paper comes with a full IB-style mark scheme you can download as a PDF.

Can I make a full IB Mathematics: Analysis & Approaches practice paper?

Yes. Choose the paper (Paper 1 SL (no calculator) · 2029 syllabus, Paper 2 SL (calculator) · 2029 syllabus, Paper 1 HL (no calculator) · 2029 syllabus, Paper 2 HL (calculator) · 2029 syllabus, Paper 3 HL problem solving · 2029 syllabus, Paper 1 SL (no calculator) · exams to Nov 2028, Paper 2 SL (calculator) · exams to Nov 2028, Paper 1 HL (no calculator) · exams to Nov 2028, Paper 2 HL (calculator) · exams to Nov 2028, Paper 3 HL problem solving · exams to Nov 2028), the topics, the marks and the difficulty, and the generator builds a fresh paper with the same number and style of questions as the real exam.

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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