100+ Free Cambridge IAL Mathematics Practice Questions

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Find the coefficient of x^3 in the expansion of (1 + 2x)^5.

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Key Facts: Cambridge IAL Mathematics Exam

9709

Cambridge syllabus code

CAIE

4 of 6

Papers required for full A-Level

CAIE 9709 syllabus

A*-E

Grading scale

CAIE

100

Free practice questions here

OpenExamPrep

Cambridge IAL Mathematics (9709) is a modular pre-university qualification. Students sit 4 papers (P1 compulsory, plus P2 or P3, then two from M1/S1/S2/P2-3). Pure mathematics, mechanics and statistics are graded A*-E, with examinations available in the May/June and October/November series worldwide.

Sample Cambridge IAL Mathematics Practice Questions

Try these sample questions to test your Cambridge IAL Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Express x^2 - 8x + 21 in the form (x - a)^2 + b.

A.(x - 4)^2 + 5

B.(x - 4)^2 + 21

C.(x - 8)^2 + 5

D.(x + 4)^2 + 5

Explanation: Complete the square: x^2 - 8x = (x - 4)^2 - 16. So x^2 - 8x + 21 = (x - 4)^2 - 16 + 21 = (x - 4)^2 + 5.

2The equation x^2 + (k - 3)x + 4 = 0 has equal roots. Find the positive value of k.

A.7

B.5

C.4

D.1

Explanation: Equal roots require discriminant = 0: (k - 3)^2 - 16 = 0, so (k - 3)^2 = 16, k - 3 = +/-4. The positive value is k = 7.

3Find the range of values of k for which x^2 + kx + 9 = 0 has two distinct real roots.

A.k < -6 or k > 6

B.-6 < k < 6

C.k > 6

D.k < 6

Explanation: Two distinct real roots require discriminant > 0: k^2 - 36 > 0, so k^2 > 36, giving k < -6 or k > 6.

4The function f is defined by f(x) = 2x - 5 for x in R. Find the inverse function f^(-1)(x).

A.(x + 5)/2

B.(x - 5)/2

C.2x + 5

D.1/(2x - 5)

Explanation: Let y = 2x - 5. Swap x and y: x = 2y - 5, so 2y = x + 5, y = (x + 5)/2. Hence f^(-1)(x) = (x + 5)/2.

5Functions f and g are defined by f(x) = x + 3 and g(x) = x^2. Find gf(2).

A.25

B.7

C.13

D.11

Explanation: gf(2) means g(f(2)). First f(2) = 2 + 3 = 5. Then g(5) = 5^2 = 25.

6Solve the equation |2x - 3| = 7.

A.x = 5 or x = -2

B.x = 5 only

C.x = -2 only

D.x = 10 or x = -7

Explanation: |2x - 3| = 7 gives 2x - 3 = 7 or 2x - 3 = -7. First: 2x = 10, x = 5. Second: 2x = -4, x = -2.

7Find the equation of the line through A(2, 5) and B(6, 13).

A.y = 2x + 1

B.y = 2x - 1

C.y = x/2 + 4

D.y = 8x - 11

Explanation: Gradient = (13 - 5)/(6 - 2) = 8/4 = 2. Using y - 5 = 2(x - 2): y = 2x - 4 + 5 = 2x + 1.

8Find the equation of the perpendicular bisector of the line segment joining A(1, 2) and B(7, 8).

A.y = -x + 9

B.y = x + 1

C.y = -x + 5

D.y = x + 9

Explanation: Midpoint M = (4, 5). Gradient of AB = (8 - 2)/(7 - 1) = 1, so perpendicular gradient = -1. Equation: y - 5 = -1(x - 4), giving y = -x + 9.

9Find the equation of the circle with centre (3, -2) and radius 5.

A.(x - 3)^2 + (y + 2)^2 = 25

B.(x + 3)^2 + (y - 2)^2 = 25

C.(x - 3)^2 + (y + 2)^2 = 5

D.(x - 3)^2 - (y + 2)^2 = 25

Explanation: A circle with centre (a, b) and radius r has equation (x - a)^2 + (y - b)^2 = r^2. Here a = 3, b = -2, r = 5, giving (x - 3)^2 + (y + 2)^2 = 25.

10An arc of a circle of radius 6 cm subtends an angle of 1.2 radians at the centre. Find the arc length.

A.7.2 cm

B.5 cm

C.3.6 cm

D.0.2 cm

Explanation: Arc length s = r*theta. So s = 6 * 1.2 = 7.2 cm.

About the Cambridge IAL Mathematics Exam

Cambridge International A-Level Mathematics (syllabus 9709) is a modular qualification offered by Cambridge Assessment International Education. Candidates take 4 of 6 papers to achieve the full A-Level: P1 Pure Mathematics 1 is compulsory, plus P2 or P3 Pure Mathematics, then a choice from M1 Mechanics, S1 Statistics, S2 Statistics. The most common combinations are P1 + P3 + S1 + M1 or P1 + P3 + S1 + S2.

Questions

100 scored questions

Time Limit

1 hour 15 minutes (P1) to 1 hour 50 minutes (P3); roughly 6 hours total across 4 papers

Passing Score

Grade E is the minimum pass; A*-E count as passing grades on the A-Level certificate

Exam Fee

Set by exam centre; typical international entry fees £85-£130 per paper (Cambridge Assessment International Education (CAIE))

Cambridge IAL Mathematics Exam Content Outline

25%

Pure Mathematics 1 (P1)

Quadratics, functions and graphs, coordinate geometry (lines and circles), circular measure (radians), trigonometry, arithmetic and geometric series, binomial expansion, differentiation, integration

25%

Pure Mathematics 3 (P3)

Algebra (modulus, polynomial division), logarithmic and exponential functions, further trigonometry (compound and double angle, R sin form), further differentiation, integration techniques (parts, substitution, partial fractions), numerical methods, 3D vectors, differential equations, complex numbers

25%

Mechanics 1 (M1)

Kinematics (suvat, motion graphs, variable acceleration), forces as vectors (equilibrium, friction, inclined planes), Newton's laws (F=ma, connected particles), work, energy and power, momentum and impulse

25%

Statistics 1 (S1)

Representation of data (histograms, box plots, cumulative frequency), measures of location and spread, permutations and combinations, probability (conditional, Venn and tree diagrams), discrete random variables, binomial distribution, normal distribution and approximation to the binomial

How to Pass the Cambridge IAL Mathematics Exam

What You Need to Know

Passing score: Grade E is the minimum pass; A*-E count as passing grades on the A-Level certificate
Exam length: 100 questions
Time limit: 1 hour 15 minutes (P1) to 1 hour 50 minutes (P3); roughly 6 hours total across 4 papers
Exam fee: Set by exam centre; typical international entry fees £85-£130 per paper

Keys to Passing

Complete 500+ practice questions
Score 80%+ consistently before scheduling
Focus on highest-weighted sections
Use our AI tutor for tough concepts

Cambridge IAL Mathematics Study Tips from Top Performers

1Practise past papers from the official 9709 archive — recent series follow the same question style year on year

2Memorise the standard formulae (e.g. R sin(theta+alpha), suvat, binomial expansion) — they are not given on the formula sheet for P1

3For Mechanics, draw a clear free-body diagram for every forces question before substituting numbers

4For Statistics, learn to use the normal distribution table (Phi(z)) for both forward and inverse lookups

Frequently Asked Questions

What is Cambridge International A-Level Mathematics (9709)?

9709 is the Cambridge Assessment International Education (CAIE) syllabus for AS and A-Level Mathematics. It is a modular qualification: students take 4 of 6 papers to achieve the full A-Level, combining Pure Mathematics with Mechanics or Statistics.

Which papers are required for the full A-Level?

P1 Pure Mathematics 1 is compulsory. Students then take either P2 or P3 Pure Mathematics, plus any 2 of the remaining papers (M1, S1, S2). The most common combination is P1 + P3 + S1 + M1.

When are 9709 exams taken?

Cambridge IAL Mathematics exams are sat in the May/June and October/November series. A March series is also offered in India. Candidates can split papers across sessions to fit their school timetable.

What calculator can I use for 9709?

A non-programmable scientific calculator is allowed in all papers. Graphical calculators and calculators with symbolic algebra are not permitted on the 9709 syllabus.

100+ Free Cambridge IAL Mathematics Practice Questions

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