CAIE A Level 9709

Mathematics 2025 · May/June

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19 Questions Q&A Available
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This CAIE A Level Mathematics paper is a broad test of advanced pure mathematics, with strong emphasis on algebra and functions, binomial expansion, complex numbers, trigonometric equations, differential equations, vectors, and calculus. Students can expect a mix of structured questions with several parts, requiring exact answers, algebraic manipulation, sketching graphs, iterative methods, and careful use of calculus and integration techniques rather than multiple-choice.…

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Questions & Model Answers 19 Qs

Q1
Structured 5 marks
Solve the equation . Give your answer correct to 3 decimal places.
Q2(a)
Structured 4 marks
Expand ()()xx61 2 in ascending powers of x, up to and including the term in x , simplifying the coefficients.
Q2(b)
Short Answer 1 mark
State the set of values of x for which the expansion is valid.
Q3
Short Answer 1 mark
On an Argand diagram shade the region whose points represent complex numbers z which satisfy both the inequalities z32iG- and ()argz12i GGrr -- . [5] * 0000800000005 *
Q5
Structured 5 marks
The square roots of 145i-- can be expressed in the Cartesian form xyi+, where x and y are real and exact. By first forming a quartic equation in x or y, find the square roots of 145i-- in exact Cartesian form.
Q6(a)
Short Answer 2 marks
By sketching a suitable pair of graphs, show that the equation sinxx22 has only one root in the interval x011 r.
Q6(b)
Short Answer 2 marks
Show by calculation that this root lies between 1 and 1.5.
Q6(c)
Short Answer 3 marks
Use the iterative formula sinxx 22 nn1 2 with an initial value of 1.03 to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Q7(a)
Short Answer 3 marks
Express sinc os72 4ii+ in the form ()cosR ia-, where R02 and r0 11a . Give the value of a correct to 4 decimal places.
Q7(b)
Structured 4 marks
Hence solve the equation .sinc osxx72 42 45 += for x011 r.
Q8
Structured 7 marks
The variables x and i satisfy the dif ferential equation ()sinc os x24 32 i=+ , and x0= when r i= . Solve the dif ferential equation and obtain an expression for x in terms of i.
Q9
Short Answer 1 mark
With respect to the origin O, the points A, B and C have position vectors given by =- fp, OB fp and OC = fp.
Q9(a)
Short Answer 2 marks
Find a vector equation for the line through A and B.
Q9(b)
Structured 4 marks
Using a scalar product, find the exact value of cos BAC.
Q9(c)
Short Answer 3 marks
Hence find the exact area of triangle ABC.
Q10(a)
Short Answer 2 marks
Find the quotient and remainder when x is divided by x14 +.
Q10(b)
Short Answer 1 mark
Find the exact value of ()tanxx x2d ; . [6] The diagram shows the graph of sincosyx x52 = for x0 GG r and its maximum point M.
Q10(a)
Structured 6 marks
Find the exact x-coordinate of M.
Q10(b)
Short Answer 1 mark
By using the substitution cosux= , find the area of the region bounded by the curve, the x-axis between x0= and x r=, and the line x r=. [5] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable ef fort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge. Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.
Paper Details
Board CAIE
Level A Level
Subject Mathematics
Code 9709
Year 2025
Session May/June
Q&A 19 questions
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