plumiCAIE A‑Level Further Maths questions by topic
Every question from the CAIE AS & A Level Further Mathematics (9231) papers Cambridge International publishes, June 2024 and the specimen papers, question by question. Cambridge keeps its other papers on its School Support Hub, for schools. Filter by topic or paper, open the mark scheme under any question, and download a set as a PDF.
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![June 2024 Paper 1, question 1: The cubic equation x 2 3 x 2 p x 5 0 , where p is a positive constant, has roots , a b c. , + - - = (a) State, in terms of p, the value of ab . [1] + bc + ca (b](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923111-jun24-01-qp.webp)
![June 2024 Paper 1, question 2: Prove by mathematical induction that 6 4 + n 38 n - is divisible by 74 for all positive integers n. 2 [6]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923111-jun24-02-qp.webp)
![June 2024 Paper 1, question 3: (a) Use standard results from the list of formulae (MF19) to show that N =/ r r ( 1 3 )( r 4 ) 1 N N ( 1 )( N 2 9 )( N 19 ). [3] + + = + + + 12 r 1 7 3 r 4 + (b](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923111-jun24-03-qp.webp)
![June 2024 Paper 1, question 6: x 2 ax + , where a 1 6 The curve C has equation y + 2 5 . = x + 2 2 (a) Find the equations of the asymptotes of C. [3] (b) Show that C has no stationary points.](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923111-jun24-06-qp.webp)
![June 2024 Paper 2, question 1: ` j, 3 = - 1 Find the roots of the equation z 108 3 108i , giving your answers in the form r cos i sin + i + i where r 2 0 and 0 1 1 2 . [5] i r](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923121-jun24-01-qp.webp)
![June 2024 Paper 2, question 2: + 2 Find the Maclaurin’s series for e 1 + x e 1 - up to and including the term in x2 . x [4]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923121-jun24-02-qp.webp)
![June 2024 Paper 2, question 3: It is given that - , for x sin - and 1 t y t cos 1 t 0 t 1 1 . = = G d y - . (a) Show that t 1 t 2 cos 1 t [3] = - + - d x 5 d 2 y (b) Find x in terms of t. [4]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923121-jun24-03-qp.webp)
![June 2024 Paper 2, question 4: = y ln 3 4 It is given that, for n H 0 , I sec h n x d x . n 0 (a) Show that, for n 2 , H ` j b 3 l n - 2 b 4 l ` j n 1 I n 2 I . [5] - = + - n 5 5 n - 2 d ` j](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923121-jun24-04-qp.webp)
![June 2024 Paper 2, question 6: ` j 1 6 (a) Show that cosh x sinh x e x 1 . [2] + 2 = 2 (b) Find the particular solution of the differential equation d 2 y d y ` j , 1 3 y 5 cosh x sinh x + +](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923121-jun24-06-qp.webp)
![June 2024 Paper 2, question 7: (a) Use the substitution u 1 x 2 to find = + y x d x . [2] 1 x 2 + (b) Find the solution of the differential equation d y x x y x 2 sinh - 1 x , - = d = ` j. gi](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923121-jun24-07-qp.webp)
![June 2024 Paper 2, question 8: (a) Find the set of values of a for which the system of equations 6 + x ay 3 , = 2 - y x 1 , = x 5 y 4 z 2 + + = has a unique solution. [2] (b) Show that the sy](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/923121-jun24-08-qp.webp)
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