plumiInduction: matrices, recurrences and inequalities: OCR A‑Level Further Maths questions
OCR A-Level Further Maths past-paper questions on 1.3 Induction: matrices, recurrences and inequalities, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 11 questions on Induction: matrices, recurrences and inequalities, as a list
- June 2025, Pure Core 2, Q8 (12 marks): A function f (x) is defined by f (x) = xsinh 2x. 2 n 0 d f d f n (a) Prove by induction that 4 ( x sinh 2 x n cosh 2 x ) for n 0 where d is defined as H = + 2 n
- June 2025, AS Pure Core, Q7 (5 marks): 20n 7 n for all integers n [5] Prove by induction that ! 2 52 . H J N 5 a 5 1 +
- June 2024, AS Pure Core, Q6 (5 marks): J N 1 a KK OO where a is a constant. 6 You are given that A = 0 1 L P J N 1 an n = KK OO for all integers n Prove by induction that A H . 1 [5] 0 1 L P
- June 2023, Pure Core 2, Q9 (9 marks): A function is defined by y f ( ) t where ( ) f t ln ( 1 at ) and a is a constant. = = + 2 3 4 n d , t y d y d y d y d y (a) By considering t , t and t , make a
- June 2022, Pure Core 1, Q6 (6 marks): Let y x cosh x . = 2 n 1 d - y Prove by induction that, for all integers n H , 1 x sinh x ( 2 n 1 ) cosh x . [6] = + - 2 n 1 d x -
- June 2022, Additional Pure Mathematics, Q3 (6 marks): ^ h plays a significant role in the sequence of Fibonacci 1 3 The irrational number 1 5 z = + 2 numbers given by F 0 , F 1 = and 1 F F F - for n H . 1 0 = = + n
- June 2022, AS Pure Core, Q4 (5 marks): n 2 4 Prove that 3 10 n for all integers n 4 . [5] H
- November 2021, Pure Core 2, Q9 (6 marks): J N 2 3 KK OO. 9 The matrix A is given by A = 0 2 L P (a) By considering A, A2, A3 and A4 make a conjecture about the form of the matrix An in terms of n for n
- November 2020, Additional Pure Mathematics, Q8 (12 marks): u 3 " , of positive real numbers is defined by u + n 8 The sequence un 1 = and u for n H . 1 1 1 = u 2 n + + n 2 (a) Prove by induction that u 3 1 0 for all pos
- November 2020, AS Pure Core, Q6 (5 marks): n 2 6 Prove that ! n 2 for all integers n H 9. [5]
- June 2019, AS Pure Core, Q8 (6 marks): In this question you must show detailed reasoning. M is the matrix 1 6 f p. 0 2 n 1 f 1 3 2 ( + 2 ) p, for any positive integer n. - n Prove that M [6] = n 0 2
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