plumiExponentials and logarithms: OCR A‑Level Maths questions
Every OCR A-Level Maths exam question on P.6 Exponentials and logarithms, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Paper 2, question 2: In this question you must show detailed reasoning. Solve the following equations. 2 3 (a) ( x 5 ) 8 [3] - = 2 3 = y (b) e 2 [2] 4 2 (c) x 3 x 4 0 [4] - - =](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun25-02-qp.webp?v=3)
![June 2025 Paper 3, question 3: (a) Sketch the graph of y x [2] 2 5 on the grid provided in the Printed Answer Booklet. = - (b) Solve the inequality x [2] 2 5 1 . - 1 (c) In this question you](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24003-jun25-03-qp.webp?v=3)
![June 2025 AS Paper 1, question 6: In this question you must show detailed reasoning. Solve the following equations. (a) log ( x 32 ) log ( 2 x ) log 2 2 [4] + - + = 3 3 3 c m 1 3 (b) log – [3] =](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun25-06-qp.webp?v=3)
![June 2024 AS Paper 1, question 1: In this question you must show detailed reasoning. ` j ` j 3 2 Solve the equation ln x x , where x 2 . [3] ln 5 ln 2 0 - =](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun24-01-qp.webp?v=3)
![June 2023 Paper 1, question 2: 1 a 2 (a) (i) Show that can be written in the form b , where a, b and c are + cx 3 2 x 3 2 x + - + constants to be determined. [2] 1 1 (ii) Hence solve the equa](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun23-02-qp.webp?v=3)
![June 2023 Paper 3, question 1: Using logarithms, solve the equation 1 = x x 2 4 + 5 , giving your answer correct to 3 significant figures. [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24003-jun23-01-qp.webp?v=3)
![June 2023 AS Paper 1, question 4: x 4 A curve has equation y e . = (a) Determine the value of x when y 10 . [2] = (b) Determine the gradient of the tangent to the curve at the point where x 2 .](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun23-04-qp.webp?v=3)
![June 2022 Paper 1, question 5: 2x 2x 4 5 (a) The graph of y can be transformed to the graph of y + either by a translation or = = by a stretch. (i) Give full details of the translation. [2] (](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun22-05-qp.webp?v=3)
All 29 questions on Exponentials and logarithms, as a list
- June 2025, Paper 2, Q2 (9 marks): In this question you must show detailed reasoning. Solve the following equations. 2 3 (a) ( x 5 ) 8 [3] - = 2 3 = y (b) e 2 [2] 4 2 (c) x 3 x 4 0 [4] - - =
- June 2025, Paper 3, Q3 (7 marks): (a) Sketch the graph of y x [2] 2 5 on the grid provided in the Printed Answer Booklet. = - (b) Solve the inequality x [2] 2 5 1 . - 1 (c) In this question you
- June 2025, AS Paper 1, Q6 (7 marks): In this question you must show detailed reasoning. Solve the following equations. (a) log ( x 32 ) log ( 2 x ) log 2 2 [4] + - + = 3 3 3 c m 1 3 (b) log – [3] =
- June 2025, AS Paper 2, Q8 (6 marks): x y x 2 y 8 The curves e 2 e = and 2 1 e 3 e 4 1 intersect at the point P. - + = 2 + y y (a) Show that the y-coordinate of P satisfies the equation 3 e 4 e 39 0
- June 2024, Paper 1, Q6 (8 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial ( )x f is defined by ( )x f 4 x 25 x 58 x 16 . = - - + 1 (a) Show that x is a root o
- June 2024, AS Paper 1, Q1 (3 marks): In this question you must show detailed reasoning. ` j ` j 3 2 Solve the equation ln x x , where x 2 . [3] ln 5 ln 2 0 - =
- June 2024, AS Paper 2, Q5 (7 marks): e x 3 5 At the point P on the curve y 21 x - the gradient of the tangent is 3. 8 = - (a) Determine the x-coordinate of P. Give your answer in the form ln a, whe
- June 2023, Paper 1, Q2 (8 marks): 1 a 2 (a) (i) Show that can be written in the form b , where a, b and c are + cx 3 2 x 3 2 x + - + constants to be determined. [2] 1 1 (ii) Hence solve the equa
- June 2023, Paper 1, Q11 (12 marks): t 11 The owners of an online shop believe that their sales can be modelled by S ab , where a and = b are both positive constants, S is the number of items sold
- June 2023, Paper 3, Q1 (3 marks): Using logarithms, solve the equation 1 = x x 2 4 + 5 , giving your answer correct to 3 significant figures. [3]
- June 2023, AS Paper 1, Q4 (4 marks): x 4 A curve has equation y e . = (a) Determine the value of x when y 10 . [2] = (b) Determine the gradient of the tangent to the curve at the point where x 2 .
- June 2023, AS Paper 2, Q5 (5 marks): y 2 x 3 2 y + = 2 x 3 y = x O + . The graph of y 2 x 3 2 x 3 2 2 x 3 The diagram shows the graphs of y and y can be = = = 2 x 3 2 transformed to the graph of y
- June 2022, Paper 1, Q5 (8 marks): 2x 2x 4 5 (a) The graph of y can be transformed to the graph of y + either by a translation or = = by a stretch. (i) Give full details of the translation. [2] (
- June 2022, Paper 1, Q8 (9 marks): (a) Substance A is decaying exponentially such that its mass is m grams at time t minutes. Find the missing values of m and t in the following table. t 0 10 50
- June 2022, Paper 3, Q6 (8 marks): In this question you must show detailed reasoning. y y 2 x 9 = + x O – 2 x y 4 e – 1 = 2 x The diagram shows the curves y 2 x + and 9 y 4 e - 1 which intersect
- June 2022, AS Paper 1, Q6 (11 marks): During some research the size, P, of a population of insects, at time t months after the start of the research, is modelled by the following formula. t P 100 e
- June 2022, AS Paper 2, Q6 (12 marks): y C x O A B The shape ABC shown in the diagram is a student’s design for the sail of a small boat. a k 3 The curve AC has equation y 2 log x and the curve BC ha
- November 2021, Paper 1, Q4 (7 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial ( ) f x is defined by ( ) f x 2 x 3 x 11 x + . 6 = - - (a) Use the factor theorem to
- November 2021, AS Paper 1, Q7 (3 marks): The relationship between the variables P and Q is modelled by the formula aPn Q = where a and n are constants. Some values of P and Q are obtained from an exper
- November 2020, Paper 1, Q6 (9 marks): A mobile phone company records their annual sales on 31st December every year. 6 t Paul thinks that the annual sales, S million, can be modelled by the equation
- November 2020, AS Paper 1, Q6 (13 marks): In this question you must show detailed reasoning. 2 (a) Solve the inequality x x 6 0 , giving your answer in set notation. [4] 2 + - 3 3 (b) Solve the equation
- November 2020, AS Paper 2, Q2 (5 marks): k 2 Two curves have equations y ln x and y , where k is a positive constant. = = x (a) Sketch the curves on a single diagram. [3] (b) Explain how your diagram s
- November 2020, AS Paper 2, Q4 (9 marks): In this question you must show detailed reasoning. 3 2 The cubic polynomial x 6 kx 57 x 20 is denoted by f (x). It is given that ( 2 x 1 ) is a factor of + + -
- June 2019, Paper 2, Q4 (5 marks): A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two p
- June 2019, AS Paper 1, Q7 (11 marks): ekx 7 (a) Write down an expression for the gradient of the curve y . [1] = 1 e x (b) The line L is a tangent to the curve y at the point where x 2 . = = 2 Show
- June 2019, AS Paper 2, Q8 (8 marks): (a) Show that the equation x kx where k is a constant, can be expressed in 2 log log ( 1 ) 3 , = - + 2 2 2 - x kx [4] the form 8 8 0 . + = (b) Given that the eq
- June 2018, Paper 1, Q11 (11 marks): In a science experiment a substance is decaying exponentially. Its mass, M grams, at time t minutes is given 0 05 . t by M e 300 - . = (i) Find the time taken f
- June 2018, Paper 2, Q6 (13 marks): In this question you must show detailed reasoning. ln x y A curve has equation . = x (i) Find the x-coordinate of the point where the curve crosses the x axis.
- June 2018, AS Paper 2, Q6 (9 marks): A pan of water is heated until it reaches 100 °C. Once the water reaches 100 °C, the heat is switched off and the temperature T °C of the water decreases. The t
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