Tangents, normals and stationary points: OCR A‑Level Maths questions

OCR A-Level Maths past-paper questions on 1.07.3 Tangents, normals and stationary points, 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.

All 31 questions on Tangents, normals and stationary points, as a list

June 2025, Paper 1, Q2 (4 marks): y x O The diagram shows the curve y ax x ( b ) , where a and b are constants and b 0 . = - 2 (a) Given that the curve has a stationary point at x = , state the
June 2025, Paper 2, Q6 (7 marks): 3 2 6 A curve has equation y 3 x 4 x 6 x 12 x . = - - + d . y (a) (i) Find x [2] d (ii) Use your answer to part (a)(i) to find the x-coordinates of the stationa
June 2025, AS Paper 1, Q8 (8 marks): 2 8 A curve with equation y x 3 x 9 x 12 has stationary points A and B. = + - - Determine the coordinates of the point where the line through A and B meets the
June 2024, Paper 1, Q5 (8 marks): 5 The line x 13 y 108 is the normal to the curve y ax b x at the point (4, 8). + = = + Determine the values of the constants a and b. [8]
June 2024, Paper 1, Q11 (12 marks): A curve has equation y 5 ln ( 1 cos 2 x ) , where x is in radians. = - (a) State the values of x for which 5 ln ( 1 cos x 2 ) is not defined. [2] - (b) P is the
June 2024, AS Paper 1, Q5 (9 marks): 2 5 A curve has equation y 4 x . = + x (a) Determine the coordinates of the stationary point on the curve. [5] 2 d y (b) Use d to determine whether this turning
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 2024, AS Paper 2, Q6 (7 marks): y x O 2 P 5 4 3 The diagram shows the curve with equation y x ax bx , where a and b are integers. The = + + curve has a minimum at the point P where x 2 . = The
June 2023, Paper 1, Q6 (9 marks): + . ex 2 3 x 6 A curve has equation y = (a) Determine the x-coordinates of any stationary points on the curve. [4] (b) Show that the curve is convex for all val
June 2023, Paper 2, Q4 (9 marks): x2 4 The diagram shows part of the graph of y . The normal to the curve at the point A(1, 1) meets = the curve again at B. Angle AOB is denoted by a. y B 2 A 1
June 2023, Paper 2, Q5 (12 marks): In this question you must show detailed reasoning. The function f is defined by f( ) x cos x 3 sin x with domain 0 x 2 r. = + G G (a) Solve the following equati
June 2023, AS Paper 1, Q6 (11 marks): 3 6 (a) Determine the two real roots of the equation x x [3] 8 7 1 0 . + - = 7 49 4 (b) Determine the coordinates of the stationary points on the curve y x x x
June 2023, AS Paper 2, Q7 (7 marks): y l2 C l1 1a k , 3 2 x O 2 The diagram shows the curve C with equation y 4 x 10 x + and two straight lines, l1 and l2. 7 = - 1a k. The line l2 is the normal to
June 2022, Paper 2, Q5 (4 marks): In this question you must show detailed reasoning. 3 2 A curve has equation y x 3 x 4 x . = - + (a) Show that the curve has no stationary points. [2] (b) Show t
June 2022, AS Paper 2, Q8 (11 marks): + 8 (a) The quadratic polynomial ax bx , where a and b are constants, is denoted by ( ) f x . Use differentiation from first principles to determine, in terms o
November 2021, Paper 1, Q5 (7 marks): (a) The graph of the function y f ( ) x passes through the point P with coordinates (2, 6), and = is a one-one function. State the coordinates of the point corr
November 2021, Paper 1, Q11 (12 marks): 4 x y 4 2 2 2 2 11 (a) Use the substitution u x + to show that 3 d x ( x 6 ) x 3 c . [5] = = - + + 3 2 x 3 + (b) In this question you must show detailed reasoni
November 2021, AS Paper 1, Q5 (8 marks): The fuel consumption of a car, C miles per gallon, varies with the speed, v miles per hour. Jamal models the fuel consumption of his car by the formula 3 12 2 C
November 2021, AS Paper 2, Q6 (6 marks): y d 2 6 A curve C has an equation which satisfies x 3 x + , for all values of x. 2 = 2 d (a) It is given that C has a single stationary point. Determine the nat
November 2021, AS Paper 2, Q8 (11 marks): y l ,16 ^ h x O 6 The diagram shows the curve y 1 x and the line l, which is the normal to the curve at the = - + x point (1, 6). (a) Determine the equation of
November 2020, Paper 1, Q3 (10 marks): A cylindrical metal tin of radius r cm is closed at both ends. It has a volume of 16000 cm3 3 . r - . (a) Show that its total surface area, Acm2, is given by A
November 2020, Paper 1, Q8 (7 marks): e x 2 8 (a) Differentiate ( 2 3 x ) with respect to x. [3] + 2 e x 2 (b) Hence show that ( 2 3 x ) is increasing for all values of x. [4] +
November 2020, AS Paper 2, Q3 (7 marks): In this question you must show detailed reasoning. Find the equation of the normal to the curve y 4 x 3 x + at the point on the curve where 1 = - x 4 . Give you
November 2020, AS Paper 2, Q7 (13 marks): E xcm B D F A C 2xcm The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism ABCDEF. The base ACFD of the
June 2019, Paper 1, Q5 (8 marks): 2 2 5 A curve has equation x 3 x y y 1 0 . - + + = 2 y xy x d 6 3 - (a) Show that x . [4] = d 2 2 y 3 x - (b) Find the equation of the normal to the curve at th
June 2019, Paper 2, Q5 (9 marks): h l r For a cone with base radius r, height h and slant height l, the following formulae are given. Curved surface area, S rl = r 1 2 Volume, V r h = r 3 A cont
June 2019, AS Paper 1, Q4 (6 marks): 2 4 (a) Find the coordinates of the stationary points on the curve y x 6 x 9 x . [4] = - + 3 2 (b) The equation x 6 x 9 x k 0 has exactly one real root. - + + =
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 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 1, Q4 (9 marks): 4 (i) It is given that y x 3 x . = + d . y (a) Find x [2] d (b) Find the values of x for which y is increasing. [2] ^ h y (ii) Find 3 4 x d x . [5] -
June 2018, AS Paper 1, Q6 (7 marks): Sketch the following curves. 2 (i) y [2] = x 3 2 (ii) y x 6 x 9 x [5] = - +

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