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

OCR MEI A-Level Maths past-paper questions on 7.3 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 42 questions on Tangents, normals and stationary points, as a list

June 2025, Paper 1, Q11 (10 marks): The curve in the graph below is defined implicitly by ( 2 x y )( y 1 ) 6 . + - = y x d = y 2 1 ( y ) - (a) Show that . [4] d x 2 y 2 x 1 + - (b) Find the equati
June 2025, Paper 2, Q14 (15 marks): 3 14 The equation of a curve is y + x . = 2 x (a) Determine the coordinates of the point where the curve cuts the x-axis. [2] d . y (b) (i) Find x [2] d 16 3 (i
June 2025, AS Paper 1, Q11 (9 marks): In this question you must show detailed reasoning. 2 f x x x x You are given that ( ) ( 2 )( 7 6 ) . = + - + (a) Solve the equation ( ) f x [3] 0 . = (b) Show t
June 2025, AS Paper 2, Q6 (6 marks): The equation of a curve is 4 3 y x 8 x 20 . = - + d , is y (a) Show that the gradient function, x d d y 2 4 x ( x 6 ) . [2] = - d x (b) (i) On the axes in the P
June 2024, Paper 1, Q13 (8 marks): 13 The curve with equation y px + , where p and q are constants, has a stationary point q = + 2 x at (2, 7). (a) Determine the values of p and q. [5] 2 d y (b)
June 2024, Paper 1, Q16 (7 marks): A block of mass m kg rests on rough horizontal ground. The coefficient of friction between the block and the ground is n. A force of magnitude T N is applied at
June 2024, Paper 2, Q8 (6 marks): The equation of a curve is 3 2 y 2 x 3 mx 9 mx + . 4 = + - Determine the range of values of m for which the curve has no stationary values. [6]
June 2024, Paper 2, Q13 (9 marks): Determine the coordinates of the turning points on the curve with equation 2 2 y xy x x = . 1 [9] + + -
June 2024, Paper 3, Q15 (6 marks): 15 (a) Show that, for the curve y ax bx + , the equation of the tangent at the point with c = + 2 x‑coordinate t is y ( 2 at b x ) at + . c [3] = + - 2 (b) Henc
June 2024, Paper 3, Q17 (3 marks): x 1 x2 t t2 is y 2 17 Show that, for the curve y , the equation of the normal at the point ( , ) t , = =- + + 2 t 2 as given in line 27. [3]
June 2024, Paper 3, Q18 (2 marks): 18 A student is investigating the intersection points of tangents to the curve y 6 x 7 x + . She 1 = - uses software to draw tangents at pairs of points with x‑
June 2024, AS Paper 1, Q5 (6 marks): In this question you must show detailed reasoning. a k is c 1 m at the point 1 31 99 , (a) Show that the gradient of the curve y x x 2 x . [4] = - - 2 4 4 2 a k
June 2024, AS Paper 1, Q12 (10 marks): The diagram shows the graph of ( ) f x k x ( p x )( q ) where k , p and q are constants. The graph = - - passes through the points (–1, 0), (0, –4) and (2, 0).
June 2024, AS Paper 2, Q13 (5 marks): 2 13 Determine the range of values of x for which y 4 x 7 x 6 x + is a decreasing function. 8 [5] = + -
June 2023, Paper 1, Q5 (5 marks): In this question you must show detailed reasoning. 16 2 (a) Find the coordinates of the two stationary points on the graph of y 15 x . [3] = - - 2 x (b) Show th
June 2023, Paper 2, Q15 (7 marks): In this question you must show detailed reasoning. The equation of a curve is 3 ln y x y 8 . + = Find the equation of the normal to the curve at the point where
June 2023, Paper 3, Q5 (8 marks): In this question you must show detailed reasoning. 3 2 This question is about the curve y x 5 x 6 x . = - + (a) Find the equation of the tangent, T, to the curv
June 2023, Paper 3, Q7 (6 marks): A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is r cm and the angle at the centre is i radians.
June 2023, AS Paper 2, Q11 (5 marks): In this question you must show detailed reasoning. 3 2 The equation of a curve is y 2 x 9 x 24 x - . 8 = + + Show that there are no stationary points on this cu
June 2023, AS Paper 2, Q14 (7 marks): In this question you must show detailed reasoning. 8 The equation of a curve is y 16 x . = + x Determine the equation of the tangent to the curve at the point w
June 2022, Paper 2, Q16 (15 marks): The equation of a curve is 4 3 2 y x x x x 6 8 21 12 6 . = + - + - (a) In this question you must show detailed reasoning. Determine • The coordinates of the sta
June 2022, Paper 3, Q5 (7 marks): 2 5 A curve is defined implicitly by the equation x 2 3 xy y 2 0 . + + + = d y 4 x 3 y + (a) Show that d . [3] =- x y x 3 2 + (b) In this question you must show
June 2022, Paper 3, Q10 (5 marks): In this question you must show detailed reasoning. 4 x ( x ) - r Fig. C2.2 indicates that the curve y sin x has a stationary point near x = . 3 = - 2 r • Verify
June 2022, AS Paper 1, Q12 (6 marks): x2 12 Below is a faulty argument that appears to show that the gradient of the curve y at the point = (3, 9) is 1. Consider the chord joining (3, 9) to the poin
June 2022, AS Paper 2, Q9 (10 marks): 9 The equation of a curve is y 12 x 4 x . = - 2 (a) State the coordinates of the intersection of the curve with the y-axis. [1] (b) Find the value of y when x 9
November 2021, Paper 1, Q6 (7 marks): (a) The diagram shows part of the graph of y cosec x , where x is in radians. = State the equations of the three vertical asymptotes that can be seen. [1] y x O
November 2021, Paper 2, Q14 (13 marks): The equation of a curve is 2 3 y x ( x 2 ) . = - d , giving your answer in factorised form. y (a) Find x [4] d (b) Determine the coordinates of the stationary p
November 2021, AS Paper 1, Q8 (12 marks): In this question you must show detailed reasoning. (a) Use differentiation to find the coordinates of the stationary point on the curve with equation 2 y 2 x 3
November 2021, AS Paper 2, Q13 (9 marks): In this question you must show detailed reasoning. 7 3 The equation of a curve is y 3 x . = + - x 2 x Determine the coordinates of the points on the curve where
November 2020, Paper 3, Q11 (2 marks): Show that ex is an increasing function for all values of x, as stated in line 39. [2]
November 2020, Paper 3, Q12 (8 marks): ln x 12 (a) Show that the only stationary point on the curve y occurs where x = e, as given in = x line 45. [3] (b) Show that the stationary point is a maximum.
November 2020, AS Paper 1, Q12 (12 marks): In this question you must show detailed reasoning. 1 2 Fig. 12 shows part of the graph of y x . = + 2 x y x O Fig. 12 1 2 17 meets the x-axis at A and meets the
November 2020, AS Paper 2, Q9 (8 marks): 9 The equation of a curve is y 24 x 8 x 16 . = - + 2 d . y (a) Find x [3] d (b) Find the coordinates of the turning point. [3] (c) Determine the nature of the t
June 2019, Paper 3, Q8 (10 marks): In this question you must show detailed reasoning. x3 A is the point (1, 0), B is the point (1, 1) and D is the point where the tangent to the curve y at = B cr
June 2019, Paper 3, Q9 (6 marks): In this question you must show detailed reasoning. 2 The curve xy y 8 is shown in Fig. 9. + = y x O Fig. 9 [6] Find the coordinates of the points on the curve a
June 2019, Paper 3, Q10 (4 marks): x e 10 Show that ( ) is an increasing function for all values of x. [4] f x = x 1 e +
June 2019, AS Paper 1, Q8 (7 marks): In this question you must show detailed reasoning. 2 Show that the only stationary point on the graph of y x 4 x is a minimum point at (,1 - 3 ) . = - [7]
June 2019, AS Paper 1, Q10 (7 marks): In this question you must show detailed reasoning. 3 2 3 (a) Sketch the gradient function for the curve y 24 x x x . [5] = - - 3 2 3 (b) Determine the set of va
June 2019, AS Paper 2, Q10 (10 marks): In this question you must show detailed reasoning. 2 x 2 The equation of a curve is y + . A tangent and a normal to the curve are drawn at the 1 = + x 4 point w
June 2018, Paper 3, Q4 (10 marks): In this question you must show detailed reasoning. 1 A curve has equation y x 5 - . The curve is shown in Fig. 4. = - + x 2 y x O Fig. 4 (i) Determine the coord
June 2018, AS Paper 1, Q9 (9 marks): ^ h maps onto the curve C1 following a stretch scale factor 2 1 2 1 in the x-direction. 9 The curve y x = - 2 (i) Show that the equation of C1 can be written as
June 2018, AS Paper 2, Q10 (9 marks): 10 (i) A curve has equation y 16 x . Find = + 2 x d , y (A) [2] x d 2 d y (B) . [2] 2 d x (ii) Hence • find the coordinates of the stationary point, • determine

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