plumiTangents, normals and stationary points: AQA A‑Level Maths questions
AQA A-Level Maths past-paper questions on G.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.
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All 27 questions on Tangents, normals and stationary points, as a list
- June 2025, Paper 1, Q11 (7 marks): The equation of a curve is x2y + 4y3 = 8x The curve has two stationary points. 11 (a) Use implicit differentiation to show that at the stationary points y = 4 x
- June 2025, Paper 1, Q13 (9 marks): A curve C has parametric equations x = 4(4t + 1)2 y = e–4t for – 1 4 ≤ t ≤ 0 13 (a) Find dy dx in terms of t [3 marks] 13 (b) Find an equation of the tangent to
- June 2025, Paper 1, Q15 (12 marks): A curve has equation y = x2 The point Q has coordinates (3, 2.5) The point P on the curve which is closest to the point Q is shown on the diagram below. x Q (3,
- June 2025, Paper 2, Q7 (9 marks): The point A lies on the curve with equation y = x3 + px2 + qx + 12 7 (a) Given that A has coordinates (–5, 37), show that 5p – q = 30 [2 marks] 7 (b) Given that
- June 2025, Paper 2, Q10 (13 marks): A curve C has equation y = xk ln x for x > 0 where k is a positive integer. 10 (a) Show that dy dx = xk–1 [A + k ln x] where A is a constant to be found. [4 mar
- June 2024, Paper 1, Q19 (7 marks): A curve has equation y3e2x + 2y – 16x = k where k is a constant. The curve has a stationary point on the y-axis. Determine the value of k [7 marks]
- June 2023, Paper 1, Q15 (9 marks): The curve with equation x2 + 2y3 − 4xy = 0 has a single stationary point at P as shown in the diagram below. O P y x 15 (a) Show that the y-coordinate of P sati
- June 2023, Paper 2, Q4 (7 marks): A curve has equation y = x2 8 + 4 x p 4 (a) Find an expression for dy dx [3 marks] 4 (b) The point P with coordinates (4, 10) lies on the curve. Find an equatio
- June 2023, Paper 3, Q3 (1 mark): A curve with equation y = f (x) passes through the point (3, 7) Given that f ’(3) = 0 find the equation of the normal to the curve at (3, 7) Circle your answer.
- June 2023, Paper 3, Q7 (14 marks): A new design for a company logo is to be made from two sectors of a circle, ORP and OQS, and a rhombus OSTR, as shown in the diagram below. θ O P R S T Q The po
- June 2023, Paper 3, Q9 (12 marks): A water slide is the shape of a curve PQ as shown in Figure 1 below. Figure 1 O P Q The curve can be modelled by the parametric equations x = t − 1 t + 4:8 y =
- June 2022, Paper 1, Q5 (3 marks): Find an equation of the tangent to the curve y = (x × 2)4 at the point where x = 0 [3 marks]
- June 2022, Paper 1, Q13 (9 marks): Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor. The entrance to the cave joins th
- June 2022, Paper 2, Q7 (9 marks): The curve y = 15 − x2 and the isosceles triangle OPQ are shown on the diagram below. x q y O P Q Vertices P and Q lie on the curve such that Q lies vertically a
- June 2022, Paper 3, Q6 (9 marks): A design for a surfboard is shown in Figure 1. Figure 1 length width The curve of the top half of the surfboard can be modelled by the parametric equations x =
- June 2022, Paper 3, Q10 (13 marks): The function f is defined by f (x) = x2 + 10 2x + 5 where f has its maximum possible domain. The curve y = f (x) intersects the line y = x at the points P and Q
- November 2021, Paper 1, Q12 (8 marks): The equation of a curve is (x + y)2 = 4y + 2x + 8 The curve intersects the positive x-axis at the point P. 12 (a) Show that the gradient of the curve at P is −
- November 2021, Paper 2, Q10 (11 marks): The function h is defined by h(x) = x p x − 3 where h has its maximum possible domain. 10 (a) Find the domain of h. Give your answer using set notation. [3 mark
- November 2021, Paper 3, Q5 (13 marks): A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost
- November 2021, Paper 3, Q9 (9 marks): A function f is defined for all real values of x as f (x) = x4 + 5x3 The function has exactly two stationary points when x = 0 and x = − 15 4 9 (a) (i) Find f 0
- November 2020, Paper 1, Q12 (13 marks): A curve C has equation x3 siny + cos y = Ax where A is a constant. C passes through the point P 3 p , p 6 12 (a) Show that A = 2 [2 marks] 12 (b) (i) Show that
- November 2020, Paper 2, Q9 (10 marks): A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height
- June 2019, Paper 1, Q13 (7 marks): A curve, C, has equation y = e3x−5 x2 Show that C has exactly one stationary point. Fully justify your answer. [7 marks]
- June 2019, Paper 2, Q7 (10 marks): (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of x3 [2 marks] x y 7 (b) The function f (x) is de
- June 2019, Paper 3, Q9 (15 marks): A curve has equation x2y2 + xy4 = 12 9 (a) Prove that the curve does not intersect the coordinate axes. [2 marks] 9 (b) (i) Show that dy dx = − 2xy + y3 2x2 + 4
- June 2018, Paper 1, Q13 (10 marks): A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company mode
- June 2018, Paper 2, Q6 (7 marks): Find the coordinates of the stationary point of the curve with equation (x + y − 2)2 = ey − 1 [7 marks]
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