Partial derivatives and stationary points: OCR A‑Level Further Maths questions

OCR A-Level Further Maths past-paper questions on 10.2 Partial derivatives 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 16 questions on Partial derivatives and stationary points, as a list

June 2025, Additional Pure Mathematics, Q4 (14 marks): In this question you must show detailed reasoning. 2 2 The surface S is given by the equation z 4 x y 3 xy 5 x for all real values of x and y. = - - (a) The poi
June 2025, AS Additional Pure Mathematics, Q3 (13 marks): 3 3 The surface S has equation z x y y x + for all real values of x and y. 2 6 3 4 = - + (a) (i) State the equation of the section of S cut by the plane y = 1.
June 2024, Additional Pure Mathematics, Q2 (5 marks): 2 A surface S has equation z 4 x y y x y for x, y 0 . = - + H Determine the equation of the tangent plane to S at the point (1, 4, 20). Give your answer in the
June 2024, Additional Pure Mathematics, Q6 (13 marks): 3 6 The surface C is given by the equation z x y axy for all real x and y, where a is a = + + non-zero real number. (a) Show that C has two stationary points, o
June 2024, AS Additional Pure Mathematics, Q3 (12 marks): 2 1 4 3 The surface S has equation z f ( , ) x y , where ( , ) f x y 4 x y 6 xy x for all real values of x = = - - 12 and y. You are given that S has a stationa
June 2023, Additional Pure Mathematics, Q6 (11 marks): y 6 The surface S has equation z x sin y for x 0 and 0 1 y . 2 1 = + r x (a) Determine, as a function of x and y, the determinant of H, the Hessian matrix of S.
June 2023, AS Additional Pure Mathematics, Q3 (6 marks): 2 3 A surface has equation z x y 3 xy 2 x + for all real values of x and y. y = - + Determine the coordinates of all stationary points of this surface. [6]
June 2022, Additional Pure Mathematics, Q1 (6 marks): 2 1 The surface E has equation z 500 3 x 2 y . = - - (a) Determine the values of z 2 and z 2 at the point P on E with coordinates (11, −8, 3). [4] x y 2 2 (b) F
June 2022, Additional Pure Mathematics, Q9 (9 marks): 2 9 For all real values of x and y the surface S has equation z 4 x 4 xy y 6 x 3 y + , where k = + + + + k is a constant and an integer. z z 2 2 . (a) Find and
June 2022, AS Additional Pure Mathematics, Q2 (6 marks): 3 2 2 2 The surface S has equation z x y 2 x 5 y 3 xy . = + - - + It is given that S has two stationary points; one at the origin, O, and the other at the point
November 2021, Additional Pure Mathematics, Q5 (8 marks): 2 2 5 The surface S has equation x y z xyz - . 1 + + = ^ h. c m z z 2 2 2 (a) Show that ( 2 z xy x x ) y 2 1 z [6] - + = + y 2 2 (b) Deduce that S has no statio
November 2021, AS Additional Pure Mathematics, Q5 (11 marks): A trading company deals in two goods. The formula used to estimate z, the total weekly cost to the company of trading the two goods, in tens of thousands of pou
November 2020, Additional Pure Mathematics, Q2 (9 marks): 7 5 4 8 2 For x, y , the function f is given by f(x, y) 2 x y 3 x y 5 x y . R ! = + - (a) Prove that x f y f n f , where n is a positive integer to be determine
November 2020, AS Additional Pure Mathematics, Q2 (11 marks): An open-topped rectangular box is to be manufactured with a fixed volume of 1000 cm3. The 2 dimensions of the base of the box are x cm by y cm. The surface area
June 2019, Additional Pure Mathematics, Q2 (11 marks): 2 A surface has equation z f ( , ) x y where ( , ) f x y x sin y 2 y cos x . = = + (a) Determine fx , fy , fxx , fyy , fxy and fyx . [5] a k. 1 1 1 2 (b) (i) Ve
June 2019, AS Additional Pure Mathematics, Q8 (13 marks): The motion of two remote controlled helicopters P and Q is modelled as two points moving along straight lines. J N J N p q 2 4 5 8 + + K O K O Helicopter P move

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