plumiMultivariable calculus: OCR A‑Level Further Maths questions
Every OCR A-Level Further Maths exam question on FP.10 Multivariable calculus, 2018–2025, with its mark scheme and examiner’s report.
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![June 2023 AS Additional Pure Mathematics, question 3: 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]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y53501-jun23-03-qp.webp?v=3)
![June 2022 Additional Pure Mathematics, question 1: 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](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54501-jun22-01-qp.webp?v=3)
![November 2021 Additional Pure Mathematics, question 5: 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](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54501-nov21-05-qp.webp?v=3)
All 25 questions on Multivariable calculus, 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, Additional Pure Mathematics, Q7 (8 marks): In this question you must show detailed reasoning. 1 3 3 A curve is defined parametrically by x cos t , y sin t , for 0 t . When the curve is = = r G G 2 rotate
- 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, Q1 (4 marks): 1 The surface S is defined for all real x and y by the equation z x 2 xy . The intersection of = + S with the plane P gives a section of the surface. On the axe
- June 2023, Additional Pure Mathematics, Q2 (6 marks): 2 A curve has equation y 1 x , for 0 x G , where both the x- and y-units are in cm. The 1 = + G area of the surface generated when this curve is rotated fully a
- 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 2023, AS Additional Pure Mathematics, Q8 (6 marks): A surface, C, is given by the equation z f ( , ) x y for all real values of x and y. You are given = that C has the following properties. • The surface is conti
- 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
- June 2022, AS Additional Pure Mathematics, Q5 (6 marks): A research student is using 3‑D graph‑plotting software to model a chain of volcanic islands in the Pacific Ocean. These islands appear above sea‑level at regul
- 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, Additional Pure Mathematics, Q10 (7 marks): B A z y O x D C A student wishes to model the saddle of a horse. They use a surface described by a function of the z x y with a saddle point at the origin O. Th
- November 2021, AS Additional Pure Mathematics, Q2 (4 marks): 2 The surface S is given by z x 4 xy for 6 x 6 and 2 y 2 . = + - - G G G G (a) (i) Write down the equation of any one section of S which is parallel to the x-z
- 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, Additional Pure Mathematics, Q5 (13 marks): A designer intends to manufacture a product using a 3-D printer. The product will take the form of a surface S which must meet a number of design specifications
- 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, Additional Pure Mathematics, Q7 (12 marks): a k and a k lie on the curve y 1 1 1 13 3 31 3 7 The points P 2 , Q 2 , x . = + 3 4 x 24 24 The area of the surface generated when arc PQ is rotated completely
- 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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