Mathematics

1. 1.

   Differentiate y = x^2 e^(3x) with respect to x.

   [3 marks] · no calculator
2. 2.

   Find the gradient of the curve y = (3x^2 + 1)^4 at the point where x = 1.

   [3 marks] · no calculator
3. 3.

   Find the exact x-coordinates of the stationary points of y = (2x + 1)/(x^2 + 1).

   [4 marks] · no calculator
4. 4.

   The curve C has equation x^2 + xy + y^2 = 7 and passes through P(1, 2). Find the equation of the tangent to C at P in the form ax + by = c.

   [5 marks] · no calculator
5. 5.

   A closed cylinder has volume 500 cm^3. Find the radius that minimises its total surface area, and the minimum surface area, giving answers to 3 significant figures. Justify that it is a minimum.

   [6 marks]
6. 6.

   Air is pumped into a spherical balloon at 100 cm^3 per second. Find (a) the rate at which the radius increases when r = 5 cm, (b) the rate at which the surface area increases at that instant. [V = 4/3 pi r^3, S = 4 pi r^2]

   [5 marks] · no calculator

Original, unofficial practice questions from SubjectScout — not official past-paper questions.