Mathematics

1. 1.

   A curve has x = t^2 + 1 and y = 3t for t > 0. Find dy/dx and the tangent equation at t = 2.

   [3 marks] · no calculator
2. 2.

   Differentiate xe^(2x). Then find its indefinite integral, showing integration by parts.

   [3 marks] · no calculator
3. 3.

   For x = t^2 and y = t^3 - 3t, with t > 0, find the horizontal tangent point and use d^2y/dx^2 to classify it.

   [4 marks] · no calculator
4. 4.

   Evaluate the exact integral from 0 to 1 of 2x/(1 + x^2) dx by substitution.

   [4 marks] · no calculator
5. 5.

   Solve dy/dx = 2x(1 + y) with y(0) = 2. State the solution's real domain and find y(1) exactly.

   [5 marks] · no calculator
6. 6.

   Use the trapezium rule with two equal strips to estimate the integral of 1/x from 1 to 3. Show that it overestimates the exact integral, using curvature, and give the error exactly.

   [5 marks] · no calculator

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