The TMUA only asks you to differentiate and integrate powers of x, but it tests whether you really understand what derivatives and integrals mean. Remember: an integral and an area are not the same thing.
Try each question without a calculator, allowing about 4 minutes, then open the solution.
Question 1. The curve y = x3 − 6x2 + 9x + 1 has a local maximum at the point
Answer: A. dy/dx = 3x2 − 12x + 9 = 3(x − 1)(x − 3), zero at x = 1 and 3. The second derivative 6x − 12 is −6 at x = 1, so that is the maximum, with y = 1 − 6 + 9 + 1 = 5: (1, 5).
Question 2. ∫02 (3x2 − 2x) dx =
Answer: B. [x3 − x2] from 0 to 2 = (8 − 4) − 0 = 4.
Question 3. The area enclosed between the curve y = 4x − x2 and the x-axis is
Answer: C. The curve meets the axis at x = 0 and 4 and lies above it in between. Area = [2x2 − x33] from 0 to 4 = 32 − 643 = 323.
Question 4. The tangent to y = x2 + 3 at the point where x = a, with a > 0, passes through the origin. What is a?
Answer: C. The tangent is y = (a2 + 3) + 2a(x − a). Putting x = 0, y = 0 gives 3 − a2 = 0, so a = √3.
The Admit Maths course has 12 modules mapped to the official specification, worked solutions for every question, and two full timed mock papers. It opens in December 2026, in time for the January sitting.
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