TMUA past papers

See three representative questions first

These are reconstructed questions from the 2024 and 2025 bank, unsimplified, so you can judge the quality before you decide to use it.

1. 1. TMUA 2025 Paper 1 · Q8

   Diagram question: geometry combined with an infinite series

   A square (side length 4) has an inscribed circle (diameter 4); the area between them is shaded. Then an inscribed square is drawn inside the circle, with another inscribed circle inside it—shade the area between this square and circle too. This process repeats indefinitely. Find the total area of all shaded regions.
   

   
   • A.

     $32-8\pi$
   • B.

     $32-4\pi$
   • C.

     $4\pi$
   • D.

     $8-2\pi$
   • E.

     $2\pi$
   Show answer

   Correct answer: A

2. 2. TMUA 2024 Paper 1 · Q9

   Paper 1: a change of variables turns the region into a rectangle

   Let R be the set of all points $(x, y)$ in the xy-plane such that:$\begin{cases} (x + y)^2 - 6(x + y) + 8 \leq 0 \\ (x - y)^2 + 4(x - y) + 3 \leq 0 \end{cases}$
   (Note: The boundaries of the region are included in R.)
   Find the area of R.
   • A.

     1
   • B.

     2
   • C.

     4
   • D.

     6
   • E.

     8
   Show answer

   Correct answer: B

3. 3. TMUA 2024 Paper 2 · Q13

   Paper 2: deciding which statements must be true

   Given that $f'(x) < 0$ for $0 \leq x \leq 1$, which of the following statements must be true?
   I. $\int_{0}^{\frac{1}{2}} f(x) \, dx > \int_{\frac{1}{2}}^{1} f(x) \, dx$
   II. $\int_{0}^{\frac{1}{2}} f(x) \, dx < \int_{0}^{\frac{1}{2}} f(x^2) \, dx$
   III. $f(0) < f(1)$
   • A.

     Only I
   • B.

     Only II
   • C.

     Only III
   • D.

     I and II
   • E.

     I and III
   • F.

     II and III
   • G.

     All of them
   Show answer

   Correct answer: D

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