Questions covered in TMUA Set A Paper 2

1. Proof & Error Analysis: Checking a proof that x + y ≥ 2 implies x² + y² ≥ 2
2. Trigonometry: Area of a quadrilateral split into two right-angled triangles
3. Proof & Error Analysis: First error in a proof about the roots of a quadratic with zero integral
4. Exponentials & Logarithms: Values of a for which logₐ(3x − 1) = 2 + logₐ(x + 1) has one solution
5. Number: The largest n = 2ᵃ3ᵇ below 200 with exactly 12 factors
6. Proof & Error Analysis: A flawed proof that a + b and ab integers force a and b to be integers
7. Geometry: Shaded areas from semicircles on the sides of a right-angled triangle
8. Coordinate Geometry: Where the tangents to y = x² at the ends of a chord meet
9. Logic: A counterexample to an integral inequality for 0 ≤ f(x) ≤ 1
10. Logic: Necessary and sufficient conditions from a quadratic, a logarithm and an exponential
11. Probability & Counting: Conditional probability for a three-digit code whose product is divisible by 6
12. Proof & Error Analysis: First error in a proof about the point of inflection of a cubic
13. Integration: The smallest n for which a sum built from ∫xⁿ dx exceeds 15
14. Logic: Negating a “for all x, if … then …” statement
15. Proof & Error Analysis: Whether a strictly increasing function must have f′(x) > 0
16. Sequences & Series: Binomial coefficients of x⁴, x⁵ and x⁶ in arithmetic progression
17. Logic: The condition on a for sin y to take every value in [−1, 1] with a ≤ y ≤ a + π
18. Logic: A necessary but not sufficient condition for a cubic to be decreasing somewhere
19. Algebra: Finding x⁶ + 1/x⁶ from x² − 3x + 1 = 0
20. Polynomials: Values of p for which x³ − 3x + p = 0 has three distinct real roots