Questions covered in TMUA Set C Paper 2

1. Trigonometry: The sum of the areas of the two triangles with angle A = 30°, BC = 2 and AC = 2√3
2. Logic: The contrapositive of an “if … and … then … or …” statement
3. Logic: Necessary and sufficient conditions on k for 4ˣ − 3 × 2ˣ + k = 0 to have real solutions
4. Logic: Conditions on a and r for the sum to infinity of a geometric series to exceed a
5. Proof & Error Analysis: Ordering the steps of a proof by contradiction that √2 is irrational
6. Sequences & Series: The sum of nine terms of a geometric progression from the sums of three and six terms
7. Logic: Necessary and sufficient conditions for sin θ + cos θ > 1 in terms of sin θ cos θ
8. Geometry: The shortest path over the surface of a 1 × 2 × 3 cuboid between opposite corners
9. Integration: Integrating ⌊2x⌋ − ⌊x⌋ from 0 to 2
10. Logic: Whether xf(x) ≥ 0 or f(0) = 0 is necessary or sufficient for ∫₀ˣ f(t) dt ≥ 0
11. Number: The sum of all k for which N/k² and k³/N are integers when N = 2⁸ × 3⁶
12. Proof & Error Analysis: Checking a solution of log₂(sin x) + log₂(cos x) = −2 for 0 < x < π
13. Number: The highest power of 2 that divides ⁶⁴C₃₂
14. Probability & Counting: Which pairs of events are independent when two cards are drawn from cards numbered 1 to 6
15. Probability & Counting: Counting the integers below 10000 whose digits strictly increase
16. Number: Which conditions on n are sufficient for 2ⁿ − 1 to be divisible by 7
17. Proof & Error Analysis: Checking a proof by contradiction that x² ≥ 1 implies x⁴ ≥ x²
18. Integration: The ratio of the areas into which y = 2x divides the region under y = 4x − x²
19. Exponentials & Logarithms: Values of m for which y = mx meets log₂(2y + 4) = 2log₂(x + 1) at exactly two points
20. Logic: The relationship between f′(4) > 0 and f(4) > f(1) for f(x) = √x + a/√x