Questions covered in TMUA Set B Paper 1

1. Sequences & Series: A geometric series whose squares and cubes sum to 2S and 36S/7
2. Algebra: Minimising x₁² + x₂² for a line through (1, 4) meeting y = x²
3. Trigonometry: Counting the solutions of cos 2x = 1 − x for 0 ≤ x ≤ π
4. Integration: The trapezium rule with four strips for the integral of ln x from 1 to 5
5. Coordinate Geometry: A tangent to a circle on diameter AB meeting the lines x = 0 and x = 10
6. Integration: Values of k for which ∫₁ˣ (t − k)/√t dt = 0 has exactly two solutions
7. Geometry: A circle touching two touching circles and their common tangent line
8. Probability & Counting: Factors of 2⁷ × 3⁵ divisible by 12 whose cofactor is a perfect square
9. Geometry: The largest perimeter of a rectangle inscribed in a semicircle
10. Polynomials: Values of k for which x² − (k + 2)x + (2k + 1) = 0 has two distinct positive roots
11. Sequences & Series: Bounding aₙ² for the recurrence aₙ₊₁ = aₙ + 1/aₙ
12. Number: Which inequalities must hold when numbers in standard form are added
13. Probability & Counting: Same-colour and different-colour draws equally likely from a bag of 30 to 50 balls
14. Functions & Graphs: Minimising the larger of two quadratics
15. Number: Factors d of 2⁶ × 3⁴ × 5³ with HCF(d, N/d) = 60
16. Polynomials: A symmetric quartic with factors (x − 2) and (x + ½)
17. Integration: Area between a cubic and the horizontal line through its local maximum
18. Algebra: The least value of (a² + b)(b² + a) when a + b = 1
19. Functions & Graphs: Counting the solutions of f(f(x)) = f(x) for f(x) = x³ − 3x
20. Inequalities: The smallest f(0) for a non-negative quadratic with f(x + 1) − f(x) ≥ 2x + 3