Questions covered in TMUA Set C Paper 1

1. Inequalities: Solving the double inequality x² < (x − 3)² < (2x + 1)²
2. Geometry: BE × EC for a right angle AED in a 3 cm by 10 cm rectangle
3. Exponentials & Logarithms: Nested logarithms: log₂(log₃x) + log₂(log₃(81/x)) = 1
4. Trigonometry: cos θ for the angle to the midpoint of a side in a 30°–60°–90° triangle
5. Sequences & Series: The largest ratio of the sum of every third term to the sum to infinity
6. Exponentials & Logarithms: The sum of the solutions of x^log₂(x²) − 18x^log₂x + 32 = 0
7. Differentiation: A common tangent to y = x² and y = 1/x, and where it crosses the x-axis
8. Sequences & Series: Five terms of a geometric sequence whose middle three have sum 14 and product 8
9. Probability & Counting: Drawing two balls, returning them only if they match, then drawing a third
10. Algebra: Possible values of xy when x + y = xy and (x − 1)² + (y − 1)² = 7
11. Probability & Counting: The coefficient of x³y² in (1 − 2x + y)⁸
12. Inequalities: The range of f(x) = (x² + 4x + 1)/(x² + 1)
13. Coordinate Geometry: Comparing the area under a chord of y = x² with the triangle formed by the tangents
14. Integration: Zeros of the area function of a piecewise linear graph
15. Number: The smallest k for which the product of the factors of 2⁴ × 3ᵏ is a fourth power
16. Polynomials: A quadratic with roots 1 and 3 for which f(f(x)) = 0 has exactly three solutions
17. Trigonometry: The longest side of a triangle with a 120° angle from its perimeter and area
18. Coordinate Geometry: Counting the lattice points strictly between (−7, −5) and (23, 45)
19. Algebra: Finding x + y from x√y + y√x = 8 and x√x + y√y = 40
20. Integration: Which conditions on ∫₀ᵗ f − ∫₋ₜ⁰ f force a cubic to be even