Questions covered in TMUA Set A Paper 1

1. Exponentials & Logarithms: Solving the inequality log₂x < logₓ2
2. Inequalities: |x − a| + |x + a| < 3|a| as distances on a number line
3. Differentiation: Two parabolas with perpendicular tangents where they meet
4. Algebra: Pairing factors of a quartic product to find x² + 7x
5. Integration: Area enclosed between a cubic and its tangent
6. Coordinate Geometry: A line through (2, 5) cutting a triangle of area 25 from the axes
7. Number: Counting the digits of 8¹⁵ × 5⁴⁰
8. Sequences & Series: A quadratic sequence from its 2nd, 5th and 8th terms
9. Geometry: Two circles touching both axes and each other
10. Sequences & Series: The recurrence aₙ₊₁ = 3aₙ² solved with logarithms
11. Number: Terminating decimals with exactly two decimal places
12. Functions & Graphs: A linear function with f(f(x)) = 4x + 3
13. Algebra: Sum and product of the roots of x² − 7x + 1 = 0
14. Probability & Counting: Right-angled triangles from the vertices of a cube
15. Probability & Counting: Same-colour and different-colour draws without replacement
16. Polynomials: Tangents to a cubic through a point on the y-axis
17. Trigonometry: A quadratic in sin x with exactly one solution
18. Polynomials: The quartic x⁴ + ax² + b with three distinct real roots
19. Functions & Graphs: |x² − 4x + 3| = kx with exactly three solutions
20. Geometry: The alternate segment theorem and similar triangles