Questions covered in TMUA Set B Paper 2

1. Polynomials: The necessary and sufficient condition for x⁴ − 4x² + k = 0 to have exactly two real solutions
2. Number: Which conditions on n are sufficient for n³ − n to be divisible by 12
3. Polynomials: Tangents to y = x³ − 3x² through the point (0, 1)
4. Algebra: Checking the solutions of 3√(x + 1) = x − 1 found by squaring
5. Trigonometry: The number of solutions of sin x cos x = p for 0 ≤ x < 2π
6. Logic: Negating the definition of a Cauchy sequence
7. Integration: The value of k that minimises the integral of |x − k| from −1 to 2
8. Proof & Error Analysis: The invalid step in solving an exponential inequality with logarithms to base 1/3
9. Trigonometry: “Necessary” and “only if” statements about the solutions of 2cos²x − k sin x = 1
10. Inequalities: Values of k for which x² + kx + 1 ≥ 0 whenever x² − 4x + 3 ≤ 0
11. Functions & Graphs: Tangents from the origin to the curve (|x| − 4)² + (y − 1)² = 1
12. Proof & Error Analysis: A flawed proof that logarithms with swapped bases are equal only when a = b
13. Geometry: Area of the triangle formed by the tangents from (a, 0) to x² + y² = 9
14. Functions & Graphs: Counting the integers x with ||x + 1| − 1| < 1
15. Geometry: The volume of a cuboid from the diagonals of its faces
16. Proof & Error Analysis: The first error in a proof that x³ − 3k²x + k = 0 has three real roots for k > 2/3
17. Polynomials: Values of p for which x² + px + 1 ≤ 0 implies x ≤ 2
18. Sequences & Series: A telescoping series built from the recurrence xₙ₊₁ = xₙ/3 + 2
19. Number: The smallest sum of two integers with HCF 18 and LCM 3780
20. Logic: Values of k for which, for every x, some y gives (x − y)² + k(x + y) > 0