Advanced level 2024 CASPA mock mathematics with mechanics 2

Advanced level 2024 CASPA mock mathematics with mechanics 2

Advanced level 2024 CASPA mock mathematics with mechanics 2

1. (i) The roots of the quadratic equation 2𝑥2 + 𝑚𝑥 + 3 = 0 are 𝑙𝑜𝑔2 𝛼 and 𝑙𝑜𝑔2 𝛽 where
0 < 𝛼 < 𝛽. Given that log8 𝛼𝛽 = 7
6
,
(a) show that 𝑚 = −7, (3 marks)
(b) determine the values of 𝛼 and 𝛽. (4 marks)
(ii) Prove by mathematical induction that ∑𝑛 𝑟=1 𝑟(𝑟 + 1) = 1
3
𝑛(𝑛 + 1)(𝑛 + 2) (4 marks)
2. (i) Given that 𝑃(𝑥) is a polynomial of degree 3 and that (𝑥2 − 1) is the factor of 𝑃(𝑥). If the polynomial
𝑃(𝑥) also passes through the points (0, 1) and (2, 9),
(a) show that 𝑃(𝑥) = 2𝑥3 − 𝑥2 − 2𝑥 + 1 (4 marks)
(b) Hence, find the range of values of 𝑥 for which 𝑃(𝑥) is increasing (3 marks)
(ii) Show, using truth tables that for any two statements 𝑝 and 𝑞, 𝑝 ∨ (𝑝 ⇒ 𝑞) is a tautology (3 marks)
3. (i) Solve the equation sin 𝑥 − sin 5𝑥 + cos 3𝑥 = 0, giving all your answers between 0° and 180°. (4 marks)
(ii) Prove that cos4 𝜃 − sin4 𝜃 = cos 2𝜃 (3 marks)
(iii) Find the range of values of 𝑥 for which |𝑥 − 1| ≥ 2𝑥 + 3 (3 marks)
4. (i) Given that 𝑥𝑦 − cos 𝑥 = 0. Show that 𝑥 𝑑2𝑦
𝑑𝑥2 + 2 𝑑𝑦 𝑑𝑥 + 𝑥𝑦 = 0 (4 marks)
(ii) Find the equation of the normal to the function 𝑦 = ln(𝑥 − 2) at the point where 𝑥 = 3 (3 marks)
5. (i) Given the Matrix 𝐴 = (−2 12 −5 24 −2 03).
(a) Find the matrix 𝐵 such that 𝐴𝐵 = 𝐵𝐴 = 𝐼, where 𝐼 is the 3 × 3 identity matrix. (4 marks)
(b) Hence solve the system
2𝑥 − 4𝑦 + 2𝑧 = 12
𝑥 + 5𝑦 − 3𝑧 = −7
2𝑥 − 2𝑦 = 4