Advanced level 2024 CASPA mock Further mathematics 2

Advanced level 2024 CASPA mock Further mathematics 2

Advanced level 2024 CASPA mock Further mathematics 2

1. (a) Find the general solution of the linear Diophantine equation 448𝑥 + 105𝑦 = 35 (4 marks)
(b) Find the smallest value of 𝑥 for which x  3mod 7 8mod13. (4 marks)
2. Let 𝑛 ∈ ℕ and 𝐼
𝑛 = ∫01 𝑥𝑛 √1 − 𝑥 𝑑𝑥
(i) Show that 0 ≤ 𝐼𝑛 ≤ 1
𝑛+1
for all 𝑛 ∈ ℕ (3 marks)
(ii) Show equally that (𝐼𝑛) is decreasing (2 marks)
(iii) Deduce that (𝐼𝑛) converges and find its limit (3 marks)
(iv) Prove that for all 𝑛 ≥ 1, (2𝑛 + 3)𝐼𝑛 = 2𝑛𝐼𝑛−1 (3 marks)
3. (a) Let 𝐺 be a group made up of 3 distinct elements 𝑎 , 𝑏 and 𝑐 and an operation ∗ defined by 𝑎 ∗ 𝑏 = 𝑐.
(i) Proof that 𝑐 is the identity element (1 mark)
(ii) Find the order of 𝑎 and hence show that 𝐺 is cyclic (3 marks)
(b) Prove by mathematical induction that 𝑥𝑛 − 1 is divisible by (𝑥 − 1) for all positive integers 𝑛 (4 marks)
4. (a) Using a suitable identity, solve the equation cosh2 𝑥 = 4 sinh 𝑥 + 6 , leaving your answer in terms of
natural logarithms. (4 marks)
(b) Given the Ellipse 𝐸 with equation (𝑥−1)2
4
+ 𝑦2 = 1. Find its vertices, centre and the length of the latus
rectum (3 marks)
5. Using the substitution 𝑥𝑦 = 𝑣, where 𝑣 is a function of 𝑥,
transform the differential equation
𝑥2 𝑑𝑦
𝑑𝑥
+ 𝑥𝑦 = 𝑦3
into a differential equation in 𝑣 and 𝑥. (2 marks)
Hence, show that for 𝑦 = 𝑥 = 1, 𝑦2 = 3𝑥
2+𝑥3 . (4 marks)
6. (a) A mapping, 𝑇 is defined by
𝑇: ℝ2 ⟶ ℝ2
(𝑥, 𝑦) ⟼ (𝑥 − 2𝑦, −3𝑥 + 6𝑦)