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 448π₯ + 105π¦ = 35 (4 marks)
(b) Find the smallest value of π₯ for which x οΊ 3ο¨mod 7ο©οΊ 8ο¨mod13ο©. (4 marks)
2. Let π β β and πΌ
π = β«01 π₯π β1 β π₯ ππ₯
(i) Show that 0 β€ πΌπ β€ 1
π+1
for all π β β (3 marks)
(ii) Show equally that (πΌπ) is decreasing (2 marks)
(iii) Deduce that (πΌπ) converges and find its limit (3 marks)
(iv) Prove that for all π β₯ 1, (2π + 3)πΌπ = 2ππΌπβ1 (3 marks)
3. (a) Let πΊ be a group made up of 3 distinct elements π , π and π and an operation β defined by π β π = π.
(i) Proof that π is the identity element (1 mark)
(ii) Find the order of π and hence show that πΊ is cyclic (3 marks)
(b) Prove by mathematical induction that π₯π β 1 is divisible by (π₯ β 1) for all positive integers π (4 marks)
4. (a) Using a suitable identity, solve the equation cosh2 π₯ = 4 sinh π₯ + 6 , leaving your answer in terms of
natural logarithms. (4 marks)
(b) Given the Ellipse πΈ with equation (π₯β1)2
4
+ π¦2 = 1. Find its vertices, centre and the length of the latus
rectum (3 marks)
5. Using the substitution π₯π¦ = π£, where π£ is a function of π₯,
transform the differential equation
π₯2 ππ¦
ππ₯
+ π₯π¦ = π¦3
into a differential equation in π£ and π₯. (2 marks)
Hence, show that for π¦ = π₯ = 1, π¦2 = 3π₯
2+π₯3 . (4 marks)
6. (a) A mapping, π is defined by
π: β2 βΆ β2
(π₯, π¦) βΌ (π₯ β 2π¦, β3π₯ + 6π¦)