Advanced level 2026 west regional mock further mathematics 2
Advanced level 2026 west regional mock further mathematics 2
1. The differential equation (x3 + 1) d2y/dx2 – 3x2 dy/dx = 2 – 4x3, is to be solved subject to the boundary conditions
y = 0, dy/dx = 4 at x = 0.
y = 0, dy/dx = 4 at x = 0.
Use the substitution u = dy/dx – 2x where u is a function of x, to show that y = x4 + x2 + 4x is a solution of the differential equation.
(8 marks)
2. (i) Given that f(x) = [(x2+1)(x2+4)] / [(x2+3)(x2-4)], x ≠ ±2, express f(x) in partial fractions.
Hence, show that ∫01 f(x) dx = 1 + 2/7 [π/6√3 – 5 ln 3].
(8 marks)
(ii) The curves (C1) and (C2) have equations y = sinh x and y = ½ sech x respectively.
The two graphs intersect at the point P.
The two graphs intersect at the point P.
a. Find the x-coordinates of P. (3 marks)
b. Hence, show that the y-coordinate of P is √[½(√2 – 1)]. (3 marks)
3. (i) Prove by mathematical induction that for every positive integer n,
∑r=1n cos[(2r – 1)x] ≡ sin(2nx) / 2sin x , x ≠ nπ
(5 marks)
∑r=1n cos[(2r – 1)x] ≡ sin(2nx) / 2sin x , x ≠ nπ
(5 marks)
(ii) Solve the equation:
P ⊗ [1 3 5 7; 3 7 1 5] = [1 3 5 7; 1 5 7 3]
(3 marks)
P ⊗ [1 3 5 7; 3 7 1 5] = [1 3 5 7; 1 5 7 3]
(3 marks)
4. (i) Find the foot of the perpendicular from the point A(5, -2, 2) to the plane 2x – 3y + 4z = -5. Hence find the image of the point A with respect to the plane.
(5 marks)
(5 marks)
(ii) The curve with polar equation r = 4√2 cos 2θ, 0 ≤ θ ≤ 2π.
Find the area of the finite region bounded by the curve and the line with polar equation θ = π/8.
(4 marks)
Find the area of the finite region bounded by the curve and the line with polar equation θ = π/8.
(4 marks)
5. Consider the map T : R3 → R3 defined by T(x,y,z) = (x + y + z, 2x + z, 2x + y).
(i) Show that T is an automorphism. (5 marks)
(ii) Find T(F) where F = { (x,y,z) ∈ R3 | 2x + y + z = 0 }.
(4 marks)
6. (i) State Rolle’s theorem. (1 mark)
