Advanced level 2025 South West Regional Mock further mathematics 2

Advanced level 2025 South West Regional Mock further mathematics 2

Advanced level 2025 South West Regional Mock further mathematics 2

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1. A particular integral to the differential equation dx2d2y​−5dxdy​+6y=2sin4x+Dcos4x is $y=C\sin 4x+D\cos 4x$. a) Find the values of the constants $C$ and $D$. b) Given that when $x=0$, $y=0$ and dxdy​=1, find the particular solution to the differential equation.

2. (i) Show that the set $C=\{1,i,-1,-i\}$, together with the operation $\times$ (multiplication), forms a group, where i2=−1. Hence, show that the group is a cyclic group. (ii) Find the remainder when 519 is divided by 59.

3. (i) Express f(x)=(x2+1)(x−2)4x−1​, x=0, in partial fractions. (ii) The line $L$ is defined by r” />=9i^−7j^​+8k^+μ(3i^−7j^​+2k^), $\mu \in \mathbb{R}$. Find: a) the coordinates of the foot of the perpendicular from the point $A(1,2,4)$ to $L$. b) the perpendicular distance from the point $A$ to $L$. c) the position vector of the reflection of $A$ in $L$.