Advanced level 2025 BAEBOC Further mathematics 2

Advanced level 2025 BAEBOC Further mathematics 2

Advanced level 2025 BAEBOC Further mathematics 2

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1. a) Given the differential equation dx2d2y​+4y=8x+4. (2mks) i) show that the particular solution is $p(x)=2x+1$. ii) find the solution of the differential equation for which dxdy​=1 and $y=1$ when $x=0$. (3mks)

b) Find the cartesian equation of the plane containing the lines l1​:r=(9i−4j+5k)+μ(4i−j+2k) and l2​:r=(4i−3j+2k)+λ(−3i+j−k). (4mks)

2. a) Given the group $(G,\ast )$ and $x\ast y=e$, where $e$ is the identity element, prove that x=y−1 and y=x−1. (3mks)

b) Construct the operation table for (G,×14​), where $G=\{2,4,6,8,10,12\}$ and ×14​ is multiplication modulo 14. (2mks) Hence, or otherwise, solve

i) completely the equation $x\ast y=e$, where $e$ is the identity element and $x\ast y$ (3mks) ii) x2=e.

3. a) A curve C has equation y=6x3​+2x1​ for $1\le x\le 3$. Show that the length of the curve C is 314​ units. (4mks)

b) Given that A=​111​011​001​​, prove by induction that An=​1n2n(n+1)​​01n​001​​, n∈Z+. (4mks)

4. a) Express f(x)=x2(1+x)1+x+x3​ in partial fractions. (6mks) Hence, show that ∫12​f(x)dx=ln25​+43​.

b) Given that In​=∫01​xn(1+x)2dx, show that (n+2)In​=nIn−1​+n−11​. (4mks) Evaluate I2​. (2mks)