Advanced level 2025 Littoral Regional Mock further mathematics 3

Advanced level 2025 Littoral Regional Mock further mathematics 3

Advanced level 2025 Littoral Regional Mock further mathematics 3

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1. Consider the differential equation dx2d2y​+x2+x​dxdy​−x2y​=2ex.

(a) Using the substitution $y=ux$, show that the above differential equation reduces to dx2d2u​+x4​dxdu​=x2ex​. (2marks) (b) Solve the above differential equation. (6marks) (c) Deduce the general solution of the differential equation dx2d2y​+x2+x​dxdy​−x2y​=2ex. (1mark)

2. (a) Given that the function f(x)=(1−x)(1+x2)x3+x2+x−2​ when expressed in partial fraction gives f(x)=1−xA​+1+x2Bx+C​ (i) Find the values of the constants A, B and C. (4marks) (ii) Hence, show that ∫01/2​f(x)dx=23​ln2−8π​−43​. (4marks)

(b) Show that the mean square value of the function $y=\tanh (2x)$, $0\le x\le \ln 2$ is given by 32ln215​. (4marks)

3. (a) Given that In​=∫0π/2​(1−x)ncosxdx for all $n\ge 1$. (i) Show that In​=(n−1)In−2​−(2π​−1)n for all $n\ge 2$. (3marks) (ii) Hence, prove by mathematical induction that I2n​=(2n)![1−(2π​−1)2/2!+…+(−1)n(2π​−1)2n/(2n)!]−2n(2π​−1). (3marks)

(b) Show that the set $G=\{1,2,4,7,8,11,13,14\}$ forms a group under multiplication modulo 15. Hence, find a subgroup of G which is isomorphic to the group ({0,1,2},+3​), where +3​ is addition modulo 3. (6marks)

4. By using the definition of sech−1x in terms of $\ln$ and ex, show that dxd​(sech−1x)=−x1−x2​1​ for $x\in (0,1]$. Hence, solve for $x$ in the equation sech−1(x)=ln2+3​2−3​​​, $x\in (0,1]$. (7marks)

5. (a) Show that the Cartesian equation of the curve (C) with polar coordinates $(r,\theta )$ such that r2(4−9sin2θ)=20 describes that of a hyperbola, stating the value of its eccentricity. (4marks) (b) Show that (C) passes through any point P with Cartesian coordinates (5cosht​,2sinht). Hence prove that the equation of the normal to (C) at P is 5cosht​2x​+2sinhty​=cosht+sinht. (5marks)