Cameroon GCE advanced level June 2025 further mathematics 2

Here are the extracted questions from the provided images:

Page 1

1. a) Given that x=uy, where u is a function of x, u=0, show that

   dxdy​=u21​(u−xdxdu​)

   Hence, use the substitution x=uy to transform the differential equation

   x3dxdy​+(4x+1)y2=3x2y

   into a differential equation involving u and x.

   b) Given the differential equation

   dx2d2y​−9y=e3x

   Find the

   i) complementary function,

   ii) particular integral of the differential equation.

2. Given two vectors

   a=xi−j+3k

   b=i+j−2yk,x,y∈Z,

   and that

   a×b=−i+7j+3k.

   i) Find the values of the real constants, x and y.

   ii) Show that a and b are linearly independent.

   iii) Find the Cartesian equation of the plane that contains a and b and passes through the point with position vector i−j.

3. Solve the equation

   cosh(lnx)−sinh(ln(x1​))=47​,x>0.

4. Given that

   f(x)=(x−1)(x+1)(x2+1)4x​,x=±1,

   express f(x) in partial fractions.

   Hence, show that

   ∫23​f(x)dx=ln(34​).

5. a) Show that the set of matrices of the form

   (ab​−ba​),where a,b∈R,a=0,

   forms a group under matrix multiplication (Assume Associativity).

   b) Solve the linear congruence

   2x+8≡5(mod11),

   giving your answer in the form x=pλ+q, where p,q∈N and λ∈Z.

Page 2

6. Show that the curve with polar coordinates (r,θ) where,

   r2=1−3sin2θ2​,

   represents a hyperbola.

   Find in polar form, the equations of the asymptotes to this hyperbola.

7. a) Solve completely the complex equation

   z4=−27i.

   b) Find the centre and scale factor of the transformation described by

   w=3z+2−i.

8. a) The parametric equations of a curve are

   x=a(tanθ−θ)andy=alnsecθ,0<θ<3π​,

   where a is a real constant.

   Find the length of arc of this curve.

   b) A transformation T is defined by matrix M, where

   M=(12​25​).

   i) Find the determinant of M.

   ii) Find the invariant point under T.

   iii) Show that the image of the line L:x=2y is the line L′:9x=4y.

   iv) Hence or otherwise, find the angle of rotation under T.

9. a) A sequence, (un​), is defined by

   u0​=3,un+1​=41​un​+3,∀n∈N.

   Consider another sequence, (vn​), defined by

   vn​=un​−4.

   i) Find (vn+1​) in terms of (vn​).

   ii) Hence, deduce that (vn​) is a geometric progression and state its common ratio.

   iii) Find expressions for (vn​) and also for (un​) in terms of n.

   iv) Show that (un​) is an increasing sequence.

   b) Find the radius of convergence of the series,

   n=0∑∞​n2+15n​xn.