Advanced level 2025 North West Regional Mock Further mathematics 2

Advanced level 2025 North West Regional Mock Further mathematics 2

Advanced level 2025 North West Regional Mock Further mathematics 2

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Problem 1:

• Consider the second order differential equation dx2d2y​+4y=sin(x).
  • a) Find the complementary function.
  • b) Find a particular solution to the non-homogeneous differential equation.
  • c) Find the general solution of the differential equation given that $y(0)=0$ and y′(0)=1.

Problem 2:

• a) Express (x−1)2(x+1)x2+3​ in partial fractions.
• b) Prove that the reduction formula for the integral In​=∫0π/2​xnexdx is given by In​=(2π​)neπ/2−nIn−1​, where $n\ge 1$. Hence, evaluate ∫0π/2​x2exdx.

Problem 3:

• a) Solve for $x$ the equation cosh(x−ln2)=45​.
• b) Let $(R,+)$ be the group of real numbers under addition. Let H=(R+,×) be the group of non-zero real numbers under multiplication. A mapping from $G$ to $H$ is defined by f(x)=2x. Prove that $f$ is a homomorphism. Also determine whether or not $f$ is an isomorphism.

Problem 4:

• a) Prove by induction that for all natural numbers $n$, the expression 9n−5n is divisible by 4.
• b) Show that the length of arc of the curve defined by the parametric equations: x=etcost and y=etsint for $0\le t\le 2\pi$ is given by L=2​(e2π−1).

Problem 5:

• The hyperbola $H$ has equation 16x2​−9y2​=1.
  • a) Find the value of the eccentricity of $H$.
  • b) Calculate the distance between the foci of $H$.
  • c) Show that the polar equation of the hyperbola is given by r2=9sec2θ−16144​.
  • d) Find the polar coordinates of the point where the hyperbola cuts the initial line.

Problem 6:

• a) Find the greatest common divisor of 456 and 789. Hence or otherwise, solve the linear congruence $456x\equiv 123(\mathrm{mod}789)$.
• b) Given that $z=(\cos \theta +i\sin \theta )$, express (z2+z21​−1)(z−z1​)2​ in terms of $\theta$. Hence or otherwise, solve the equation z6+z3−2=0.
• c) Using De Moivre’s theorem or otherwise, solve the equation z4=1−i.