Advanced level 2026 Littoral regional mock further mathematics 2

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Question 3

A curve $C$ has polar equation $r(1 + \cos\theta) = 2$.

• (i) Find the Cartesian equation of $C$, is a parabola. (3 marks)

• (ii) Show that the point $(0, 2)$ lies on the $C$. (2 marks)

• (iii) State the coordinates and equation of directrix of the parabola. (2 marks)

• The straight line with equation $4r = 3\sec\theta$ intersects the curve $C$ at the points $P$ and $Q$.

• (iv) Find the length of $PQ$. (3 marks)

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Question 4

A complex function $f$ is defined by $f(z) = \cosh\left(\ln \frac{1}{z}\right) – \sinh(\ln z) – z^2$.

• (i) Show that $f(z) = \frac{1 – z^3}{z}$. (2 marks)

• (ii) Determine in algebraic form the solution of the equation $f(z) = 0$. (4 marks)

• Let $W$, where $W = \{1, \omega, \omega^2\}$, with $\omega^3 = 1$ be the solution set for $f(z) = 0$.

• (iii) Show that the set $W$ under multiplication forms a group. You may assume associativity. (4 marks)

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Question 5

Let $g: \mathbb{R} \to \mathbb{R}$ be given by $g(x) = 3 + 4x$. Prove by induction that for all positive integers $n$, $g^n(x) = (4^n – 1) + 4^n(x)$. If for every positive integer $k$, we interpret $g^{-k}$ as the inverse of the function $g^k$, prove that the above formula holds also for all negative integers $n$.

(a)

Let $g: \mathbb{R} \to \mathbb{R}$ be given by $g(x) = 3 + 4x$.

• (i) Prove by mathematical induction that for all positive integers $n$, $g^n(x) = (4^n – 1) + 4^n(x)$. (4 marks)

• If for every positive integer $k$, we interpret $g^{-k}$ as the inverse of $g^k$,

• (ii) Prove that the above formula holds also for all negative integers $n$. (2 marks)

(b)

Determine the remainder when the number $5^n, \forall n \in \mathbb{N}$, is divided by 3, when $n$ is an even number. (2 marks)