Section A [40 Marks] Answer all questions. 1 (a) Simplify ... (b) Y is inversely proportional to the cube root of x + 1.

Recall that $\sqrt{x} = x^{1/2}$ and $x^a \cdot x^b = x^{a+b}$.

1 (b)

$Y$ is inversely proportional to the cube root of $x+1$.

Given $Y=2$ when $x=63$.

Substitute the value of $k$ back into the proportionality equation.

2 (a)

Given $U_{n+1} = 3 - U_n$ and $U_1 = 2$.
For $n=1$: $U_2 = 3 - U_1 = 3 - 2 = 1$.
For $n=2$: $U_3 = 3 - U_2 = 3 - 1 = 2$.
For $n=3$: $U_4 = 3 - U_3 = 3 - 2 = 1$.

The sequence terms are $2, 1, 2, 1, \dots$.
The sequence is oscillating between the values 2 and 1.

2 (b)

We have $U_{n+1} = 3 - U_n$.
To find $U_{n+2}$, we replace $n$ with $n+1$ in the recurrence relation:

Now substitute the expression for $U_{n+1}$:

This shows that the sequence repeats every two terms.

Using the recurrence relation:

From Step 1, we know $U_{n+2} = U_n$. Substitute this into the equation for $U_{n+3}$:

3 (a)

The payments are $20, 22, \dots, 114$.
This is an arithmetic progression with:
First term $a = 20$.
Common difference $d = 22 - 20 = 2$.
Last term $L = 114$.

The formula for the $n$-th term is $L = a + (n-1)d$.

3 (b)

The formula for the sum of an arithmetic progression is $S_n = \frac{n}{2}(a+L)$.
We have $n=48$, $a=20$, and $L=114$.

$S_{48} = \boxed{\$3216}$

4 (a) (i)

Given $f(x) = 2 - \frac{1}{x}$ and $g(x) = \frac{1}{2x+6}$.
To find $gf(x)$, substitute $f(x)$ into $g(x)$:

To remove the fraction in the denominator, multiply the numerator and denominator by $x$:

4 (a) (ii)

Let $y = f(x)$. So, $y = 2 - \frac{1}{x}$.
To find the inverse, swap $x$ and $y$:

Multiply both sides by $-1$:

Take the reciprocal of both sides:

So, $f^{-1}(x) = \boxed{\frac{1}{2 - x}}$

4 (b)
The graphs of a function $g(x)$ and its inverse $g^{-1}(x)$ are reflections of each other in the line $y=x$.