[Maximum number: 5]

Consider the functions f(x)=x-3 and $g(x)=x^{2}+k^{2}$ , where k is a real constant.

(a)

Write down an expression for $(g \circ f)(x)$ .

[ 2 ]

(b)

Given that $(g \circ f)(2)=10$ , find the possible values of k.

[ 3 ]

[Maximum number: 3]

The functions f and g are both defined for $-1 \leq x \leq 0$ by

\begin{aligned} & f(x)=1-x^{2} \\ & g(x)=\mathrm{e}^{2 x} \end{aligned}

The graphs of f and g intersect at x=a and x=b, where a<b.

(a)

Find the value of a and the value of b.

[ 3 ]

[Maximum number: 5]

The graph of y=f(x) for $-4 \leq x \leq 6$ is shown in the following diagram.

(a)

Write down the value of

[ 2 ]

(i)

f(2);

(ii)

$(f \circ f)(2)$ .

[ 2 ]

(b)

Let $g(x)=\frac{1}{2} f(x)+1$ for $-4 \leq x \leq 6$ . On the axes above, sketch the graph of g.

[ 3 ]

[Maximum number: 3]

Consider the graph of $y=x+\sin (x-3),-\pi \leq x \leq \pi$ .

(a)

Sketch the graph, clearly labelling the x and y intercepts with their values.

[ 3 ]

[Maximum number: 12]

This question asks you to explore properties of a family of curves of the type $y^{2}=x^{3}+a x+b$ for various values of a and b, where $a, b \in \mathbb{N}$ .

(a)

On the same set of axes, sketch the following curves for $-2 \leq x \leq 2$ and $-2 \leq y \leq 2$ , clearly indicating any points of intersection with the coordinate axes.

[ 4 ]

(i)

$y^{2}=x^{3}, x \geq 0$

[ 2 ]

(ii)

$y^{2}=x^{3}+1, x \geq-1$

[ 2 ]

(b)

(i)

By considering each curve from part (a), identify two key features that would distinguish one curve from the other.

Now, consider curves of the form $y^{2}=x^{3}+b$ , for $x \geq-\sqrt[3]{b}$ , where $b \in \mathbb{Z}^{+}$ .

[ 1 ]

(c)

By varying the value of b, suggest two key features common to these curves.

Next, consider the curve $y^{2}=x^{3}+x, x \geq 0$ .

[ 2 ]

(d)

The point S(-1,1) also lies on C. The line $[\mathrm{QS}]$ intersects C at a further point. Determine the coordinates of this point.

[ 5 ]

[Maximum number: 7]

Consider the function $f(x)=11 \sqrt{x}-2 x-11$ , where $0 \leq x \leq 20$ .

(a)

Find the value of

[ 2 ]

(i)

f(0);

(ii)

f(20).

[ 2 ]

(b)

Find the two roots of f(x)=0.

[ 2 ]

(c)

Sketch the graph of y=f(x) on the following grid.

[ 3 ]

[Maximum number: 1]

In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

(a)

The following table gives values of $n^{2}$ and $\sum_{i=1}^{n} i^{2}$ for n=1,2,3.

[ 1 ]

(i)

Write down the value of p.

[ 1 ]

[Maximum number: 6]

The quadratic function $f(x)=p+q x-x^{2}$ has a maximum value of 5 when x=3.

(a)

Find the value of p and the value of q.

[ 4 ]

(b)

The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the equation of the new graph.

[ 2 ]

[Maximum number: 11]

This question asks you to explore self-composite linear functions, of the form f(x)=m x+c, for varying values of m.
A function composed with itself is called a self-composite function.
For a function f, the function composition with itself is given by $(f \circ f)(x)=f(f(x))$ .
Let $f^{n}$ denote the nth composition of f with itself where $f^{n}(x)=\underbrace{(f \circ f \circ \cdots \circ f)}_{n \text { times }}(x)$ .
Hence, for example, $f^{2}(x)=(f \circ f)(x)$ and $f^{3}(x)=(f \circ f \circ f)(x)=f(f(f(x)))$ .
Consider the linear function f(x)=m x+c, where $x \in \mathbb{R}$ and $m, c \in \mathbb{R}$ .

(a)

Show that

[ 5 ]

(i)

$\quad f^{2}(x)=m^{2} x+c(1+m)$ ;

[ 3 ]

(ii)

$\quad f^{3}(x)=m^{3} x+c\left(1+m+m^{2}\right)$ .

[ 2 ]

(b)

(i)

Write down an expression for $f^{4}(x)$ .

[ 1 ]

(ii)

Suggest a similar expression for $f^{n}(x), n \in \mathbb{Z}^{+}$ .

[ 2 ]

(iii)

By using your expression from part (b)(ii), or otherwise, find an expression in terms of n for $f^{n}(x)$ when m=1.

[ 3 ]

[Maximum number: 2]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

(a)

(i)

Sketch the curve of y for $-10 \leq x \leq 10$ .

[ 1 ]

(ii)

State the coordinates of the points where the curve crosses the x-axis.

[ 1 ]

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