[Maximum number: 6]

This question asks you to explore the behaviour and key features of cubic polynomials of the form $x^{3}-3 c x+d$ .
Consider the function $f(x)=x^{3}-3 c x+2$ for $x \in \mathbb{R}$ and where c is a parameter, $c \in \mathbb{R}$ .
The graphs of y=f(x) for c=-1 and c=0 are shown in the following diagrams.

c=-1

c=0

(a)

On separate axes, sketch the graph of y=f(x) showing the value of the y-intercept and the coordinates of any points with zero gradient, for

[ 6 ]

(i)

$\quad c=1$ ;

[ 3 ]

(ii)

$\quad c=2$ .

[ 3 ]

[Maximum number: 5]

Let $f(x)=x^{4}+p x^{3}+q x+5$ where p, q are constants.
The remainder when f(x) is divided by (x+1) is 7 , and the remainder when f(x) is divided by ( x-2 ) is 1 . Find the value of p and the value of q.

[Maximum number: 11]

This question asks you to investigate lines normal to curves of the form $y=\frac{k^{2}}{x}$ .
The curve H has equation $y=\frac{1}{x}$ where $x \in \mathbb{R}, x \neq 0$ .

(a)

The equation for N given in part (a)(ii) is of the form y=m x+c.

[ 5 ]

(i)

Show that either $c=\frac{1}{\sqrt{m}}\left(1-m^{2}\right)$ or $c=\frac{1}{\sqrt{m}}\left(m^{2}-1\right)$ .

[ 4 ]

(ii)

Determine the set of values of m for which there exists at least one line normal to H.

[ 1 ]

(b)

Hence, or otherwise, determine the set of values of m for which there exists exactly

[ 3 ]

(i)

one line normal to H;

[ 1 ]

(ii)

two lines normal to H.

[ 2 ]

(c)

The equation of the line normal to F at A is given by $y=t^{2} x-k t^{3}+\frac{k}{t}$ .

[ 3 ]

(i)

Show that the x-coordinates of A and B satisfy the quadratic equation

x^{2}-k\left(t-\frac{1}{t^{3}}\right) x-\frac{k^{2}}{t^{2}}=0

(ii)

Hence, by considering either the sum or product of the roots of this quadratic equation, or otherwise, determine the coordinates of B.

From A , the line passing through the origin O intersects F again at point C .
Points A, B and C form triangle ABC as shown in the following diagram.

[ 3 ]

[Maximum number: 4]

Find the set of values of x for which |x-1|>|2 x-1|.

[Maximum number: 5]

Given that $A x^{3}+B x^{2}+x+6$ is exactly divisible by (x+1)(x-2), find the value of A and the value of B.

[Maximum number: 4]

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)

(i)

Show that $f^{n}(x)=-x+c$ when n is odd.

[ 2 ]

(ii)

Find an expression for $f^{n}(x)$ when n is even.

[ 2 ]

[Maximum number: 4]

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)

State whether the function $f(x)=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ is odd, even or neither. Justify your answer. [2]

Now consider the general curve given by $y=\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ , where A is a positive constant
and $x \in \mathbb{R}$ . and $x \in \mathbb{R}$ .

[ 2 ]

(b)

(i)

Hence, determine the equation of the oblique asymptote to the curve.

[ 1 ]

(ii)

Write down the coordinates of a point on the curve where the oblique asymptote is parallel to the tangent to the curve at that point.

Now consider the differential equation $x^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=x(x+y)-y^{2}$ , where $x \neq 0, y \neq \pm x$ .
Using the substitution y=v x, the differential equation can be written as $x \frac{\mathrm{~d} v}{\mathrm{~d} x}=1-v^{2}$ .

[ 1 ]

[Maximum number: 5]

The cubic polynomial $3 x^{3}+p x^{2}+q x-2$ has a factor ( x+2 ) and leaves a remainder 4 when divided by (x+1). Find the value of p and the value of q.

[Maximum number: 4]

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by

f(x)=2 \mathrm{e}^{x}-\mathrm{e}^{-x}

(a)

Show that f is a bijection.

[ 4 ]

[Maximum number: 6]

The same remainder is found when $2 x^{3}+k x^{2}+6 x+32$ and $x^{4}-6 x^{2}-k^{2} x+9$ are divided by x+1. Find the possible values of k.

Question 1

Question 1(a)

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