[Maximum number: 7]

The following question explores features of composed trigonometric functions, such as $\sin (\sin x), \sin (\sin (\sin x))$ .
Suppose $S_{n}(x)$ denotes the function $\sin x$ composed within itself n-1 times, defined for $n \geq 1, n \in \mathbb{Z}^{+}$ , where $0 \leq x \leq 2 \pi$ .
For example, $S_{1}(x)=\sin x$ and $S_{2}(x)=\sin (\sin x)$ where $0 \leq x \leq 2 \pi$ .

(a)

By considering the equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$ , show that there are exactly two points of zero gradient, one at $x=\frac{\pi}{2}$ and one at $x=\frac{3 \pi}{2}$ .

The derivative $S_{n}^{\prime}(x)=\frac{\mathrm{d}}{\mathrm{d} x}\left(S_{n}(x)\right)$ can be expressed as a product of cosine functions, as follows:

S_{n}^{\prime}(x)=\cos \left(S_{n-1}(x)\right) \cos \left(S_{n-2}(x)\right) \ldots \cos \left(S_{1}(x)\right) \cos x

[ 6 ]

(b)

Hence, show that $S_{3}^{\prime}(x)=\cos (\sin (\sin x)) \cos (\sin x) \cos x$ .

[ 1 ]

[Maximum number: 16]

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)

(i)

Write down the coordinates of the two points of inflexion on the curve $y^{2}=x^{3}+1$ .

[ 1 ]

(b)

(i)

Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}= \pm \frac{3 x^{2}+1}{2 \sqrt{x^{3}+x}}$ , for x>0.

[ 3 ]

(ii)

Hence deduce that the curve $y^{2}=x^{3}+x$ has no local minimum or maximum points.

The curve $y^{2}=x^{3}+x$ has two points of inflexion. Due to the symmetry of the curve these points have the same x-coordinate.

[ 1 ]

(c)

Find the value of this x-coordinate, giving your answer in the form $x=\sqrt{\frac{p \sqrt{3}+q}{r}}$ ,
where $p, q, r \in \mathbb{Z}$ . where $p, q, r \in \mathbb{Z}$ .
P(x, y) is defined to be a rational point on a curve if x and y are rational numbers.
The tangent to the curve $y^{2}=x^{3}+a x+b$ at a rational point P intersects the curve at another rational point Q .

Let C be the curve $y^{2}=x^{3}+2$ , for $x \geq-\sqrt[3]{2}$ . The rational point P(-1,-1) lies on C.

[ 7 ]

(d)

(i)

Find the equation of the tangent to C at P .

[ 2 ]

(ii)

Hence, find the coordinates of the rational point Q where this tangent intersects C, expressing each coordinate as a fraction.

[ 2 ]

[Maximum number: 20]

This question asks you to explore the behaviour and some key features of the function $f_{n}(x)=x^{n}(a-x)^{n}$ , where $a \in \mathbb{R}^{+}$ and $n \in \mathbb{Z}^{+}$ .
In parts (a) and (b), only consider the case where a=2.
Consider $f_{1}(x)=x(2-x)$ .

(a)

Use your graphic display calculator to explore the graph of $y=f_{n}(x)$ for
- the odd values n=3 and n=5;
- the even values n=2 and n=4.

Hence, copy and complete the following table.

Now consider $f_{n}(x)=x^{n}(a-x)^{n}$ where $a \in \mathbb{R}^{+}$ and $n \in \mathbb{Z}^{+}, n>1$ .

[ 6 ]

(b)

Show that $f_{n}^{\prime}(x)=n x^{n-1}(a-2 x)(a-x)^{n-1}$ .

[ 5 ]

(c)

State the three solutions to the equation $f_{n}^{\prime}(x)=0$ .

[ 2 ]

(d)

Hence, or otherwise, show that $f_{n}^{\prime}\left(\frac{a}{4}\right)>0$ , for $n \in \mathbb{Z}^{+}$ .

[ 2 ]

(e)

By using the result from part (f) and considering the sign of $f_{n}^{\prime}(-1)$ , show that the point (0,0) on the graph of $y=f_{n}(x)$ is

[ 5 ]

(i)

a local minimum point for even values of n, where n>1 and $a \in \mathbb{R}^{+}$ ;

[ 3 ]

(ii)

a point of inflexion with zero gradient for odd values of n, where n>1 and $a \in \mathbb{R}^{+}$ .

Consider the graph of $y=x^{n}(a-x)^{n}-k$ , where $n \in \mathbb{Z}^{+}, a \in \mathbb{R}^{+}$ and $k \in \mathbb{R}$ .

[ 2 ]

[Maximum number: 4]

Given that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\cos \left(x-\frac{\pi}{4}\right)$ and y=2 when $x=\frac{3 \pi}{4}$ , find y in terms of x.

[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 area of the region enclosed by the graphs of f and g.

[ 3 ]

[Maximum number: 7]

In this question, you will be investigating the family of functions of the form $f(x)=x^{n} \mathrm{e}^{-x}$ .
Consider the family of functions $f_{n}(x)=x^{n} \mathrm{e}^{-x}$ , where $x \geq 0$ and $n \in \mathbb{Z}^{+}$ .
When n=1, the function $f_{1}(x)=x \mathrm{e}^{-x}$ , where $x \geq 0$ .

(a)

Sketch the graph of $y=f_{1}(x)$ , stating the coordinates of the local maximum point.

[ 4 ]

(b)

Use your graphic display calculator, and an appropriate value for the upper limit, to determine the value of

[ 3 ]

(i)

$A_{4}$ ;

[ 2 ]

(ii)

$A_{5}$ .

[ 1 ]

[Maximum number: 6]

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)

State the coordinates of the local maximum point and the coordinates of the local minimum point.

[ 2 ]

(b)

Given $f(x)=\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ , prove that $f^{\prime}(\sqrt{A})$ is independent of A.

[ 4 ]

[Maximum number: 5]

Find the value of $\int_{1}^{9}\left(\frac{3 \sqrt{x}-5}{\sqrt{x}}\right) \mathrm{d} x$ .

[Maximum number: 14]

This question asks you to examine the number and nature of intersection points of the graph of $y=\log _{a} x$ where $a \in \mathbb{R}^{+}, a \neq 1$ and the line y=x for particular sets of values of a.
In this question you may either use the change of logarithm base formula $\log _{a} x=\frac{\ln x}{\ln a}$ or a
graphic display alculator "logarithm to all " graphic display calculator "logarithm to any base feature".
The function f is defined by

f(x)=\log _{a} x \text { where } x \in \mathbb{R}^{+} \text {and } a \in \mathbb{R}^{+}, a \neq 1

(a)

Use calculus to find the minimum value of the expression $x-\ln x$ , justifying that this value is a minimum.

[ 5 ]

(b)

Hence deduce that $x>\ln x$ .

[ 1 ]

(c)

Find the exact coordinates of P and the exact value of a.

[ 8 ]

[Maximum number: 2]

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

(a)

Find the area of the region bounded by the graph and the x and y axes.

[ 2 ]

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