From the set of five sketches, select all the relations that are functions. Give a brief reason using the vertical line test.

From those that are functions, select any one that is not one-to-one and explain why using the horizontal line test.

For one graph that is a one-to-one function, state its domain and range as suggested by the sketch and describe how the graph of its inverse would relate to it. No sketch required.

Find the equation of the normal to $y = x^2$ at $A(1, 1)$.

The normal meets the parabola again at $B$. Find the coordinates of $B$.

Let $\theta$ be the angle $A\hat{O}B$ (the angle between $OA$ and $OB$). Show that $\tan \theta = 5$.

Write down, in terms of $r$ and $\theta$ (radians), the length of arc $AB$ and the area of sector $OAB$.

Consider the smaller, concentric sector $OMB$ of radius $\frac{r}{2}$ and angle $\theta$. Show that $\mathrm{area}(OMB) : \mathrm{area}(MAB) = 1 : 3$, where $MAB$ denotes the annular sector between radii $OM$ and $OA$.

If $r = 8\ \mathrm{cm}$ and arc $AB$ has length $10\ \mathrm{cm}$, find $\theta$ and hence the area of region $MAB$.

Solve for $x$: $3^{2x} = 27 \cdot 9^{x-1}$.

Solve for $x$: $2\cdot \log_{2}(x+1) - \log_{2}(x-3) = 2$, given $x > 3$.

Find the median and the interquartile range (IQR).

Using the $1.5\times \mathrm{IQR}$ rule, decide whether $25\ \mathrm{mm}$ is an outlier. Show working.

If all values were decreased by 3, state the new mean and state what happens to the standard deviation.

Write $f(x)$ in vertex form and then in expanded form.

Find the discriminant and hence the number of real roots.

Solve $f(x) = 0$ exactly.

Show that the curves intersect at $(0, 3)$.

Find the $x$-intercepts of each curve.

Write an exact expression, as a sum of definite integrals, for the total shaded area.

Hence find the exact value of the shaded area.

State the amplitude, the midline, and the period of $f$.

Find the maximum value of $f$ and one $x$-value in $[0, 2\pi]$ at which it occurs.

Solve $2\cos!\left(3\left(x - \frac{\pi}{6}\right)\right) + 1 = \frac{1}{2}$ for $x$ in $[0, 2\pi]$. Give exact values.

Show that if the right-hand upper vertex is $(x, 9 - x^2)$ with $x > 0$, then the area $A$ of the rectangle is $A = 2x(9 - x^2)$.

Find the value of $x$ that maximises $A$ and the maximum area.

Determine the $y$-coordinate of the rectangle’s upper side at maximum area.