Write down the value of the discriminant of $f\prime$.

Write down the values of $h$ and $k$.

Write down the equation of the axis of symmetry.

Sketch a graph of $v$ against $t$, clearly showing any points of intersection with the axes.

Sketch the graph of $y=g\left(t\right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.

Explain why $f$ has no inverse on the given domain.

find the $x$-coordinates of the $x$-intercepts.

With reference to your graph, explain why $f$ is a function on the given domain.

Explain why $f$ is not a function for $-\frac{3\pi }{4}⩽x⩽\frac{\pi }{4}$.

Sketch the graph of $y=f\left(x\right)$ for $-\frac{5\pi }{8}⩽x⩽\frac{\pi }{8}$.

Hence find the values of $x$ where the graph of $f$ is both negative and increasing.

Show that $g\left(t\right)={\left(\frac{1+t}{1-t}\right)}^2$.

Hence or otherwise, find the value of $p$.

Find the value of the gradient of the graph of $f\prime$ at $x=0$.

The following table shows the probability distribution of a discrete random variable $X$ where $x=1, 2, 3, 4$.

Find the value of $k$, justifying your answer.

Point $\text{Q}$ has coordinates $(5, 12)$. Find the value of $a$.

Given that $\left(h\circ g\right)\left(a\right)=\frac{\mathrm{\pi }}{4}$, find the value of $a$.

Give your answer in the form $p+\frac{q}{2}\sqrt{r}$, where $p, q, r\in {\mathrm{\mathbb{Z}}}^{+}$.

Complete the table below placing a tick (✔) to show whether the unknown parameters $a$ and $b$ are positive, zero or negative. The row for $c$ has been completed as an example.

Find the value of $p$ and the value of $q$.

(i)     Write down the value of $h$.

(ii)     Find the value of $k$.

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

Find the maximum value of $S_n$.

Find the range of $f$.

Given that $\left(g\circ f\right)\left(x\right)\le 0$ for all $x\in \mathrm{ℝ}$, determine the set of possible values for $c$.

The graph of $f\left(x\right)=x^2$ is transformed to obtain the graph of $g$.

Describe this transformation.

Show that $m=4$.

Show that $\alpha$ + β < −2.

The function $f$ can be written in the form $f\left(x\right)=-2{\left(x-h\right)}^2+k$.

Write down the value of $h$ and the value of $k$.

Find the total distance travelled by $P$.

Show that the distance of $P$ from $\mathrm{O}$ at this time is $\frac{88}{27}$ metres.

Find the value of $q$.

Find the value of $p$.

Find the value of $a$.

Find the value of $b$.

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

Find the value of $h$ and the value of $k$.

Find $\alpha$ and β in terms of $k$.

Write down the other solution of $f\left(x\right)=k$.

State the values of $x$ for which $f\left(x\right)$ is decreasing.

Find the value of $t$ when $P$ reaches its maximum velocity.