Find the coordinates of P, given that $x_{P}>0$.

Find the equation of the tangent at P. Give your answer correct to $3$ significant figures.

State the horizontal asymptote of the curve.

Given that the area of the sector is less than $(44x-7)\text{ cm}^{2}$, find the set of possible values of $x$.

For $x=1$, find the length of arc AB.

Using technology, find $\bar{x}$, $\bar{y}$ and the population standard deviations of $x$ and $y$, each correct to $3$ significant figures.

Using technology, find the Pearson correlation coefficient $r$ and the regression line of $y$ on $x$.

Interpret the slope of your regression line in this context.

Use technology to find the solution in the interval $[0,1]$, correct to $3$ significant figures.

Let $f(x)=e^{2x}-1-x^{2}$. Show that $f'(x)>0$ on $[0,1]$ and hence justify that the solution is unique.

In terms of $k$, determine the discriminant and hence the values of $k$ for which the equation has

(i) two distinct real roots;
(ii) a repeated real root;
(iii) no real roots.

For $k=1$, solve the equation.

Find the exact area of the region enclosed by the two curves.

Find the equation of the tangent to $y=x^{2}$ that is parallel to the line $y=2x$.

Express $x$ as a function of $y$ and hence show that the y-coordinates of P and Q satisfy $2y^{2}-10y+k=0$.

Deduce that, at P and Q, $x=2y$.

Hence find all positive integers $k$ for which there are two distinct points where the tangent is parallel to the y-axis, and give the two y-values in terms of $k$.

For $k=9$, find the coordinates of P and Q, and the equations of the vertical tangents at these points.

For $k=9$, write down the equation of the normal at P.

Show that $f$ has a unique maximum on $[0,\infty)$.

Find the x-coordinate of the maximum and the maximum value.

Solve $f(x)=\dfrac{1}{2}$ for $x\ge 0$, correct to $3$ significant figures, using technology.

State the amplitude, midline, period and phase shift of $g$.

Solve $g(x)=1$ for $0\le x\le 2\pi$.

Solve $2\sin^{2}x=3\cos x$ for $0\le x\le 2\pi$.