Review practice

Sketch the curve with equation $16y^2-16y-4x^2=-3$, indicating clearly the equation of any asymptote (s) and the coordinates of any stationary point (s).4 marks

The curve $C$ has equation $y=\dfrac{x^2+ax+b}{c-x}$. The vertical asymptote of $C$ is $x=-2$, and the coordinates of the turning points are $(-4,2)$ and $(0,-6)$.

(i) Find the values of $a$, $b$ and $c$.3 marks

(ii) Sketch $C$, stating the equations of the asymptotes.2 marks

(iii) By drawing an appropriate graph on the sketch of $C$, find the range of values of $k$ $(k>0)$ such that the equation $(x+4)^2+\left(\dfrac{x^2+ax+b}{c-x}\right)^2=k^2$ has no real roots.2 marks

(iv) When $C$ undergoes a translation in the direction of the $y$-axis by $p$ units, $C$ intersects the line $y=-1$ at one negative value of $x$. State the set of values for $p$.2 marks

The circle with equation $x^2+(y-k)^2=25$, where $k$ is a real constant, intersects the parabola with equation $4y=x^2$ at four distinct points. Determine the range of values of $k$.4 marks

The curve $C_1$ has equation $(x-1)^2-y^2=1$ and the curve $C_2$ has equation $y=\dfrac12\sqrt{16-x^2}$.

(i) Sketch $C_1$ and $C_2$ on the same diagram, indicating clearly the axial intercepts and the equations of any asymptotes.5 marks

(ii) Find the $x$-coordinate(s) of the point (s) of intersection of $C_1$ and $C_2$.1 mark

(iii) Hence solve the inequality $2\sqrt{(x-1)^2-1}<\sqrt{16-x^2}$.2 marks

The graph of $(x-p)^2+(y-p)^2=p^2$, $p>0$, underwent exactly two simple transformations, involving either reflection, stretching or translation only, to become the ellipse that passes through $(0,p)$, $(2p,p)$, $(p,3p)$ and $(p,-p)$.

(i) Write down the equation of the ellipse.2 marks

(ii) Describe clearly (in words) a sequence of the two transformations.1 mark

(a) The curve $C_1$ has equation $\dfrac{(x+2)^2}{2^2}-\dfrac{y^2}{3^2}=1$. Sketch $C_1$, giving the equations of any asymptotes, coordinates of its centre and the vertices.3 marks

(b) The curve $C_2$ has equation $y=\dfrac{3x^2+12x+\lambda}{2(x+2)}$, $x\ne-2$, $\lambda\in\mathbb R$, $\lambda\ne12$. Find the range of values of $\lambda$ for which the graph of $C_2$ is strictly increasing.4 marks

A tunnel is built to facilitate the transportation of goods by trains between Country $X$ and Country $Y$. Due to differences in the rail systems between the two countries, two types of tracks are used: the international track with track gauge of 1435 mm and the narrow track with track gauge of 1000 mm (in rail transport, track gauge is the spacing on a railway track). It is known that the cross-section of the tunnel is a half ellipse with centre $O$ and width $MN$. The maximum height of the tunnel is 2000 mm. To standardize the volumes of the goods to be transported, the areas $ABCD$ and $EFGH$ are made equal. Find the width of the tunnel $MN$, giving your answer to the nearest mm.

In the diagram, $O$ is the midpoint of the base $MN$. $A$ and $B$ lie on the base, 1000 mm apart and symmetric about $O$, with $D$ and $C$ on the tunnel wall directly above them. $E$ and $F$ lie on the base, 1435 mm apart and symmetric about $O$, with $H$ and $G$ on the tunnel wall directly above them.