Curve $S$ has equation $y = 2 - \frac{5}{2x+1}$ where $x \neq -\frac{1}{2}$

(a) Write down an equation of the asymptote to $S$ that is parallel to

(i) the $y$-axis

(ii) the $x$-axis

(b) Find the coordinates of the point where $S$ crosses

(i) the $y$-axis

(ii) the $x$-axis

(c) Using the axes on the page opposite, sketch $S$, showing clearly the asymptotes and the coordinates of the points where $S$ crosses the coordinate axes.

The curve $C$ has equation $y = \frac{ax - 5}{b - x}$ where $a$ and $b$ are integers and $x \neq b$
One intersection of $C$ with the coordinate axes is at the point with coordinates $\left(\frac{5}{4}, 0\right)$

The asymptote parallel to the $y$-axis has equation $x = 3$

(a) Find the value of $a$ and the value of $b$

(b) Sketch $C$, showing clearly the asymptotes with their equations and the coordinates of the points of intersection with the coordinate axes.

The straight line $l$ with equation $4y - 7x = k$ has no points of intersection with $C$

(c) Show, using algebra, that the range of possible values of $k$ can be written as

where $m$ and $n$ are integers to be found.

Figure 2 shows part of the curve with equation $y = \frac{x^2}{3} - \frac{1}{2x}$ for $-4 < x < 0$

By drawing a suitable straight line on the grid, obtain estimates, to one decimal place,

of the roots of the equation $4x^3 + 3x^2 - 36x - 6 = 0$ in the interval $-4 < x < 0$

Figure 2 shows a sketch of part of the curve $C$ with equation

(a) Write down the equation of the asymptote to $C$ that is parallel to the $y$-axis.

The line $l$ is the normal to $C$ at the point where $x = -1$

(b) Find an equation of $l$

The line $l$ meets $C$ again at the point $D$

(c) Find the coordinates of $D$

Figure 1 shows part of the curve with equation $y = \frac{x}{2} + \frac{4}{x^2}$ in the interval $0.8 < x < 7$

By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the roots of the equation $3x^3 - 12x^2 + 8 = 0$ in the interval $0.8 < x < 7$

Given that $(2x - 1)$ is a factor of $f(x)$

show that the curve with equation $y = f(x)$ has only one intersection with the $x$-axis.

A curve $C$ has equation

where $a$ is a constant and $x\neq-5$.

The gradient of $C$ at the point on the curve where $x=2$ is $\frac{18}{49}$.

(a) Show that $a=3$.

Hence

(b) write down an equation of the asymptote to $C$ that is

(i) parallel to the $x$-axis,

(ii) parallel to the $y$-axis,

(c) find the coordinates of the point where $C$ crosses

(i) the $x$-axis,

(ii) the $y$-axis.

(d) Sketch the curve $C$, showing clearly its asymptotes and the coordinates of the points where $C$ crosses the coordinate axes.

Figure 2 shows the graph of

for $1\leq x\leq4$, drawn on a grid.

(a) By drawing a suitable straight line on the grid, obtain estimates, to one decimal place, for the roots of

in the interval $1\leq x\leq4$.

(b) By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, for the root of

in the interval $1\leq x\leq4$.

A curve $C$ has equation

(a) Write the equation of $C$ in the form

where $a$, $b$ and $c$ are integers whose values are to be found.

(b) Hence show that $x$ is real when $y\leq2$ and when $y\geq3$.

(c) Find the coordinates of the stationary points on $C$.

(d) Sketch $C$, showing clearly

(i) the exact coordinates of the points where $C$ crosses the $x$-axis,

(ii) the asymptote to $C$ that is parallel to the $y$-axis,

(iii) the coordinates of the stationary points.

(a) Show that $f(x)=2x^{3}+11x^{2}-x-3$.

(b) Hence use algebra to solve the equation $2x^{3}+11x^{2}-x-3=0$.

Give your roots to 3 decimal places where appropriate.

A curve $C$ has equation

(a) Write down an equation of the asymptote to $C$ that is

(i) parallel to the $y$-axis,

(ii) parallel to the $x$-axis.

(b) Find the coordinates of the points of intersection of $C$ with the coordinate axes.

(c) Using calculus show that at every point on the curve, the gradient of $C$ is positive.

(d) Using the axes on the opposite page, sketch $C$, showing clearly the asymptotes and the coordinates of the points of intersection of $C$ with the coordinate axes.

The line $l$ is the tangent to $C$ at the point on the curve where $x=1$.

(e) Find an equation of $l$, giving your answer in the form $y=mx+c$.

Figure 1 shows part of the curve $S$ with equation $y=\dfrac{ax+b}{x+c}$ where $a$, $b$ and $c$ are integers.

The asymptote to $S$ that is parallel to the $x$-axis has equation $y=-2$

The asymptote to $S$ that is parallel to the $y$-axis has equation $x=-3$

The curve crosses the $x$-axis at the point with coordinates $(4,0)$

The curve crosses the $y$-axis at the point with coordinates $(0,p)$ where $p$ is a rational number.

Find

(i) the value of $a$,

(ii) the value of $b$,

(iii) the value of $c$,

(iv) the value of $p$.

Figure 2 shows the graph of $y=x-\dfrac1{2x^2}$ for $0.4\le x\le5$ drawn on a grid.

(a) (i) Express $x-\dfrac1{2x^2}$ as a single fraction.

(ii) Hence use the graph to obtain, to one significant figure, an estimate for the value of $\sqrt[3]{0.5}$. (3)

(b) By drawing a suitable straight line on the grid, find an estimate to 2 significant figures, for the root of the equation

in the interval $0.4\le x\le5$. (3)

The curve $C$ has equation

(a) Write down an equation of the asymptote to $C$ which is parallel to

(i) the $x$-axis, (ii) the $y$-axis. (2)

(b) Find the coordinates of the point where $C$ crosses

(i) the $x$-axis, (ii) the $y$-axis. (2)

(c) Sketch $C$, showing clearly the asymptotes and the coordinates of the points where $C$ crosses the coordinate axes. (3)

Figure 3 shows a sketch of the curve with equation

where $a$, $b$ and $c$ are integers.

The equations of the asymptotes to the curve are $x=-2$ and $y=3$.

The curve crosses the $y$-axis at $(0,3.5)$.

(a) Write down the value of $a$ and the value of $b$. (2)

(b) Find the value of $c$. (2)

Given that the curve crosses the $x$-axis at $(s,0)$

(c) find the value of $s$. (2)

(a) Complete the table of values for $y=x+\dfrac6{x^2}$. Give your answers to 2 decimal places where necessary.

(b) On the grid opposite, draw the graph of $y=x+\dfrac6{x^2}$ for $1\le x\le3$. (2)

(c) By drawing a suitable straight line on the grid, obtain estimates, to 1 decimal place, for the solutions of the equation $x^3-3x^2+3=0$ in the interval $1\le x\le3$. (4)