Graphs (Edexcel IGCSE Further Pure Maths): Exam Questions

The curve $C$ with equation

$y=\frac{ax-5}{x-b}$

where $a$ and $b$ are integers, crosses the $x$-axis at the point $(2.5,0)$.

The asymptote to $C$ which is parallel to the $y$-axis has equation $x=1$

(i) Show that $a=2$

(ii) Find the value of $b$.

Find the coordinates of the point where $C$ crosses the $y$-axis.

Find the equation of the asymptote to $C$ which is parallel to the $x$-axis.

Using the axes below, sketch the curve $C$ showing clearly the asymptotes and the coordinates of the points where $C$ crosses the coordinate axes.

Figure 1 shows a sketch of part of the curve $C$ with equation

$y=\frac{x^2}{4}-3\sqrt{x}+8$

The point $P$ lies on $C$ and has coordinates $(4,a)$

Show that $a=6$

The line $L$is the normal to $C$ at the point $P$

Show that an equation of $L$ is $5y+4x-46=0$

The finite region $R$ is bounded by the curve $C$, the line $L$, the $x$-axis and the line with equation $x=1$

Use calculus to find the exact area of $R$

The curve $C$ has equation $y=\frac{ax-5}{b-x}$ where $a$ and $b$ are integers and $x\ne b$

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 $x=3$

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

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 $C$

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

$m<k<n$

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 $-4<x<0$

Complete the table of values for $y={\log }_{10}\left(6x-1\right)-x$ giving your answers to 2 decimal places.

On the grid opposite, draw the graph of $y={\log }_{10}\left(6x-1\right)-x$ for $0.25⩽x⩽3$

By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

${10}^{\frac{3x-4}{2}}=6x-1$ in the interval $0.25⩽x⩽3$

A curve $C$ has equation

$y=\frac{5x-2}{3x+2}x\ne \frac{2}{3}$

Find the coordinates of the point where $C$ intersects the

(i) $x$-axis

[1]

(ii) $y$-axis

[1]

A curve $C$ has equation

$y=\frac{5x-2}{3x+2}x\ne \frac{2}{3}$

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

(i) parallel to the $x$-axis

[1]

(ii) parallel to the $y$-axis

[1]

Sketch $C$ on the opposite page.
Show and label the asymptotes and the coordinates of the points where $C$ crosses the coordinate axes.

Point $A$ lies on $C$ such that the gradient of $C$ at $A$is parallel to the line with equation $4y-x-7$

The normal to $C$ at $A$intersects the $x$-axis at point $D$and the $y$-axis at point $E$
Given that the $x$ coordinate of $A$ is positive,

find, in its simplified form, the exact length of line $DE$

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 $0.8<x<7$

The curve $C$ with equation $y=\frac{6-3x}{x-4}$ where $x$≠ 4 , crosses the $x$-axis at the point $P$ and the $y$-axis at the point $Q$

Find the coordinates of

(i) $P$

(ii) $Q$

The curve $C$ with equation $y=\frac{6-3x}{x-4}$ where $x$≠ 4 , crosses the $x$-axis at the point $P$ and the $y$-axis at the point $Q$

Write down an equation of the asymptote to $C$ which is

(i) parallel to the $y$-axis

(ii) parallel to the $x$-axis

The curve $C$ with equation $y=\frac{6-3x}{x-4}$ where $x$≠ 4 , crosses the $x$-axis at the point $P$ and the $y$-axis at the point $Q$

Sketch $C$ showing clearly the asymptotes and the coordinates of the points $P$ and $Q$

The line $L$ is the normal to $C$ at the point on $C$ where $x=2$

Find an equation of $L$

The line $L$ intersects $C$ again at the point $R$

Find the $x$ coordinate of $R$

Curve $C$ has equation $y=\frac{ax+3}{1-2x}$ where $x\ne \frac{1}{2}$and $a$ is a constant.

The asymptote to $C$ that is parallel to the $x$‑axis has equation $y=4$

Find the value of $a$

Write down the equation of the asymptote to $C$ that is parallel to the $y$‑axis.

Find the coordinates of the point where $C$ crosses

(i) the $x$‑axis,

(ii)  the $y$‑axis.

Using the axes below, sketch $C$, showing clearly the asymptotes and the coordinates of the points where $C$ crosses the coordinate axes.

Complete the table of values for

$y=0.5^{\left(\frac{x}{3}+1\right)}+2$

giving each value to 2 decimal places where appropriate.

On the grid opposite, draw the graph of $y=0.5^{\left(\frac{x}{3}+1\right)}+2$ for $-6⩽x⩽0$

By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation

${\log }_2(2x+2{)}^3+x+3=0$ in the interval $-6⩽x⩽0$