Coordinate Geometry (Edexcel IGCSE Further Pure Maths): Exam Questions

Figure 1 shows the curve $M$ with equation $y=x^3–13x–12$

The point $P$, with $x$ coordinate −2, lies on $M$ and line$l_1$ is the tangent to $M$ at the point $P$.

Find an equation for $l_1$

The point $Q$ lies on $M$ and the line $l_2$ is the tangent to $M$ at the point $Q$.

Given that $l_1$ and $l_2$ are parallel,

find an equation for $l_2$

The normal to $M$ at $P$ meets $l_2$ at the point $R$.

Find the coordinates of $R$.

Find the exact length of the line $PR$.

The tangent and normal at $P$and the tangent and normal at $Q$ form a rectangle.

Find the exact area of this rectangle.

The line $l$ passes through the point $A$ with coordinates $(–2,2)$ and the point $B$with coordinates $(3,12)$

The point $C$ with coordinates $(p,q)$ lies on $l$ such that $AC:CB=3:2$

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

The line $k$ is perpendicular to $l$and passes through the point $C$

Show that an equation of $k$ is $2y+x-17=0$

The line $k$ crosses the $x$-axis at the point $D$

Find the exact length of $CD$

The point $X$ with coordinates $(m,n)$ lies on $l$such that

area of triangle $DXC$= 80 units²

Given that $m>0$

find the value of $m$ and the value of $n$

On the grid below, draw the line with equation

(i) $3x+4y=24$

(ii) $2x-5y+10=0$

Show, by shading on the grid, the region $R$ defined by the inequalities

$3x+4y⩽242x-5y+10⩾0y⩽5x⩾-1$

Label the region $R$

The curve $S$ with equation $y=\frac{x^2}{4}+2$ where $x⩾0$ and the line $l$ with equation $2y-x-4=0$ where $x⩾0$ intersect at the points $A$and $B$, as shown in Figure 2.

(i) Show that the coordinates of point $A$ are (0, 2)

(ii) Find the coordinates of the point $B$

The finite region bounded by $S$ and $l$, shown shaded in Figure 2, is rotated through $2\pi$ radians about the $y$-axis.

Use algebraic integration to find the volume of the solid generated.
Give your answer in terms of $\pi$

On the grid opposite draw the line with equation

(i) $y=2x+5$

(ii) $4y=x-8$

(iii) $5y+3x=30$

Show, by shading, the region $R$defined by the inequalities

$y⩽2x+54y⩾x-85y+3x⩽30$

For all points in $R$ with coordinates $\left(x,y\right)$

$P=2x-5y$

Using your graph, find the least value of $P$

Expand ${\left(1+\frac{x}{3}\right)}^{-3}$ in ascending powers of $x$ up to and including the term in $x^3$

Where appropriate express each coefficient as an exact fraction in its lowest terms.

Write down the range of values of $x$ for which your expression is valid.

Express ${\left(3+x\right)}^{-3}$ in the form $P{\left(1+Qx\right)}^{-3}$ where $P$ and $Q$ are rational numbers whose values should be stated.

$\mathrm{f}\left(x\right)=\frac{1+4x}{(3+x{)}^3}$

Obtain a series expansion for $\mathrm{f}\left(x\right)$ in ascending powers of $x$ up to and including the term in $x^2$

$\mathrm{f}\left(x\right)=\frac{1+4x}{(3+x{)}^3}$

Hence, using algebraic integration, obtain an estimate of ${\int }_0^{0.2}\mathrm{f}\left(x\right) \mathrm{d}x$

Give your answer to 5 significant figures.

The point $A$ with coordinates $\left(12,14\right)$ and the point $B$ with coordinates $\left(q,2\right)$ where $q$ is a constant, lie on the straight line with equation $3y-2x-p=0$ where $p$ is a constant.

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

The line $L$ is perpendicular to $AB$ and passes through the point $X$, which lies on $AB$ such that $AX:XB=1:2$

Find an equation for $L$ in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers to be found.

On the axes opposite, draw the line with equation

(i) $y=-x-1$

(ii) $y-3x+8=0$

(iii) $2y=x+8$

Show, by shading on your graph, the region $R$ defined by the inequalities

$y⩾-x-1$ and $y⩾3x-8$ and $2y⩽x+8$

For all points in $R$, with coordinates $\left(x,y\right)$

$P=2y-3x$

Find

(i) the greatest value of $P$

(ii) the least value of $P$

The point $A$ has coordinates (–5, 3), the point $B$ has coordinates (4, 0) and the point $C$ has coordinates (–1, 5).

The line $l$ passes through $C$ and is perpendicular to $AB$.

Find an equation of $l$ .
Give your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

The line $l$ intersects $AB$ at the point $D$.

Show that the coordinates of $D$ are (-2, 2).

The point $A$ has coordinates (–5, 3), the point $B$ has coordinates (4, 0) and the point $C$ has coordinates (–1, 5).

Show that $l$ is not the perpendicular bisector of $AB$.

Find the value of $\tan \angle ABC$.
Give your answer in its simplest form.