The point $\displaystyle A$ has coordinates $\displaystyle (3, 2)$, the point $\displaystyle B$ has coordinates $\displaystyle (8, 3)$ and the point $\displaystyle C$ has coordinates $\displaystyle (4, 7)$

(a) Show that $\displaystyle ABC$ is an isosceles triangle.

The midpoint of $\displaystyle BC$ is $\displaystyle M$

(b) Find an equation of the line that passes through $\displaystyle A$ and $\displaystyle M$

Give your answer in the form $\displaystyle y = mx + c$

The points $\displaystyle A$, $\displaystyle C$ and $\displaystyle D$ are collinear such that $\displaystyle AD = kAC$ ($\displaystyle k > 1$)

Given that $\displaystyle \angle ABD = 90^\circ$

(c) find the coordinates of $\displaystyle D$

Show your working clearly.

The equation of the line $\displaystyle L_1$ is $\displaystyle y - 3x + a = 0$

The point $\displaystyle A$ with coordinates $\displaystyle (a, 10)$ lies on $\displaystyle L_1$

(a) Show that $\displaystyle a = 5$

Line $\displaystyle L_1$ crosses the $\displaystyle y$-axis at the point $\displaystyle B$

(b) Write down the coordinates of $\displaystyle B$

The point $\displaystyle C$ with coordinates $\displaystyle (50, -5)$ lies on $\displaystyle L_2$

Given that $\displaystyle L_2$ passes through $\displaystyle A$

(c) (i) show that $\displaystyle L_1$ and $\displaystyle L_2$ are perpendicular

(ii) hence find an equation for $\displaystyle L_2$ giving your answer in the form $\displaystyle y = px + q$

The point $\displaystyle D$ has coordinates $\displaystyle (m, n)$

The length of $\displaystyle CD$ is $\displaystyle 15\sqrt{10}$ and the gradient of $\displaystyle BD$ is $\displaystyle -3$

(d) Find the value of $\displaystyle m$ and the value of $\displaystyle n$

(e) Find the area of quadrilateral $\displaystyle ABCD$

Given that $\displaystyle f(x) = (1+x^2)^4$

(a) show that $\displaystyle f'(x) = 8x(1+x^2)^3$

The curve $\displaystyle C$ has equation $\displaystyle y = \frac{\mathrm{sin} 3x}{(1+x^2)^4}$

The point $\displaystyle A$ on $\displaystyle C$ has $\displaystyle x$ coordinate $\displaystyle \frac{2\pi}{3}$

(b) Show that the gradient of the normal to $\displaystyle C$ at $\displaystyle A$ is

A curve $\displaystyle C$ has equation

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

(i) parallel to the $\displaystyle y$-axis

(ii) parallel to the $\displaystyle x$-axis

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

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

$\displaystyle C$ passes through the point $\displaystyle A$ with coordinates $\displaystyle \left(\frac{1}{2}, \frac{13}{8}\right)$

(d) Show that the gradient of $\displaystyle C$ at $\displaystyle A$ is $\displaystyle -\frac{17}{16}$

$\displaystyle C$ also passes through the point $\displaystyle P$

Given that the tangent to $\displaystyle C$ at $\displaystyle P$ is parallel to the tangent to $\displaystyle C$ at $\displaystyle A$

(e) find an equation of the tangent to $\displaystyle C$ at $\displaystyle P$

Give your answer in the form $\displaystyle ax + by + c = 0$ where $\displaystyle a, b$ and $\displaystyle c$ are integers.

(a) Use algebra to find the $\displaystyle x$ coordinates of the points where the curve with equation $\displaystyle y = 3x^2 + 9x - 17$ intersects the line with equation $\displaystyle y = 3 - 2x$

(b) Hence, or otherwise, find the set of values of $\displaystyle x$ for which $\displaystyle 3 - 2x \leq 3x^2 + 9x - 17$

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

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

(i) the $\displaystyle y$-axis

(ii) the $\displaystyle x$-axis

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

(i) the $\displaystyle y$-axis

(ii) the $\displaystyle x$-axis

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

The point $\displaystyle P$ has coordinates $\displaystyle (-7, 3)$ and the point $\displaystyle Q$ has coordinates $\displaystyle (3, 18)$

The point $\displaystyle R$ divides the line $\displaystyle PQ$ in the ratio $\displaystyle 2:3$

The line $\displaystyle l$ passes through $\displaystyle R$ and is perpendicular to the line $\displaystyle PQ$

(a) Find an equation of line $\displaystyle l$

Give your answer in the form $\displaystyle ay + bx + c = 0$ where $\displaystyle a, b$ and $\displaystyle c$ are integers.

The line $\displaystyle l$ crosses the $\displaystyle x$-axis at the point $\displaystyle S$

(b) Find the area of triangle $\displaystyle QRS$

The curve $\displaystyle C$ has equation

The asymptote to $\displaystyle C$ that is parallel to the $\displaystyle y$-axis has equation $\displaystyle x = -\frac{4}{3}$

$\displaystyle C$ crosses the $\displaystyle y$-axis at the point with coordinates $\displaystyle \left(0, -\frac{1}{4}\right)$

(a) (i) Show that $\displaystyle b = 8$

(ii) Find the value of $\displaystyle a$

(b) Write down the equation of the asymptote to $\displaystyle C$ that is parallel to the $\displaystyle x$-axis.

(c) Find the coordinates of the point where $\displaystyle C$ crosses the $\displaystyle x$-axis.

(d) Using calculus, show that at every point on $\displaystyle C$, the gradient is positive.

(e) Using the axes on the next page, sketch $\displaystyle C$

Clearly label the asymptotes and the coordinates where $\displaystyle C$ crosses the coordinate axes.

The gradient of $\displaystyle C$ is $\displaystyle \frac{13}{100}$ at the point $\displaystyle E$ and at the point $\displaystyle F$

(f) Find the length of the line $\displaystyle EF$, giving your answer to 2 decimal places.

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

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

(a) Find the value of $\displaystyle p$ and the value of $\displaystyle q$

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

(b) Show that an equation of $\displaystyle k$ is $\displaystyle 2y + x - 17 = 0$

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

(c) Find the exact length of $\displaystyle CD$

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

area of triangle $\displaystyle DXC = 80$ units2

Given that $\displaystyle m > 0$

(d) find the value of $\displaystyle m$ and the value of $\displaystyle n$

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

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

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

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

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

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

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

The point $\displaystyle A$ has coordinates $\displaystyle (-4, 3)$ and the point $\displaystyle B$ has coordinates $\displaystyle (6, 8)$
The points $\displaystyle A$ and $\displaystyle B$ lie on the line $\displaystyle k$

(a) Find an equation of $\displaystyle k$

The point $\displaystyle C$, on $\displaystyle k$, is such that $\displaystyle AC : CB = 4:1$

(b) Find the coordinates of point $\displaystyle C$

The point $\displaystyle D$ with coordinates $\displaystyle (p, q)$, where $\displaystyle p < 0$, lies on the line $\displaystyle l$ through $\displaystyle C$ that is perpendicular to $\displaystyle k$

The length of $\displaystyle CD$ is $\displaystyle 8\sqrt{5}$

(c) Find the coordinates of $\displaystyle D$

(d) Find the area of triangle $\displaystyle ACD$

A curve $\displaystyle C$ has equation

(a) Find the coordinates of the point where $\displaystyle C$ intersects the

(i) $\displaystyle x$-axis

(ii) $\displaystyle y$-axis

(b) Write down an equation of the asymptote to $\displaystyle C$ that is

(i) parallel to the $\displaystyle x$-axis

(ii) parallel to the $\displaystyle y$-axis

(c) Sketch $\displaystyle C$ on the opposite page.

Show and label the asymptotes and the coordinates of the points where $\displaystyle C$ crosses the coordinate axes.

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

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

(d) find, in its simplified form, the exact length of line $\displaystyle DE$

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

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

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

(b) Find an equation of $\displaystyle l$

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

(c) Find the coordinates of $\displaystyle D$

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

(a) Find the value of $\displaystyle p$ and the value of $\displaystyle q$

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

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

The points $\displaystyle A$ and $\displaystyle B$ have coordinates $\displaystyle (-6, 8)$ and $\displaystyle (12, 2)$ respectively.

(a) Find an equation of the straight line passing through $\displaystyle A$ and $\displaystyle B$ in the form $\displaystyle ax + by + c = 0$, where $\displaystyle a, b$ and $\displaystyle c$ are integers to be found.

(b) Find the exact length of $\displaystyle AB$

The point $\displaystyle X$ with coordinates $\displaystyle (m, n)$ lies on $\displaystyle AB$ such that $\displaystyle AX:XB = 1:2$

(c) Find the value of $\displaystyle m$ and the value of $\displaystyle n$

The line $\displaystyle L$ passes through the point $\displaystyle X$ and is perpendicular to $\displaystyle AB$

The point $\displaystyle C$ with coordinates $\displaystyle (p, q)$ lies on $\displaystyle L$ where $\displaystyle p > 0$ and $\displaystyle q > 0$

Given that $\displaystyle AB$ is a diameter of a circle and $\displaystyle C$ also lies on the circumference of the circle,

(d) find

(i) the exact value of $\displaystyle p$

(ii) the exact value of $\displaystyle q$

(e) Find the exact area of triangle $\displaystyle ABC$

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

(a) Find the coordinates of

(i) $\displaystyle P$

(ii) $\displaystyle Q$

(b) Write down an equation of the asymptote to $\displaystyle C$ which is

(i) parallel to the $\displaystyle y$-axis

(ii) parallel to the $\displaystyle x$-axis

(c) Sketch $\displaystyle C$ showing clearly the asymptotes and the coordinates of the points $\displaystyle P$ and $\displaystyle Q$

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

(d) Find an equation of $\displaystyle L$

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

(e) Find the $\displaystyle x$ coordinate of $\displaystyle R$

The points $\displaystyle A$ and $\displaystyle B$ have coordinates $\displaystyle (1, 5)$ and $\displaystyle (9, 9)$ respectively.

(a) Find an equation of line $\displaystyle AB$, giving your answer in the form $\displaystyle ax + by + c = 0$, where $\displaystyle a, b$ and $\displaystyle c$ are integers to be found.

The line $\displaystyle l$ is perpendicular to $\displaystyle AB$ and passes through the point $\displaystyle X$ which lies on $\displaystyle AB$ such that $\displaystyle AX : XB = 3:1$

(b) Show that an equation of $\displaystyle l$ is $\displaystyle y = -2x + 22$

The point $\displaystyle C$ has coordinates $\displaystyle (6, p)$

Given that $\displaystyle C$ lies on $\displaystyle l$

(c) find the value of $\displaystyle p$

$\displaystyle ABCD$ is a parallelogram where the $\displaystyle x$ coordinate of $\displaystyle D$ is negative.

(d) Find the coordinates of the point $\displaystyle D$

(e) Find the area of the parallelogram $\displaystyle ABCD$

A curve $\displaystyle C$ has equation $\displaystyle y = \frac{3 - 2x}{x + 6}$ where $\displaystyle x \neq -6$

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

(i) $\displaystyle x$-axis (ii) $\displaystyle y$-axis

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

(i) $\displaystyle x$-axis (ii) $\displaystyle y$-axis

(c) Using the axes opposite, sketch the graph of $\displaystyle C$, showing clearly its asymptotes and the coordinates of the points where $\displaystyle C$ crosses the coordinate axes.

(d) Show that the gradient of the tangent to $\displaystyle C$ is always negative.

A tangent to $\displaystyle C$ has equation $\displaystyle y = -\frac{3}{5}x + k$ where $\displaystyle k > 0$

(e) Find the value of $\displaystyle k$

An equation of the straight line $\displaystyle l$ is

The point $\displaystyle A$ on $\displaystyle l$ lies on the $\displaystyle y$-axis. The point $\displaystyle B$ on $\displaystyle l$ has coordinates $\displaystyle (10,b)$, where $\displaystyle b$ is an integer. The point $\displaystyle C$ divides $\displaystyle AB$ in the ratio $\displaystyle 2:3$.

The straight line $\displaystyle k$ passes through $\displaystyle C$ and is perpendicular to $\displaystyle l$.

(a) Show that an equation of $\displaystyle k$ is

The point $\displaystyle D$ with coordinates $\displaystyle (p,q)$, where $\displaystyle q$ is positive, is such that $\displaystyle AD$ is parallel to $\displaystyle k$ and the length of $\displaystyle AD$ is $\displaystyle 12\sqrt{10}$.

(b) Find the coordinates of $\displaystyle D$.

The point $\displaystyle E$ lies on $\displaystyle k$ such that $\displaystyle DE$ is parallel to the $\displaystyle y$-axis. The point $\displaystyle F$ lies on $\displaystyle l$ such that $\displaystyle DF$ is parallel to the $\displaystyle y$-axis.

(c) Find the exact area of triangle $\displaystyle ECF$.

The straight line $\displaystyle L_1$ passes through the point $\displaystyle A$ with coordinates $\displaystyle (4,7)$ and has gradient $\displaystyle m$, where $\displaystyle m<0$.

Another straight line $\displaystyle L_2$ is perpendicular to $\displaystyle L_1$ and passes through the point $\displaystyle B$ with coordinates $\displaystyle (4,k)$, where $\displaystyle k\neq7$.

The lines $\displaystyle L_1$ and $\displaystyle L_2$ intersect at the point $\displaystyle C$. Given that the $\displaystyle y$ coordinate of $\displaystyle C$ is $\displaystyle Y$,

(a) show that

Given that the triangle $\displaystyle ABC$ is isosceles,

(b) find the value of $\displaystyle m$.

A curve $\displaystyle C$ has equation

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

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

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

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

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

(d) Using the axes on the opposite page, sketch $\displaystyle C$. Show clearly and label with their equation any asymptotes and the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

The straight line $\displaystyle l$ is the normal to $\displaystyle C$ at the point $\displaystyle A$. The $\displaystyle x$ coordinate of $\displaystyle A$ is positive and the gradient of $\displaystyle l$ is $\displaystyle 17$. The line $\displaystyle l$ also intersects $\displaystyle C$ at the point $\displaystyle B$.

(e) Find the exact coordinates of $\displaystyle B$.

A curve $\displaystyle C$ has equation

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

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

(a) Show that $\displaystyle a=3$.

Hence

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

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

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

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

(i) the $\displaystyle x$-axis,

(ii) the $\displaystyle y$-axis.

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

The point $\displaystyle A$ has coordinates $\displaystyle (-4,-10)$ and the point $\displaystyle B$ has coordinates $\displaystyle (3,11)$. The line $\displaystyle l$ passes through $\displaystyle A$ and $\displaystyle B$.

(a) Find an equation of $\displaystyle l$.

The point $\displaystyle P$ lies on $\displaystyle l$ such that $\displaystyle AP:PB=3:4$.

(b) Find the coordinates of $\displaystyle P$.

The point $\displaystyle Q$ with coordinates $\displaystyle (m,n)$, where $\displaystyle m<0$, lies on the line through $\displaystyle P$ that is perpendicular to $\displaystyle l$. Given that the length of $\displaystyle PQ$ is $\displaystyle \sqrt{10}$,

(c) find the coordinates of $\displaystyle Q$.

The point $\displaystyle R$ has coordinates $\displaystyle (-11,-21)$.

(d) Show that

(i) $\displaystyle AB$ and $\displaystyle RQ$ are equal in length,

(ii) $\displaystyle AB$ and $\displaystyle RQ$ are parallel.

(e) Find the area of the quadrilateral $\displaystyle ABQR$.

Figure 2 shows the graph of

for $\displaystyle 1\leq x\leq4$, 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 $\displaystyle 1\leq x\leq4$.

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

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

The points $\displaystyle P$, $\displaystyle Q$, $\displaystyle R$ and $\displaystyle S$ have coordinates $\displaystyle (4,7)$, $\displaystyle (3,0)$, $\displaystyle (10,1)$ and $\displaystyle (11,8)$ respectively.

(a) Show, by calculation, that the lines $\displaystyle PR$ and $\displaystyle QS$ are perpendicular.

(b) Find the exact lengths of

(i) $\displaystyle PR$,

(ii) $\displaystyle QS$.

(c) Find the area of the quadrilateral $\displaystyle PQRS$.

A curve $\displaystyle C$ has equation

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

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

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

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

(d) Sketch $\displaystyle C$, showing clearly

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

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

(iii) the coordinates of the stationary points.

The point $\displaystyle A$ has coordinates $\displaystyle (-3,-6)$ and the point $\displaystyle B$ has coordinates $\displaystyle (5,-2)$.

The line $\displaystyle l$ passes through the point $\displaystyle A$ and the point $\displaystyle B$.

(a) Find an equation of $\displaystyle l$, giving your answer in the form

The point $\displaystyle P$ has coordinates $\displaystyle (k,-2)$. The line through $\displaystyle A$ and $\displaystyle P$ is perpendicular to $\displaystyle l$.

(b) Show that $\displaystyle k=-5$.

The point $\displaystyle Q$ has coordinates $\displaystyle (e,f)$. The line through $\displaystyle B$ and $\displaystyle Q$ is also perpendicular to $\displaystyle l$.

Given that the length of $\displaystyle PQ$ is $\displaystyle \sqrt{85}$ and that $\displaystyle f>0$,

(c) find the coordinates of $\displaystyle Q$.

(d) Calculate the area of quadrilateral $\displaystyle ABQP$.

The points $\displaystyle A$ and $\displaystyle B$ have coordinates $\displaystyle (-1,3)$ and $\displaystyle (5,6)$ respectively.

(a) Find an equation for the line $\displaystyle AB$.

The point $\displaystyle P$ divides $\displaystyle AB$ in the ratio $\displaystyle 2:1$.

(b) Show that the coordinates of $\displaystyle P$ are $\displaystyle (3,5)$.

The point $\displaystyle C$ with coordinates $\displaystyle (m,n)$, where $\displaystyle m>0$, is such that $\displaystyle CP$ is perpendicular to the line $\displaystyle AB$.

Given that the radius of the circle which passes through $\displaystyle A$, $\displaystyle P$ and $\displaystyle C$ is 5,

(c) find the value of $\displaystyle m$ and the value of $\displaystyle n$.

The point $\displaystyle D$ with coordinates $\displaystyle (p,q)$ is such that the line $\displaystyle AD$ is perpendicular to the line $\displaystyle AB$ and the line $\displaystyle DC$ is parallel to the line $\displaystyle AB$.

(d) Find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(e) Find the area of trapezium $\displaystyle ABCD$.

The point $\displaystyle A$ has coordinates $\displaystyle (2,6)$, the point $\displaystyle B$ has coordinates $\displaystyle (6,8)$ and the point $\displaystyle C$ has coordinates $\displaystyle (4,2)$.

(a) Find the exact length of

(i) $\displaystyle AB$,

(ii) $\displaystyle BC$,

(iii) $\displaystyle AC$.

(b) Find the size of each angle of triangle $\displaystyle ABC$ in degrees.

The points $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$ lie on a circle with centre $\displaystyle P$.

(c) Find the coordinates of $\displaystyle P$.

(d) Find the exact length of the radius of the circle in the form $\displaystyle \sqrt{a}$, where $\displaystyle a$ is an integer.