OCR MEI C1 (Core Mathematics 1)

1 \begin{figure}[h] \end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).

1. Find the equation of the line through \(\mathbf { A }\) and \(\mathbf { B }\).
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.

2

1. Find the coordinates of the point where the line \(5 x + 2 y = 20\) intersects the \(x\)-axis.
2. Find the coordinates of the point of intersection of the lines \(5 x + 2 y = 20\) and \(y = 5 - x\).

3 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).

4 \begin{figure}[h] \end{figure} Fig. 10 shows a trapezium ABCD . The coordinates of its vertices are \(\mathrm { A } ( - 2 , - 1 ) , \mathrm { B } ( 6,3 ) , \mathrm { C } ( 3,5 )\) and \(\mathrm { D } ( - 1,3 )\).

1. Verify that the lines AB and DC are parallel.
2. Prove that the trapezium is not isosceles.
3. The diagonals of the trapezium meet at M . Find the exact coordinates of M .
4. Show that neither diagonal of the trapezium bisects the other.

5 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes. \begin{figure}[h] \end{figure} Fig. 11 shows the line joining the points \(\mathrm { A } ( 0,3 )\) and \(\mathrm { B } ( 6,1 )\).

1. Find the equation of the line perpendicular to AB that passes through the origin, O .
2. Find the coordinates of the point where this perpendicular meets AB .
3. Show that the perpendicular distance of AB from the origin is \(\frac { 9 \sqrt { 10 } } { 10 }\).
4. Find the length of AB , expressing your answer in the form \(a \sqrt { 10 }\).
5. Find the area of triangle OAB .