OCR MEI C1 (Core Mathematics 1)

A line \(L\) is parallel to \(y = 4x + 5\) and passes through the point \((-1, 6)\). Find the equation of the line \(L\) in the form \(y = ax + b\). Find also the coordinates of its intersections with the axes. [5]

Find the coordinates of the point of intersection of the lines \(y = 5x - 2\) and \(x + 3y = 8\). [4]

A is the point \((1, 5)\) and B is the point \((6, -1)\). M is the midpoint of AB. Determine whether the line with equation \(y = 2x - 5\) passes through M. [3]

Find the equation of the line which is perpendicular to the line \(y = 2x - 5\) and which passes through the point \((4, 1)\). Give your answer in the form \(y = ax + b\). [3]

Find the equation of the line with gradient \(-2\) which passes through the point \((3, 1)\). Give your answer in the form \(y = ax + b\). Find also the points of intersection of this line with the axes. [3]

\includegraphics{figure_8} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).

1. Show that AB = BC. [3]
2. Show that the diagonals AC and BD are perpendicular. [3]
3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]

Find the equation of the line which is perpendicular to the line \(y = 5x + 2\) and which passes through the point \((1, 6)\). Give your answer in the form \(y = ax + b\). [3]