OCR MEI C1 (Core Mathematics 1)

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

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

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

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

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

7 Find the set of values of \(k\) for which the graph of \(y = x ^ { 2 } + 2 k x + 5\) does not intersect the \(x\)-axis.

8 \begin{figure}[h] \end{figure} Fig. 10 is a sketch of quadrilateral ABCD with vertices \(\mathrm { A } ( 1,5 ) , \mathrm { B } ( - 1,1 ) , \mathrm { C } ( 3 , - 1 )\) and \(\mathrm { D } ( 11,5 )\).

1. Show that \(\mathrm { AB } = \mathrm { BC }\).
2. Show that the diagonals AC and BD are perpendicular.
3. Find the midpoint of AC . Show that BD bisects AC but AC does not bisect BD .

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