OCR MEI C1 (Core Mathematics 1)

Find the equation of the line passing through \((-1, -9)\) and \((3, 11)\). Give your answer in the form \(y = mx + c\). [3]

1. Find the points of intersection of the line \(2x + 3y = 12\) with the axes. [2]
2. Find also the gradient of this line. [2]

1. Express \(x^2 - 6x + 2\) in the form \((x - a)^2 - b\). [3]
2. State the coordinates of the turning point on the graph of \(y = x^2 - 6x + 2\). [2]
3. Sketch the graph of \(y = x^2 - 6x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis. [2]
4. Solve the simultaneous equations \(y = x^2 - 6x + 2\) and \(y = 2x - 14\). Hence show that the line \(y = 2x - 14\) is a tangent to the curve \(y = x^2 - 6x + 2\). [5]

Find, algebraically, the coordinates of the point of intersection of the lines \(y = 2x - 5\) and \(6x + 2y = 7\). [4]

1. Find the gradient of the line \(4x + 5y = 24\). [2]
2. A line parallel to \(4x + 5y = 24\) passes through the point \((0, 12)\). Find the coordinates of its point of intersection with the \(x\)-axis. [3]

1. \includegraphics{figure_1} Fig. 10 shows a sketch of the graph of \(y = \frac{1}{x}\). Sketch the graph of \(y = \frac{1}{x-2}\), showing clearly the coordinates of any points where it crosses the axes. [3]
2. Find the value of \(x\) for which \(\frac{1}{x-2} = 5\). [2]
3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac{1}{x-2}\). Give your answers in the form \(a \pm \sqrt{b}\). Show the position of these points on your graph in part (i). [6]

Find, in the form \(y = ax + b\), the equation of the line through \((3, 10)\) which is parallel to \(y = 2x + 7\). [3]