OCR MEI C2 (Core Mathematics 2)

\includegraphics{figure_1} Fig. 9 shows a sketch of the graph of \(y = x^3 - 10x^2 + 12x + 72\).

1. Write down \(\frac{dy}{dx}\). [2]
2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\). [4]
3. Show that the curve crosses the \(x\)-axis at \(x = -2\). Show also that the curve touches the \(x\)-axis at \(x = 6\). [3]
4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9. [4]

Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_2}

1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C (16, 0). [6]
2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]

The point A has \(x\)-coordinate 5 and lies on the curve \(y = x^2 - 4x + 3\).

1. Sketch the curve. [2]
2. Use calculus to find the equation of the tangent to the curve at A. [4]
3. Show that the equation of the normal to the curve at A is \(x + 6y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. [6]

\includegraphics{figure_3} A is the point with coordinates (1, 4) on the curve \(y = 4x^2\). B is the point with coordinates (0, 1), as shown in Fig. 10.

1. The line through A and B intersects the curve again at the point C. Show that the coordinates of C are \(\left(-\frac{1}{4}, \frac{1}{4}\right)\). [4]
2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = -2x - \frac{1}{4}\). [6]
3. The two tangents intersect at the point D. Find the \(y\)-coordinate of D. [2]

Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]