OCR MEI C2 (Core Mathematics 2)

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

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 \(\mathrm { C } ( 16,0 )\).
2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis.

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

1. Sketch the curve.
2. Use calculus to find the equation of the tangent to the curve at A .
3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65dd0efe-5c99-4814-b741-16e368c3469e-3_641_791_240_714} \captionsetup{labelformat=empty} \caption{Fig. 10}
   \end{figure} A is the point with coordinates \(( 1,4 )\) on the curve \(y = 4 x ^ { 2 }\). B is the point with coordinates \(( 0,1 )\), as shown in Fig. 10.
4. The line through A and B intersects the curve again at the point C . Show that the coordinates of C are \(\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)\).
5. 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 \(y = - 2 x - \frac { 1 } { 4 }\).
6. The two tangents intersect at the point D . Find the \(y\)-coordinate of D .

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