10 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-3_680_800_1146_628}
\captionsetup{labelformat=empty}
\caption{Fig. 10}
\end{figure}
- Write down the coordinates of B .
- Find the radius of the circle and hence write down the equation of the circle.
- AD is a diameter of the circle. Find the coordinates of D .
- Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-4_643_853_269_589}
\captionsetup{labelformat=empty}
\caption{Fig. 11}
\end{figure}
Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\). - Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
- Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
- The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
- Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-5_775_1461_317_296}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{figure}
Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\). - Use the intersections with both axes to express the equation of the curve in a factorised form.
- Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
- Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
- Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation
$$2 x ^ { 2 } + 7 x - 20 = 0 .$$
Hence find the exact values of the \(x\)-coordinates of the other points of intersection.
\section*{END OF QUESTION PAPER}