Computer Science and Computer Science \& Philosophy applicants should turn to page 20. Below is a sketch of the curve $S$ with equation $y ^ { 2 } - y = x ^ { 3 } - x$. The curve crosses the $x$-axis at the origin and at $( a , 0 )$ and at $( b , 0 )$ for some real numbers $a < 0$ and $b > 0$. The curve only exists for $\alpha \leqslant x \leqslant \beta$ and for $x \geqslant \gamma$. The three points with coordinates $( \alpha , \delta ) , ( \beta , \delta )$, and $( \gamma , \delta )$ are all on the curve. [Figure] (i) What are the values of $a$ and $b$ ? (ii) By completing the square, or otherwise, find the value of $\delta$. (iii) Explain why the curve is symmetric about the line $y = \delta$. (iv) Find a cubic equation in $x$ which has roots $\alpha , \beta , \gamma$. (Your expression for the cubic should not involve $\alpha , \beta$, or $\gamma$ ). Justify your answer. (v) By considering the factorization of this cubic, find the value of $\alpha + \beta + \gamma$. (vi) Let $C$ denote the circle which has the points $( \alpha , \delta )$ and $( \beta , \delta )$ as ends of a diameter. Write down the equation of $C$. Show that $C$ intersects $S$ at two other points and find their common $x$-co-ordinate in terms of $\gamma$. This page has been intentionally left blank
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\section*{3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.}
Computer Science and Computer Science \& Philosophy applicants should turn to page 20.
Below is a sketch of the curve $S$ with equation $y ^ { 2 } - y = x ^ { 3 } - x$. The curve crosses the $x$-axis at the origin and at $( a , 0 )$ and at $( b , 0 )$ for some real numbers $a < 0$ and $b > 0$. The curve only exists for $\alpha \leqslant x \leqslant \beta$ and for $x \geqslant \gamma$. The three points with coordinates $( \alpha , \delta ) , ( \beta , \delta )$, and $( \gamma , \delta )$ are all on the curve.\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-12_967_814_1014_587}\\
(i) What are the values of $a$ and $b$ ?\\
(ii) By completing the square, or otherwise, find the value of $\delta$.\\
(iii) Explain why the curve is symmetric about the line $y = \delta$.\\
(iv) Find a cubic equation in $x$ which has roots $\alpha , \beta , \gamma$. (Your expression for the cubic should not involve $\alpha , \beta$, or $\gamma$ ). Justify your answer.\\
(v) By considering the factorization of this cubic, find the value of $\alpha + \beta + \gamma$.\\
(vi) Let $C$ denote the circle which has the points $( \alpha , \delta )$ and $( \beta , \delta )$ as ends of a diameter. Write down the equation of $C$. Show that $C$ intersects $S$ at two other points and find their common $x$-co-ordinate in terms of $\gamma$.
This page has been intentionally left blank\\