Computer Science and Computer Science \& Philosophy applicants should turn to page 20. (i) Sketch $y = \left( x ^ { 2 } - 1 \right) ^ { n }$ for $n = 2$ and for $n = 3$ on the same axes, labelling any points that lie on both curves, or that lie on either the $x$-axis or the $y$-axis. (ii) Without calculating the integral explicitly, explain why there is no positive value of $a$ such that $\int _ { 0 } ^ { a } \left( x ^ { 2 } - 1 \right) ^ { n } \mathrm {~d} x = 0$ if $n$ is even. If $n > 0$ is odd we will write $n = 2 m - 1$ and define $a _ { m } > 0$ to be the positive real number that satisfies $$\int _ { 0 } ^ { a _ { m } } \left( x ^ { 2 } - 1 \right) ^ { 2 m - 1 } \mathrm {~d} x = 0$$ if such a number exists. (iii) Explain why such a number $a _ { m }$ exists for each whole number $m \geqslant 1$. (iv) Find $a _ { 1 }$. (v) Prove that $\sqrt { 2 } < a _ { 2 } < \sqrt { 3 }$. (vi) Without calculating further integrals, find the approximate value of $a _ { m }$ when $m$ is a very large positive whole number. You may use without proof the fact that $\int _ { 0 } ^ { \sqrt { 2 } } \left( x ^ { 2 } - 1 \right) ^ { 2 m - 1 } \mathrm {~d} x < 0$ for any sufficiently large whole number $m$. 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.\\
(i) Sketch $y = \left( x ^ { 2 } - 1 \right) ^ { n }$ for $n = 2$ and for $n = 3$ on the same axes, labelling any points that lie on both curves, or that lie on either the $x$-axis or the $y$-axis.\\
(ii) Without calculating the integral explicitly, explain why there is no positive value of $a$ such that $\int _ { 0 } ^ { a } \left( x ^ { 2 } - 1 \right) ^ { n } \mathrm {~d} x = 0$ if $n$ is even.
If $n > 0$ is odd we will write $n = 2 m - 1$ and define $a _ { m } > 0$ to be the positive real number that satisfies
$$\int _ { 0 } ^ { a _ { m } } \left( x ^ { 2 } - 1 \right) ^ { 2 m - 1 } \mathrm {~d} x = 0$$
if such a number exists.\\
(iii) Explain why such a number $a _ { m }$ exists for each whole number $m \geqslant 1$.\\
(iv) Find $a _ { 1 }$.\\
(v) Prove that $\sqrt { 2 } < a _ { 2 } < \sqrt { 3 }$.\\
(vi) Without calculating further integrals, find the approximate value of $a _ { m }$ when $m$ is a very large positive whole number. You may use without proof the fact that $\int _ { 0 } ^ { \sqrt { 2 } } \left( x ^ { 2 } - 1 \right) ^ { 2 m - 1 } \mathrm {~d} x < 0$ for any sufficiently large whole number $m$.
This page has been intentionally left blank\\