Computer Science and Computer Science \& Philosophy applicants should turn to page 14. For each positive integer $k$, let $f _ { k } ( x ) = x ^ { 1 / k }$ for $x \geqslant 0$. (i) On the same axes (provided below), labelling each curve clearly, sketch $y = f _ { k } ( x )$ for $k = 1,2,3$, indicating the intersection points. (ii) Between the two points of intersection in (i), the curves $y = f _ { k } ( x )$ enclose several regions. What is the area of the region between $y = f _ { k } ( x )$ and $y = f _ { k + 1 } ( x )$ ? Verify that the area of the region between $y = f _ { 1 } ( x )$ and $y = f _ { 2 } ( x )$ is $\frac { 1 } { 6 }$. Let $c$ be a constant where $0 < c < 1$. (iii) Find the $x$-coordinates of the points of intersection of the line $y = c$ with $y = f _ { 1 } ( x )$ and of $y = c$ with $y = f _ { 2 } ( x )$. (iv) The constant $c$ is chosen so that the line $y = c$ divides the region between $y = f _ { 1 } ( x )$ and $y = f _ { 2 } ( x )$ into two regions of equal area. Show that $c$ satisfies the cubic equation $4 c ^ { 3 } - 6 c ^ { 2 } + 1 = 0$. Hence find $c$. [Figure]
(i) [3 marks]
\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 14.
For each positive integer $k$, let $f _ { k } ( x ) = x ^ { 1 / k }$ for $x \geqslant 0$.\\
(i) On the same axes (provided below), labelling each curve clearly, sketch $y = f _ { k } ( x )$ for $k = 1,2,3$, indicating the intersection points.\\
(ii) Between the two points of intersection in (i), the curves $y = f _ { k } ( x )$ enclose several regions. What is the area of the region between $y = f _ { k } ( x )$ and $y = f _ { k + 1 } ( x )$ ? Verify that the area of the region between $y = f _ { 1 } ( x )$ and $y = f _ { 2 } ( x )$ is $\frac { 1 } { 6 }$.
Let $c$ be a constant where $0 < c < 1$.\\
(iii) Find the $x$-coordinates of the points of intersection of the line $y = c$ with $y = f _ { 1 } ( x )$ and of $y = c$ with $y = f _ { 2 } ( x )$.\\
(iv) The constant $c$ is chosen so that the line $y = c$ divides the region between $y = f _ { 1 } ( x )$ and $y = f _ { 2 } ( x )$ into two regions of equal area. Show that $c$ satisfies the cubic equation $4 c ^ { 3 } - 6 c ^ { 2 } + 1 = 0$. Hence find $c$.\\
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