(a) The curve $C_1$ and $C_2$ have equations $y=\frac{x}{x^2+1}$ and $y=\sqrt{\frac{5}{4}-x^2}$ respectively.

(i) Sketch $C_1$ and $C_2$ on the same diagram, stating the exact coordinates of any points of intersection with the axes and stationary points, and the equation(s) of any asymptote(s).

[4]

(ii) State the coordinates of the point of intersection of $C_1$ and $C_2$.

[1]

(iii) Hence solve the inequality $\frac{x}{x^2+1} \geq \sqrt{\frac{5}{4}-x^2}$.

[1]

(b) The diagram below shows a sketch of the graph of $y=\mathrm{f}(x)$. The graph meets the origin $(0,0)$, has a turning point at $(1,1)$ and the equation of the asymptote is $y=0$.

On separate diagrams, draw sketches of the graphs of

(i) $y=\mathrm{f}(|x|)$

[2]

(ii) $y=\mathrm{f}^{\prime}(x)$,

[2]

stating the coordinates of the turning point(s), point(s) of intersection with the $x$-axis and equation(s) of asymptote(s) when it is possible to do so.

(a)(i)

(ii) $\left(1, \frac{1}{2}\right)$

(iii) $1 \leq x \leq \frac{\sqrt{5}}{2}$

(b)(i)

(ii)

(b)(i)

(ii)