RI/2021/JC2/Prelim/P1/Q08

(i) The curve $G$ has equation $y=\frac{1}{1+x^2}$. Sketch the graph of $G$, stating the equation(s) of any asymptote(s) and the coordinates of any turning point(s).

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(ii) The line $l$ intersects $G$ at $x=0$ and is tangential to $G$ at the point $(c, d)$, where $c>0$. Find $c$ and $d$, and determine the equation of $l$.

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Let $R$ denote the region bounded by $G$, the $x$-axis and the lines $x=0$ and $x=1$.
(iii) By comparing the area of $R$ and the area of the trapezoidal region between $l$ and the $x$-axis for $0 \leq x \leq 1$, show that $\pi>3$.

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(iv) By considering the volume of revolution of a suitable region rotated through $2 \pi$ radians about the $y$-axis, show that $\ln 2>\frac{2}{3}$.

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