A curve is defined by the parametric equations $$x=e^t+e^{-t},\quad y=e^t-e^{-t}+d,$$ where \(t

Question

A curve is defined by the parametric equations $$x=e^t+e^{-t},\quad y=e^t-e^{-t}+d,$$ where $t\in\mathbb{R}$ and $d>0$. By considering $x^2$ and $(y-d)^2$, show that the cartesian equation of the curve is $x^2-(y-d)^2=4$ and state the restriction on the values of $x$.[3] A solid is made by rotating the region bounded by the curve, the $x$-axis, the $y$-axis and the line $y=10$, completely about the $y$-axis. Find the volume of the solid in terms of $d$.[3] The diagram below shows region $R$ bounded by parts of the curves with equations $x^2-(y-4)^2=4$ and $y=(x-9)^3$. Find the area of region $R$, leaving your answer correct to $3$ decimal places.[4]

Accepted Answer

(a)(i) $x^2-(y-d)^2=4,\ x\geq2$ (a)(ii) $\pi\left(10d^2-100d+\dfrac{1120}{3}\right)$ (b) $90.862$ square units.

2026 HCI Promo Q11

2026 HCI Promo Q11

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