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Question 12 A particle $P$ travels in a straight line so that, $t$ seconds after passing through a fixed point $O$, its velocity, $v \mathrm{~ms}^{-1}$, is given by $v=\dfrac{t}{2 \mathrm{e}}$ for $0 \leqslant t \leqslant 2$, $v=\mathrm{e}^{-\dfrac{t}{2}} \quad$ for $t>2$. Given that, after leaving $O$, particle $P$ is never at rest, find the distance it travels between $t=1$ and $t=3 .$
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Question $\begin{array}{lll}\mathbf{11} & \text { (a) (i) Find } & \int\dfrac{1}{(10 x-1)^{6}} \mathrm{~d} x \text { . }\end{array}$ (ii) Find $\int \dfrac{\left(2 x^{3}+5\right)^{2}}{x} \mathrm{~d} x$. (b) (i) Differentiate $y=\tan (3 x+1)$ with respect to $x$. (ii) Hence find $\int_{\dfrac{\pi}{12}}^{\dfrac{\pi}{10}}\left(\dfrac{\sec ^{2}(3 x+1)}{2}-\sin x\right) \mathrm{d} x$.
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Question 12 A particle $P$ travels in a straight line so that, $t$ seconds after passing through a fixed point $O$, its velocity, $v \mathrm{~ms}^{-1}$, is given by $v=\dfrac{t}{2 \mathrm{e}}$ for $0 \leqslant t \leqslant 2$, $v=\mathrm{e}^{-\dfrac{t}{2}} \quad$ for $t>2$. Given that, after leaving $O$, particle $P$ is never at rest, find the distance it travels between $t=1$ and $t=3 .$
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Question $\begin{array}{lll}\mathbf{11} & \text { (a) (i) Find } & \int\dfrac{1}{(10 x-1)^{6}} \mathrm{~d} x \text { . }\end{array}$ (ii) Find $\int \dfrac{\left(2 x^{3}+5\right)^{2}}{x} \mathrm{~d} x$. (b) (i) Differentiate $y=\tan (3 x+1)$ with respect to $x$. (ii) Hence find $\int_{\dfrac{\pi}{12}}^{\dfrac{\pi}{10}}\left(\dfrac{\sec ^{2}(3 x+1)}{2}-\sin x\right) \mathrm{d} x$.
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