Polar curve with exponential function

6 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { - \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
2. Find the exact value of the area of the region bounded by \(C\) and the initial line.
3. Show that, at the point on \(C\) furthest from the initial line, $$1 - e ^ { \theta - \frac { 1 } { 2 } \pi } - \tan \theta = 0$$ and verify that this equation has a root between 0.56 and 0.57 .

7 The diagram shows the curve \(C _ { 1 }\) with polar equation $$r = 1 + 6 \mathrm { e } ^ { - \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-4_300_513_1414_760}

1. Find, in terms of \(\pi\) and e , the area of the shaded region bounded by \(C _ { 1 }\) and the initial line.
2. The polar equation of a curve \(C _ { 2 }\) is $$r = \mathrm { e } ^ { \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Sketch the curve \(C _ { 2 }\) and state the polar coordinates of the end-points of this curve.
3. The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point \(P\). Find the polar coordinates of \(P\).

\includegraphics{figure_1} The curve \(C\) has polar equation $$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$ where \(0 \leq \theta \leq \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]

The equation of a curve, in polar coordinates, is $$r = e^{-2\theta}, \quad \text{for } 0 \leq \theta \leq \pi.$$

1. Sketch the curve, stating the polar coordinates of the point at which \(r\) takes its greatest value. [2]
2. The pole is \(O\) and points \(P\) and \(Q\), with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\) respectively, lie on the curve. Given that \(\theta_2 > \theta_1\), show that the area of the region enclosed by the curve and the lines \(OP\) and \(OQ\) can be expressed as \(k(r_1^2 - r_2^2)\), where \(k\) is a constant to be found. [5]

The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \((r, \theta)\) where \(0 \leq \theta \leq \frac{\pi}{2}\) The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole. \(TPQ\) is the tangent to the curve at \(P\). \includegraphics{figure_15}

1. Show that the gradient of \(TPQ\) is equal to $$\frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$ [4 marks]
2. The curve has polar equation $$r = e^{(\cot b)\theta}$$ where \(b\) is a constant such that \(0 < b < \frac{\pi}{2}\) Use the result of part (a) to show that the angle between the line \(OP\) and the tangent \(TPQ\) does not depend on \(\theta\). [7 marks]

In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin \theta} e^{\frac{1}{2}\cos \theta}\) for \(0 \leqslant \theta \leqslant \pi\). \includegraphics{figure_13}

1. Find the exact area enclosed by the curve. [4]
2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}} e^{\frac{1}{6}}\). [7]

In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin\theta}e^{\cos\theta}\) for \(0 \leq \theta < \pi\). \includegraphics{figure_5}

1. Find the exact area enclosed by the curve. [4]
2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}}e^{\frac{1}{2}}\). [7]

A curve has polar equation \(r = \frac{3}{10}e^{3\theta}\) for \(\theta \geq 0\). The length of the arc of this curve between \(\theta = 0\) and \(\theta = \alpha\) is denoted by \(L(\alpha)\).

1. Show that \(L(\alpha) = \frac{1}{3}(e^{3\alpha} - 1)\). [5]
2. The point \(P\) on the curve corresponding to \(\theta = \beta\) is such that \(L(\beta) = OP\), where \(O\) is the pole. Find the value of \(\beta\). [2]