A curve \(C\) has polar equation \(r^{2}=8 \operatorname{cosec} 2 \theta\) for \(0\lt \theta\lt \frac{1}{2} \pi\). Find a cartesian equation of \(C\).
Sketch \(C\).
Determine the exact area of the sector bounded by the arc of \(C\) between \(\theta=\frac{1}{6} \pi\) and \(\theta=\frac{1}{3} \pi\), the half-line \(\theta=\frac{1}{6} \pi\) and the half-line \(\theta=\frac{1}{3} \pi\).
[It is given that \(\int \operatorname{cosec} x \mathrm{~d} x=\ln \left|\tan \frac{1}{2} x\right|+c\).]
OR
The polar equation of a curve \(C\) is \(r=a(1+\cos\theta)\), for \(0\leq\theta\lt 2\pi\), where \(a\) is a positive constant.
(i) Sketch \(C\).
(ii) Show that the Cartesian equation of \(C\) is
\(x^2+y^2=a\left(x+\sqrt{x^2+y^2}\right).\)
(iii) Find the area of the sector of \(C\) between \(\theta=0\) and \(\theta=\frac{\pi}{3}\).
(iv) Find the arc length of \(C\) between \(\theta=0\) and \(\theta=\frac{\pi}{3}\).
A circle has polar equation \(r=a\), for \(0 \leqslant \theta\lt 2 \pi\), and a cardioid has polar equation \(r=a(1-\cos \theta)\), for \(0 \leqslant \theta\lt 2 \pi\), where \(a\) is a positive constant. Draw sketches of the circle and the cardioid on the same diagram.
Write down the polar coordinates of the points of intersection of the circle and the cardioid.
Show that the area of the region that is both inside the circle and inside the cardioid is
\(\left(\frac{5}{4} \pi-2\right) a^{2} .\)
A circle has polar equation \(r=a\), for \(0 \leqslant \theta\lt 2 \pi\), and a cardioid has polar equation \(r=a(1-\cos \theta)\), for \(0 \leqslant \theta\lt 2 \pi\), where \(a\) is a positive constant. Draw sketches of the circle and the cardioid on the same diagram.
Write down the polar coordinates of the points of intersection of the circle and the cardioid.
Show that the area of the region that is both inside the circle and inside the cardioid is
\(\left(\frac{5}{4} \pi-2\right) a^{2} .\)
Find the area of the region enclosed by the curve with polar equation \(r=2(1+\cos \theta)\), for \(0 \leqslant \theta\lt 2 \pi\).
Use the identity \(2 \sin P \cos Q \equiv \sin (P+Q)+\sin (P-Q)\) to show that
\(2 \sin \theta \cos \left(\theta-\frac{1}{4} \pi\right) \equiv \cos \left(2 \theta-\frac{3}{4} \pi\right)+\frac{1}{\sqrt{2}}\)
A curve has polar equation \(r=2 \sin \theta \cos \left(\theta-\frac{1}{4} \pi\right)\), for \(0 \leqslant \theta \leqslant \frac{3}{4} \pi\). Sketch the curve and state the polar equation of its line of symmetry, justifying your answer.
Show that the area of the region enclosed by the curve is \(\frac{3}{8}(\pi+1)\).
[Question 11 is printed on the next page.]
The curve \(C\) has polar equation \(r=2 \mathrm{e}^{\theta}\), for \(\frac{1}{6} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\). Find
(i) the area of the region bounded by the half-lines \(\theta=\frac{1}{6} \pi, \theta=\frac{1}{2} \pi\) and \(C\),
(ii) the length of \(C\).
The curve \(C\) has polar equation \(r=2 \mathrm{e}^{\theta}\), for \(\frac{1}{6} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\). Find
(i) the area of the region bounded by the half-lines \(\theta=\frac{1}{6} \pi, \theta=\frac{1}{2} \pi\) and \(C\),
(ii) the length of \(C\).
The curve \(C\) has polar equation \(r=2 \sin \theta(1-\cos \theta)\), for \(0 \leqslant \theta \leqslant \pi\). Find \(\frac{\mathrm{d} r}{\mathrm{~d} \theta}\) and hence find the polar coordinates of the point of \(C\) that is furthest from the pole.
Sketch \(C\).
Find the exact area of the sector from \(\theta=0\) to \(\theta=\frac{1}{4} \pi\).
[Question 11 is printed on the next page.]
The curves \(C_{1}\) and \(C_{2}\) have polar equations
\(r=\theta+2 \quad \text { and } \quad r=\theta^{2}\)
respectively, where \(0 \leqslant \theta \leqslant \pi\).
(i) Find the polar coordinates of the point of intersection of \(C_{1}\) and \(C_{2}\).
(ii) Sketch \(C_{1}\) and \(C_{2}\) on the same diagram.
(iii) Find the area bounded by \(C_{1}, C_{2}\) and the line \(\theta=0\).
The curve \(C\) has polar equation
\(r=\left(\frac{1}{2} \pi-\theta\right)^{2},\)
where \(0 \leqslant \theta \leqslant \frac{1}{2} \pi\). Draw a sketch of \(C\).
Find the area of the region bounded by \(C\) and the initial line, leaving your answer in terms of \(\pi\).
The curve \(C\) has polar equation \(r=2 \cos 2 \theta\). Sketch the curve for \(0 \leqslant \theta\lt 2 \pi\).
Find the exact area of one loop of the curve.
The curve \(C\) has polar equation \(r=a(1+\sin \theta)\), where \(a\) is a positive constant and \(0 \leqslant \theta\lt 2 \pi\). Draw a sketch of \(C\).
Find the exact value of the area of the region enclosed by \(C\) and the half-lines \(\theta=\frac{1}{3} \pi\) and \(\theta=\frac{2}{3} \pi\).
The curve \(C\) has polar equation \(r=1+2 \cos \theta\). Sketch the curve for \(-\frac{2}{3} \pi \leqslant \theta\lt \frac{2}{3} \pi\).
Find the area bounded by \(C\) and the half-lines \(\theta=-\frac{1}{3} \pi, \theta=\frac{1}{3} \pi\).
The curve \(C\) has polar equation
\(r=a\left(1-\mathrm{e}^{-\theta}\right),\)
where \(a\) is a positive constant and \(0 \leqslant \theta\lt 2 \pi\).
(i) Draw a sketch of \(C\).
(ii) Show that the area of the region bounded by \(C\) and the lines \(\theta=\ln 2\) and \(\theta=\ln 4\) is
\(\frac{1}{2} a^{2}\left(\ln 2-\frac{13}{32}\right) .\)
The curves \(C_{1}\) and \(C_{2}\) have polar equations
\(\begin{array}{ll} C_{1}: & r=a \\ C_{2}: & r=2 a \cos 2 \theta, \text { for } 0 \leqslant \theta \leqslant \frac{1}{4} \pi \end{array}\)
where \(a\) is a positive constant. Sketch \(C_{1}\) and \(C_{2}\) on the same diagram.
The curves \(C_{1}\) and \(C_{2}\) intersect at the point with polar coordinates \((a, \beta)\). State the value of \(\beta\).
Show that the area of the region bounded by the initial line, the arc of \(C_{1}\) from \(\theta=0\) to \(\theta=\beta\), and the \(\operatorname{arc}\) of \(C_{2}\) from \(\theta=\beta\) to \(\theta=\frac{1}{4} \pi\) is
\(a^{2}\left(\frac{1}{6} \pi-\frac{1}{8} \sqrt{ } 3\right) .\)
The curve \(C\) has polar equation \(r=3+2 \cos \theta\), for \(-\pi\lt \theta \leqslant \pi\). The straight line \(l\) has polar equation \(r \cos \theta=2\). Sketch both \(C\) and \(l\) on a single diagram.
Find the polar coordinates of the points of intersection of \(C\) and \(l\).
The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\).
[Question 11 is printed on the next page.]
The curve \(C\) has polar equation \(r=1+\sin \theta\) for \(-\frac{1}{2} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\). Draw a sketch of \(C\).
The area of the region enclosed by the initial line, the half-line \(\theta=\frac{1}{2} \pi\), and the part of \(C\) for which \(\theta\) is positive, is denoted by \(A_{1}\). The area of the region enclosed by the initial line, and the part of \(C\) for which \(\theta\) is negative, is denoted by \(A_{2}\). Find the ratio \(A_{1}: A_{2}\), giving your answer correct to 1 decimal place.
Draw a sketch of the curve \(C\) whose polar equation is \(r=\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2} \pi\).
On the same diagram draw the line \(\theta=\alpha\), where \(0\lt \alpha\lt \frac{1}{2} \pi\).
The region bounded by \(C\) and the line \(\theta=\frac{1}{2} \pi\) is denoted by \(R\). Find the exact value of \(\alpha\) for which the line \(\theta=\alpha\) divides \(R\) into two regions of equal area.
The curve \(C\) has polar equation
\(r=\frac{a}{1+\theta},\)
where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac{1}{2} \pi\).
(i) Show that \(r\) decreases as \(\theta\) increases.
(ii) The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\),
\(\tan \theta=1+\theta,\)
and verify that this equation has a root between 1.1 and 1.2.
(iii) Draw a sketch of \(C\).
(iv) Find the area of the region bounded by the initial line, the line \(\theta=\frac{1}{2} \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).