Past Exam Questions

The curve C has polar equation \(r = \frac{1}{\pi - \theta} - \frac{1}{\pi}\), where \(0 \leq \theta \leq \frac{1}{2}\pi\).

(a) Sketch C.

(b) Show that the area of the region bounded by the half-line \(\theta = \frac{1}{2}\pi\) and C is \(\frac{3 - 4 \ln 2}{4\pi}\).

The curve C has polar equation \(r = 2 \cos \theta (1 + \sin \theta)\), for \(0 \leq \theta \leq \frac{1}{2} \pi\).

1. Find the polar coordinates of the point on C that is furthest from the pole.
2. Sketch C.
3. Find the area of the region bounded by C and the initial line, giving your answer in exact form.

The curve C has polar equation \(r = 3 + 2 \sin \theta\), for \(-\pi < \theta \leq \pi\).

(a) The diagram shows part of C. Sketch the rest of C on the diagram.

The straight line l has polar equation \(r \sin \theta = 2\).

(b) Add l to the diagram in part (a) and find the polar coordinates of the points of intersection of C and l.

(c) The region R is enclosed by C and l, and contains the pole. Find the area of R, giving your answer in exact form.

7 (a) Show that the curve with Cartesian equation
\(\left(x^{2}+y^{2}\right)^{\frac{5}{2}}=4 x y\left(x^{2}-y^{2}\right)\)
has polar equation \(r=\sin 4 \theta\).

The curve \(C\) has polar equation \(r=\sin 4 \theta\), for \(0 \leqslant \theta \leqslant \frac{1}{4} \pi\).
(b) Sketch \(C\) and state the equation of the line of symmetry.

(c) Find the exact value of the area of the region enclosed by \(C\).

(d) Using the identity \(\sin 4 \theta \equiv 4 \sin \theta \cos ^{3} \theta-4 \sin ^{3} \theta \cos \theta\), find the maximum distance of \(C\) from the line \(\theta=\frac{1}{2} \pi\). Give your answer correct to 2 decimal places.

7 The curve \(C_{1}\) has polar equation \(r=\theta \cos \theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2} \pi\).
(a) The point on \(C_{1}\) furthest from the line \(\theta=\frac{1}{2} \pi\) is denoted by \(P\). Show that, at \(P\),
\(2 \theta \tan \theta-1=0\)
and verify that this equation has a root between 0.6 and 0.7 .

The curve \(C_{2}\) has polar equation \(r=\theta \sin \theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2} \pi\). The curves \(C_{1}\) and \(C_{2}\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
(b) Find the polar coordinates of \(Q\), giving your answers in exact form.

(c) Sketch \(C_{1}\) and \(C_{2}\) on the same diagram.

(d) Find, in terms of \(\pi\), the area of the region bounded by the \(\operatorname{arc} O Q\) of \(C_{1}\) and the \(\operatorname{arc} O Q\) of \(C_{2}\). [7]

11 Answer only one of the following two alternatives.

EITHER

The curve \(C_1\) has polar equation \(r^2=2\theta\), for \(0\leq \theta\leq \dfrac{\pi}{2}\).

(i) The point on \(C_1\) furthest from the line \(\theta=\dfrac{\pi}{2}\) is denoted by \(P\). Show that, at \(P\), \(2\theta\tan\theta=1\), and verify that this equation has a root between \(0.6\) and \(0.7\).

The curve \(C_2\) has polar equation \(r^2=\theta\sec^2\theta\), for \(0\leq\theta\leq\dfrac{\pi}{4}\). The curves \(C_1\) and \(C_2\) intersect at the pole, denoted by \(O\), and at another point \(Q\).

(ii) Find the exact value of \(\theta\) at \(Q\).

(iii) The diagram below shows the curve \(C_2\). Sketch \(C_1\) on this diagram.

(iv) Find, in exact form, the area of the region \(OPQ\) enclosed by \(C_1\) and \(C_2\).

2 The curve \(C\) has polar equation \(r^{2}=\ln (1+\theta)\), for \(0 \leqslant \theta \leqslant 2 \pi\).
(i) Sketch \(C\).

(ii) Using the substitution \(u=1+\theta\), or otherwise, find the area of the region bounded by \(C\) and the initial line, leaving your answer in an exact form.

The curve \(C\) has polar equation \(r=\cos 2\theta\), for \(-\frac{\pi}{4}\leq \theta\leq \frac{\pi}{4}\).

(i) Sketch \(C\).

(ii) Find the area of the region enclosed by \(C\), showing full working.

(iii) Find a Cartesian equation of \(C\).

The curves \(C_{1}\) and \(C_{2}\) have polar equations, for \(0 \leqslant \theta \leqslant \pi\), as follows:
\(\begin{array}{l}
C_{1}: r=a \\
C_{2}: r=2 a|\cos \theta|
\end{array}\)
where \(a\) is a positive constant. The curves intersect at the points \(P_{1}\) and \(P_{2}\).
(i) Find the polar coordinates of \(P_{1}\) and \(P_{2}\).

(ii) In a single diagram, sketch \(C_{1}, C_{2}\) and their line of symmetry.

(iii) The region \(R\) enclosed by \(C_{1}\) and \(C_{2}\) is bounded by the arcs \(O P_{1}, P_{1} P_{2}\) and \(P_{2} O\), where \(O\) is the pole. Find the area of \(R\), giving your answer in exact form.

The curve \(C\) has polar equation

\(r=5\sqrt{\cot\theta},\qquad 0.01\le \theta\le \frac12\pi.\)

(i) Find the area of the finite region bounded by \(C\) and the line \(\theta=0.01\), showing full working. Give your answer correct to 1 decimal place.

Let \(P\) be the point on \(C\) where \(\theta=0.01\).

(ii) Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.

(iii) Find the maximum distance of \(C\) from the initial line.

(iv) Sketch \(C\).

(a) The curve \(C\) has polar equation \(r=\sin3\theta\), for \(0\le\theta\le\frac13\pi\). Sketch \(C\) and state the equation of the line of symmetry.

(b) Find the exact value of the area of the region enclosed by \(C\).

(c) Using \(\sin3\theta=3\sin\theta-4\sin^3\theta\), find the maximum distance of a point on \(C\) from the initial line.

(d) Using \(\sin3\theta=3\sin\theta-4\sin^3\theta\), find a Cartesian equation for \(C\).

(a) The curve \(C\) has polar equation \(r^2=\tan 2\theta\), where \(0\le\theta\le\frac18\pi\). Sketch \(C\) and state the greatest distance of a point on \(C\) from the pole.

(b) Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta=\frac18\pi\).

(c) Show that \(C\) has Cartesian equation \(x^4-2xy-y^4=0\), given the first-quadrant restrictions from \(0\le\theta\le\frac18\pi\).

(d) Using your answer to part (b), deduce the exact value of the area bounded by \(C\), the \(x\)-axis and the line \(x=\cos\frac18\pi\).

The curve \(C\) has polar equation \(r=\mathrm{e}^{\frac{3}{4} \theta}\) for \(0 \leqslant \theta \leqslant \alpha\).
Given that the length of \(C\) is \(s\), find \(\alpha\) in terms of \(s\).

The curve \(C\) has polar equation \(r=a \cos 3 \theta\), for \(-\frac{1}{6} \pi \leqslant \theta \leqslant \frac{1}{6} \pi\), where \(a\) is a positive constant.
(i) Sketch \(C\).

(ii) Find the area of the region enclosed by \(C\), showing full working.

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(iii) Using the identity \(\cos 3 \theta \equiv 4 \cos ^{3} \theta-3 \cos \theta\), find a cartesian equation of \(C\).

The curve \(C\) has cartesian equation \(\left(x^{2}+y^{2}\right)^{2}=2 a^{2} x y\), where \(a\) is a positive constant. Show that the polar equation of \(C\) is \(r^{2}=a^{2} \sin 2 \theta\).

Sketch \(C\) for \(-\pi\lt \theta \leqslant \pi\).

Find the area enclosed by one loop of \(C\).

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

\(r=a(1-\cos\theta),\qquad 0\leq \theta\lt 2\pi.\)

Sketch \(C\).

Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{2}\pi\leq\theta\leq\frac{3}{2}\pi\), the half-line \(\theta=\frac{1}{2}\pi\), and the half-line \(\theta=\frac{3}{2}\pi\).

Show that

\(\left(\frac{ds}{d\theta}\right)^2=4a^2\sin^2\left(\frac{\theta}{2}\right),\)

where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{2}\pi\leq\theta\leq\frac{3}{2}\pi\).

The curve \(C\) has polar equation \(r=\mathrm{e}^{4 \theta}\) for \(0 \leqslant \theta \leqslant \alpha\), where \(\alpha\) is measured in radians. The length of \(C\) is 2015 . Find the value of \(\alpha\).

The curves \(C_1\) and \(C_2\) have polar equations

\(C_1:r=\frac1{\sqrt2}\), for \(0\leqslant\theta\lt2\pi\), and \(C_2:r=\sqrt{\sin\frac12\theta}\), for \(0\leqslant\theta\leqslant\pi\).

Find the polar coordinates of the point of intersection of \(C_1\) and \(C_2\).

Sketch \(C_1\) and \(C_2\) on the same diagram.

Find the exact value of the area of the region enclosed by \(C_1\), \(C_2\), and the half-line \(\theta=0\).

A curve has polar equation \(r=\dfrac{1}{1-\cos\theta}\), for \(0\lt\theta\lt 2\pi\). Find, in the form \(y^2=f(x)\), the cartesian equation of the curve.

Hence sketch the curve, and shade the region whose area is given by \(\dfrac12\int_{\pi/2}^{3\pi/2}\dfrac{1}{(1-\cos\theta)^2}\,d\theta\).

Using the cartesian equation of the curve, find the area of this region.