4.09c Area enclosed: by polar curve

1

1. 1. Given that \(\mathrm { f } ( x ) = \arctan x\), write down an expression for \(\mathrm { f } ^ { \prime } ( x )\). Assuming that \(x\) is small, use a binomial expansion to express \(\mathrm { f } ^ { \prime } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 4 }\).
   2. Hence express \(\arctan x\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).
2. Find, in exact form, the value of the following integral. $$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
3. A curve has polar equation \(r = \frac { a } { \sqrt { \theta } }\) where \(a > 0\).
   1. Sketch the curve for \(\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi\).
   2. State what happens to \(r\) as \(\theta\) tends to zero.
   3. Find the area of the region enclosed by the part of the curve sketched in part (i) and the lines \(\theta = \frac { \pi } { 4 }\) and \(\theta = 2 \pi\). Give your answer in an exact simplified form.
      1. (i) Express \(2 \sin \frac { 1 } { 2 } \theta \left( \sin \frac { 1 } { 2 } \theta - \mathrm { j } \cos \frac { 1 } { 2 } \theta \right)\) in terms of \(z\) where \(z = \cos \theta + \mathrm { j } \sin \theta\).
         (ii) The series \(C\) and \(S\) are defined as follows. $$\begin{aligned} C & = 1 - \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \cos n \theta \\ S & = - \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \sin n \theta \end{aligned}$$ Show that $$C + \mathrm { j } S = \left\{ - 2 \mathrm { j } \sin \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { j } \sin \frac { 1 } { 2 } \theta \right) \right\} ^ { n } .$$ Hence show that, for even values of \(n\), $$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$
      2. Write the complex number \(z = \sqrt { 6 } + \mathrm { j } \sqrt { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), expressing \(r\) and \(\theta\) as simply as possible. Hence find the cube roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Show the points representing \(z\) and its cube roots on an Argand diagram.
         1. Find the eigenvalues and eigenvectors of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { l l } \frac { 1 } { 2 } & \frac { 1 } { 2 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right)$$ Hence express \(\mathbf { M }\) in the form \(\mathbf { P D P } ^ { - 1 }\) where \(\mathbf { D }\) is a diagonal matrix.
         2. Write down an equation for \(\mathbf { M } ^ { n }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence obtain expressions for the elements of \(\mathbf { M } ^ { n }\).
            Show that \(\mathbf { M } ^ { n }\) tends to a limit as \(n\) tends to infinity. Find that limit.
         3. Express \(\mathbf { M } ^ { - 1 }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence determine whether or not \(\left( \mathbf { M } ^ { - 1 } \right) ^ { n }\) tends to a limit as \(n\) tends to infinity. Section B (18 marks)
            1. Given that \(y = \cosh x\), use the definition of \(\cosh x\) in terms of exponential functions to prove that $$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$
            2. Solve the equation $$\cosh x + \cosh 2 x = 5$$ giving the roots in an exact logarithmic form.
            3. Sketch the curve with equation \(y = \cosh x + \cosh 2 x\). Show on your sketch the line \(y = 5\). Find the area of the finite region bounded by the curve and the line \(y = 5\). Give your answer in an exact form that does not involve hyperbolic functions. \section*{END OF QUESTION PAPER}

The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\). Sketch \(C\). Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\). Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) ,$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). {www.cie.org.uk} after the live examination series.
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9 The curve \(C\) has polar equation $$r = 5 \sqrt { } ( \cot \theta ) ,$$ where \(0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. 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 \(\theta = 0.01\).
2. Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
3. Find the maximum distance of \(C\) from the initial line.
4. Sketch \(C\).

7 A curve \(C\) has polar equation \(r = 2 + \cos \theta\) for \(- \pi < \theta \leqslant \pi\).

1. The point \(P\) on \(C\) corresponds to \(\theta = \alpha\), and the point \(Q\) on \(C\) is such that \(P O Q\) is a straight line, where \(O\) is the pole. Show that the length \(P Q\) is independent of \(\alpha\).
2. Find, in an exact form, the area of the region enclosed by \(C\).
3. Show that \(\left( x ^ { 2 } + y ^ { 2 } - x \right) ^ { 2 } = 4 \left( x ^ { 2 } + y ^ { 2 } \right)\) is a cartesian equation for \(C\). Identify the coordinates of the point which is included in this cartesian equation but is not on \(C\).

2 Find the area enclosed by the curve with polar equation \(r = \sin \theta + \cos \theta , 0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

11 A curve has polar equation \(r = \mathrm { e } ^ { \sin \theta }\) for \(- \pi < \theta \leqslant \pi\).

1. State the polar coordinates of the point where the curve crosses the initial line.
2. State also the polar coordinates of the points where \(r\) takes its least and greatest values.
3. Sketch the curve.
4. By deriving a suitable Maclaurin series up to and including the term in \(\theta ^ { 2 }\), find an approximation, to 3 decimal places, for the area of the region enclosed by the curve, the initial line and the line \(\theta = 0.3\).

10

1. Sketch the curve with polar equation \(r = \left| \frac { 1 } { 2 } + \sin \theta \right|\), for \(0 \leqslant \theta < 2 \pi\).
2. Find in an exact form the total area enclosed by the curve.

2 A curve has polar equation \(r = \sin \theta + \cos \theta\). Find the area enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\).

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

3

1. Sketch the curve with polar equation \(r = \frac { 1 } { 1 + \theta } , 0 \leqslant \theta \leqslant 2 \pi\).
2. Find, in terms of \(\pi\), the area of the region enclosed by the curve and the part of the initial line between the endpoints of the curve.

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 } .$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

12

1. Let \(I _ { \mathrm { n } } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

8 The curve \(C\) has polar equation \(r = \theta ^ { 2 } + 2 \theta\) for \(0 \leq \theta \leq 3\).

1. Find the area of the region enclosed by \(C\) and the half-lines \(\theta = 0\) and \(\theta = 3\).
2. Determine the length of \(C\).

12 \includegraphics[max width=\textwidth, alt={}, center]{0f5edc87-cb14-4583-a54d-badec47741d1-08_414_659_804_744} The diagram shows a sketch of the curve \(C\) with polar equation \(r = 4 \cos ^ { 2 } \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\).
2. The equation of \(C\) may be expressed in the form \(r = k ( 1 + \cos 2 \theta )\). State the value of \(k\) and use this form to show that the area of the region enclosed by \(C\) is given by $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } ( 3 + 4 \cos 2 \theta + \cos 4 \theta ) d \theta ,$$ and hence find this area.
3. The length of \(C\) is denoted by \(L\). Show that $$L = 8 \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos \theta \sqrt { 1 + 3 \sin ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution \(\sinh x = \sqrt { 3 } \sin \theta\) to determine \(L\) in an exact form.

4

1. Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
2. On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\).
3. Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.

1. Show that the curve with Cartesian equation $$\left(x^2+y^2\right)^2 = 6xy$$ has polar equation \(r^2 = 3\sin 2\theta\). [2]

The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
2. Find the area of the region enclosed by \(C\). [2]
3. Find the maximum distance of a point on \(C\) from the initial line. [6]

1. Show that the curve with Cartesian equation \(\left(x^2 + y^2\right)^2 = 6xy\) has polar equation \(r^2 = 3\sin 2\theta\). [2]

The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
2. Find the area of the region enclosed by \(C\). [2]
3. Find the maximum distance of a point on \(C\) from the initial line. [6]

\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]

Answer only one of the following two alternatives. **EITHER** Show that \(\left(n + \frac{1}{2}\right)^3 - \left(n - \frac{1}{2}\right)^3 \equiv 3n^2 + \frac{1}{4}\). [1] Use this result to prove that \(\sum_{n=1}^N n^2 = \frac{1}{6}N(N + 1)(2N + 1)\). [2] The sums \(S\), \(T\) and \(U\) are defined as follows: \begin{align} S &= 1^2 + 2^2 + 3^2 + 4^2 + \ldots + (2N)^2 + (2N + 1)^2,
T &= 1^2 + 3^2 + 5^2 + 7^2 + \ldots + (2N - 1)^2 + (2N + 1)^2,
U &= 1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2N)^2 + (2N + 1)^2. \end{align} Find and simplify expressions in terms of \(N\) for each of \(S\), \(T\) and \(U\). [5] Hence

1. describe the behaviour of \(\frac{S}{T}\) as \(N \to \infty\), [1]
2. prove that if \(\frac{S}{U}\) is an integer then \(\frac{T}{U}\) is an integer. [3]

**OR** The curves \(C_1\) and \(C_2\) have polar equations $$r = 4\cos\theta \quad \text{and} \quad r = 1 + \cos\theta$$ respectively, where \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Show that \(C_1\) and \(C_2\) meet at the points \(A\left(\frac{4}{3}, \alpha\right)\) and \(B\left(\frac{4}{3}, -\alpha\right)\), where \(\alpha\) is the acute angle such that \(\cos\alpha = \frac{1}{3}\). [2]
2. In a single diagram, draw sketch graphs of \(C_1\) and \(C_2\). [3]
3. Show that the area of the region bounded by the arcs \(OA\) and \(OB\) of \(C_1\), and the arc \(AB\) of \(C_2\), is $$4\pi - \frac{1}{3}\sqrt{2} - \frac{13}{2}\alpha.$$ [7]

Answer only one of the following two alternatives. EITHER The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}\), \(2\mathbf{j}\) and \(4\mathbf{k}\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(ABC\). The point \(P\) on the line-segment \(ON\) is such that \(OP = \frac{3}{4}ON\). The line \(AP\) meets the plane \(OBC\) at \(Q\). Find a vector perpendicular to the plane \(ABC\) and show that the length of \(ON\) is \(\frac{1}{\sqrt{(21)}}\). [4] Find the position vector of the point \(Q\). [5] Show that the acute angle between the planes \(ABC\) and \(ABQ\) is \(\cos^{-1}\left(\frac{4}{5}\right)\). [5] OR The curve \(C\) has polar equation \(r = a(1 - \cos\theta)\) for \(0 \leqslant \theta < 2\pi\). Sketch \(C\). [2] Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), the half-line \(\theta = \frac{1}{3}\pi\) and the half-line \(\theta = \frac{2}{3}\pi\). [5] Show that $$\left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^2 = 4a^2\sin^2\left(\frac{1}{2}\theta\right),$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\). [7]

The curve \(C\) has polar equation $$r = 5\sqrt{\cot \theta},$$ where \(0.01 \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. 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. [3]

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

1. Find the distance of \(P\) from the initial line, giving your answer correct to \(1\) decimal place. [2]
2. Find the maximum distance of \(C\) from the initial line. [3]
3. Sketch \(C\). [2]

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.

1. Sketch \(C\). [2]
2. Find the area of the region enclosed by \(C\), showing full working. [3]
3. Using the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\), find a cartesian equation of \(C\). [3]

Answer only one of the following two alternatives. **EITHER** It is given that \(w = \cos y\) and $$\tan y \frac{d^2 y}{dx^2} + \left( \frac{dy}{dx} \right)^2 + 2 \tan y \frac{dy}{dx} = 1 + e^{-2x} \sec y.$$

1. Show that $$\frac{d^2 w}{dx^2} + 2 \frac{dw}{dx} + w = -e^{-2x}.$$ [4]
2. Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0\), \(y = \frac{1}{4}\pi\) and \(\frac{dy}{dx} = \frac{1}{\sqrt{3}}\). [10]

**OR** The curves \(C_1\) and \(C_2\) have polar equations, for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\), as follows: \begin{align} C_1 : r &= 2(e^\theta + e^{-\theta}),
C_2 : r &= e^{2\theta} - e^{-2\theta}. \end{align} The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(e^{2\alpha} - 2e^\alpha - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4\sqrt{2}\). [6]
2. Sketch \(C_1\) and \(C_2\) on the same diagram. [3]
3. Find the area of the region enclosed by \(C_1\), \(C_2\) and the initial line, giving your answer correct to 3 significant figures. [5]

Answer only one of the following two alternatives. **EITHER** It is given that \(w = \cos y\) and $$\tan y \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + 2\tan y \frac{dy}{dx} = 1 + e^{-2x} \sec y.$$

1. Show that $$\frac{d^2 w}{dx^2} + 2\frac{dw}{dx} + w = -e^{-2x}.$$ [4]
2. Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0\), \(y = \frac{1}{4}\pi\) and \(\frac{dy}{dx} = \frac{1}{\sqrt{3}}\). [10]

**OR** The curves \(C_1\) and \(C_2\) have polar equations, for \(0 \leq \theta \leq \frac{1}{2}\pi\), as follows: \begin{align} C_1: r &= 2(e^\theta + e^{-\theta}),
C_2: r &= e^{2\theta} - e^{-2\theta}. \end{align} The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(e^{2\alpha} - 2e^\alpha - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4\sqrt{2}\). [6]
2. Sketch \(C_1\) and \(C_2\) on the same diagram. [3]
3. Find the area of the region enclosed by \(C_1\), \(C_2\) and the initial line, giving your answer correct to 3 significant figures. [5]