Maximum/minimum distance from pole or line

6
\includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_405_914_260_612} The diagram shows the curve \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 \left( x ^ { 2 } - y ^ { 2 } \right)\) and one of its maximum points \(M\). Find the coordinates of \(M\).

7 The curve \(C _ { 1 }\) has polar equation \(r = \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. 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 \(Q\).
2. Find the polar coordinates of \(Q\), giving your answers in exact form.
3. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
4. Find, in terms of \(\pi\), the area of the region bounded by the arc \(O Q\) of \(C _ { 1 }\) and the arc \(O Q\) of \(C _ { 2 }\). [7]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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

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 half-line \(\theta = 2\).
   Now consider the part of \(C\) where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
3. Show that, at the point furthest from the half-line \(\theta = \frac { 1 } { 2 } \pi\), $$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$ and verify that this equation has a root between 0.6 and 0.7 .
   \(7 \quad\) The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{array} \right)\).
4. Find the set of values of \(k\) for which \(\mathbf { A }\) is non-singular.
5. Given that \(\mathbf { A }\) is non-singular, find, in terms of \(k\), the entries in the top row of \(\mathbf { A } ^ { - 1 }\).
6. Given that \(\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right)\), give an example of a matrix \(\mathbf { C }\) such that \(\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\).
7. Find the set of values of \(k\) for which the transformation in the \(x - y\) plane represented by \(\left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\) has two distinct invariant lines through the origin.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Find the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \pi\).
3. Show that, at the point of \(C\) furthest from the initial line, $$\left( \theta + \frac { 1 } { \theta } \right) \cot \theta - 1 = 0$$ and verify that this equation has a root between 1.1 and 1.2.

11 The curve \(C\) has polar equation $$r = \frac { a } { 1 + \theta }$$ where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(r\) decreases as \(\theta\) increases.
2. 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.
3. Draw a sketch of \(C\).
4. 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 \(a\).

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

1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta = 1$$ 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 \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
2. Find the exact value of \(\theta\) at \(Q\).
3. The diagram below shows the curve \(C _ { 2 }\). Sketch \(C _ { 1 }\) on this diagram.
4. Find, in exact form, the area of the region \(O P Q\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).

10 The curve \(C\) has polar equation $$r = a \sin 3 \theta$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).

1. Show that the area of the region enclosed by \(C\) is \(\frac { 1 } { 12 } \pi a ^ { 2 }\).
2. Show that, at the point of \(C\) at maximum distance from the initial line, $$\tan 3 \theta + 3 \tan \theta = 0 .$$
3. Use the formula $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ to find this maximum distance.
4. Draw a sketch of \(C\).

10 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 \(\theta = \frac { 1 } { 4 } \pi\).

14 A curve has polar equation \(\mathrm { r } = \mathrm { a } ( \cos \theta + 2 \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \pi\).

1. Determine the polar coordinates of the point on the curve which is furthest from the pole.
2. 1. Show that the curve is a circle whose radius should be specified.
   2. Write down the polar coordinates of the centre of the circle.

5 In this question you must show detailed reasoning.
The diagram below shows the curve \(r = \sqrt { \sin \theta } \mathrm { e } ^ { \frac { 1 } { 3 } \cos \theta }\) for \(0 \leqslant \theta \leqslant \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{58789f16-bfc3-4f21-b7c4-abca4f549fc7-03_574_878_276_477}

1. Find the exact area enclosed by the curve.
2. Show that the greatest value of \(r\) on the curve is \(\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\). Total Marks for Question Set 4: 37 \section*{Mark scheme} \section*{Marking Instructions} a An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not always be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed.
   b The following types of marks are available.
   M
   A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified.
   A method mark may usually be implied by a correct answer unless the question includes the DR statement, the command words "Determine" or "Show that", or some other indication that the method must be given explicitly. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. B
   Mark for a correct result or statement independent of Method marks. \section*{E} A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
   c When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
   d The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'.
   e We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so.
   • When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value.
   • When a value is not given in the paper accept any answer that agrees with the correct value to \(\mathbf { 3 ~ s } . \mathbf { f }\). unless a different level of accuracy has been asked for in the question, or the mark scheme specifies an acceptable range.
   Follow through should be used so that only one mark in any question is lost for each distinct accuracy error.
   Candidates using a value of \(9.80,9.81\) or 10 for \(g\) should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is given in the rubric.
   f Rules for replaced work and multiple attempts:
   • If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others.
   • If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete.
   • if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately.
     g For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors.
     If a candidate corrects the misread in a later part, do not continue to follow through. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
     h If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers, provided that there is nothing in the wording of the question specifying that analytical methods are required such as the bold "In this question you must show detailed reasoning", or the command words "Show" or "Determine". Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero.
   Abbreviations

   Question		Answer	Marks	AO	Guidance
   \multirow[t]{7}{*}{2}	\multirow{7}{*}{}	DR	\multirow{6}{*}{ B1 М1 A1 B1 A1 }	\multirow{4}{*}{ 1.1a 2.1 1.1 }	\multirow[b]{3}{*}{Consideration of a finite upper limit}	\multirow{7}{*}{Can be seen as part of limit of both terms, but must be explicitly shown as zero}
   		\(\int ( x - 1 ) ^ { - \frac { 3 } { 2 } } \mathrm {~d} x = - 2 ( x - 1 ) ^ { - \frac { 1 } { 2 } } ( + c )\)
   		\(\int _ { 5 } ^ { N } ( x - 1 ) ^ { - \frac { 3 } { 2 } } \mathrm {~d} x = \left[ - 2 ( x - 1 ) ^ { - \frac { 1 } { 2 } } \right] _ { 5 } ^ { N }\) \(- \frac { 2 } { \sqrt { N - 1 } } + \frac { 2 } { \sqrt { 5 - 1 } }\)
   		\(\lim _ { N \rightarrow \infty } \frac { 1 } { \sqrt { N - 1 } } = 0\) oe		2.1	Not just eg \(\frac { 1 } { \infty } = 0\)
   		\(\int _ { 5 } ^ { \infty } ( x - 1 ) ^ { - \frac { 3 } { 2 } } \mathrm {~d} x = \lim _ { N \rightarrow \infty } \left\{ - \frac { 2 } { \sqrt { N - 1 } } + \frac { 2 } { \sqrt { 5 - 1 } } \right\} = 1\)		2.2a	AG. Convincing argument equating improper integral to solution
   			[5]

   Question			Answer	Marks	AO	Guidance
   3	(a)		\includegraphics[max width=\textwidth, alt={}]{58789f16-bfc3-4f21-b7c4-abca4f549fc7-10_450_671_93_559}	B1 B1 B1 B1	1.2 3.4 3.4 3.1b	At least one cycle of \(A \cos \omega t\) graph Amplitude 0.1 Period 0.4 Intersect with the \(t\)-axis at 0.1 and 0.3 (values must be indicated or implied unambiguously (eg by tickmarks and a single value))	graph must instantaneously horizontal at top/bottom, continuous and not vertical at any point. Ignore any graph outside [0, 0.4] Non-inverted cos graph can still get 4/4
   	(b)		So \(x = \pm 0.1 \cos ( 5 \pi \mathrm { t } )\) or \(x = 0.1 \sin \left( 5 \pi t \pm \frac { 1 } { 2 } \pi \right)\) when \(t = 0.75\) or 0.35 \(x = - \frac { \sqrt { 2 } } { 20 } ( = - 0.0707 \text { to } 3 \mathrm { sf } )\)	M1 A1 [2]	3.1b 3.4	or by argument from sketch	Condone amplitude of 0.2 for M1
   4	(a)		\(\begin{aligned}	( 2 + 3 i ) - ( 1 - i ) ( = \pm ( 1 + 4 i ) ) \text { soi }
   	| 1 + 4 i | = \sqrt { 1 ^ { 2 } + 4 ^ { 2 } }
   	17 \end{aligned}\)	B1 M1 A1 [3]	1.1 2.2a 1.1	Either way round Can be implied by vector Or finding the square of their side
   	(b)		\(( \pm \mathrm { i } ) \times ( \pm ( 1 + 4 \mathrm { i } ) )\) \(( 2 + 3 i ) \pm i ( 1 + 4 i )\) and \(( 1 - i ) \pm i ( 1 + 4 i )\) So vertices at - 3 and \(- 2 + 4 \mathrm { i }\) Or at \(5 - 2 \mathrm { i }\) and \(6 + 2 \mathrm { i }\)	*M1 dep*M1 A1 A1 [4]	3.1a 2.2a 3.2a 3.2a	Method to find a complex number representing perpendicular side. Can be implied by \(\pm ( 4 - \mathrm { i } )\) Method to find both pairs of numbers Both clearly paired and in complex number form for final A1	Or vector form if take geometric approach If M1M0A0A0 then add SC1 for any two correct vertices

   Question			Answer	Marks	AO	Guidance
   5	(a)		DR \(\begin{aligned}	\frac { 1 } { 2 } \int \left( \sqrt { \sin \theta } e ^ { \frac { 1 } { 3 } \cos \theta } \right) ^ { 2 } \mathrm {~d} \theta
   	\mathrm {~A} = \frac { 1 } { 2 } \int _ { 0 } ^ { \pi } \sin \theta \mathrm { e } ^ { \frac { 2 } { 3 } \cos \theta } \mathrm {~d} \theta
   	= \frac { 1 } { 2 } \times - \frac { 3 } { 2 } \left[ \mathrm { e } ^ { \frac { 2 } { 3 } \cos \theta } \right] _ { 0 } ^ { \pi }
   	\frac { 3 } { 4 } \left( \mathrm { e } ^ { \frac { 2 } { 3 } } - \mathrm { e } ^ { - \frac { 2 } { 3 } } \right) \end{aligned}\)	M1 *A1 dep*M1 A1 [4]	3.1a 2.1 1.1a 1.1	Correct form, in terms of \(\theta\), Integrand has been squared out. Must include limits (can be seen later) Might be as result of substitution Allow coefficient error for M1 isw	M1 can be implied by 1.0757 ... BC eg \(\frac { 3 } { 4 } \left[ \mathrm { e } ^ { u } \right] _ { - \frac { 2 } { 3 } } ^ { \frac { 2 } { 3 } }\) or \(\frac { 3 } { 4 } \left[ \mathrm { e } ^ { \frac { 2 } { 3 } u } \right] _ { - 1 } ^ { 1 }\) oe
   	(b)		DR \(\frac { \mathrm { d } r } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \cos \theta ( \sin \theta ) ^ { - \frac { 1 } { 2 } } e ^ { \frac { 1 } { 3 } \cos \theta } +\) \(( \sin \theta ) ^ { \frac { 1 } { 2 } } \left( - \frac { 1 } { 3 } \sin \theta \right) e ^ { \frac { 1 } { 3 } \cos \theta }\) \(\frac { \mathrm { d } r } { \mathrm {~d} \theta } = \frac { 1 } { 6 } ( \sin \theta ) ^ { - \frac { 1 } { 2 } } e ^ { \frac { 1 } { 3 } \cos \theta } \left( 3 \cos \theta - 2 \sin ^ { 2 } \theta \right)\) \(\frac { \mathrm { d } r } { \mathrm {~d} \theta } = 0 \Rightarrow 3 \cos \theta - 2 \sin ^ { 2 } \theta = 0\) \(2 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0\) \(\cos \theta = \frac { 1 } { 2 } , - 2\) \(\cos \theta \neq - 2\) \(\Rightarrow \sin \theta = \frac { \sqrt { 3 } } { 2 } \Rightarrow r = \sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 3 } \times \frac { 1 } { 2 } } = \sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\)	*M1 A1 dep*M1 M1 *A1 dep*A1	3.1a 1.1 2.2a 2.1 1.1 2.3 2.2a	Attempt to differentiate using product and chain rules. Setting \(r ^ { \prime }\) to zero and factorising/cancelling to produce a quadratic equation in \(\cos\) and/or sin Use of \(\cos ^ { 2 } + \sin ^ { 2 } = 1\) to find 3 term quadratic equation in \(\cos \theta\). Solving quadratic correctly Explicitly rejecting root AG. At least one intermediate step must be seen.	Must be in the form \(u v ^ { \prime } + u ^ { \prime } v\) with at most one of \(u , v , u ^ { \prime }\) or \(v ^ { \prime }\) incorrect or omitted Or could be in \(\sin ^ { 2 } \theta\); \(4 \sin ^ { 4 } \theta + 9 \sin ^ { 2 } \theta - 9 = 0\) \(\sin ^ { 2 } \theta = \frac { 3 } { 4 } , - 3\) Rejects \(\sin ^ { 2 } \theta = - 3\) and \(\sin \theta =\) \(- \frac { \sqrt { 3 } } { 2 }\) Can be awarded even if rejection of root(s) was implicit.