Particular solution with initial conditions

3. At time \(t\) seconds, the position vector \(\mathbf { r }\) metres of a particle \(P\), relative to a fixed origin \(O\), satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } + 3 \mathbf { r } = \mathbf { 0 }$$ At time \(t = 0 , P\) is at the point with position vector \(2 \mathbf { i } \mathrm {~m}\) and is moving with velocity \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the position vector of \(P\) when \(t = \ln 2\).
(10 marks)

8. A particle \(P\) moves in the \(x - y\) plane and has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\) at time \(t \mathrm {~s}\). Given that \(\mathbf { r }\) satisfies the vector differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 9 \mathbf { r } = 8 \sin t \mathbf { i }$$ and that when \(t = 0 \mathrm {~s} , P\) is at \(O\) and moving with velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\),

1. find \(\mathbf { r }\) at time \(t\).
2. Hence find when \(P\) next returns to \(O\).

14

1. Find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } - 2 y = 12 e ^ { - x }\). You are given that \(y\) tends to zero as \(x\) tends to infinity, and that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0\) when \(x = 0\).
2. Find the exact value of \(x\) for which \(y = 0\).

6. Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 7 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 0$$ where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 8\) when \(x = 0\).

12. Find the solution of the differential equation $$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 8 + 6 x - 2 x ^ { 2 }$$ where \(y = 6\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\) when \(x = 0\).

1. An engineer is investigating the motion of a sprung diving board at a swimming pool.

Let \(E\) be the position of the end of the diving board when it is at rest in its equilibrium position and when there is no diver standing on the diving board.
A diver jumps from the diving board.
The vertical displacement, \(h \mathrm {~cm}\), of the end of the diving board above \(E\) is modelled by the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} h } { \mathrm {~d} t } + 37 h = 0$$ where \(t\) seconds is the time after the diver jumps.

1. Find a general solution of the differential equation. When \(t = 0\), the end of the diving board is 20 cm below \(E\) and is moving upwards with a speed of \(55 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
2. Find, according to the model, the maximum vertical displacement of the end of the diving board above \(E\).
3. Comment on the suitability of the model for large values of \(t\).

1. A scientist is investigating the concentration of antibodies in the bloodstream of a patient following a vaccination.
   The concentration of antibodies, \(x\), measured in micrograms ( \(\mu \mathrm { g }\) ) per millilitre ( ml ) of blood, is modelled by the differential equation

$$100 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 60 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 26$$ where \(t\) is the number of weeks since the vaccination was given.

1. Find a general solution of the differential equation. Initially,
   • there are no antibodies in the bloodstream of the patient
   • the concentration of antibodies is estimated to be increasing at \(10 \mu \mathrm {~g} / \mathrm { ml }\) per week
   • Find, according to the model, the maximum concentration of antibodies in the bloodstream of the patient after the vaccination.
   A second dose of the vaccine has to be given to try to ensure that it is fully effective. It is only safe to give the second dose if the concentration of antibodies in the bloodstream of the patient is less than \(5 \mu \mathrm {~g} / \mathrm { ml }\).
2. Determine whether, according to the model, it is safe to give the second dose of the vaccine to the patient exactly 10 weeks after the first dose.

1. The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation

$$2 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 4 t + 12$$ where \(P\) is \(x\) metres from the origin \(O\) at time \(t\) seconds, \(t \geqslant 0\)

1. Determine the general solution of the differential equation.
2. Hence determine the particular solution for which \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2\) when \(t = 0\)
3. 1. Show that, according to the model, the minimum distance between \(O\) and \(P\) is \(( 2 + \ln 2 )\) metres.
   2. Justify that this distance is a minimum. For large values of \(t\) the particle is expected to move with constant speed.
4. Comment on the suitability of the model in light of this information.

8. $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 6 x = 2 \mathrm { e } ^ { - t }$$ Given that \(x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\) at \(t = 0\),

1. find \(x\) in terms of \(t\). The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0 , P\) is \(x\) metres from the origin \(O\).
2. Show that the maximum distance between \(O\) and \(P\) is \(\frac { 2 \sqrt { 3 } } { 9 } \mathrm {~m}\) and justify that this distance is a maximum.
   (7) \section*{TOTAL FOR PAPER: 75 MARKS} \section*{6668/01} \section*{Further Pure Mathematics FP2 Advanced Subsidiary} \section*{Thursday 24 June 2010 - Morning} Mathematical Formulae (Pink) Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper reference (6668), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
   Full marks may be obtained for answers to ALL questions.
   There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
   You must show sufficient working to make your methods clear to the Examiner.
   Answers without working may not gain full credit.
   1. (a) Express \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) in partial fractions.
   2. Using your answer to part (a) and the method of differences, show that
   $$\sum _ { r = 1 } ^ { n } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { 3 n } { 2 ( 3 n + 2 ) }$$
3. Evaluate \(\sum _ { r = 100 } ^ { 1000 } \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer to 3 significant figures.
   2. The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + x + \cos x = 0$$ When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }\).
   Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t ^ { 3 }\).
   3. (a) Find the set of values of \(x\) for which $$x + 4 > \frac { 2 } { x + 3 } .$$
4. Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \frac { 2 } { | x + 3 | }$$ 4. $$z = - 8 + ( 8 \sqrt { } 3 ) \mathrm { i }$$
5. Find the modulus of \(z\) and the argument of \(z\). Using de Moivre's theorem,
6. find \(z ^ { 3 }\),
7. find the values of \(w\) such that \(w ^ { 4 } = z\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
   5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3caa00c9-8a77-4987-888d-054eba7712ee-04_339_488_687_511} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows the curves given by the polar equations $$\begin{array} { c c } r = 2 , & 0 \leq \theta \leq \frac { \pi } { 2 } \\ \text { and } r = 1.5 + \sin 3 \theta , & 0 \leq \theta \leq \frac { \pi } { 2 } \end{array}$$
8. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r > 2\) and for which \(r < ( 1.5 + \sin 3 \theta )\), is shown shaded in Figure 1.
9. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a \pi + b \sqrt { } 3\), where \(a\) and \(b\) are simplified fractions.
   6. A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
10. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
11. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
12. Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.
    7. (a) Show that the transformation \(z = y ^ { \frac { 1 } { 2 } }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 4 y \tan x = 2 y ^ { \frac { 1 } { 2 } }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 2 z \tan x = 1$$
13. Solve the differential equation (II) to find \(z\) as a function of \(x\).
14. Hence obtain the general solution of the differential equation (I).
    8. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$
15. Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
16. find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
17. Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leq x \leq \pi\).

1

1. Find the roots of the equation \(m ^ { 2 } + 2 m + 2 = 0\) in the form \(a + i b\).
   (2 marks)
2. 1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x$$
   2. Hence express \(y\) in terms of \(x\), given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) when \(x = 0\).

19 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 45 y = 21 \mathrm { e } ^ { 5 x } - 0.3 x + 27 x ^ { 2 }$$ given that \(y = \frac { 37 } { 225 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\) [0pt] [10 marks]

• \(\begin{gathered} \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \\ \text { - } \end{gathered}\)

3 The equation of a plane is \(4 x + 2 y + z = 7\).
The point \(A\) has coordinates \(( 9,6,1 )\) and the point \(B\) is the reflection of \(A\) in the plane.
Find the coordinates of the point \(B\). You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
3. For each of the values of \(a\) found in part (b) determine whether the equations have
   • a unique solution, which should be found, or
   • an infinite set of solutions or
   • no solution.
   A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line. The displacement of the particle may be modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t\) where \(x\) is the displacement of the particle below the equilibrium position at time \(t\).
   When \(t = 0\) the particle is stationary and its displacement is 2 .
4. Find the particular solution of the differential equation.
5. Write down an approximate equation for the displacement when \(t\) is large. \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. \section*{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. \section*{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.
   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. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Abbreviations}

   Abbreviations used in the mark scheme	Meaning
   dep*	Mark dependent on a previous mark, indicated by *. The * may be omitted if only one previous M mark
   cao	Correct answer only
   ое	Or equivalent
   rot	Rounded or truncated
   soi	Seen or implied
   www	Without wrong working
   AG	Answer given
   awrt	Anything which rounds to
   BC	By Calculator
   DR	This question included the instruction: In this question you must show detailed reasoning.

   \end{table}
   \includegraphics[max width=\textwidth, alt={}]{5392d682-c940-41bb-bcaf-10e7a7440c80-06_1740_2523_141_242}

   Question		Answer	Marks	AO	Guidance
   4	(a)	\(\operatorname { det } \mathbf { A } = a ^ { 2 } - 10 a + 16\)	M1 A1 [2]	1.1a 1.1	Attempt to work out the determinant
   	(b)	\(a ^ { 2 } - 10 a + 16 = 0 \Rightarrow ( a - 2 ) ( a - 8 ) = 0 \Rightarrow a = 2,8\)	M1 A1 [2]	1.1a 1.1	Solving their quadratic soi
   	(c)	For both values there is no unique solution as \(\operatorname { det } \mathbf { A } = 0\) For \(a = 2\), equations are: \(\begin{aligned} p _ { 1 } : 2 x + 2 y = 6 p _ { 2 } : 2 y + 2 z = 8 p _ { 3 } : 4 x + 5 y + z = 16 2 p _ { 1 } + \frac { 1 } { 2 } p _ { 2 } = p _ { 3 } \end{aligned}\) So there is an infinite set of solutions. For \(a = 8\), equations are: \(\begin{aligned} p _ { 1 } : 8 x + 2 y = 6 p _ { 2 } : 8 y + 2 z = 8 p _ { 3 } : 4 x + 5 y + z = 16 \frac { 1 } { 2 } p _ { 1 } + \frac { 1 } { 2 } p _ { 2 } \neq p _ { 3 } \text { as it gives } 4 x + 5 y + z = 7 \end{aligned}\) and \(16 \neq 7\) so no solution	B1 М1 A1 A1 М1 A1 A1 [7]	2.4 2.1 1.1 2.2a	Soi by correct answers Substitute one of their values and solve	"correct answers" means solns are either infinite or nonexistent.

   \includegraphics[max width=\textwidth, alt={}]{5392d682-c940-41bb-bcaf-10e7a7440c80-09_1653_2523_141_242}

   (b)	\(\begin{aligned}	x = \mathrm { e } ^ { - t } ( A \cos 2 t + B \sin 2 t ) + 2 \sin t - \cos t
   	\text { As } t \rightarrow \infty , \quad \mathrm { e } ^ { - t } \rightarrow 0
   	\Rightarrow x \approx 2 \sin t - \cos t \end{aligned}\)	М1 A1 [2]	3.4 3.4	Consider behaviour of \(\mathrm { e } ^ { - \mathrm { t } }\) Accept = ft their equation from (a)	Accept this final line for 2 marks

3 A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geqslant 0 , Q\) can be modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } Q } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} Q } { \mathrm {~d} t } - 15 Q = 0\). Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.

1. Determine the charge on the capacitor when \(t = 0.5\).
2. Determine the finite limit of \(Q\) as \(t\) tends to infinity. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4 \\ - 0.8 & 1.8 \end{array} \right)\).
3. Find \(\operatorname { det } \mathbf { A }\). The matrix \(\mathbf { A }\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
4. By considering the determinants of these transformations, determine the scale factor of the stretch.
5. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer.
6. Find the angle of rotation.