Questions Mechanics 1 (36 questions)

6
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-06_544_1232_251_260} The diagram shows the curve with parametric equations \(x = \ln \left( t ^ { 2 } - 4 \right) , \quad y = \frac { 4 } { t ^ { 2 } } , \quad\) where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).

1. Show that the area of \(R\) is given by
   \(\int _ { a } ^ { b } \frac { 8 } { t \left( t ^ { 2 } - 4 \right) } \mathrm { d } t\),
   where \(a\) and \(b\) are constants to be determined.
2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined.
3. Find a cartesian equation of the curve in the form \(y = \mathrm { f } ( x )\).

7 A particle \(P\) moves with constant acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the displacement vector of \(P\) at time \(t = 4\) seconds.
2. Find the speed of \(P\) at time \(t = 0\) seconds.

8 A uniform ladder \(A B\), of weight 150 N and length 4 m , rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 3\). A man of weight 750 N is standing on the ladder at a distance \(x \mathrm {~m}\) from \(A\).

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac { 25 } { 2 } ( 2 + 5 x ) \mathrm { N }\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\).
2. Find the greatest value of \(x\) for which equilibrium is possible.

9 A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = 2 t ^ { 4 } + k t ^ { 2 } - 4\). The acceleration of \(P\) when \(t = 2\) is \(28 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(k = - 9\).
2. Show that the velocity of \(P\) has its minimum value when \(t = 1.5\). When \(t = 1 , P\) is at the point \(( - 6.4125,0 )\).
3. Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity.
   \includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-08_698_1009_260_246}
   \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is \(20 \mathrm {~m} . M\) is a point on the plane between \(A\) and \(B\). The surface of the plane is smooth between \(A\) and \(M\), and rough between \(M\) and \(B\). A particle \(P\) is projected with speed \(4.2 \mathrm {~ms} ^ { - 1 }\) from \(A\) down the line of greatest slope (see diagram). \(P\) moves down the plane and reaches \(B\) with speed \(12.6 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between \(P\) and the rough part of the plane is \(\frac { \sqrt { 3 } } { 6 }\).
4. Find the distance \(A M\).
5. Find the angle between the contact force and the downward direction of the line of greatest slope when \(P\) is in motion between \(M\) and \(B\).

11 A ball \(B\) is projected with speed \(V\) at an angle \(\alpha\) above the horizontal from a point \(O\) on horizontal ground. The greatest height of \(B\) above \(O\) is \(H\) and the horizontal range of \(B\) is \(R\). The ball is modelled as a particle moving freely under gravity.

1. Show that
   1. \(H = \frac { V ^ { 2 } } { 2 g } \sin ^ { 2 } \alpha\),
   2. \(R = \frac { V ^ { 2 } } { g } \sin 2 \alpha\).
2. Hence show that \(16 H ^ { 2 } - 8 R _ { 0 } H + R ^ { 2 } = 0\), where \(R _ { 0 }\) is the maximum range for the given speed of projection.
3. Given that \(R _ { 0 } = 200 \mathrm {~m}\) and \(R = 192 \mathrm {~m}\), find
   1. the two possible values of the greatest height of \(B\),
   2. the corresponding values of the angle of projection.
4. State one limitation of the model that could affect your answers to part (iii). \section*{OCR} Oxford Cambridge and RSA

8 In this question you must show detailed reasoning. The diagram shows triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-06_735_1383_456_342} The angles \(C A B\) and \(A B C\) are each \(45 ^ { \circ }\), and angle \(A C B = 90 ^ { \circ }\).
The points \(D\) and \(E\) lie on \(A C\) and \(A B\) respectively. \(A E = D E = 1 , D B = 2\).
Angle \(B E D = 90 ^ { \circ }\), angle \(E B D = 30 ^ { \circ }\) and angle \(D B C = 15 ^ { \circ }\).

1. Show that \(B C = \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 }\).
2. By considering triangle \(B C D\), show that \(\sin 15 ^ { \circ } = \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }\).

9 Two forces, of magnitudes 2 N and 5 N , act on a particle in the directions shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-07_323_755_548_283} \section*{(a) Calculate the magnitude of the resultant force on the particle.
(b) Calculate the angle between this resultant force and the force of magnitude 5 N .}

10 A body of mass 20 kg is on a rough plane inclined at angle \(\alpha\) to the horizontal. The body is held at rest on the plane by the action of a force of magnitude \(P \mathrm {~N}\). The force is acting up the plane in a direction parallel to a line of greatest slope of the plane. The coefficient of friction between the body and the plane is \(\mu\).

1. When \(P = 100\), the body is on the point of sliding down the plane. Show that \(g \sin \alpha = g \mu \cos \alpha + 5\).
2. When \(P\) is increased to 150, the body is on the point of sliding up the plane. Use this, and your answer to part (a), to find an expression for \(\alpha\) in terms of \(g\).

12 A girl is practising netball.
She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop.
The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection. The situation is modelled as follows.

• The initial velocity of the ball has magnitude \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
• The angle of projection is \(40 ^ { \circ }\).
• The ball is modelled as a particle.
• The hoop is modelled as a point.

This is shown on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-09_378_1210_1119_278}

1. For \(U = 10\), find
   1. the greatest height above the ground reached by the ball
   2. the distance between the ball and the hoop when the ball is vertically above the hoop.
2. Calculate the value of \(U\) which allows her to hit the hoop.
3. How appropriate is this model for predicting the path of the ball when it is thrown by the girl?
4. Suggest one improvement that might be made to this model.

13 Particle \(A\), of mass \(m \mathrm {~kg}\), lies on the plane \(\Pi _ { 1 }\) inclined at an angle of \(\tan ^ { - 1 } \frac { 3 } { 4 }\) to the horizontal.
Particle \(B\), of \(4 m \mathrm {~kg}\), lies on the plane \(\Pi _ { 2 }\) inclined at an angle of \(\tan ^ { - 1 } \frac { 4 } { 3 }\) to the horizontal.
The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\).
The coefficient of friction between particle \(A\) and \(\Pi _ { 1 }\) is \(\frac { 1 } { 3 }\) and plane \(\Pi _ { 2 }\) is smooth.
Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-10_403_851_865_303}

1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac { 7 g } { 15 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac { 1 } { 4 } \mathrm {~m}\) when its speed is \(\sqrt { \frac { 7 g } { 30 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   [0pt] [2]

14 A uniform ladder \(A B\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall.
The ladder is inclined at an angle of \(45 ^ { \circ }\) to the horizontal.
A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\).
The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\) and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 2 }\).
The system is in limiting equilibrium. Find \(x\).
[0pt] [8] \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{...day June 20XX - Morning/Afternoon} A Level Mathematics A
H240/03 Pure Mathematics and Mechanics SAMPLE MARK SCHEME Duration: 2 hours MAXIMUM MARK 100
\includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-13_259_1320_1242_826} This document consists of 20 pages \section*{Text Instructions} \section*{1. Annotations and abbreviations} \section*{2. Subject-specific Marking Instructions for A Level Mathematics A} Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded. 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 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. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
If you are in any doubt whatsoever you should contact your Team Leader.
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. \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} Mark for explaining a result or establishing a given result. 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.
d 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.
e 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. If this is not the case please, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
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'. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question.
f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question. (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km , when this would be assumed to be the unspecified unit.) 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. This rule should be applied to each case. When a value is not given in the paper accept any answer that agrees with the correct value to 2 s .f. Follow through should be used so that only one mark is lost for each distinct accuracy error, except for errors due to premature approximation which should be penalised only once in the examination. There is no penalty for using a wrong value for \(g\). E marks will be lost except when results agree to the accuracy required in the question.
g Rules for replaced work: if a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests; if there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.
h 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 mark in the question. Marks designated as cao may be awarded as long as there are no other errors. E marks are lost unless, by chance, the given results are established by equivalent working. 'Fresh starts' will not affect an earlier decision about a misread. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers (provided, of course, that there is nothing in the wording of the question specifying that analytical methods are required). Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader.
j If in any case the scheme operates with considerable unfairness consult your Team Leader. \end{table} PS = Problem Solving
M = Modelling \section*{Summary of Updates} \section*{A Level Mathematics A} \section*{H240/03 Pure Mathematics and Mechanics
Printed Answer Booklet} \section*{Date - Morning/Afternoon} \section*{Time allowed: \(\mathbf { 2 }\) hours} \section*{You must have:}

• Question Paper H240/03 (inserted)

\section*{You may use:}

• a scientific or graphical calculator
  \includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-33_122_473_1052_1248}
  \includegraphics[max width=\textwidth, alt={}, center]{77c6e712-be31-47fb-baa0-4931b7defbf4-33_301_1454_1295_328}

\section*{INSTRUCTIONS}

• The Question Paper will be found inside the Printed Answer Booklet.
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• The acceleration due to gravity is denoted by \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).

\section*{INFORMATION}

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