Questions — OCR H240/03 (134 questions)

4 For a small angle \(\theta\), where \(\theta\) is in radians, show that \(1 + \cos \theta - 3 \cos ^ { 2 } \theta \approx - 1 + \frac { 5 } { 2 } \theta ^ { 2 }\).

5

1. Find the first three terms in the expansion of \(( 1 + p x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. The expansion of \(( 1 + q x ) ( 1 + p x ) ^ { \frac { 1 } { 3 } }\) is \(1 + x - \frac { 2 } { 9 } x ^ { 2 } + \ldots\). Find the possible values of the constants \(p\) and \(q\).

6 A curve has equation \(y = x ^ { 2 } + k x - 4 x ^ { - 1 }\) where \(k\) is a constant. Given that the curve has a minimum point when \(x = - 2\)

• find the value of \(k\)
• show that the curve has a point of inflection which is not a stationary point.

7

1. Find \(\int 5 x ^ { 3 } \sqrt { x ^ { 2 } + 1 } \mathrm {~d} x\).
2. Find \(\int \theta \tan ^ { 2 } \theta \mathrm {~d} \theta\). You may use the result \(\int \tan \theta \mathrm { d } \theta = \ln | \sec \theta | + c\).

8 In this question you must show detailed reasoning. The diagram shows triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-06_737_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]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-07_323_755_548_283}

1. Calculate the magnitude of the resultant force on the particle.
2. 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 \(g\).

11 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively. A particle of mass 0.12 kg is moving so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds is given by \(\mathbf { r } = 2 t ^ { 3 } \mathbf { i } + \left( 5 t ^ { 2 } - 4 t \right) \mathbf { j }\).

1. Show that when \(t = 0.7\) the bearing on which the particle is moving is approximately \(044 ^ { \circ }\).
2. Find the magnitude of the resultant force acting on the particle at the instant when \(t = 0.7\).
3. Determine the times at which the particle is moving on a bearing of \(045 ^ { \circ }\).

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]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-09_375_1207_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]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-10_398_844_868_306}

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 \(x\). 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 \section*{...day June 20XX - Morning/Afternoon} A Level Mathematics A
H240/03 Pure Mathematics and Mechanics \section*{SAMPLE MARK SCHEME} MAXIMUM MARK 100 \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-13_259_1320_1242_826} \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. PS = Problem Solving
M = Modelling

\includegraphics{figure_1} The diagram shows triangle \(ABC\), with \(AC = 13.5\) cm, \(BC = 8.3\) cm and angle \(ABC = 32°\). Find angle \(CAB\). [2]

A circle with centre \(C\) has equation \(x^2 + y^2 - 6x + 4y + 4 = 0\).

1. Find
   1. the coordinates of \(C\), [2]
   2. the radius of the circle. [1]
2. Determine the set of values of \(k\) for which the line \(y = kx - 3\) does not intersect or touch the circle. [5]

\includegraphics{figure_4} The diagram shows the part of the curve \(y = 3x \sin 2x\) for which \(0 \leqslant x \leqslant \frac{1}{2}\pi\). The maximum point on the curve is denoted by \(P\).

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\tan 2x + 2x = 0\). [3]
2. Use the Newton-Raphson method, with a suitable initial value, to find the \(x\)-coordinate of \(P\), giving your answer correct to 4 decimal places. Show the result of each iteration. [4]
3. The trapezium rule, with four strips of equal width, is used to find an approximation to $$\int_0^{\frac{1}{2}\pi} 3x \sin 2x \, dx.$$ Show that the result can be expressed as \(k\pi^2(\sqrt{2} + 1)\), where \(k\) is a rational number to be determined. [4]
4. 1. Evaluate \(\int_0^{\frac{1}{2}\pi} 3x \sin 2x \, dx\). [1]
   2. Hence determine whether using the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region enclosed by the curve \(y = 3x \sin 2x\) and the \(x\)-axis for \(0 \leqslant x \leqslant \frac{1}{2}\pi\). [1]
   3. Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region in this case. [1]

In this question you must show detailed reasoning.

1. Prove that \((\cot \theta + \cosec \theta)^2 = \frac{1 + \cos \theta}{1 - \cos \theta}\). [4]
2. Hence solve, for \(0 < \theta < 2\pi\), \(3(\cot \theta + \cosec \theta)^2 = 2 \sec \theta\). [5]

\includegraphics{figure_6} The diagram shows part of the curve \(y = \frac{2x - 1}{(2x + 3)(x + 1)^2}\). Find the exact area of the shaded region, giving your answer in the form \(p + q \ln r\), where \(p\) and \(q\) are positive integers and \(r\) is a positive rational number. [10]

A cyclist starting from rest accelerates uniformly at \(1.5 \text{ m s}^{-2}\) for \(4\) s and then travels at constant speed.

1. Sketch a velocity-time graph to represent the first \(10\) seconds of the cyclist's motion. [2]
2. Calculate the distance travelled by the cyclist in the first \(10\) seconds. [2]

A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after \(2.4\) seconds. The horizontal component of the initial velocity of \(P\) is \(\frac{5}{3}d \text{ m s}^{-1}\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground. [1]
2. Find the vertical component of the initial velocity of \(P\). [2]

\(P\) just clears a vertical wall which is situated at a horizontal distance \(d\) m from \(O\).

1. Find the height of the wall. [3]

The speed of \(P\) as it passes over the wall is \(16 \text{ m s}^{-1}\).

1. Find the value of \(d\) correct to \(3\) significant figures. [4]

\includegraphics{figure_9} The diagram shows a small block \(B\), of mass \(0.2\) kg, and a particle \(P\), of mass \(0.5\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of \(\theta\) with the horizontal where \(\tan \theta = \frac{3}{4}\). The system is released from rest. In the first \(0.4\) seconds of the motion \(P\) moves \(0.3\) m down the plane and \(B\) does not reach the pulley.

1. Find the tension in the string during the first \(0.4\) seconds of the motion. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(R\) of mass \(2\) kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, the velocity \(\mathbf{v} \text{ m s}^{-1}\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf{v} = (pt^2 - 3t)\mathbf{i} + (8t + q)\mathbf{j}\), where \(p\) and \(q\) are constants and \(p < 0\).

1. Given that when \(t = 0.5\) the magnitude of \(\mathbf{F}\) is \(20\), find the value of \(p\). [6]

When \(t = 0\), \(R\) is at the point with position vector \((2\mathbf{i} - 3\mathbf{j})\) m.

1. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). [4]

When \(t = 1\), \(R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2\mathbf{i} - 8\mathbf{j}\).

1. Find the value of \(q\). [3]

The diagram shows a ladder \(AB\), of length \(2a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30°\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2m\) is placed on the ladder at a point \(C\) where \(AC = d\). The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.

1. Show that the normal contact force between the ladder and the wall is \(\frac{mg(a + 2d)\sqrt{3}}{4h}\). [4]

It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac{1}{3}\sqrt{3}\).

1. Show that \(h = k(a + 2d)\), where \(k\) is a constant to be determined. [7]
2. Hence find, in terms of \(a\), the greatest possible value of \(d\). [2]
3. State one improvement that could be made to the model. [1]

Triangle \(ABC\) has \(AB = 8.5\) cm, \(BC = 6.2\) cm and angle \(B = 35°\). Calculate the area of the triangle. [2]

A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]

The functions f and g are defined for all real values of x by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).

1. Find the range of f. [3]
2. Give a reason why f has no inverse. [1]
3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]