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OCR Stats 1 2018 September Q14
7 marks Standard +0.3
14 A counter is initially at point \(O\) on the \(x\)-axis. A fair coin is thrown 6 times. Each time the coin shows heads, the counter is moved one unit in the positive \(x\)-direction. Each time the coin shows tails, the counter is moved one unit in the negative \(x\)-direction. The final distance of the counter from \(O\), in either direction, is denoted by \(D\). Determine the most probable value of \(D\). \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}
OCR Mechanics 1 2018 September Q1
5 marks Easy -1.8
1
  1. Show that \(4 x ^ { 2 } - 12 x + 3 = 4 \left( x - \frac { 3 } { 2 } \right) ^ { 2 } - 6\).
  2. State the coordinates of the minimum point of the curve \(y = 4 x ^ { 2 } - 12 x + 3\).
OCR Mechanics 1 2018 September Q2
6 marks Standard +0.3
2 A curve has equation \(y = a x ^ { 4 } + b x ^ { 3 } - 2 x + 3\).
  1. Given that the curve has a stationary point where \(x = 2\), show that \(16 a + 6 b = 1\).
  2. Given also that this stationary point is a point of inflection, determine the values of \(a\) and \(b\).
OCR Mechanics 1 2018 September Q3
7 marks Standard +0.3
3
  1. Given that \(\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta\), show that \(6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0\).
  2. In this question you must show detailed reasoning. Solve the equation $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\) correct to 1 decimal place.
  3. Explain why not all the solutions from part (ii) are solutions of the equation $$\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta$$
OCR Mechanics 1 2018 September Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-05_787_892_267_568} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = \begin{cases} 4 - 4 x , & x \leqslant a , \\ \ln ( b x - 8 ) - 2 , & x \geqslant a . \end{cases}$$ The range of f is \(\mathrm { f } ( x ) \geqslant - 2\).
  1. Show that \(a = \frac { 3 } { 2 }\).
  2. Find the value of \(b\).
  3. Find the exact value of \(\mathrm { ff } ( - 1 )\).
  4. Explain why the function f does not have an inverse.
OCR Mechanics 1 2018 September Q5
8 marks Challenging +1.2
5 The curve \(C\) has equation $$3 x ^ { 2 } - 5 x y + \mathrm { e } ^ { 2 y - 4 } + 6 = 0$$ The point \(P\) with coordinates \(( 1,2 )\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\) and the normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(B\). Find the exact area of triangle \(A B P\).
OCR Mechanics 1 2018 September Q6
16 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-06_463_702_264_685} The diagram shows the curve \(C\) with parametric equations $$x = \frac { 1 } { 4 } \sin t , \quad y = t \cos t$$ where \(0 \leqslant t \leqslant k\).
  1. Find the value of \(k\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) in terms of \(t\). The maximum point on \(C\) is denoted by \(P\).
  3. Using your answer to part (ii) and the standard small angle approximations, find an approximation for the \(x\)-coordinate of \(P\).
  4. (a) Show that the area of the finite region bounded by \(C\) and the \(x\)-axis is given by $$b \int _ { 0 } ^ { a } t ( 1 + \cos 2 t ) \mathrm { d } t$$ where \(a\) and \(b\) are constants to be determined.
    (b) In this question you must show detailed reasoning. Hence find the exact area of the finite region bounded by \(C\) and the \(x\)-axis.
OCR Mechanics 1 2018 September Q7
5 marks Moderate -0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-07_512_1072_484_502} The diagram shows the velocity-time graph for a train travelling on a straight level track between stations \(A\) and \(B\) that are 2 km apart. The train leaves \(A\), accelerating uniformly from rest for 400 m until reaching a speed of \(32 \mathrm {~ms} ^ { - 1 }\). The train then travels at this steady speed for \(T\) seconds before decelerating uniformly at \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\). Find the total time for the journey.
OCR Mechanics 1 2018 September Q8
8 marks Standard +0.3
8 At time \(t\) seconds a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When \(t = 2 , P\) is moving parallel to the vector \(\mathbf { i }\).
  1. Show that \(k = - 5\).
  2. Find the values of \(t\) when the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\).
OCR Mechanics 1 2018 September Q9
8 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-08_302_992_260_539} The diagram shows a plank of wood \(A B\), of mass 10 kg and length 6 m , resting with its end \(A\) on rough horizontal ground and its end \(B\) in contact with a fixed cylindrical oil drum. The plank is in a vertical plane perpendicular to the axis of the drum, and the line \(A B\) is a tangent to the circular cross-section of the drum, with the point of contact at \(B\). The plank is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The plank is modelled as a uniform rod and the oil drum is modelled as being smooth.
  1. Find, in terms of \(g\), the normal contact force between the drum and the plank.
  2. Given that the plank is in limiting equilibrium, find the coefficient of friction between the plank and the ground.
OCR Mechanics 1 2018 September Q10
14 marks Standard +0.3
10 A small ball \(P\) is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) and moves freely under gravity. The horizontal and vertically upwards displacements of the ball from \(O\) at any subsequent time \(t\) seconds are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The ball is modelled as a particle and the acceleration due to gravity is taken to be \(10 \mathrm {~ms} ^ { - 2 }\).
  1. Show that the equation of the trajectory of \(P\) is $$y = x \tan \theta - \frac { x ^ { 2 } } { 5 } \left( 1 + \tan ^ { 2 } \theta \right)$$ It is given that \(\tan \theta = 3\).
  2. Using part (i), find the maximum height above the level of \(O\) of \(P\) in the subsequent motion.
  3. Find the values of \(t\) when \(P\) is moving at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 1 } { 3 }\).
  4. Give two possible reasons why the values of \(t\) found in part (iii) may not be accurate. \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-09_435_714_267_678} Two particles \(P\) and \(Q\), with masses 2 kg and 8 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. Plane \(\Pi _ { 1 }\) is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and plane \(\Pi _ { 2 }\) is inclined at an angle of \(\theta\) to the horizontal. Particle \(P\) is on \(\Pi _ { 1 }\) and \(Q\) is on \(\Pi _ { 2 }\) with the string taut (see diagram). \(\Pi _ { 1 }\) is rough and the coefficient of friction between \(P\) and \(\Pi _ { 1 }\) is \(\frac { \sqrt { 3 } } { 3 }\). \(\Pi _ { 2 }\) is smooth.
    The particles are released from rest and \(P\) begins to move towards the pulley with an acceleration of \(g \cos \theta \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  5. Show that \(\theta\) satisfies the equation $$4 \sin \theta - 5 \cos \theta = 1 .$$
  6. By expressing \(4 \sin \theta - 5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), find, correct to 3 significant figures, the tension in the string.
OCR Pure 1 2018 December Q1
4 marks Easy -1.2
1 In this question you must show detailed reasoning. Andrea is comparing the prices charged by two different taxi firms.
Firm A charges \(\pounds 20\) for a 5 mile journey and \(\pounds 30\) for a 10 mile journey, and there is a linear relationship between the price and the length of the journey.
Firm B charges a pick-up fee of \(\pounds 3\) and then \(\pounds 2.40\) for each mile travelled.
Find the length of journey for which both firms would charge the same amount.
OCR Pure 1 2018 December Q2
6 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-4_661_579_831_246} The diagram shows a patio. The perimeter of the patio has to be less than 44 m .
The area of the patio has to be at least \(45 \mathrm {~m} ^ { 2 }\).
  1. Write down, in terms of \(x\), an inequality satisfied by
    1. the perimeter of the patio,
    2. the area of the patio.
  2. Hence determine the set of possible values of \(x\).
OCR Pure 1 2018 December Q3
6 marks Moderate -0.3
3 In this question you must show detailed reasoning.
  1. Given that \(\sin \alpha = \frac { 2 } { 3 }\), find the exact values of \(\cos \alpha\).
  2. Given that \(2 \tan ^ { 2 } \beta - 7 \sec \beta + 5 = 0\), find the exact value of \(\sec \beta\).
OCR Pure 1 2018 December Q4
5 marks Standard +0.8
4 In this question you must show detailed reasoning. Solve the simultaneous equations \(\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 3\) \(\mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { 2 y } = 33\). Give your answer in an exact form.
OCR Pure 1 2018 December Q5
8 marks Moderate -0.8
5
  1. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 4\).
  2. Find the equation of the curve through \(( 2,7 )\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - 4\).
OCR Pure 1 2018 December Q6
10 marks Moderate -0.3
6 In this question you must show detailed reasoning.
A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) defined by \(u _ { 1 } = 500\) and \(u _ { n + 1 } = 0.8 u _ { n }\).
  1. State whether \(S\) is an arithmetic sequence or a geometric sequence, giving a reason for your answer.
  2. Find \(u _ { 20 }\).
  3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
  4. Given that \(\sum _ { n = k } ^ { \infty } u _ { n } = 1024\), find the value of \(k\).
OCR Pure 1 2018 December Q7
9 marks Moderate -0.3
7 As a spherical snowball melts its volume decreases. The rate of decrease of the volume of the snowball at any given time is modelled as being proportional to its volume at that time. Initially the volume of the snowball is \(500 \mathrm {~cm} ^ { 3 }\) and the rate of decrease of its volume is \(20 \mathrm {~cm} ^ { 3 }\) per hour.
  1. Find the time that this model would predict for the snowball's volume to decrease to \(250 \mathrm {~cm} ^ { 3 }\).
  2. Write down one assumption made when using this model.
  3. Comment on how realistic this model would be in the long term.
OCR Pure 1 2018 December Q8
9 marks Standard +0.3
8
  1. Expand \(\sqrt { 1 + 2 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. Hence expand \(\frac { \sqrt { 1 + 2 x } } { 1 + 9 x ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  3. Determine the range of values of \(x\) for which the expansion in part (b) is valid.
OCR Pure 1 2018 December Q9
9 marks Standard +0.3
9 A function f is defined for \(x > 0\) by \(\mathrm { f } ( x ) = \frac { 6 } { x ^ { 2 } + a }\), where \(a\) is a positive constant.
  1. Show that f is a decreasing function.
  2. Find, in terms of \(a\), the coordinates of the point of inflection on the curve \(y = \mathrm { f } ( x )\).
OCR Pure 1 2018 December Q10
13 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-7_524_714_274_246} The diagram shows the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\), which crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).
  1. Determine the coordinates of the points \(A\) and \(B\).
  2. Give full details of a sequence of three geometrical transformations which transform the graph of \(y = \tan ^ { - 1 } x\) to the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\). The equation \(x = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\) has only one root.
  3. Show by calculation that this root lies between \(x = 0\) and \(x = 1\).
  4. Use the iterative formula \(x _ { n + 1 } = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } - \frac { 1 } { 3 } \pi \right)\), with a suitable starting value, to find the root correct to 3 significant figures. Show the result of each iteration.
  5. Using the diagram in the Printed Answer Booklet, show how the iterative process converges to the root.
OCR Pure 1 2018 December Q11
12 marks Standard +0.3
11 In this question you must show detailed reasoning. A function f is given by \(\mathrm { f } ( x ) = \frac { x - 4 } { ( x + 2 ) ( x - 1 ) } + \frac { 3 x + 1 } { ( x + 3 ) ( x - 1 ) }\).
  1. Show that \(\mathrm { f } ( x )\) can be written as \(\frac { 2 ( 2 x + 5 ) } { ( x + 2 ) ( x + 3 ) }\).
  2. Given that \(\int _ { a } ^ { a + 4 } \mathrm { f } ( x ) \mathrm { d } x = 2 \ln 3\), find the value of the positive constant \(a\).
OCR Pure 1 2018 December Q12
9 marks Challenging +1.2
12
  1. By first writing \(\tan 3 \theta\) as \(\tan ( 2 \theta + \theta )\), show that \(\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }\).
  2. Hence show that there are always exactly two different values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) which satisfy the equation \(3 \tan 3 \theta = \tan \theta + k\),
    where \(k\) is a non-zero constant. \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR Stats 1 2018 December Q1
4 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-04_515_732_447_664} The diagram shows the curve \(y = \sqrt { x } - 3\). The shaded region is bounded by the curve and the two axes. Find the exact area of the shaded region. \(2 \mathrm { f } ( x )\) is a cubic polynomial in which the coefficient of \(x ^ { 3 }\) is 1 . The equation \(\mathrm { f } ( x ) = 0\) has exactly two roots.
  1. Sketch a possible graph of \(y = \mathrm { f } ( x )\). It is now given that the two roots are \(x = 2\) and \(x = 3\).
  2. Find, in expanded form, the two possible polynomials \(\mathrm { f } ( x )\).
OCR Stats 1 2018 December Q3
4 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-05_796_1653_260_205} The diagram shows the graph of \(y = \mathrm { g } ( x )\).
In the printed answer booklet, using the same scale as in this diagram, sketch the curves
  1. \(\quad y = \frac { 3 } { 2 } \mathrm {~g} ( x )\),
  2. \(y = \mathrm { g } \left( \frac { 1 } { 2 } x \right)\).