Questions — OCR Mechanics 1 (36 questions)

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OCR Mechanics 1 2018 March Q1
1 Show in a sketch the region of the \(x - y\) plane within which all three of the following inequalities are satisfied. $$3 y \geqslant 4 x \quad y - x \leqslant 1 \quad y \geqslant ( x - 1 ) ^ { 2 }$$ You should indicate the region for which the inequalities hold by labelling the region \(R\).
OCR Mechanics 1 2018 March Q2
2 The first term of a geometric progression is 12 and the second term is 9 .
  1. Find the fifth term. The sum of the first \(N\) terms is denoted by \(S _ { N }\) and the sum to infinity is denoted by \(S _ { \infty }\). It is given that the difference between \(S _ { \infty }\) and \(S _ { N }\) is at most 0.0096 .
  2. Show that \(\left( \frac { 3 } { 4 } \right) ^ { N } \leqslant 0.0002\).
  3. Use logarithms to find the smallest possible value of \(N\).
OCR Mechanics 1 2018 March Q3
3 A sequence of three transformations maps the curve \(y = \ln x\) to the curve \(y = \mathrm { e } ^ { 3 x } - 5\). Give details of these transformations.
OCR Mechanics 1 2018 March Q4
4 A curve is defined, for \(t \geqslant 0\), by the parametric equations $$x = t ^ { 2 } , \quad y = t ^ { 3 }$$
  1. Show that the equation of the tangent at the point with parameter \(t\) is $$2 y = 3 t x - t ^ { 3 } .$$
  2. In this question you must show detailed reasoning. It is given that this tangent passes through the point \(A \left( \frac { 19 } { 12 } , - \frac { 15 } { 8 } \right)\) and it meets the \(x\)-axis at the point \(B\).
    Find the area of triangle \(O A B\), where \(O\) is the origin.
OCR Mechanics 1 2018 March Q5
5 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}]{467d7747-6a07-40ea-bb47-41ea3117f233-5_392_1102_319_466}
The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 3 x \right) \mathrm { e } ^ { - x }$$ The diagram shows the curve \(y = \mathrm { f } ( x )\).
  1. Find the range of f.
  2. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm { g } ( x ) = \left( 2 x ^ { 2 } - 3 x \right) \mathrm { e } ^ { - x } .$$ Given that g is a one-one function, state the least possible value of \(k\).
  3. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis.
  4. Determine the values of \(p\) and \(q\) for which $$x ^ { 2 } - 6 x + 10 \equiv ( x - p ) ^ { 2 } + q .$$
  5. Use the substitution \(x - p = \tan u\), where \(p\) takes the value found in part (i), to evaluate $$\int _ { 3 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 6 x + 10 } \mathrm {~d} x .$$
  6. Determine the value of $$\int _ { 3 } ^ { 4 } \frac { x } { x ^ { 2 } - 6 x + 10 } \mathrm {~d} x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants to be determined.
OCR Mechanics 1 2018 March Q7
7 Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle are given by $$\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - 2 a \mathbf { j } ) \mathrm { N } , \quad \mathbf { F } _ { 2 } = ( 2 b \mathbf { i } + 3 a \mathbf { j } ) \mathrm { N } \quad \text { and } \quad \mathbf { F } _ { 3 } = ( - 2 \mathbf { i } + b \mathbf { j } ) \mathrm { N } .$$ The particle is in equilibrium under the action of these three forces.
Find the value of \(a\) and the value of \(b\).
OCR Mechanics 1 2018 March Q8
8 A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The jogger then runs at a constant velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before decelerating at a constant rate of \(0.08 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) back to rest. The jogger runs a total distance of 880 m in 250 s .
  1. Sketch the velocity-time graph for the jogger's journey.
  2. Show that \(3 V ^ { 2 } - 100 V + 352 = 0\).
  3. Hence find the value of \(V\), giving a reason for your answer.
OCR Mechanics 1 2018 March Q9
9 Two particles \(A\) and \(B\) have position vectors \(\mathbf { r } _ { A }\) metres and \(\mathbf { r } _ { B }\) metres at time \(t\) seconds, where $$\mathbf { r } _ { A } = t ^ { 2 } \mathbf { i } + ( 3 t - 1 ) \mathbf { j } \quad \text { and } \quad \mathbf { r } _ { B } = \left( 1 - 2 t ^ { 2 } \right) \mathbf { i } + \left( 3 t - 2 t ^ { 2 } \right) \mathbf { j } , \quad \text { for } t \geqslant 0$$
  1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed.
  2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d ^ { 2 } = 13 t ^ { 4 } - 10 t ^ { 2 } + 2$$
  3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion.
OCR Mechanics 1 2018 March Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{467d7747-6a07-40ea-bb47-41ea3117f233-7_442_1006_251_532} A uniform \(\operatorname { rod } A B\), of weight \(W \mathrm {~N}\) and length \(2 a \mathrm {~m}\), rests with the end \(A\) on a rough horizontal table. A small object of weight \(2 W \mathrm {~N}\) is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).
  1. Find the least possible value of the coefficient of friction between the rod and the table.
  2. Given that the magnitude of the contact force at \(A\) is \(\sqrt { 39 } \mathrm {~N}\), find the value of \(W\).
OCR Mechanics 1 2018 March Q11
11 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{467d7747-6a07-40ea-bb47-41ea3117f233-7_239_1164_1299_452} A football \(P\) is kicked with speed \(25 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\) from a point \(A\) on horizontal ground. At the same instant a second football \(Q\) is kicked with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(2 \alpha\) from a point \(B\) on the same horizontal ground, where \(A B = 72 \mathrm {~m}\). The footballs are modelled as particles moving freely under gravity in the same vertical plane and they collide with each other at the point \(C\) (see diagram).
  1. Calculate the height of \(C\) above the ground.
  2. Find the direction of motion of \(P\) at the moment of impact.
  3. Suggest one improvement that could be made to the model. \section*{OCR} Oxford Cambridge and RSA
OCR Mechanics 1 2018 September Q1
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
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
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
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
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
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
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 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
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
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 Mechanics 1 2018 December Q1
1 Use logarithms to solve the equation \(2 ^ { 3 x - 1 } = 3 ^ { x + 4 }\), giving your answer correct to 3 significant figures.
OCR Mechanics 1 2018 December Q2
2 In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac { 2 } { 3 } x ^ { 3 } + \frac { 5 } { 2 } x ^ { 2 } - 3 x + 7\) is positive. Give your answer in set notation.
OCR Mechanics 1 2018 December Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-04_695_714_1087_248} The diagram shows a circle with centre \(( a , - a )\) that passes through the origin.
  1. Write down an equation for the circle in terms of \(a\).
  2. Given that the point \(( 1 , - 5 )\) lies on the circle, find the exact area of the circle.
OCR Mechanics 1 2018 December Q4
4 The first three terms of an arithmetic series are \(9 p , 8 p - 3,5 p\) respectively, where \(p\) is a constant.
Given that the sum of the first \(n\) terms of this series is - 1512 , find the value of \(n\).
OCR Mechanics 1 2018 December Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-05_444_757_548_251} The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 3 }\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { - 2 x }\). The diagram shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real root \(\alpha\).
  1. Show that \(\alpha\) satisfies the equation \(\mathrm { h } ( x ) = 0\), where \(\mathrm { h } ( x ) = x ^ { 2 } + 3 - x \mathrm { e } ^ { 2 x }\).
  2. Hence show that a Newton-Raphson iterative formula for finding \(\alpha\) can be written in the form $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } \left( 1 - 2 \mathrm { e } ^ { 2 x _ { n } } \right) - 3 } { 2 x _ { n } - \left( 1 + 2 x _ { n } \right) \mathrm { e } ^ { 2 x _ { n } } } .$$
  3. Use this iterative formula, with a suitable initial value, to find \(\alpha\) correct to 3 decimal places. Show the result of each iteration. \section*{(d) In this question you must show detailed reasoning.} Find the exact value of \(x\) for which \(\operatorname { fg } ( x ) = \frac { 2 } { 13 }\).