Questions — OCR (4619 questions)

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OCR M1 2007 June Q1
Easy -1.2
1
\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_415_823_264_660} Two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) act at the origin O of rectangular coordinates Oxy (see diagram). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 14 N and 5 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 9 N and 7 N respectively.
  1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis.
OCR M1 2007 June Q2
Moderate -0.8
2
\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_714_1048_1231_552} A particle starts from the point A and travels in a straight line. The diagram shows the ( \(\mathrm { t } , \mathrm { v }\) ) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leqslant t \leqslant 290\).
  1. Find the value of ther which the distance of the particle from A is greatest.
  2. Find the displacement of the particle from A when \(\mathrm { t } = 290\).
  3. Find the total distance travelled by the particle during the interval \(0 \leqslant \mathrm { t } \leqslant 290\).
OCR M1 2007 June Q3
Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_437_846_274_651} A block of mass 50 kg is in equilibrium on smooth horizontal ground with one end of a light wire attached to its upper surface. The other end of the wire is attached to an object of mass mkg . The wire passes over a small smooth pulley, and the object hangs vertically below the pulley. The part of the wire between the block and the pulley makes an angle of \(72 ^ { \circ }\) with the horizontal. A horizontal force of magnitude X N acts on the block in the vertical plane containing the wire (see diagram). The tension in the wire is T N and the contact force exerted by the ground on the block is R N.
  1. By resolving forces on the block vertically, find a relationship between T and R . It is given that the block is on the point of lifting off the ground.
  2. Show that \(\mathrm { T } = 515\), correct to 3 significant figures, and hence find the value of m .
  3. By resolving forces on the block horizontally, write down a relationship between T and X , and hence find the value of \(X\).
OCR M1 2007 June Q4
Standard +0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_149_606_1626_772} Two particles of masses 0.18 kg and m kg move on a smooth horizontal plane. They are moving towards each other in the same straight line when they collide. Immediately before the impact the speeds of the particles are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).
  1. Given that the particles are brought to rest by the impact, find m .
  2. Given instead that the particles move with equal speeds of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after the impact, find
    (a) the value of m , assuming that the particles move in opposite directions after the impact,
    (b) the two possible values of m , assuming that the particles coalesce.
OCR M1 2007 June Q5
Moderate -0.3
5 A particle \(P\) is projected vertically upwards, from horizontal ground, with speed \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the greatest height above the ground reached by P is 3.6 m . A particle Q is projected vertically upwards, from a point 2 m above the ground, with speed \(\mathrm { um } \mathrm { s } ^ { - 1 }\). The greatest height abovetheground reached by Q is also 3.6 m .
  2. Find the value of \(u\). It is given that P and Q are projected simultaneously.
  3. Show that, at the instant when P and Q are at the same height, the particles have the same speed and are moving in opposite directions.
OCR M1 2007 June Q6
Standard +0.3
6 A particle starts from rest at the point A and travels in a straight line. The displacement sm of the particle from A at time ts after leaving A is given by $$s = 0.001 t ^ { 4 } - 0.04 t ^ { 3 } + 0.6 t ^ { 2 } , \quad \text { for } 0 \leqslant t \leqslant 10 .$$
  1. Show that the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(\mathrm { t } = 10\). The acceleration of the particle for \(t \geqslant 10\) is \(( 0.8 - 0.08 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Show that the velocity of the particle is zero when \(\mathrm { t } = 20\).
  3. Find the displacement from A of the particle when \(\mathrm { t } = 20\).
OCR M1 2007 June Q7
Standard +0.3
7
\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-5_488_739_269_703} One end of a light inextensible string is attached to a block of mass 1.5 kg . The other end of the string is attached to an object of mass 1.2 kg . The block is held at rest in contact with a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The string is taut and passes over a small smooth pulley at the bottom edge of the plane. The part of the string above the pulley is parallel to a line of greatest slope of the plane and the object hangs freely below the pulley (see diagram). The block is released and the object moves vertically downwards with acceleration \(\mathrm { am } \mathrm { s } ^ { - 2 }\). The tension in the string is TN . The coefficient of friction between the block and the plane is 0.8 .
  1. Show that the frictional force acting on the block has magnitude 10.98 N , correct to 2 decimal places.
  2. By applying Newton's second law to the block and to the object, find a pair of simultaneous equations in T and a .
  3. Hence show that \(\mathrm { a } = 2.24\), correct to 2 decimal places.
  4. Given that the object is initially 2 m above a horizontal floor and that the block is 2.8 m from the pulley, find the speed of the block at the instant when
    (a) the object reaches the floor,
    (b) the block reaches the pulley. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
    7
OCR M1 2010 June Q1
Moderate -0.8
1 A block \(B\) of mass 3 kg moves with deceleration \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in a straight line on a rough horizontal surface. The initial speed of \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
  1. the time for which \(B\) is in motion,
  2. the distance travelled by \(B\) before it comes to rest,
  3. the coefficient of friction between \(B\) and the surface.
OCR M1 2010 June Q2
Moderate -0.3
2 Two particles \(P\) and \(Q\) are moving in opposite directions in the same straight line on a smooth horizontal surface when they collide. \(P\) has mass 0.4 kg and speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 } . Q\) has mass 0.6 kg and speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the speed of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Given that \(P\) and \(Q\) are moving in the same direction after the collision, find the speed of \(Q\).
  2. Given instead that \(P\) and \(Q\) are moving in opposite directions after the collision, find the distance between them 3 s after the collision.
OCR M1 2010 June Q3
Moderate -0.3
3
\includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-02_570_495_1114_826} Three horizontal forces of magnitudes \(12 \mathrm {~N} , 5 \mathrm {~N}\), and 9 N act along bearings \(000 ^ { \circ } , 150 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram).
  1. Show that the component of the resultant of the three forces along bearing \(270 ^ { \circ }\) has magnitude 6.5 N .
  2. Find the component of the resultant of the three forces along bearing \(000 ^ { \circ }\).
  3. Hence find the magnitude and bearing of the resultant of the three forces.
OCR M1 2010 June Q4
Moderate -0.3
4 A particle \(P\) moving in a straight line has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after passing through a fixed point \(O\). It is given that \(v = 3.2 - 0.2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 5\). Calculate
  1. the value of \(t\) when \(P\) is at instantaneous rest,
  2. the acceleration of \(P\) when it is at instantaneous rest,
  3. the greatest distance of \(P\) from \(O\).
OCR M1 2010 June Q5
Moderate -0.8
5
\includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-03_508_1397_255_374} The diagram shows the ( \(t , v\) ) graph for a lorry delivering waste to a recycling centre. The graph consists of six straight line segments. The lorry reverses in a straight line from a stationary position on a weighbridge before coming to rest. It deposits its waste and then moves forwards in a straight line accelerating to a maximum speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It maintains this speed for 4 s and then decelerates, coming to rest at the weighbridge.
  1. Calculate the distance from the weighbridge to the point where the lorry deposits the waste.
  2. Calculate the time which elapses between the lorry leaving the weighbridge and returning to it.
  3. Given that the acceleration of the lorry when it is moving forwards is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), calculate its final deceleration.
OCR M1 2010 June Q6
Standard +0.3
6 A block \(B\) of mass 0.85 kg lies on a smooth slope inclined at \(30 ^ { \circ }\) to the horizontal. \(B\) is attached to one end of a light inextensible string which is parallel to the slope. At the top of the slope, the string passes over a smooth pulley. The other end of the string hangs vertically and is attached to a particle \(P\) of mass 0.55 kg . The string is taut at the instant when \(P\) is projected vertically downwards.
  1. Calculate
    (a) the acceleration of \(B\) and the tension in the string,
    (b) the magnitude of the force exerted by the string on the pulley. The initial speed of \(P\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after moving \(1.5 \mathrm {~m} P\) reaches the ground, where it remains at rest. \(B\) continues to move up the slope and does not reach the pulley.
  2. Calculate the total distance \(B\) moves up the slope before coming instantaneously to rest.
OCR M1 2010 June Q7
Standard +0.8
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_305_748_260_699} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A rectangular block \(B\) of weight 12 N lies in limiting equilibrium on a horizontal surface. A horizontal force of 4 N and a coplanar force of 5 N inclined at \(60 ^ { \circ }\) to the vertical act on \(B\) (see Fig. 1).
  1. Find the coefficient of friction between \(B\) and the surface. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \(B\) is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is \(\mu\).
  2. Given that the upper block is in limiting equilibrium, find \(\mu\).
  3. Given instead that \(\mu = 0.1\), find the accelerations of the two blocks.
OCR M1 2010 June Q8
Moderate -0.8
8 \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
6 (ii)
\href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR M1 2010 June Q10
Moderate -0.8
10
7
7
  • 7
  		• (continued)
    \multirow[t]{26}{*}{7
  • }
    • \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT
    OCR M1 Specimen Q1
    Easy -1.2
    1
    \includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_99_812_310_635} An engine pulls a truck of mass 6000 kg along a straight horizontal track, exerting a constant horizontal force of magnitude \(E\) newtons on the truck (see diagram). The resistance to motion of the truck has magnitude 400 N , and the acceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(E\).
    OCR M1 Specimen Q2
    Moderate -0.8
    2 \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_166_518_824_351} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-2_168_755_822_1043} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Forces of magnitudes 8 N and 5 N act on a particle. The angle between the directions of the two forces is \(30 ^ { \circ }\), as shown in Fig. 1. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at an angle \(\theta ^ { \circ }\) to the force of magnitude 8 N , as shown in Fig. 2. Find \(R\) and \(\theta\).
    OCR M1 Specimen Q3
    Moderate -0.8
    3 A particle is projected vertically upwards, from the ground, with a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring air resistance, find
    1. the maximum height reached by the particle,
    2. the speed of the particle when it is 30 m above the ground,
    3. the time taken for the particle to fall from its highest point to a height of 30 m ,
    4. the length of time for which the particle is more than 30 m above the ground. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_569_1132_258_516} \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{figure} A woman runs from \(A\) to \(B\), then from \(B\) to \(A\) and then from \(A\) to \(B\) again, on a straight track, taking 90 s . The woman runs at a constant speed throughout. Fig. 1 shows the \(( t , v )\) graph for the woman.
    5. Find the total distance run by the woman.
    6. Find the distance of the woman from \(A\) when \(t = 50\) and when \(t = 80\), \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-3_424_1135_1233_513} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} At time \(t = 0\), a child also starts to move, from \(A\), along \(A B\). The child walks at a constant speed for the first 50 s and then at an increasing speed for the next 40 s . Fig. 2 shows the ( \(t , v\) ) graph for the child; it consists of two straight line segments.
    7. At time \(t = 50\), the woman and the child pass each other, moving in opposite directions. Find the speed of the child during the first 50 s .
    8. At time \(t = 80\), the woman overtakes the child. Find the speed of the child at this instant.
    OCR M1 Specimen Q5
    Standard +0.3
    5 A particle \(P\) moves in a straight line so that, at time \(t\) seconds after leaving a fixed point \(O\), its acceleration is \(- \frac { 1 } { 10 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find, by integration, an expression in terms of \(t\) and \(V\) for the velocity of \(P\).
    2. Find the value of \(V\), given that \(P\) is instantaneously at rest when \(t = 10\).
    3. Find the displacement of \(P\) from \(O\) when \(t = 10\).
    4. Find the speed with which the particle returns to \(O\).
    OCR M1 Specimen Q6
    Standard +0.8
    6
    \includegraphics[max width=\textwidth, alt={}, center]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_168_1032_292_552} Three uniform spheres \(A , B\) and \(C\) have masses \(0.3 \mathrm {~kg} , 0.4 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres lie in a smooth horizontal groove with \(B\) between \(A\) and \(C\). Sphere \(B\) is at rest and spheres \(A\) and \(C\) are each moving with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) (see diagram). Air resistance may be ignored.
    1. \(A\) collides with \(B\). After this collision \(A\) continues to move in the same direction as before, but with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed with which \(B\) starts to move.
    2. \(B\) and \(C\) then collide, after which they both move towards \(A\), with speeds of \(3.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).
    3. The next collision is between \(A\) and \(B\). Explain briefly how you can tell that, after this collision, \(A\) and \(B\) cannot both be moving towards \(C\).
    4. When the spheres have finished colliding, which direction is \(A\) moving in? What can you say about its speed? Justify your answers.
    OCR M1 Specimen Q7
    Standard +0.3
    7 A sledge of mass 25 kg is on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the sledge and the plane is 0.2 .
    1. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_289_488_1493_849} \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{figure} The sledge is pulled up the plane, with constant acceleration, by means of a light cable which is parallel to a line of greatest slope (see Fig. 1). The sledge starts from rest and acquires a speed of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after being pulled for 10 s . Ignoring air resistance, find the tension in the cable.
    2. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{463347e9-b850-4f4a-b2d2-423cf142e30f-4_291_490_2149_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} On a subsequent occasion the cable is not in use and two people of total mass 150 kg are seated in the sledge. The sledge is held at rest by a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. Find the least value of \(P\) which will prevent the sledge from sliding down the plane.
    OCR C3 Q1
    Standard +0.3
    1. Evaluate
    $$\int _ { 2 } ^ { 15 } \frac { 1 } { \sqrt [ 3 ] { 2 x - 3 } } d x$$
    OCR C3 Q2
    Standard +0.3
    2.
    \includegraphics[max width=\textwidth, alt={}]{49d985bf-7c94-4a54-88c1-c0084cd94000-1_563_833_532_513}
    The diagram shows the curve with equation \(y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0\).
    The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
    Find the volume of the solid formed when the shaded region is rotated through four right angles about the \(x\)-axis, giving your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are integers.
    OCR C3 Q3
    Standard +0.3
    3. A curve has the equation \(y = ( 3 x - 5 ) ^ { 3 }\).
    1. Find an equation for the tangent to the curve at the point \(P ( 2,1 )\). The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
    2. Find the coordinates of \(Q\).