Questions — OCR (4907 questions)

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OCR M1 2007 June Q4
10 marks 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
    1. the value of m , assuming that the particles move in opposite directions after the impact,
    2. the two possible values of m , assuming that the particles coalesce.
OCR M1 2007 June Q5
11 marks 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
14 marks 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
16 marks 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
    1. the object reaches the floor,
    2. the block reaches the pulley. {}
      7
OCR M1 2010 June Q1
8 marks 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
9 marks 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
9 marks 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
10 marks 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
9 marks 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
13 marks 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
    1. the acceleration of \(B\) and the tension in the string,
    2. 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.
    3. Calculate the total distance \(B\) moves up the slope before coming instantaneously to rest.
OCR M1 2010 June Q7
14 marks 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 {}
6 (ii)
{}
OCR M1 2010 June Q10
Moderate -0.8
10
7
  • 7 (ii)(continued)
    \multirow[t]{26}{*}{7 (iii)}
    \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT
    OCR C3 Q1
    5 marks Standard +0.3
    1. Evaluate
    $$\int _ { 2 } ^ { 15 } \frac { 1 } { \sqrt [ 3 ] { 2 x - 3 } } d x$$
    OCR C3 Q2
    6 marks 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
    7 marks 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 \(Q\).
    OCR C3 Q4
    7 marks Standard +0.8
    4. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0 \\ & \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$
    OCR C3 Q5
    8 marks Standard +0.3
    1. Find the exact value of \(x\) such that $$3 \tan ^ { - 1 } ( x - 2 ) + \pi = 0$$
    2. Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).
    OCR C3 Q6
    8 marks Moderate -0.3
    6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 3 x - 4 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \frac { 2 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$
    1. Evaluate fg(1).
    2. Solve the equation \(\operatorname { gf } ( x ) = 6\).
    3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
    OCR C3 Q7
    8 marks Standard +0.3
    7.
    1. Express \(2 \sin x ^ { \circ } - 3 \cos x ^ { \circ }\) in the form \(R \sin ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
    2. Show that the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ can be written in the form $$2 \sin x ^ { \circ } - 3 \cos x ^ { \circ } = 1$$
    3. Solve the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place.
    OCR C3 Q8
    10 marks Standard +0.3
    8. The functions f and g are defined for all real values of \(x\) by $$\begin{aligned} & \mathrm { f } : x \rightarrow | x - 3 a | \\ & \mathrm { g } : x \rightarrow | 2 x + a | \end{aligned}$$ where \(a\) is a positive constant.
    1. Evaluate fg(-2a).
    2. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), showing the coordinates of any points where each graph meets the coordinate axes.
    3. Solve the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$
    OCR C3 Q9
    13 marks Standard +0.3
    9.
    \includegraphics[max width=\textwidth, alt={}]{49d985bf-7c94-4a54-88c1-c0084cd94000-3_485_945_1119_447}
    The diagram shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).
    1. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
    2. Show that \(1 < q < 2\).
    3. Show that \(q\) is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
    4. Use an iterative process based on the equation above with a starting value of 1.5 to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.
    OCR C3 Q1
    5 marks Standard +0.3
    1. Find the set of values of \(x\) such that
    $$| 2 x - 3 | > | x + 2 |$$
    OCR C3 Q2
    6 marks Standard +0.3
    1. Find, to 2 decimal places, the solutions of the equation
    $$3 \cot ^ { 2 } x - 4 \operatorname { cosec } x + \operatorname { cosec } ^ { 2 } x = 0$$ in the interval \(0 \leq x \leq 2 \pi\).
    OCR C3 Q3
    6 marks Moderate -0.3
    3. A curve has the equation \(x = y ^ { 2 } - 3 \ln 2 y\).
    1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { 2 y ^ { 2 } - 3 }$$
    2. Find an equation for the tangent to the curve at the point where \(y = \frac { 1 } { 2 }\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.