Questions — OCR (4907 questions)

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OCR C2 Q2
6 marks Standard +0.8
2. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593} The diagram shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).
OCR C2 Q3
7 marks Standard +0.3
3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n + 1 } = \left( u _ { n } \right) ^ { 2 } - 1 , \quad n \geq 1 .$$ Given that \(u _ { 1 } = k\), where \(k\) is a constant,
  1. find expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given also that \(u _ { 2 } + u _ { 3 } = 11\),
  2. find the possible values of \(k\).
OCR C2 Q4
7 marks Moderate -0.8
4. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-2_465_844_246_516} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } + 1 }\).
The shaded region \(R\) is bounded by the curve, the coordinate axes and the line \(x = 2\).
  1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
  2. find an estimate for the volume of the support.
OCR C2 Q5
7 marks Moderate -0.3
5. (i) Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$ (ii) Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.
OCR C2 Q6
9 marks Moderate -0.8
6. (i) Evaluate $$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$ (ii) Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1$$ and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).
OCR C2 Q7
9 marks Moderate -0.3
7. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-3_499_721_248_552} The diagram shows part of the curve \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { 1 - 8 x ^ { 3 } } { x ^ { 2 } } , x \neq 0\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
  3. Find the area of the shaded region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = 2\).
OCR C2 Q8
10 marks Moderate -0.3
8. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month. In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, forming an arithmetic sequence. Given that sales total \(\pounds 8100\) during the first six months, use the model to
  1. find the value of \(x\),
  2. find the expected value of sales in the eighth month,
  3. show that the expected total of sales in pounds during the first \(n\) months is given by \(k n ( 51 - n )\), where \(k\) is an integer to be found.
  4. Explain why this model cannot be valid over a long period of time.
OCR C2 Q9
11 marks Standard +0.3
9. \(f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + x + 2\).
  1. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\).
  4. Find, in terms of \(\pi\), the values of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) for which $$2 \sin ^ { 3 } \theta - 5 \sin ^ { 2 } \theta + \sin \theta + 2 = 0$$
OCR M1 2005 June Q1
7 marks Standard +0.3
1
[diagram]
A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m \mathrm {~kg}\) is threaded on the string and is pulled by a force of magnitude 1.6 N acting upwards at \(45 ^ { \circ }\) to the horizontal. The section \(A R\) of the string makes an angle of \(30 ^ { \circ }\) with the downward vertical and the section \(B R\) is horizontal (see diagram). The ring is in equilibrium with the string taut.
  1. Give a reason why the tension in the part \(A R\) of the string is the same as that in the part \(B R\).
  2. Show that the tension in the string is 0.754 N , correct to 3 significant figures.
  3. Find the value of \(m\).
OCR M1 2005 June Q2
7 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-2_643_289_1475_927} Particles \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.25 N (see diagram). The downward acceleration of each of the particles is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the string is \(T \mathrm {~N}\).
  1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\).
  2. Find the values of \(a\) and \(T\).
OCR M1 2005 June Q3
8 marks Moderate -0.8
3 Two small spheres \(P\) and \(Q\) have masses 0.1 kg and 0.2 kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) with speed \(( 3.5 - u ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the value of \(u\). After the collision the spheres both move with deceleration of magnitude \(5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until they come to rest on the plane.
  2. Find the distance \(P Q\) when both \(P\) and \(Q\) are at rest.
OCR M1 2005 June Q4
9 marks Standard +0.3
4 A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u \mathrm {~ms} ^ { - 1 }\). The particle travels 2 m during the first 0.8 s after passing through \(P\), then a further 6 m in the next 1.2 s . Find
  1. the value of \(u\) and the acceleration of the particle,
  2. the value of \(\alpha\) in degrees.
OCR M1 2005 June Q5
12 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-3_697_579_1238_781} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(A P\) and \(B P\) of the string are taut. The system is in equilibrium with angle \(B A P = \alpha\) and angle \(A B P = \beta\) (see diagram). The weight of \(A\) is 2 N and the tensions in the parts \(A P\) and \(B P\) of the string are 7 N and \(T \mathrm {~N}\) respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.
  1. Find the coefficient of friction between the wire and the ring \(A\).
  2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\).
  3. Given that there is no frictional force acting on \(B\), find the mass of \(B\).
OCR M1 2005 June Q6
12 marks Moderate -0.3
6 A particle of mass 0.04 kg is acted on by a force of magnitude \(P \mathrm {~N}\) in a direction at an angle \(\alpha\) to the upward vertical.
  1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20 ^ { \circ }\) find
    1. the value of \(P\),
    2. the magnitude of the resultant,
    3. the magnitude of the acceleration of the particle.
    4. It is given instead that \(P = 0.08\) and \(\alpha = 90 ^ { \circ }\). Find the magnitude and direction of the resultant force on the particle.
OCR M1 2005 June Q7
17 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-4_634_1127_934_507} A car \(P\) starts from rest and travels along a straight road for 600 s . The \(( t , v )\) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find
  1. the distance travelled by \(P\),
  2. the deceleration of \(P\) during the interval \(500 < t < 600\). Another car \(Q\) starts from rest at the same instant as \(P\) and travels in the same direction along the same road for 600 s . At time \(t \mathrm {~s}\) after starting the velocity of \(Q\) is \(\left( 600 t ^ { 2 } - t ^ { 3 } \right) \times 10 ^ { - 6 } \mathrm {~ms} ^ { - 1 }\).
  3. Find an expression in terms of \(t\) for the acceleration of \(Q\).
  4. Find how much less \(Q\) 's deceleration is than \(P\) 's when \(t = 550\).
  5. Show that \(Q\) has its maximum velocity when \(t = 400\).
  6. Find how much further \(Q\) has travelled than \(P\) when \(t = 400\).
OCR M1 2006 June Q1
5 marks Moderate -0.8
1 Each of two wagons has an unloaded mass of 1200 kg . One of the wagons carries a load of mass \(m \mathrm {~kg}\) and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(m\).
OCR M1 2006 June Q2
6 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-2_620_711_543_717} Forces of magnitudes 6.5 N and 2.5 N act at a point in the directions shown. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at right angles to the force of magnitude 2.5 N (see diagram).
  1. Show that \(\theta = 22.6 ^ { \circ }\), correct to 3 significant figures.
  2. Find the value of \(R\).
OCR M1 2006 June Q3
11 marks Moderate -0.8
3 A man travels 360 m along a straight road. He walks for the first 120 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), runs the next 180 m at \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then walks the final 60 m at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The man's displacement from his starting point after \(t\) seconds is \(x\) metres.
  1. Sketch the \(( t , x )\) graph for the journey, showing the values of \(t\) for which \(x = 120,300\) and 360 . A woman jogs the same 360 m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.
  2. Draw a dotted line on your ( \(t , x\) ) graph to represent the woman's journey.
  3. Calculate the value of \(t\) at which the man overtakes the woman.
OCR M1 2006 June Q4
10 marks Moderate -0.8
4 A cyclist travels along a straight road. Her velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds after starting from a point \(O\), is given by $$\begin{aligned} & v = 2 \quad \text { for } 0 \leqslant t \leqslant 10 \\ & v = 0.03 t ^ { 2 } - 0.3 t + 2 \quad \text { for } t \geqslant 10 . \end{aligned}$$
  1. Find the displacement of the cyclist from \(O\) when \(t = 10\).
  2. Show that, for \(t \geqslant 10\), the displacement of the cyclist from \(O\) is given by the expression \(0.01 t ^ { 3 } - 0.15 t ^ { 2 } + 2 t + 5\).
  3. Find the time when the acceleration of the cyclist is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Hence find the displacement of the cyclist from \(O\) when her acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
OCR M1 2006 June Q5
11 marks Moderate -0.3
5 A block of mass \(m \mathrm {~kg}\) is at rest on a horizontal plane. The coefficient of friction between the block and the plane is 0.2 .
  1. When a horizontal force of magnitude 5 N acts on the block, the block is on the point of slipping. Find the value of \(m\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-3_312_711_1244_758} When a force of magnitude \(P \mathrm {~N}\) acts downwards on the block at an angle \(\alpha\) to the horizontal, as shown in the diagram, the frictional force on the block has magnitude 6 N and the block is again on the point of slipping. Find
    1. the value of \(\alpha\) in degrees,
    2. the value of \(P\).
OCR M1 2006 June Q6
14 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-4_314_997_267_577} A train of total mass 80000 kg consists of an engine \(E\) and two trucks \(A\) and \(B\). The engine \(E\) and truck \(A\) are connected by a rigid coupling \(X\), and trucks \(A\) and \(B\) are connected by another rigid coupling \(Y\). The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on \(E , A\) and \(B\) are \(10500 \mathrm {~N} , 3000 \mathrm {~N}\) and 1500 N respectively (see diagram).
  1. By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than 15000 N .
  2. Show that, when the magnitude of the driving force is 35000 N , the acceleration of the train is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  3. Hence find the mass of \(E\), given that the tension in the coupling \(X\) is 8500 N when the magnitude of the driving force is 35000 N . The driving force is replaced by a braking force of magnitude 15000 N acting on the engine. The force exerted by the coupling \(Y\) is zero.
  4. Find the mass of \(B\).
  5. Show that the coupling \(X\) exerts a forward force of magnitude 1500 N on the engine.
OCR M1 2006 June Q7
15 marks Standard +0.3
7 A particle of mass 0.1 kg is at rest at a point \(A\) on a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The particle is given an initial velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .
  1. Find the coefficient of friction between the particle and the plane.
  2. Show that, after coming to instantaneous rest, the particle moves down the plane.
  3. Find the speed with which the particle passes through \(A\) during its downward motion.
OCR M1 2007 June Q1
6 marks 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
7 marks 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
8 marks 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 \(X\).