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CAIE M1 2012 November Q5
8 marks Standard +0.3
  1. Find the value of \(\theta\). At time 4.8 s after leaving \(A\), the particle comes to rest at \(C\).
  2. Find the coefficient of friction between \(P\) and the rough part of the plane.
CAIE M1 2014 November Q6
9 marks Standard +0.3
  1. the work done against the frictional force acting on \(B\),
  2. the loss of potential energy of the system,
  3. the gain in kinetic energy of the system. At the instant when \(B\) has moved 0.9 m the string breaks. \(A\) is at a height of 0.54 m above a horizontal floor at this instant.
    (ii) Find the speed with which \(A\) reaches the floor. \(6 \quad A B C\) is a line of greatest slope of a plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). The point \(A\) is at the top of the plane, the point \(C\) is at the bottom of the plane and the length of \(A C\) is 5 m . The part of the plane above the level of \(B\) is smooth and the part below the level of \(B\) is rough. A particle \(P\) is released from rest at \(A\) and reaches \(C\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between \(P\) and the part of the plane below \(B\) is 0.5 . Find
    1. the acceleration of \(P\) while moving
      1. from \(A\) to \(B\),
      2. from \(B\) to \(C\),
      3. the distance \(A B\),
      4. the time taken for \(P\) to move from \(A\) to \(C\).
CAIE M2 2010 June Q5
8 marks Standard +0.8
  1. It is given that when the ball moves with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the tension in the string \(Q B\) is three times the tension in the string \(P B\). Calculate the radius of the circle. The ball now moves along this circular path with the minimum possible speed.
  2. State the tension in the string \(P B\) in this case, and find the speed of the ball.
CAIE M2 2019 March Q6
8 marks Challenging +1.2
  1. Find, in terms of \(r\), the distance of the centre of mass of the prism from the centre of the cylinder.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-11_633_729_258_708} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism has weight \(W \mathrm {~N}\) and is placed with its curved surface on a rough horizontal plane. The axis of symmetry of the cross-section makes an angle of \(30 ^ { \circ }\) with the vertical. A horizontal force of magnitude \(P \mathrm {~N}\) acting in the plane of the cross-section through the centre of mass is applied to the cylinder at the highest point of this cross-section (see Fig. 2). The prism rests in limiting equilibrium.
  2. Find the coefficient of friction between the prism and the plane.
CAIE M2 2003 November Q4
10 marks Standard +0.3
  1. Show that the distance of the centre of mass of the lamina from the side \(B C\) is 6.37 cm . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-3_671_608_1050_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lamina is smoothly hinged to a wall at \(A\) and is supported, with \(A B\) horizontal, by a light wire attached at \(B\). The other end of the wire is attached to a point on the wall, vertically above \(A\), such that the wire makes an angle of \(30 ^ { \circ }\) with \(A B\) (see Fig. 2). The mass of the lamina is 8 kg . Find
  2. the tension in the wire,
  3. the magnitude of the vertical component of the force acting on the lamina at \(A\).
CAIE M2 2008 November Q4
7 marks Standard +0.3
  1. the base of the cylinder,
  2. the curved surface of the cylinder.
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_348_745_1183_740} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Sphere \(A\) is now attached to one end of a light inextensible string. The string passes through a small smooth hole in the middle of the base of the cylinder. Another small sphere \(B\), of mass 0.25 kg , is attached to the other end of the string. \(B\) hangs in equilibrium below the hole while \(A\) is moving in a horizontal circle of radius 0.2 m (see Fig. 2). Find the angular speed of \(A\).
CAIE M2 2012 November Q4
8 marks Challenging +1.2
  1. Find \(r\). The upper cylinder is now fixed to the lower cylinder to create a uniform object.
  2. Show that the centre of mass of the object is $$\frac { 25 h ^ { 2 } + 180 h + 81 } { 50 h + 180 } \mathrm {~m}$$ from \(A\). The object is placed with the plane face containing \(A\) in contact with a rough plane inclined at \(\alpha ^ { \circ }\) to the horizontal, where \(\tan \alpha = 0.5\). The object is on the point of toppling without sliding.
  3. Calculate \(h\).
CAIE Further Paper 3 2022 November Q3
7 marks Challenging +1.2
  1. Show that \(\mathrm { N } = \frac { 8 } { 15 } \mathrm {~W} ( 1 + 2 \mathrm { k } )\).
  2. Find the value of \(k\).
CAIE S1 2006 June Q6
9 marks Easy -1.2
  1. How many teams play in only 1 match?
  2. How many teams play in exactly 2 matches?
  3. Draw up a frequency table for the numbers of matches which the teams play.
  4. Calculate the mean and variance of the numbers of matches which the teams play.
CAIE S2 2019 June Q6
9 marks Standard +0.3
  1. Show that \(b = \frac { a } { a - 1 }\).
  2. Given that the median of \(X\) is \(\frac { 3 } { 2 }\), find the values of \(a\) and \(b\).
  3. Use your values of \(a\) and \(b\) from part (ii) to find \(\mathrm { E } ( X )\).
CAIE Further Paper 4 2023 November Q4
9 marks Standard +0.3
  1. Given that \(\mathrm { P } ( X \leqslant 2 ) = \frac { 1 } { 3 }\), show that \(m = \frac { 1 } { 6 }\) and find the values of \(k\) and \(c\).
  2. Find the exact numerical value of the interquartile range of \(X\).
Edexcel C12 2018 January Q8
6 marks Moderate -0.5
  1. \(y = \mathrm { f } ( - x )\)
  2. \(y = \mathrm { f } ( 2 x )\) On each diagram, show clearly the coordinates of any points of intersection of the curve with the two coordinate axes and the coordinates of the stationary points.
Edexcel C1 2006 January Q6
9 marks Moderate -0.8
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = 2 \mathrm { f } ( x )\),
  3. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). On each diagram show clearly the coordinates of all the points where the curve meets the axes.
Edexcel C2 2013 June Q7
9 marks Moderate -0.3
  1. Find by calculation the \(x\)-coordinate of \(A\) and the \(x\)-coordinate of \(B\). The shaded region \(R\) is bounded by the line with equation \(y = 10\) and the curve as shown in Figure 1.
  2. Use calculus to find the exact area of \(R\).
Edexcel P3 2022 October Q8
9 marks Standard +0.3
  1. Express \(8 \sin x - 15 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures. $$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
  2. Find
    1. the minimum value of \(\mathrm { f } ( x )\)
    2. the smallest value of \(x\) at which this minimum value occurs.
  3. State the \(y\) coordinate of the minimum points on the curve with equation $$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
  4. State the smallest value of \(x\) at which a maximum point occurs for the curve with equation $$y = - \mathrm { f } ( 2 x ) \quad x > 0$$ \section*{8. In this question you must show all stages of your working.
    In this question you must show all stages of your working.}
Edexcel P4 2021 October Q8
7 marks Standard +0.8
  1. Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\) Figure 3 shows a sketch of part of the curve with equation $$y = x \ln x \quad x > 0$$ The region \(R\), shown shaded in Figure 3, lies entirely above the \(x\)-axis and is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Find the exact volume of the solid formed, giving your answer in simplest form. \section*{8. In this question you must show all stages of your working.
    In this question you must show all stages of your working.}
Edexcel C4 2013 January Q5
15 marks Moderate -0.3
  1. Show that \(A\) has coordinates \(( 0,3 )\).
  2. Find the \(x\) coordinate of the point \(B\).
  3. Find an equation of the normal to \(C\) at the point \(A\). The region \(R\), as shown shaded in Figure 2, is bounded by the curve \(C\), the line \(x = - 1\) and the \(x\)-axis.
  4. Use integration to find the exact area of \(R\).
Edexcel M3 2003 January Q3
10 marks Challenging +1.2
  1. Show that the distance \(d\) of the centre of mass of the toy from its lowest point \(O\) is given by $$d = \frac { h ^ { 2 } + 2 h r + 5 r ^ { 2 } } { 2 ( h + 4 r ) } .$$ When the toy is placed with any point of the curved surface of the hemisphere resting on the plane it will remain in equilibrium.
  2. Find \(h\) in terms of \(r\).
    (3)
Edexcel D1 2002 June Q4
10 marks Moderate -0.3
  1. Use Dijkstra's algorithm to find the shortest route from \(A\) to \(I\). Show all necessary working in the boxes in the answer booklet and state your shortest route and its length.
    (5) The park warden wishes to check each of the paths to check for frost damage. She has to cycle along each path at least once, starting and finishing at \(A\).
    1. Use an appropriate algorithm to find which paths will be covered twice and state these paths.
    2. Find a route of minimum length.
    3. Find the total length of this shortest route.
      (5)
Edexcel D1 2002 November Q4
7 marks Standard +0.3
  1. Use the Route Inspection algorithm to find which paths, if any, need to be traversed twice. It is decided to start the inspection at node \(C\). The inspection must still traverse each pipe at least once but may finish at any node.
  2. Explaining your reasoning briefly, determine the node at which the inspection should finish if the route is to be minimised. State the length of your route.
    (3)
Edexcel D1 2004 November Q5
10 marks Standard +0.3
  1. find the route the driver should follow, starting and ending at \(F\), to clear all the roads of snow. Give the length of this route. The local authority decides to build a road bridge over the river at \(B\). The snowplough will be able to cross the road bridge.
  2. Reapply the algorithm to find the minimum distance the snowplough will have to travel (ignore the length of the new bridge). \section*{6.} \section*{Figure 3}
    \includegraphics[max width=\textwidth, alt={}]{4bbe6272-3900-42de-b287-599638ca75e4-07_1131_1118_347_502}
    Peter wishes to minimise the time spent driving from his home \(H\), to a campsite at \(G\). Figure 3 shows a number of towns and the time, in minutes, taken to drive between them. The volume of traffic on the roads into \(G\) is variable, and so the length of time taken to drive along these roads is expressed in terms of \(x\), where \(x \geq 0\).
    1. On the diagram in the answer book, use Dijkstra's algorithm to find two routes from \(H\) to \(G\) (one via \(A\) and one via \(B\) ) that minimise the travelling time from \(H\) to \(G\). State the length of each route in terms of \(x\).
    2. Find the range of values of \(x\) for which Peter should follow the route via \(A\). \section*{7.} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{4bbe6272-3900-42de-b287-599638ca75e4-08_1495_1335_322_392}
      \end{figure} The company EXYCEL makes two types of battery, X and Y . Machinery, workforce and predicted sales determine the number of batteries EXYCEL make. The company decides to use a graphical method to find its optimal daily production of X and Y . The constraints are modelled in Figure 4 where $$\begin{aligned} & x = \text { the number (in thousands) of type } \mathrm { X } \text { batteries produced each day, } \\ & y = \text { the number (in thousands) of type } \mathrm { Y } \text { batteries produced each day. } \end{aligned}$$ The profit on each type X battery is 40 p and on each type Y battery is 20 p . The company wishes to maximise its daily profit.
    3. Write this as a linear programming problem, in terms of \(x\) and \(y\), stating the objective function and all the constraints.
    4. Find the optimal number of batteries to be made each day. Show your method clearly.
    5. Find the daily profit, in \(\pounds\), made by EXYCEL.
AQA C1 2014 June Q7
14 marks Moderate -0.5
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
    1. Write down the coordinates of \(C\).
    2. Show that the circle has radius \(n \sqrt { 5 }\), where \(n\) is an integer.
  3. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(x + p y = q\), where \(p\) and \(q\) are integers.
  4. The point \(B\) lies on the tangent to the circle at \(A\) and the length of \(B C\) is 6. Find the length of \(A B\).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}]{f2124c89-79de-4758-b7b8-ff273345b9dd-8_1421_1709_1286_153}
AQA C1 2015 June Q4
11 marks Standard +0.3
  1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
    1. State the coordinates of \(C\).
    2. Find the radius of the circle, giving your answer in the form \(n \sqrt { 2 }\).
  3. The point \(P\) with coordinates \(( 4 , k )\) lies on the circle. Find the possible values of \(k\).
  4. The points \(Q\) and \(R\) also lie on the circle, and the length of the chord \(Q R\) is 2 . Calculate the shortest distance from \(C\) to the chord \(Q R\).
    [0pt] [2 marks]
OCR C1 2007 January Q9
12 marks Moderate -0.3
  1. Find the equation of the line through \(A\) parallel to the line \(y = 4 x - 5\), giving your answer in the form \(y = m x + c\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find the equation of the line which passes through the mid-point of \(A B\) and which is perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2009 June Q9
8 marks Moderate -0.8
  1. Calculate the length of \(A B\).
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the equation of the line through \(( 1,3 )\) which is parallel to \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.