Questions — OCR MEI (4455 questions)

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OCR MEI M2 2007 January Q2
20 marks Standard +0.8
\includegraphics{figure_2} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods AB, BC, AD, BD, BE, CE and DE. [The triangles ABD, BDE and BCE are all equilateral.] The rods AB, BC and DE are horizontal. The rods are freely pin-jointed to each other at A, B, C, D and E. The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD. The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(LN\) at C; the normal reaction force \(RN\) of the plane on the framework at D; the horizontal and vertical forces \(XN\) and \(YN\), respectively, acting at A.
  1. Write down equations for the horizontal and vertical equilibrium of the framework. [3]
  2. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt{3}L\) and \(Y = 0\). [4]
  3. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods. [2]
  4. Show that the internal force in the rod AD is zero. [2]
  5. Find the forces internal to AB, CE and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.] [7]
  6. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust. [2]
OCR MEI M2 2007 January Q3
18 marks Standard +0.8
A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \includegraphics{figure_3.1}
  1. The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. [1]
The base OABC is added to the vertical faces.
  1. Write down the \(x\)- and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875. [5]
The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.
  1. Show that the coordinates of the centre of mass of the box in this situation are \((10, 2.4, 2.1)\). [6]
[This question is continued on the facing page.] The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30°\) to the horizontal, as shown in Fig. 3.2. \includegraphics{figure_3.2} The weight of the box is 40 N. A force \(P\) N acts parallel to the plane and is applied to the mid-point of FG at \(90°\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO.
  1. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm. [4]
  2. Calculate the value of \(P\). [2]
OCR MEI M2 2007 January Q4
17 marks Standard +0.3
Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}
  1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
  2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
    1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
    2. Calculate the work done against friction per tile. [2]
    3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
  3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]
OCR MEI M2 2008 January Q1
19 marks Moderate -0.3
  1. A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The battering-ram has a mass of 4000 kg. \includegraphics{figure_1} Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N.
    1. At what speed does the battering-ram hit the wall? [3]
    The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2.
    1. Calculate the speeds of the stone block and of the battering-ram immediately after the impact. [6]
    2. Calculate the energy lost in the impact. [3]
  2. Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed 18 m s\(^{-1}\) in the \(\mathbf{i}\) direction. B has mass 8 kg and speed 9 m s\(^{-1}\) in the direction shown in Fig. 1.2, where \(\mathbf{i}\) and \(\mathbf{j}\) are the standard unit vectors. \includegraphics{figure_2}
    1. Write down the linear momentum of A and show that the linear momentum of B is \((36\mathbf{i} + 36\sqrt{3}\mathbf{j})\) N s. [2]
    After the objects meet they stick together (coalesce) and move with a common velocity of \((u\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\).
    1. Calculate \(u\) and \(v\). [3]
    2. Find the angle between the direction of motion of the combined object and the \(\mathbf{i}\) direction. Make your method clear. [2]
OCR MEI M2 2008 January Q2
17 marks Moderate -0.3
A cyclist and her bicycle have a combined mass of 80 kg.
  1. Initially, the cyclist accelerates from rest to 3 m s\(^{-1}\) against negligible resistances along a horizontal road.
    1. How much energy is gained by the cyclist and bicycle? [2]
    2. The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle? [2]
  2. While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from 4 m s\(^{-1}\) to 10 m s\(^{-1}\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h\) m. Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\). [5]
  3. The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N.
    1. When the power of the driving force on the bicycle is a constant 200 W, what constant speed can the cyclist maintain? [3]
    2. Find the power of the driving force on the bicycle when travelling at a speed of 0.5 m s\(^{-1}\) with an acceleration of 2 m s\(^{-2}\). [5]
OCR MEI M2 2008 January Q3
18 marks Standard +0.3
A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. \includegraphics{figure_3}
  1. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. [5]
The rectangles BCJA and FGHI are folded through 90° about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.
  1. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). [4]
The \(x\)- and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force \(P\) N is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH.
  1. Calculate the value of \(P\). [5]
The coefficient of friction between the fire-screen and the floor is \(\mu\).
  1. For what values of \(\mu\) does the fire-screen slide before it tips? [4]
OCR MEI M2 2008 January Q4
18 marks Standard +0.3
Fig. 4.1 shows a uniform beam, CE, of weight 2200 N and length 4.5 m. The beam is freely pivoted on a fixed support at D and is supported at C. The distance CD is 2.75 m. \includegraphics{figure_4} The beam is horizontal and in equilibrium.
  1. Show that the anticlockwise moment of the weight of the beam about D is 1100 N m. Find the value of the normal reaction on the beam of the support at C. [6]
The support at C is removed and spheres at P and Q are suspended from the beam by light strings attached to the points C and R. The sphere at P has weight 440 N and the sphere at Q has weight \(W\) N. The point R of the beam is 1.5 m from D. This situation is shown in Fig. 4.2.
  1. The beam is horizontal and in equilibrium. Show that \(W = 1540\). [3]
The sphere at P is changed for a lighter one with weight 400 N. The sphere at Q is unchanged. The beam is now held in equilibrium at an angle of 20° to the horizontal by means of a light rope attached to the beam at E. This situation (but without the rope at E) is shown in Fig. 4.3. \includegraphics{figure_5}
  1. Calculate the tension in the rope when it is
    1. at 90° to the beam, [6]
    2. horizontal. [3]
OCR MEI M2 2011 January Q1
19 marks Standard +0.3
Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.8\). The coefficient of friction between A and the slope is 0.85. \includegraphics{figure_1}
  1. Calculate the maximum possible frictional force between A and the slope. Show that A will remain at rest. [6]
With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has a speed of 16 m s\(^{-1}\) when it collides with A. In this collision the coefficient of restitution is 0.4, the impulses are parallel to the slope and linear momentum parallel to the slope is conserved.
  1. Show that the velocity of A immediately after the collision is 8.4 m s\(^{-1}\) down the slope. Find the velocity of B immediately after the collision. [6]
  2. Calculate the impulse on B in the collision. [3]
The blocks do not collide again.
  1. For what length of time after the collision does A slide before it comes to rest? [4]
OCR MEI M2 2011 January Q2
17 marks Standard +0.3
  1. A firework is instantaneously at rest in the air when it explodes into two parts. One part is the body B of mass 0.06 kg and the other a cap C of mass 0.004 kg. The total kinetic energy given to B and C is 0.8 J. B moves off horizontally in the \(\mathbf{i}\) direction. By considering both kinetic energy and linear momentum, calculate the velocities of B and C immediately after the explosion. [8]
  2. A car of mass 800 kg is travelling up some hills. In one situation the car climbs a vertical height of 20 m while its speed decreases from 30 m s\(^{-1}\) to 12 m s\(^{-1}\). The car is subject to a resistance to its motion but there is no driving force and the brakes are not being applied.
    1. Using an energy method, calculate the work done by the car against the resistance to its motion. [4]
    In another situation the car is travelling at a constant speed of 18 m s\(^{-1}\) and climbs a vertical height of 20 m in 25 s up a uniform slope. The resistance to its motion is now 750 N.
    1. Calculate the power of the driving force required. [5]
OCR MEI M2 2011 January Q3
19 marks Standard +0.8
\includegraphics{figure_3} Fig. 3 shows a framework in equilibrium in a vertical plane. The framework is made from the equal, light, rigid rods AB, AD, BC, BD and CD so that ABD and BCD are equilateral triangles of side 2 m. AD and BC are horizontal. The rods are freely pin-jointed to each other at A, B, C and D. The pin-joint at A is fixed to a wall and the pin-joint at B rests on a smooth horizontal support. Fig. 3 also shows the external forces acting on the framework: there is a vertical load of 45 N at C and a horizontal force of 50 N applied at D; the normal reaction of the support on the framework at B is \(R\) N; horizontal and vertical forces \(X\) N and \(Y\) N act at A.
  1. Write down equations for the horizontal and vertical equilibrium of the framework. [2]
  2. Show that \(R = 135\) and \(Y = 90\). [3]
  3. On the diagram in your printed answer book, show the forces internal to the rods acting on the pin-joints. [2]
  4. Calculate the forces internal to the five rods, stating whether each rod is in tension or compression (thrust). [You may leave your answers in surd form. Your working in this part should correspond to your diagram in part (iii).] [10]
  5. Suppose that the force of magnitude 50 N applied at D is no longer horizontal, and the system remains in equilibrium in the same position. By considering the equilibrium at C, show that the forces in rods CD and BC are not changed. [2]
OCR MEI M2 2011 January Q4
17 marks Standard +0.3
You are given that the centre of mass, G, of a uniform lamina in the shape of an isosceles triangle lies on its axis of symmetry in the position shown in Fig. 4.1. \includegraphics{figure_4_1} Fig. 4.2 shows the cross-section OABCD of a prism made from uniform material. OAB is an isosceles triangle, where OA = AB, and OBCD is a rectangle. The distance OD is \(h\) cm, where \(h\) can take various positive values. All coordinates refer to the axes Ox and Oy shown. The units of the axes are centimetres. \includegraphics{figure_4_2}
  1. Write down the coordinates of the centre of mass of the triangle OAB. [1]
  2. Show that the centre of mass of the region OABCD is \(\left(\frac{12-h^2}{2(h+3)}, 2.5\right)\). [6]
The \(x\)-axis is horizontal. The prism is placed on a horizontal plane in the position shown in Fig. 4.2.
  1. Find the values of \(h\) for which the prism would topple. [3]
The following questions refer to the case where \(h = 3\) with the prism held in the position shown in Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism is 15 N. You should assume that the prism does not slide.
  1. Suppose that the prism is held in this position by a vertical force applied at A. Given that the prism is on the point of tipping clockwise, calculate the magnitude of this force. [3]
  2. Suppose instead that the prism is held in this position by a force in the plane of the cross-section OABCD, applied at 30° below the horizontal at C, as shown in Fig. 4.3. Given that the prism is on the point of tipping anti-clockwise, calculate the magnitude of this force. [4]
\includegraphics{figure_4_3}
OCR MEI FP1 2006 June Q1
4 marks Easy -1.3
  1. State the transformation represented by the matrix \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\). [1]
  2. Write down the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin. [1]
  3. Find the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin, followed by reflection in the \(x\)-axis. [2]
OCR MEI FP1 2006 June Q2
5 marks Easy -1.2
Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2x^3 - 3x^2 + x - 2 \equiv (x + 2)(Ax^2 + Bx + C) + D.$$ [5]
OCR MEI FP1 2006 June Q3
6 marks Moderate -0.3
The cubic equation \(z^3 + 4z^2 - 3z + 1 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
  1. Write down the values of \(\alpha + \beta + \gamma\), \(\alpha\beta + \beta\gamma + \gamma\alpha\) and \(\alpha\beta\gamma\). [3]
  2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 22\). [3]
OCR MEI FP1 2006 June Q4
8 marks Moderate -0.8
Indicate, on separate Argand diagrams,
  1. the set of points \(z\) for which \(|z-(3-\mathrm{j})| \leqslant 3\), [3]
  2. the set of points \(z\) for which \(1 < |z-(3-\mathrm{j})| \leqslant 3\), [2]
  3. the set of points \(z\) for which \(\arg(z-(3-\mathrm{j})) = \frac{1}{4}\pi\). [3]
OCR MEI FP1 2006 June Q5
6 marks Moderate -0.3
  1. The matrix \(\mathbf{S} = \begin{pmatrix} -1 & 2 \\ -3 & 4 \end{pmatrix}\) represents a transformation.
    1. Show that the point \((1, 1)\) is invariant under this transformation. [1]
    2. Calculate \(\mathbf{S}^{-1}\). [2]
    3. Verify that \((1, 1)\) is also invariant under the transformation represented by \(\mathbf{S}^{-1}\). [1]
  2. Part (i) may be generalised as follows. If \((x, y)\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf{T}\), it is also invariant under the transformation represented by \(\mathbf{T}^{-1}\). Starting with \(\mathbf{T}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\), or otherwise, prove this result. [2]
OCR MEI FP1 2006 June Q6
7 marks Standard +0.3
Prove by induction that \(3 + 6 + 12 + \ldots + 3 \times 2^{n-1} = 3(2^n - 1)\) for all positive integers \(n\). [7]
OCR MEI FP1 2006 June Q7
13 marks Standard +0.3
A curve has equation \(y = \frac{x^2}{(x-2)(x+1)}\).
  1. Write down the equations of the three asymptotes. [3]
  2. Determine whether the curve approaches the horizontal asymptote from above or from below for
    1. large positive values of \(x\),
    2. large negative values of \(x\). [3]
  3. Sketch the curve. [4]
  4. Solve the inequality \(\frac{x^2}{(x-2)(x+1)} > 0\). [3]
OCR MEI FP1 2006 June Q8
10 marks Moderate -0.3
  1. Verify that \(2 + \mathrm{j}\) is a root of the equation \(2x^3 - 11x^2 + 22x - 15 = 0\). [5]
  2. Write down the other complex root. [1]
  3. Find the third root of the equation. [4]
OCR MEI FP1 2006 June Q9
13 marks Standard +0.3
  1. Show that \(r(r+1)(r+2) - (r-1)r(r+1) \equiv 3r(r+1)\). [2]
  2. Hence use the method of differences to find an expression for \(\sum_{r=1}^{n} r(r+1)\). [6]
  3. Show that you can obtain the same expression for \(\sum_{r=1}^{n} r(r+1)\) using the standard formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\). [5]
OCR MEI FP1 2007 June Q1
3 marks Moderate -0.8
You are given the matrix \(\mathbf{M} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}\).
  1. Find the inverse of \(\mathbf{M}\). [2]
  2. A triangle of area 2 square units undergoes the transformation represented by the matrix \(\mathbf{M}\). Find the area of the image of the triangle following this transformation. [1]
OCR MEI FP1 2007 June Q2
3 marks Easy -1.2
Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. [3] \includegraphics{figure_2}
OCR MEI FP1 2007 June Q3
5 marks Easy -1.2
Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]
OCR MEI FP1 2007 June Q4
7 marks Moderate -0.8
Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 - 2\mathrm{j}\) and \(\beta = -2 - \mathrm{j}\).
  1. Represent \(\beta\) and its complex conjugate \(\beta^*\) on an Argand diagram. [2]
  2. Express \(\alpha\beta\) in the form \(a + b\mathrm{j}\). [2]
  3. Express \(\frac{\alpha + \beta}{\beta}\) in the form \(a + b\mathrm{j}\). [3]
OCR MEI FP1 2007 June Q5
6 marks Standard +0.3
The roots of the cubic equation \(x^3 + 3x^2 - 7x + 1 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\). Find the cubic equation whose roots are \(3\alpha\), \(3\beta\) and \(3\gamma\), expressing your answer in a form with integer coefficients. [6]