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OCR M1 2013 June Q4
10 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-3_298_540_262_735} The diagram shows the \(( t , v )\) graph of a car moving along a straight road, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t \mathrm {~s}\) after it passes through the point \(A\). The car passes through \(A\) with velocity \(18 \mathrm {~ms} ^ { - 1 }\), and moves with constant acceleration \(2.4 \mathrm {~ms} ^ { - 2 }\) until \(t = 5\). The car subsequently moves with constant velocity until it is 300 m from \(A\). When the car is more than 300 m from \(A\), it has constant deceleration \(6 \mathrm {~ms} ^ { - 2 }\), until it comes to rest.
  1. Find the greatest speed of the car.
  2. Calculate the value of \(t\) for the instant when the car begins to decelerate.
  3. Calculate the distance from \(A\) of the car when it is at rest.
OCR M1 2013 June Q5
10 marks Standard +0.3
5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) from the top of a smooth inclined plane of length \(2 d\) metres. After its projection \(P\) moves downwards along a line of greatest slope with acceleration \(4 \mathrm {~ms} ^ { - 2 }\). At the instant 3 s after projection \(P\) has moved half way down the plane. \(P\) reaches the foot of the plane 5 s after the instant of projection.
  1. Form two simultaneous equations in \(u\) and \(d\), and hence calculate the speed of projection of \(P\) and the length of the plane.
  2. Find the inclination of the plane to the horizontal.
  3. Given that the contact force exerted on \(P\) by the plane has magnitude 6 N , calculate the mass of \(P\).
OCR M1 2013 June Q6
14 marks Standard +0.3
6 A particle \(P\) moves in a straight line. At time \(t\) s after passing through a point \(O\) of the line, the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). Given that \(x = 0.06 t ^ { 3 } - 0.45 t ^ { 2 } - 0.24 t\), find
  1. the velocity and the acceleration of \(P\) when \(t = 0\),
  2. the value of \(x\) when \(P\) has its minimum velocity, and the speed of \(P\) at this instant,
  3. the positive value of \(t\) when the direction of motion of \(P\) changes.
OCR M1 2013 June Q7
15 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-4_310_579_255_721} A block \(B\) is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.6 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth pulley fixed to the top of the plane. A particle \(Q\) of mass 0.5 kg is attached to the other end of the string. The portion of the string attached to \(P\) is parallel to a line of greatest slope of the plane, the portion of the string attached to \(Q\) is vertical and the string is taut. The particles are released from rest and start to move with acceleration \(1.4 \mathrm {~ms} ^ { - 2 }\) (see diagram). It is given that \(B\) is in equilibrium while \(P\) moves on its upper surface.
  1. Find the tension in the string while \(P\) and \(B\) are in contact.
  2. Calculate the coefficient of friction between \(P\) and \(B\).
  3. Given that the weight of \(B\) is 7 N , calculate the set of possible values of the coefficient of friction between \(B\) and the plane.
OCR M1 2015 June Q1
7 marks Moderate -0.8
1 A particle \(P\) is projected vertically downwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 30 m above the ground.
  1. Calculate the speed of \(P\) when it reaches the ground.
  2. Find the distance travelled by \(P\) in the first 0.4 s of its motion.
  3. Calculate the time taken for \(P\) to travel the final 15 m of its descent.
OCR M1 2015 June Q2
8 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_138_1118_680_463} Three particles \(P , Q\) and \(R\) with masses \(0.4 \mathrm {~kg} , 0.3 \mathrm {~kg}\) and \(m \mathrm {~kg}\) are moving along the same straight line on a smooth horizontal surface. \(P\) and \(Q\) are moving towards each other with speeds \(u \mathrm {~ms} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(R\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the same direction as \(Q\) (see diagram).
  1. Immediately after the collision between \(P\) and \(Q\) their directions of motion have been reversed, but their speeds are unchanged. Calculate \(u\). The next collision is between \(Q\) and \(R\). After the collision between \(Q\) and \(R\), particle \(Q\) is at rest and \(R\) has speed \(9 \mathrm {~ms} ^ { - 1 }\).
  2. Calculate \(m\). \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-2_547_1506_1521_251} Two travellers \(A\) and \(B\) make the same journey on a long straight road. Each traveller walks for part of the journey and rides a bicycle for part of the journey. They start their journeys at the same instant, and they end their journeys simultaneously after travelling for \(T\) hours. \(A\) starts the journey cycling at a steady \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) for 1 hour. \(A\) then leaves the bicycle at the side of the road, and completes the journey walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \(B\) begins the journey walking at a steady \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). When \(B\) finds the bicycle where \(A\) left it, \(B\) cycles at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) to complete the journey (see diagram).
OCR M1 2015 June Q4
9 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).
  1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
  2. the magnitude of the force exerted on \(P\) by the surface,
  3. the angle between the surface and the 10 N force.
OCR M1 2015 June Q5
11 marks Moderate -0.3
5 A particle \(P\) of mass 0.4 kg is at rest on a horizontal surface. The coefficient of friction between \(P\) and the surface is 0.2 . A force of magnitude 1.2 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal is then applied to \(P\). Find the acceleration of \(P\) in each of the following cases:
  1. \(\theta = 0\);
  2. \(\theta = 20\);
  3. \(\theta = 70\);
  4. \(\theta = 90\).
OCR M1 2015 June Q6
14 marks Standard +0.3
6 A particle \(P\) moves in a straight line on a horizontal surface. \(P\) passes through a fixed point \(O\) on the line with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(( 4 + 12 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  1. Calculate the velocity of \(P\) when \(t = 3\).
  2. Find the distance \(O P\) when \(t = 3\). A second particle \(Q\), having the same mass as \(P\), moves along the same straight line. The displacement of \(Q\) from \(O\) is \(\left( k - 2 t ^ { 3 } \right) \mathrm { m }\), where \(k\) is a constant. When \(t = 3\) the particles collide and coalesce.
  3. Find the value of \(k\).
  4. Find the common velocity of the particles immediately after their collision.
OCR M1 2015 June Q7
15 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-4_392_1192_255_424} \(A B\) and \(B C\) are lines of greatest slope on a fixed triangular prism, and \(M\) is the mid-point of \(B C . A B\) and \(B C\) are inclined at \(30 ^ { \circ }\) to the horizontal. The surface of the prism is smooth between \(A\) and \(B\), and between \(B\) and \(M\). Between \(M\) and \(C\) the surface of the prism is rough. A small smooth pulley is fixed to the prism at \(B\). A light inextensible string passes over the pulley. Particle \(P\) of mass 0.3 kg is fixed to one end of the string, and is placed at \(A\). Particle \(Q\) of mass 0.4 kg is fixed to the other end of the string and is placed next to the pulley on \(B C\). The particles are released from rest with the string taut. \(P\) begins to move towards the pulley, and \(Q\) begins to move towards \(M\) (see diagram).
  1. Show that the initial acceleration of the particles is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and find the tension in the string. The particle \(Q\) reaches \(M 1.8 \mathrm {~s}\) after being released from rest.
  2. Find the speed of the particles when \(Q\) reaches \(M\). After \(Q\) passes through \(M\), the string remains taut and the particles decelerate uniformly. \(Q\) comes to rest between \(M\) and \(C 1.4 \mathrm {~s}\) after passing through \(M\).
  3. Find the deceleration of the particles while \(Q\) is moving from \(M\) towards \(C\).
  4. (a) By considering the motion of \(P\), find the tension in the string while \(Q\) is moving from \(M\) towards \(C\).
    (b) Calculate the magnitude of the frictional force which acts on \(Q\) while it is moving from \(M\) towards \(C\). \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR FP1 2009 January Q1
4 marks Easy -1.2
1 Express \(\frac { 2 + 3 \mathrm { i } } { 5 - \mathrm { i } }\) in the form \(x + \mathrm { i } y\), showing clearly how you obtain your answer.
OCR FP1 2009 January Q2
4 marks Moderate -0.8
2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ a & 5 \end{array} \right)\). Find
  1. \(\mathbf { A } ^ { - 1 }\),
  2. \(2 \mathbf { A } - \left( \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right)\).
OCR FP1 2009 January Q3
6 marks Moderate -0.8
3 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 6 r ^ { 2 } + 2 r \right)\), expressing your answer in a fully factorised form.
OCR FP1 2009 January Q4
4 marks Standard +0.3
4 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are \(2 \times 2\) non-singular matrices and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, simplify $$\mathbf { B } ( \mathbf { A B } ) ^ { - 1 } \mathbf { A } - \mathbf { I } .$$
OCR FP1 2009 January Q5
5 marks Standard +0.3
5 By using the determinant of an appropriate matrix, or otherwise, find the value of \(k\) for which the simultaneous equations $$\begin{aligned} 2 x - y + z & = 7 \\ 3 y + z & = 4 \\ x + k y + k z & = 5 \end{aligned}$$ do not have a unique solution for \(x , y\) and \(z\).
OCR FP1 2009 January Q6
9 marks Moderate -0.8
6
  1. The transformation P is represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\). Give a geometrical description of transformation P .
  2. The transformation Q is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\). Give a geometrical description of transformation Q.
  3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
  4. Give a geometrical description of the single transformation that is represented by your answer to part (iii).
OCR FP1 2009 January Q7
7 marks Standard +0.3
7 It is given that \(u _ { n } = 13 ^ { n } + 6 ^ { n - 1 }\), where \(n\) is a positive integer.
  1. Show that \(u _ { n } + u _ { n + 1 } = 14 \times 13 ^ { n } + 7 \times 6 ^ { n - 1 }\).
  2. Prove by induction that \(u _ { n }\) is a multiple of 7 .
OCR FP1 2009 January Q8
10 marks Standard +0.3
8
  1. Show that \(( \alpha - \beta ) ^ { 2 } \equiv ( \alpha + \beta ) ^ { 2 } - 4 \alpha \beta\). The quadratic equation \(x ^ { 2 } - 6 k x + k ^ { 2 } = 0\), where \(k\) is a positive constant, has roots \(\alpha\) and \(\beta\), with \(\alpha > \beta\).
  2. Show that \(\alpha - \beta = 4 \sqrt { 2 } k\).
  3. Hence find a quadratic equation with roots \(\alpha + 1\) and \(\beta - 1\).
OCR FP1 2009 January Q9
9 marks Standard +0.8
9
  1. Show that \(\frac { 1 } { 2 r - 3 } - \frac { 1 } { 2 r + 1 } = \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }\).
  2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 2 } ^ { n } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$$
  3. Show that \(\sum _ { r = 2 } ^ { \infty } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 } = \frac { 4 } { 3 }\).
  1. Use an algebraic method to find the square roots of the complex number \(2 + \mathrm { i } \sqrt { 5 }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
  2. Hence find, in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are exact real numbers, the roots of the equation $$z ^ { 4 } - 4 z ^ { 2 } + 9 = 0$$
  3. Show, on an Argand diagram, the roots of the equation in part (ii).
  4. Given that \(\alpha\) is the root of the equation in part (ii) such that \(0 < \arg \alpha < \frac { 1 } { 2 } \pi\), sketch on the same Argand diagram the locus given by \(| z - \alpha | = | z |\).
OCR FP1 2010 January Q1
5 marks Moderate -0.8
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } a & 2 \\ 3 & 4 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  1. Find A-4I.
  2. Given that \(\mathbf { A }\) is singular, find the value of \(a\).
OCR FP1 2010 January Q2
5 marks Standard +0.3
2 The cubic equation \(2 x ^ { 3 } + 3 x - 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = u - 1\) to find a cubic equation in \(u\) with integer coefficients.
  2. Hence find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
OCR FP1 2010 January Q3
5 marks Moderate -0.3
3 The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). Find \(z\), giving your answer in the form \(x + \mathrm { i } y\).
OCR FP1 2010 January Q4
6 marks Standard +0.3
4 Find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 2 )\), expressing your answer in a fully factorised form.
OCR FP1 2010 January Q5
6 marks Moderate -0.3
5
  1. The transformation T is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\). Give a geometrical description of T .
  2. The transformation T is equivalent to a reflection in the line \(y = - x\) followed by another transformation S . Give a geometrical description of S and find the matrix that represents S .
OCR FP1 2010 January Q6
7 marks Standard +0.3
6 One root of the cubic equation \(x ^ { 3 } + p x ^ { 2 } + 6 x + q = 0\), where \(p\) and \(q\) are real, is the complex number 5-i.
  1. Find the real root of the cubic equation.
  2. Find the values of \(p\) and \(q\).