Questions — OCR MEI (4301 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI C2 Q6
4 marks Moderate -0.8
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-3_819_1370_271_383} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Fig. 3 shows sketches of three graphs, A, B and C. The equation of graph A is \(y = \mathrm { f } ( x )\). State the equation of
  1. graph B ,
  2. graph C .
OCR MEI C2 Q7
5 marks Moderate -0.8
7
  1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
OCR MEI C2 Q8
4 marks Moderate -0.8
8
  1. The point \(\mathrm { P } ( 4 , - 2 )\) lies on the curve \(y = \mathrm { f } ( x )\). Find the coordinates of the image of P when the curve is transformed to \(y = \mathrm { f } ( 5 x )\).
  2. Describe fully a single transformation which maps the curve \(y = \sin x ^ { \circ }\) onto the curve \(y = \sin ( x - 90 ) ^ { \circ }\).
OCR MEI C2 Q9
3 marks Moderate -0.8
9 Figs. 5.1 and 5.2 show the graph of \(y = \sin x\) for values of \(x\) from \(0 ^ { \circ }\) to \(360 ^ { \circ }\) and two transformations of this graph. State the equation of each graph after it has been transformed.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-4_511_941_828_586} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
    \end{figure}
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-4_517_937_1508_584} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
OCR MEI C2 Q10
4 marks Moderate -0.8
10 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C2 Q11
4 marks Moderate -0.8
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-5_546_989_828_596} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Fig. 5 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { P } , \mathrm { Q }\) and R .
  1. \(y = \mathrm { f } ( 2 x )\)
  2. \(y = \frac { 1 } { 4 } \mathrm { f } ( x )\)
OCR MEI C2 Q13
4 marks Moderate -0.8
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-7_618_867_267_679} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { A } , \mathrm { B }\) and C .
  1. \(y = 2 \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } ( x + 3 )\)
OCR MEI C2 Q14
5 marks Moderate -0.8
14
  1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).
OCR MEI M1 2016 June Q1
6 marks Moderate -0.3
1 Fig. 1 shows a block of mass \(M \mathrm {~kg}\) being pushed over level ground by means of a light rod. The force, \(T \mathrm {~N}\), this exerts on the block is along the line of the rod. The ground is rough.
The rod makes an angle \(\alpha\) with the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-2_307_876_621_593} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Draw a diagram showing all the forces acting on the block.
  2. You are given that \(M = 5 , \alpha = 60 ^ { \circ } , T = 40\) and the acceleration of the block is \(1.5 \mathrm {~ms} ^ { - 2 }\). Find the frictional force.
OCR MEI M1 2016 June Q2
7 marks Moderate -0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-2_117_1162_1486_438} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A particle moves on the straight line shown in Fig. 2. The positive direction is indicated on the diagram. The time, \(t\), is measured in seconds. The particle has constant acceleration, \(a \mathrm {~ms} ^ { - 2 }\). Initially it is at the point O and has velocity \(u \mathrm {~ms} ^ { - 1 }\).
When \(t = 2\), the particle is at A where OA is 12 m . The particle is also at A when \(t = 6\).
  1. Write down two equations in \(u\) and \(a\) and solve them.
  2. The particle changes direction when it is at B . Find the distance AB .
OCR MEI M1 2016 June Q3
8 marks Moderate -0.3
3 Fig. 3.1 shows a block of mass 8 kg on a smooth horizontal table.
This block is connected by a light string passing over a smooth pulley to a block of mass 4 kg which hangs freely. The part of the string between the 8 kg block and the pulley is parallel to the table. The system has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_330_809_525_628} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Write down two equations of motion, one for each block.
  2. Find the value of \(a\). The table is now tilted at an angle of \(\theta\) to the horizontal as shown in Fig. 3.2. The system is set up as before; the 4 kg block still hangs freely. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_410_727_1324_669} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  3. The system is now in equilibrium. Find the value of \(\theta\).
OCR MEI M1 2016 June Q4
8 marks Moderate -0.8
4 A particle is initially at the origin, moving with velocity \(\mathbf { u }\). Its acceleration \(\mathbf { a }\) is constant. At time \(t\) its displacement from the origin is \(\mathbf { r } = \binom { x } { y }\), where \(\binom { x } { y } = \binom { 2 } { 6 } t - \binom { 0 } { 4 } t ^ { 2 }\).
  1. Write down \(\mathbf { u }\) and \(\mathbf { a }\) as column vectors.
  2. Find the speed of the particle when \(t = 2\).
  3. Show that the equation of the path of the particle is \(y = 3 x - x ^ { 2 }\).
OCR MEI M1 2016 June Q5
7 marks Standard +0.3
5 Mr McGregor is a keen vegetable gardener. A pigeon that eats his vegetables is his great enemy.
One day he sees the pigeon sitting on a small branch of a tree. He takes a stone from the ground and throws it. The trajectory of the stone is in a vertical plane that contains the pigeon. The same vertical plane intersects the window of his house. The situation is illustrated in Fig. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-4_400_1221_1078_411} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  • The stone is thrown from point O on level ground. Its initial velocity is \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(8 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.
  • The pigeon is at point P which is 4 m above the ground.
  • The house is 22.5 m from O .
  • The bottom of the window is 0.8 m above the ground and the window is 1.2 m high.
Show that the stone does not reach the height of the pigeon. Determine whether the stone hits the window.
OCR MEI C2 Q2
5 marks Moderate -0.3
2 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.
OCR MEI C2 Q3
2 marks Easy -1.3
3 The points \(\mathrm { P } ( 2,3.6 )\) and \(\mathrm { Q } ( 2.2,2.4 )\) lie on the curve \(y = \mathrm { f } ( x )\). Use P and Q to estimate the gradient of the curve at the point where \(x = 2\).
OCR MEI C2 Q4
5 marks Easy -1.8
4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when
  1. \(y = 2 x ^ { - 5 }\),
  2. \(y = \sqrt [ 3 ] { x }\).
OCR MEI C2 Q5
5 marks Easy -1.2
5 The equation of a curve is \(y = \sqrt { 1 + 2 x }\).
  1. Calculate the gradient of the chord joining the points on the curve where \(x = 4\) and \(x = 4\). Give your answer correct to 4 decimal places.
  2. Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when \(x = 4\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f540b962-ee6b-409a-a2a1-cd7ad4945514-2_1031_1113_273_499} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} Fig. 5 shows the graph of \(y = 2 ^ { x }\).
OCR MEI C2 Q7
3 marks Easy -1.2
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { x } + \frac { 3 } { x }\).
OCR MEI C2 Q8
5 marks Moderate -0.8
8 The gradient of a curve is \(6 x ^ { 2 } + 12 x ^ { \frac { 1 } { 2 } }\). The curve passes through the point \(( 4,10 )\). Find the equation of the curve.
OCR MEI C2 Q9
3 marks Moderate -0.5
9 Use calculus to find the set of values of \(x\) for which \(\mathrm { f } ( x ) = 12 x - x ^ { 3 }\) is an increasing function.
OCR MEI C2 Q10
5 marks Easy -1.3
10 Given tha \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).
OCR MEI C2 Q11
4 marks Moderate -0.8
11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - x ^ { 2 }\). The curve passes through the point \(( 6,1 )\). Find the equation of the curve.
OCR MEI C2 Q12
5 marks Easy -1.2
12 Given tha \(y = 6 x ^ { 3 } + \sqrt { x } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR MEI C2 Q1
4 marks Easy -1.2
1 Differentiate \(x + \sqrt { x ^ { 3 } }\).
OCR MEI C2 Q2
5 marks Moderate -0.8
2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { x ^ { 3 } }\). The curve passes through \(( 1,4 )\).
Find the equation of the curve.