Questions — OCR MEI C4 (332 questions)

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OCR MEI C4 Q4
4 Fig. 4 shows a sketch of the region enclosed by the curve \(\sqrt { 1 + \mathrm { e } ^ { - 2 x } }\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88eadcf5-4335-4016-baf5-d3f74513bbb8-02_517_755_1576_649} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find the volume of the solid generated when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in an exact form.
OCR MEI C4 2007 January Q1
1 Solve the equation \(\frac { 1 } { x } + \frac { x } { x + 2 } = 1\).
OCR MEI C4 2007 January Q2
2 marks
2 Fig. 2 shows part of the curve \(y = \sqrt { 1 + x ^ { 3 } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5dcd4f44-4c61-4384-be1b-a8d63cb6b5aa-2_540_648_662_712} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Use the trapezium rule with 4 strips to estimate \(\int _ { 0 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer correct to 3 significant figures.
  2. Chris and Dave each estimate the value of this integral using the trapezium rule with 8 strips. Chris gets a result of 3.25, and Dave gets 3.30. One of these results is correct. Without performing the calculation, state with a reason which is correct.
    [0pt] [2]
OCR MEI C4 2007 January Q3
3
  1. Use the formula for \(\sin ( \theta + \phi )\), with \(\theta = 45 ^ { \circ }\) and \(\phi = 60 ^ { \circ }\), to show that \(\sin 105 ^ { \circ } = \frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }\).
  2. In triangle ABC , angle \(\mathrm { BAC } = 45 ^ { \circ }\), angle \(\mathrm { ACB } = 30 ^ { \circ }\) and \(\mathrm { AB } = 1\) unit (see Fig. 3). Fig. 3 Using the sine rule, together with the result in part (i), show that \(\mathrm { AC } = \frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }\).
OCR MEI C4 2007 January Q4
4 Show that \(\frac { 1 + \tan ^ { 2 } \theta } { 1 - \tan ^ { 2 } \theta } = \sec 2 \theta\).
Hence, or otherwise, solve the equation \(\frac { 1 + \tan ^ { 2 } \theta } { 1 - \tan ^ { 2 } \theta } = 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
OCR MEI C4 2007 January Q5
5 Find the first four terms in the binomial expansion of \(( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\).
State the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 2007 January Q6
6
  1. Express \(\frac { 1 } { ( 2 x + 1 ) ( x + 1 ) }\) in partial fractions.
  2. A curve passes through the point \(( 0,2 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { ( 2 x + 1 ) ( x + 1 ) }$$ Show by integration that \(y = \frac { 4 x + 2 } { x + 1 }\). Section B (36 marks)
OCR MEI C4 2007 January Q7
7 Fig. 7 shows the curve with parametric equations $$x = \cos \theta , y = \sin \theta - \frac { 1 } { 8 } \sin 2 \theta , 0 \leqslant \theta < 2 \pi$$ The curve crosses the \(x\)-axis at points \(\mathrm { A } ( 1,0 )\) and \(\mathrm { B } ( - 1,0 )\), and the positive \(y\)-axis at C . D is the maximum point of the curve, and E is the minimum point. The solid of revolution formed when this curve is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is used to model the shape of an egg. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5dcd4f44-4c61-4384-be1b-a8d63cb6b5aa-4_744_1207_776_431} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Show that, at the point \(\mathrm { A } , \theta = 0\). Write down the value of \(\theta\) at the point B , and find the coordinates of C .
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Hence show that, at the point D, $$2 \cos ^ { 2 } \theta - 4 \cos \theta - 1 = 0 .$$
  3. Solve this equation, and hence find the \(y\)-coordinate of D , giving your answer correct to 2 decimal places. The cartesian equation of the curve (for \(0 \leqslant \theta \leqslant \pi\) ) is $$y = \frac { 1 } { 4 } ( 4 - x ) \sqrt { 1 - x ^ { 2 } } .$$
  4. Show that the volume of the solid of revolution of this curve about the \(x\)-axis is given by $$\frac { 1 } { 16 } \pi \int _ { - 1 } ^ { 1 } \left( 16 - 8 x - 15 x ^ { 2 } + 8 x ^ { 3 } - x ^ { 4 } \right) \mathrm { d } x .$$ Evaluate this integral.
OCR MEI C4 2007 January Q8
8 A pipeline is to be drilled under a river (see Fig. 8). With respect to axes Oxyz, with the \(x\)-axis pointing East, the \(y\)-axis North and the \(z\)-axis vertical, the pipeline is to consist of a straight section AB from the point \(\mathrm { A } ( 0 , - 40,0 )\) to the point \(\mathrm { B } ( 40,0 , - 20 )\) directly under the river, and another straight section BC . All lengths are in metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5dcd4f44-4c61-4384-be1b-a8d63cb6b5aa-5_744_1068_495_500} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Calculate the distance AB . The section BC is to be drilled in the direction of the vector \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\).
  2. Find the angle ABC between the sections AB and BC . The section BC reaches ground level at the point \(\mathrm { C } ( a , b , 0 )\).
  3. Write down a vector equation of the line BC . Hence find \(a\) and \(b\).
  4. Show that the vector \(6 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the cartesian equation of this plane.
OCR MEI C4 2008 January Q1
1 Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
Hence solve the equation \(3 \cos \theta + 4 \sin \theta = 2\) for \(- \pi \leqslant \theta \leqslant \pi\).
OCR MEI C4 2008 January Q2
2
  1. Find the first three terms in the binomial expansion of \(\frac { 1 } { \sqrt { 1 - 2 x } }\). State the set of values of \(x\) for which the expansion is valid.
  2. Hence find the first three terms in the series expansion of \(\frac { 1 + 2 x } { \sqrt { 1 - 2 x } }\).
OCR MEI C4 2008 January Q3
3 Fig. 3 shows part of the curve \(y = 1 + x ^ { 2 }\), together with the line \(y = 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a8332ec-2216-4e1f-9768-ef175c9e159b-2_568_721_1034_712} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} The region enclosed by the curve, the \(y\)-axis and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the volume of the solid generated, giving your answer in terms of \(\pi\).
OCR MEI C4 2008 January Q4
4 The angle \(\theta\) satisfies the equation \(\sin \left( \theta + 45 ^ { \circ } \right) = \cos \theta\).
  1. Using the exact values of \(\sin 45 ^ { \circ }\) and \(\cos 45 ^ { \circ }\), show that \(\tan \theta = \sqrt { 2 } - 1\).
  2. Find the values of \(\theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR MEI C4 2008 January Q5
5 Express \(\frac { 4 } { x \left( x ^ { 2 } + 4 \right) }\) in partial fractions.
OCR MEI C4 2008 January Q6
6 Solve the equation \(\operatorname { cosec } \theta = 3\), for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR MEI C4 2008 January Q7
7 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a8332ec-2216-4e1f-9768-ef175c9e159b-3_632_1102_486_520} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
  2. Find the length of the edge CD.
  3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE .
  4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
    The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
  5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.
OCR MEI C4 2008 January Q8
8 A curve has equation $$x ^ { 2 } + 4 y ^ { 2 } = k ^ { 2 } ,$$ where \(k\) is a positive constant.
  1. Verify that $$x = k \cos \theta , \quad y = \frac { 1 } { 2 } k \sin \theta ,$$ are parametric equations for the curve.
  2. Hence or otherwise show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { 4 y }\).
  3. Fig. 8 illustrates the curve for a particular value of \(k\). Write down this value of \(k\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9a8332ec-2216-4e1f-9768-ef175c9e159b-4_657_1071_938_577} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
  4. Copy Fig. 8 and on the same axes sketch the curves for \(k = 1 , k = 3\) and \(k = 4\). On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.
  5. Explain why the path of the stream is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y } { x } .$$
  6. Solve this differential equation. Given that the path of the stream passes through the point \(( 2,1 )\), show that its equation is \(y = \frac { x ^ { 4 } } { 16 }\).
OCR MEI C4 2007 June Q1
1 Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C4 2007 June Q2
2 Write down normal vectors to the planes \(2 x + 3 y + 4 z = 10\) and \(x - 2 y + z = 5\).
Hence show that these planes are perpendicular to each other.
OCR MEI C4 2007 June Q3
3 Fig. 3 shows the curve \(y = \ln x\) and part of the line \(y = 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9296c786-a42a-4aa5-b326-39adbb544cbc-02_250_550_979_753} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.
  1. Show that the volume of the solid of revolution formed is given by \(\int _ { 0 } ^ { 2 } \pi \mathrm { e } ^ { 2 y } \mathrm {~d} y\).
  2. Evaluate this, leaving your answer in an exact form.
OCR MEI C4 2007 June Q4
4 A curve is defined by parametric equations $$x = \frac { 1 } { t } - 1 , y = \frac { 2 + t } { 1 + t }$$ Show that the cartesian equation of the curve is \(y = \frac { 3 + 2 x } { 2 + x }\).
OCR MEI C4 2007 June Q5
5 Verify that the point \(( - 1,6,5 )\) lies on both the lines $$\mathbf { r } = \left( \begin{array} { r } 1
2
- 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 0
OCR MEI C4 2007 June Q6
6
3 \end{array} \right) + \mu \left( \begin{array} { r } 1
0
- 2 \end{array} \right)$$ Find the acute angle between the lines. 6 Two students are trying to evaluate the integral \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\).
Sarah uses the trapezium rule with 2 strips, and starts by constructing the following table.
\(x\)11.52
\(\sqrt { 1 + \mathrm { e } ^ { - x } }\)1.16961.10601.0655
  1. Complete the calculation, giving your answer to 3 significant figures. Anish uses a binomial approximation for \(\sqrt { 1 + \mathrm { e } ^ { - x } }\) and then integrates this.
  2. Show that, provided \(\mathrm { e } ^ { - x }\) is suitably small, \(\left( 1 + \mathrm { e } ^ { - x } \right) ^ { \frac { 1 } { 2 } } \approx 1 + \frac { 1 } { 2 } \mathrm { e } ^ { - x } - \frac { 1 } { 8 } \mathrm { e } ^ { - 2 x }\).
  3. Use this result to evaluate \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\) approximately, giving your answer to 3 significant figures.
OCR MEI C4 2007 June Q7
7 Data suggest that the number of cases of infection from a particular disease tends to oscillate between two values over a period of approximately 6 months.
  1. Suppose that the number of cases, \(P\) thousand, after time \(t\) months is modelled by the equation \(P = \frac { 2 } { 2 - \sin t }\). Thus, when \(t = 0 , P = 1\).
    1. By considering the greatest and least values of \(\sin t\), write down the greatest and least values of \(P\) predicted by this model.
    2. Verify that \(P\) satisfies the differential equation \(\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } P ^ { 2 } \cos t\).
  2. An alternative model is proposed, with differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } \left( 2 P ^ { 2 } - P \right) \cos t$$ As before, \(P = 1\) when \(t = 0\).
    1. Express \(\frac { 1 } { P ( 2 P - 1 ) }\) in partial fractions.
    2. Solve the differential equation (*) to show that $$\ln \left( \frac { 2 P - 1 } { P } \right) = \frac { 1 } { 2 } \sin t$$ This equation can be rearranged to give \(P = \frac { 1 } { 2 - \mathrm { e } ^ { \frac { 1 } { 2 } \sin t } }\).
    3. Find the greatest and least values of \(P\) predicted by this model. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{9296c786-a42a-4aa5-b326-39adbb544cbc-05_609_622_301_719} \captionsetup{labelformat=empty} \caption{Fig. 8}
      \end{figure} In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations $$x = 10 \cos \theta + 5 \cos 2 \theta , \quad y = 10 \sin \theta + 5 \sin 2 \theta , \quad ( 0 \leqslant \theta < 2 \pi )$$ where \(x\) and \(y\) are in metres.
    4. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { \cos \theta + \cos 2 \theta } { \sin \theta + \sin 2 \theta }\). Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(\theta = \frac { 1 } { 3 } \pi\). Hence find the exact coordinates of the highest point A on the path of C .
    5. Express \(x ^ { 2 } + y ^ { 2 }\) in terms of \(\theta\). Hence show that $$x ^ { 2 } + y ^ { 2 } = 125 + 100 \cos \theta$$
    6. Using this result, or otherwise, find the greatest and least distances of C from O . You are given that, at the point B on the path vertically above O , $$2 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$
    7. Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. \section*{ADVANCED GCE UNIT MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) \section*{Paper B: Comprehension} \section*{THURSDAY 14 JUNE 2007} Afternoon
      Time: Up to 1 hour
      Additional materials:
      Rough paper
      MEI Examination Formulae and Tables (MF2) Candidate
      Name □
      Centre
      Number sufficient detail of the working to indicate that a correct method is being used. 1 This basic cycloid has parametric equations $$x = a \theta - a \sin \theta , \quad y = a - a \cos \theta$$
      \includegraphics[max width=\textwidth, alt={}]{9296c786-a42a-4aa5-b326-39adbb544cbc-10_307_1138_445_411}
      Find the coordinates of the points M and N , stating the value of \(\theta\) at each of them. Point M Point N 2 A sea wave has parametric equations (in suitable units) $$x = 7 \theta - 0.25 \sin \theta , \quad y = 0.25 \cos \theta$$ Find the wavelength and height of the wave.
      3 The graph below shows the profile of a wave.
    8. Assuming that it has parametric equations of the form given on line 68 , find the values of \(a\) and \(b\).
    9. Investigate whether the ratio of the trough length to the crest length is consistent with this shape.
      \includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-11_312_1141_623_415}
    10. \(\_\_\_\_\)
    11. \(\_\_\_\_\)
      4 This diagram illustrates two wave shapes \(U\) and \(V\). They have the same wavelength and the same height.
      \includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-12_423_1552_356_205} One of the curves is a sine wave, the other is a curtate cycloid.
    12. State which is which, justifying your answer.
    13. \(\_\_\_\_\)
      The parametric equations for the curves are: $$x = a \theta , \quad y = b \cos \theta ,$$ and $$x = a \theta - b \sin \theta , \quad y = b \cos \theta .$$
    14. Show that the distance marked \(d\) on the diagram is equal to \(b\).
    15. Hence justify the statement in lines 109 to 111: "In such cases, the curtate cycloid and the sine curve with the same wavelength and height are very similar and so the sine curve is also a good model."
    16. \(\_\_\_\_\)
    17. \(\_\_\_\_\)
      5 The diagram shows a curtate cycloid with scales given. Show that this curve could not be a scale drawing of the shape of a stable sea wave.
      \includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-13_289_1310_397_331}
OCR MEI C4 2010 June Q1
1 Express \(\frac { x } { x ^ { 2 } - 1 } + \frac { 2 } { x + 1 }\) as a single fraction, simplifying your answer.