Questions — OCR MEI C4 (354 questions)

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OCR MEI C4 2013 January Q6
5 marks Moderate -0.3
6 In Fig. 6, \(\mathrm { ABC } , \mathrm { ACD }\) and AED are right-angled triangles and \(\mathrm { BC } = 1\) unit. Angles CAB and CAD are \(\theta\) and \(\phi\) respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9bceee25-35bd-448b-a4a2-1a5667be5f11-03_440_524_504_753} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Find AC and AD in terms of \(\theta\) and \(\phi\).
  2. Hence show that \(\mathrm { DE } = 1 + \frac { \tan \phi } { \tan \theta }\). Section B (36 marks)
OCR MEI C4 2013 January Q7
17 marks Standard +0.3
7 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(\mathrm { O } x y\) plane is horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9bceee25-35bd-448b-a4a2-1a5667be5f11-03_547_987_1580_539} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
  2. Show that the vector \(\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\) is normal to the plane through \(\mathrm { A } , \mathrm { D }\) and E . Hence find the equation of this plane. Given that B lies in this plane, find \(a\).
  3. Verify that the equation of the plane BCD is \(x + z = 8\). Hence find the acute angle between the planes ABDE and BCD .
OCR MEI C4 2013 January Q8
19 marks Standard +0.3
8 The growth of a tree is modelled by the differential equation $$10 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 20 - h ,$$ where \(h\) is its height in metres and the time \(t\) is in years. It is assumed that the tree is grown from seed, so that \(h = 0\) when \(t = 0\).
  1. Write down the value of \(h\) for which \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 0\), and interpret this in terms of the growth of the tree.
  2. Verify that \(h = 20 \left( 1 - \mathrm { e } ^ { - 0.1 t } \right)\) satisfies this differential equation and its initial condition. The alternative differential equation $$200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 400 - h ^ { 2 }$$ is proposed to model the growth of the tree. As before, \(h = 0\) when \(t = 0\).
  3. Using partial fractions, show by integration that the solution to the alternative differential equation is $$h = \frac { 20 \left( 1 - \mathrm { e } ^ { - 0.2 t } \right) } { 1 + \mathrm { e } ^ { - 0.2 t } } .$$
  4. What does this solution indicate about the long-term height of the tree?
  5. After a year, the tree has grown to a height of 2 m . Which model fits this information better?
OCR MEI C4 2015 June Q2
7 marks Moderate -0.3
2 Express \(6 \cos 2 \theta + \sin \theta\) in terms of \(\sin \theta\).
Hence solve the equation \(6 \cos 2 \theta + \sin \theta = 0\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C4 2015 June Q3
8 marks Standard +0.3
3
  1. Find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }\). State the set of values of \(x\) for which
    the expansion is valid. the expansion is valid.
  2. Hence find \(a\) and \(b\) such that \(\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots\).
OCR MEI C4 2015 June Q4
8 marks Moderate -0.3
4 You are given that \(\mathrm { f } ( x ) = \cos x + \lambda \sin x\) where \(\lambda\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving \(R\) and \(\alpha\) in terms of \(\lambda\).
  2. Given that the maximum value (as \(x\) varies) of \(\mathrm { f } ( x )\) is 2 , find \(R , \lambda\) and \(\alpha\), giving your answers in exact form.
OCR MEI C4 2015 June Q5
8 marks Standard +0.3
5 A curve has parametric equations \(x = \sec \theta , y = 2 \tan \theta\).
  1. Given that the derivative of \(\sec \theta\) is \(\sec \theta \tan \theta\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta\).
  2. Verify that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4\). Fig. 5 shows the region enclosed by the curve and the line \(x = 2\). This region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-02_545_853_1738_607} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  3. Find the volume of revolution produced, giving your answer in exact form.
OCR MEI C4 2015 June Q6
18 marks Standard +0.3
6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-03_785_1283_424_392} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
  2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
    (B) Hence find the cartesian equation of this plane.
    (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
  3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
  4. Find the height of Q above the ground. \section*{Question 7 begins on page 4.}
OCR MEI C4 2015 June Q7
18 marks Standard +0.3
7 A drug is administered by an intravenous drip. The concentration, \(x\), of the drug in the blood is measured as a fraction of its maximum level. The drug concentration after \(t\) hours is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k \left( 1 + x - 2 x ^ { 2 } \right) ,$$ where \(0 \leqslant x < 1\), and \(k\) is a positive constant. Initially, \(x = 0\).
  1. Express \(\frac { 1 } { ( 1 + 2 x ) ( 1 - x ) }\) in partial fractions.
  2. Hence solve the differential equation to show that \(\frac { 1 + 2 x } { 1 - x } = \mathrm { e } ^ { 3 k t }\).
  3. After 1 hour the drug concentration reaches \(75 \%\) of its maximum value and so \(x = 0.75\). Find the value of \(k\), and the time taken for the drug concentration to reach \(90 \%\) of its maximum value.
  4. Rearrange the equation in part (ii) to show that \(x = \frac { 1 - \mathrm { e } ^ { - 3 k t } } { 1 + 2 \mathrm { e } ^ { - 3 k t } }\). Verify that in the long term the drug concentration approaches its maximum value. \section*{END OF QUESTION PAPER} \section*{Tuesday 16 J une 2015 - Afternoon} \section*{A2 GCE MATHEMATICS (MEI)} 4754/01B Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{QUESTION PAPER} \section*{Candidates answer on the Question Paper.} \section*{OCR supplied materials:}
    • Insert (inserted)
    • MEI Examination Formulae and Tables (MF2)
    \section*{Other materials required:}
    • Scientific or graphical calculator
    • Rough paper
    Duration: Up to 1 hour \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-05_117_495_1014_1308} PLEASE DO NOT WRITE IN THIS SPACE 2 In line 79 it says "For most journeys, more than half the journey time is composed of load time and transfer time". For what percentage of the journey time for the round trip made by car A in Table 4 is the car stationary?
    \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_645_1746_388_164}
    3 Using the expression on line 51, work out the answer to the question on lines 39 and 40 for the case where there are 10 upper floors and 7 people. Give your answer to 2 decimal places.
    \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_488_1746_1233_164}
    4 In lines 89 and 90 it says "... on average there will be approximately 8 stops per trip. A round trip with 8 stops would take between 188 and 200 seconds". Explain how the figure of 188 seconds has been derived. 5
  5. Referring to Strategy 3 and lines 99 to 101, complete the table below for car C .
  6. Calculate the time car C will take to transport all the people who work on floors 7 and 8 , and return to the ground floor.
    5
  		7. \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-08_1095_816_484_700}
    68 people make independent visits to any one of the upper floors of a building with 10 upper floors. What is the probability that at least one of the visitors goes to the top floor?
    6
    7 On lines 94 and 95 it says "Table 4 gives the timings for round trips in which the cars are required to stop at every floor they serve; Table 2 suggests this is a common occurrence in this case". Explain how Table 2 is used to make this claim. \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-09_1093_1740_1238_166} END OF QUESTION PAPER
OCR MEI C4 Q1
Easy -2.5
1 Explain why the number 1836.108 for the ratio Rest mass of electron would be suitable for communication with other civilisations whereas neither the rest mass of the proton nor that of the electron would be.
OCR MEI C4 Q2
Standard +0.8
2 A civilisation which works in base 5 sends out the first 6 digits of \(\pi\) as 3.032 32. Convert this to base 10.
OCR MEI C4 Q3
Easy -1.8
3 Complete this table to show the next 3 values of the iteration $$x _ { n + 1 } = k x _ { n } \left( 1 - x _ { n } \right)$$ in the case when \(k = 3.2\) and \(x _ { 0 } = 0.5\). Give your answers to calculator accuracy.
\(n\)\(x _ { n }\)
00.5
10.8
20.512
3
4
5
OCR MEI C4 Q4
Moderate -0.5
4 Justify the statement that the equation in line 83, $$\frac { \phi } { 1 } = \frac { 1 } { \phi - 1 }$$ has the solution \(\phi = \frac { 1 \pm \sqrt { 5 } } { 2 }\).
OCR MEI C4 Q5
Standard +0.8
5 Justify the statement in line 87 that $$\frac { 1 } { \phi } = \frac { \sqrt { 5 } - 1 } { 2 }$$
OCR MEI C4 Q6
4 marks Challenging +1.2
6 A sequence is defined by $$a _ { n + 1 } = 2 a _ { n } + 3 a _ { n - 1 } \quad \text { with } a _ { 1 } = 1 \text { and } a _ { 2 } = 1 .$$ Using the method on page 5, show that the value to which the ratio of successive terms converges is 3 .
[0pt] [4]
OCR MEI C4 Q8
Standard +0.3
8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{070e9904-12b9-4458-b8f2-60c89b31b828-093_1013_1399_488_372} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)
  1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
  2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
  3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
  4. Write down a vector equation of the line RS. Calculate the coordinates of S.
OCR MEI C4 2005 June Q2
6 marks Moderate -0.5
2 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 2005 June Q3
4 marks Easy -1.2
3 Solve the equation $$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$ giving your answers in terms of \(\pi\).
OCR MEI C4 2005 June Q4
5 marks Standard +0.3
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]{7a1123f8-53cd-4b24-bec6-8c3bccc22653-3_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 2005 June Q5
7 marks Moderate -0.3
5 Solve the equation \(2 \cos 2 x = 1 + \cos x\), for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
OCR MEI C4 2005 June Q6
8 marks Standard +0.3
6 A curve has cartesian equation \(y ^ { 2 } - x ^ { 2 } = 4\).
  1. Verify that $$x = t - \frac { 1 } { t } , \quad y = t + \frac { 1 } { t } ,$$ are parametric equations of the curve.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - 1 ) ( t + 1 ) } { t ^ { 2 } + 1 }\). Hence find the coordinates of the stationary points of the curve. Section B (36 marks)
OCR MEI C4 2005 June Q7
18 marks Standard +0.3
7 In a chemical process, the mass \(M\) grams of a chemical at time \(t\) minutes is modelled by the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = \frac { M } { t \left( 1 + t ^ { 2 } \right) }$$
  1. Find \(\int \frac { t } { 1 + t ^ { 2 } } \mathrm {~d} t\).
  2. Find constants \(A , B\) and \(C\) such that $$\frac { 1 } { t \left( 1 + t ^ { 2 } \right) } = \frac { A } { t } + \frac { B t + C } { 1 + t ^ { 2 } } .$$
  3. Use integration, together with your results in parts (i) and (ii), to show that $$M = \frac { K t } { \sqrt { 1 + t ^ { 2 } } } ,$$ where \(K\) is a constant.
  4. When \(t = 1 , M = 25\). Calculate \(K\). What is the mass of the chemical in the long term?
OCR MEI C4 2010 June Q5
8 marks Standard +0.3
  1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
  2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
  3. Find the coordinates of the point where the pipeline meets the layer of rock.
  4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer. 8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \caption{Fig. 8}
    \end{figure}
  5. Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
  6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
  7. Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
  8. Express \(\cos \theta - 4 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\). Hence solve the equation \(\cos \theta - 4 \sin \theta = 2\) for \(0 \leqslant \theta \leqslant 2 \pi\). {www.ocr.org.uk}) after the live examination series.
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    OCR is part of the \section*{ADVANCED GCE
    MATHEMATICS (MEI)} 4754B
    Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
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    \section*{Other Materials Required:}
    • Rough paper
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    Wednesday 9 June 2010 Afternoon \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
    2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
    [0pt] [An answer obtained from the graph will be given no marks.]
    3
  9. In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
  10. The text then goes on to state that the emissions per extra passenger on this journey are less than \(\frac { 1 } { 2 } \mathrm {~kg}\). Justify this figure.
  11. \(\_\_\_\_\)
  12. \(\_\_\_\_\) 4 The daily number of trains, \(n\), on a line in another country may be modelled by the function defined below, where \(P\) is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 } \\ & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 } \\ & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 } \\ & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 } \\ & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 } \\ & \ldots \text { and so on } \ldots \end{aligned}$$
  13. Sketch the graph of \(n\) against \(P\).
  14. Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
  15. \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
  16. \(\_\_\_\_\)
OCR MEI C4 Q6
4 marks Moderate -0.8
6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.
OCR MEI C4 2006 January Q1
5 marks Moderate -0.3
1 Solve the equation \(\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3\).