Questions C4 (1162 questions)

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OCR MEI C4 2006 June Q4
4
  1. The number of bacteria in a colony is increasing at a rate that is proportional to the square root of the number of bacteria present. Form a differential equation relating \(x\), the number of bacteria, to the time \(t\).
  2. In another colony, the number of bacteria, \(y\), after time \(t\) minutes is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = \frac { 10000 } { \sqrt { y } } .$$ Find \(y\) in terms of \(t\), given that \(y = 900\) when \(t = 0\). Hence find the number of bacteria after 10 minutes.
OCR MEI C4 2006 June Q5
5
  1. Show that \(\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 x } ( 1 + 2 x ) + c\). A vase is made in the shape of the volume of revolution of the curve \(y = x ^ { 1 / 2 } \mathrm { e } ^ { - x }\) about the \(x\)-axis between \(x = 0\) and \(x = 2\) (see Fig. 5). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-3_716_741_1233_662} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  2. Show that this volume of revolution is \(\frac { 1 } { 4 } \pi \left( 1 - \frac { 5 } { \mathrm { e } ^ { 4 } } \right)\). Fig. 6 shows the arch ABCD of a bridge. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-4_378_1630_461_214} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure} The section from B to C is part of the curve OBCE with parametric equations $$x = a ( \theta - \sin \theta ) , y = a ( 1 - \cos \theta ) \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ where \(a\) is a constant.
  3. Find, in terms of \(a\),
    (A) the length of the straight line OE,
    (B) the maximum height of the arch.
  4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). The straight line sections AB and CD are inclined at \(30 ^ { \circ }\) to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the \(x\)-axis. BF is parallel to the \(y\)-axis.
  5. Show that at the point B the parameter \(\theta\) satisfies the equation $$\sin \theta = \frac { 1 } { \sqrt { 3 } } ( 1 - \cos \theta )$$ Verify that \(\theta = \frac { 2 } { 3 } \pi\) is a solution of this equation.
    Hence show that \(\mathrm { BF } = \frac { 3 } { 2 } a\), and find OF in terms of \(a\), giving your answer exactly.
  6. Find BC and AF in terms of \(a\). Given that the straight line distance AD is 20 metres, calculate the value of \(a\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-5_748_1306_319_367} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .
  7. Find the length AE .
  8. Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
  9. Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30$$ Write down a vector normal to this plane.
  10. Show that the vector \(\left( \begin{array} { l } 4
    3
    5 \end{array} \right)\) is normal to the plane ABDE . Hence find the equation of the plane ABDE .
  11. Find the angle between the planes ABC and ABDE . RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4) \section*{Paper B: Comprehension} Monday 12 JUNE 2006 Afternoon Up to 1 hour Additional materials:
    Rough paper
    MEI Examination Formulae and Tables (MF2) TIME Up to 1 hour
    • Write your name, centre number and candidate number in the spaces at the top of this page.
    • Answer all the questions.
    • Write your answers in the spaces provided on the question paper.
    • You are permitted to use a graphical calculator in this paper.
    • The number of marks is given in brackets [ ] at the end of each question or part question.
    • The insert contains the text for use with the questions.
    • You may find it helpful to make notes and do some calculations as you read the passage.
    • You are not required to hand in these notes with your question paper.
    • You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
    • The total number of marks for this paper is 18.
    For Examiner's Use
    Qu.Mark
    1
    2
    3
    4
    5
    6
    Total
    1 The marathon is 26 miles and 385 yards long ( 1 mile is 1760 yards). There are now several men who can run 2 miles in 8 minutes. Imagine that an athlete maintains this average speed for a whole marathon. How long does the athlete take?
    2 According to the linear model, in which calendar year would the record for the men's mile first become negative?
    3 Explain the statement in line 93 "According to this model the 2-hour marathon will never be run."
    4 Explain how the equation in line 49, $$R = L + ( U - L ) \mathrm { e } ^ { - k t } ,$$ is consistent with Fig. 2
  12. initially,
  13. for large values of \(t\).
  15. \(\_\_\_\_\)
    5 A model for an athletics record has the form $$R = A - ( A - B ) \mathrm { e } ^ { - k t } \text { where } A > B > 0 \text { and } k > 0 .$$
  16. Sketch the graph of \(R\) against \(t\), showing \(A\) and \(B\) on your graph.
  17. Name one event for which this might be an appropriate model.

  18. \includegraphics[max width=\textwidth, alt={}, center]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-9_803_808_721_575}
  19. \(\_\_\_\_\)
OCR MEI C4 2006 June Q6
6 A number of cases of the general exponential model for the marathon are given in Table 6. One of these is $$R = 115 + ( 175 - 115 ) \mathrm { e } ^ { - 0.0467 t ^ { 0.797 } }$$
  1. What is the value of \(t\) for the year 2012?
  2. What record time does this model predict for the year 2012?
  3. \(\_\_\_\_\)
  4. \(\_\_\_\_\)
OCR MEI C4 2008 June Q1
1 Express \(\frac { x } { x ^ { 2 } - 4 } + \frac { 2 } { x + 2 }\) as a single fraction, simplifying your answer.
OCR MEI C4 2008 June Q2
2 Fig. 2 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-02_432_873_587_635} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of the solid of revolution produced is \(\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)\).
OCR MEI C4 2008 June Q3
3 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).
OCR MEI C4 2008 June Q4
4 Given that \(x = 2 \sec \theta\) and \(y = 3 \tan \theta\), show that \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1\).
OCR MEI C4 2008 June Q5
5 A curve has parametric equations \(x = 1 + u ^ { 2 } , y = 2 u ^ { 3 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\).
  2. Hence find the gradient of the curve at the point with coordinates \(( 5,16 )\).
OCR MEI C4 2008 June Q6
6
  1. Find the first three non-zero terms of the binomial series expansion of \(\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }\), and state the set of values of \(x\) for which the expansion is valid.
  2. Hence find the first three non-zero terms of the series expansion of \(\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }\).
OCR MEI C4 2008 June Q7
7 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).
OCR MEI C4 2008 June Q8
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]{8ad99e2a-4cef-40b3-af8d-673b97536227-03_1004_1397_493_374} \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 2008 June Q9
9 A skydiver drops from a helicopter. Before she opens her parachute, her speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after time \(t\) seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When \(t = 0 , v = 0\).
  1. Find \(v\) in terms of \(t\).
  2. According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Her speed \(t\) seconds after this is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
  3. Express \(\frac { 1 } { ( w - 4 ) ( w + 5 ) }\) in partial fractions.
  4. Using this result, show that \(\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }\).
  5. According to this model, what is the speed of the skydiver in the long term? RECOGNISING ACHIEVEMENT \section*{ADVANCED GCE} \section*{4754/01B} \section*{MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) Paper B: Comprehension
    WEDNESDAY 21 MAY 2008
    Afternoon
    Time: Up to 1 hour
    Additional materials: Rough paper
    MEI Examination Formulae and Tables (MF 2) \section*{Candidate Forename}
    \includegraphics[max width=\textwidth, alt={}]{8ad99e2a-4cef-40b3-af8d-673b97536227-05_125_547_986_516}
    This document consists of \(\mathbf { 6 }\) printed pages, \(\mathbf { 2 }\) blank pages and an insert. 1 Complete these Latin square puzzles.
  6. 213
    3

  7. \includegraphics[max width=\textwidth, alt={}, center]{8ad99e2a-4cef-40b3-af8d-673b97536227-06_391_419_836_854} 2 In line 51, the text says that the Latin square
    1234
    3142
    2413
    4321
    could not be the solution to a Sudoku puzzle.
    Explain this briefly.
    3 On lines 114 and 115 the text says "It turns out that there are 16 different ways of filling in the remaining cells while keeping to the Sudoku rules. One of these ways is shown in Fig.10." Complete the grid below with a solution different from that given in Fig. 10.
    1234
    4 Lines 154 and 155 of the article read "There are three other embedded Latin squares in Fig. 14; one of them is illustrated in Fig. 16." Indicate one of the other two embedded Latin squares on this copy of Fig. 14.
    4231
    24
    42
    2413
    5 The number of \(9 \times 9\) Sudokus is given in line 121 .
    Without doing any calculations, explain why you would expect 9! to be a factor of this number.
    6 In the table below, \(M\) represents the maximum number of givens for which a Sudoku puzzle may have no unique solution (Investigation 3 in the article). \(s\) is the side length of the Sudoku grid and \(b\) is the side length of its blocks.
    Block side
    length, \(b\)
    Sudoku,
    \(s \times s\)
    		\(M\)
    1\(1 \times 1\)-
    2\(4 \times 4\)12
    3\(9 \times 9\)
    4\(16 \times 16\)
    5
  8. Complete the table.
  9. Give a formula for \(M\) in terms of \(b\).
    7 A man is setting a Sudoku puzzle and starts with this solution.
    123456789
    456897312
    789312564
    231564897
    564978123
    897123645
    312645978
    645789231
    978231456
    He then removes some of the numbers to give the puzzles in parts (i) and (ii). In each case explain briefly, and without trying to solve the puzzle, why it does not have a unique solution.
    [0pt] [2,2]
  10. 12469
    4891
    86
    2147
    647812
    8924
    16497
    64791
    982146
  11. 123456789
    456897312
    789564
    231564897
    564978123
    897645
    312645978
    645789231
    978456
  12. \(\_\_\_\_\)
  13. \(\_\_\_\_\)