Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-003_305_897_310_795} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).
  1. Find \(\theta\).
  2. Hence find the perimeter of the sector.
OCR MEI C2 Q9
9
  1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
    \end{figure} Using this model,
    (A) find the greatest height of the tunnel,
    (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
  2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
    Hence estimate the volume of earth removed when the tunnel is re-shaped.
OCR MEI C2 Q11
11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.
Time (minutes)1020304050
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)6853423631
The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.
  1. Give a physical interpretation for the constant \(z _ { 0 }\).
  2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
  3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
    Hence estimate the temperature of the drink 70 minutes after it is made. \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Concepts for Advanced Mathematics (C2)
    INSERT
    Wednesday
OCR MEI C2 2005 January Q1
1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).
OCR MEI C2 2005 January Q2
2 Find \(\int \left( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } \right) \mathrm { d } x\).
OCR MEI C2 2005 January Q3
3 Sketch the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.2\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2005 January Q4
4 Fig. 4 For triangle ABC shown in Fig. 4, calculate
  1. the length of BC ,
  2. the area of triangle ABC .
OCR MEI C2 2005 January Q5
5 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2.
Find also the sum to infinity of this progression.
OCR MEI C2 2005 January Q6
6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4
a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.
OCR MEI C2 2005 January Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-4_305_897_310_795} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).
  1. Find \(\theta\).
  2. Hence find the perimeter of the sector.
OCR MEI C2 2005 January Q8
8
  1. Solve the equation \(10 ^ { x } = 316\).
  2. Simplify \(\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)\).
OCR MEI C2 2005 January Q9
9
  1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
    \end{figure} Using this model,
    (A) find the greatest height of the tunnel,
    (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
  2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
    Hence estimate the volume of earth removed when the tunnel is re-shaped.
OCR MEI C2 2005 January Q10
10 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
  1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
  2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).
OCR MEI C2 2005 January Q11
11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.
Time (minutes)1020304050
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)6853423631
The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.
  1. Give a physical interpretation for the constant \(z _ { 0 }\).
  2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
  3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
    Hence estimate the temperature of the drink 70 minutes after it is made.
OCR MEI C2 2006 January Q1
1 Given that \(140 ^ { \circ } = k \pi\) radians, find the exact value of \(k\).
OCR MEI C2 2006 January Q2
2 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).
OCR MEI C2 2006 January Q3
3 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
OCR MEI C2 2006 January Q4
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-2_615_971_1457_539} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a curve which passes through the points shown in the following table.
\(x\)11.522.533.54
\(y\)8.26.45.55.04.74.44.2
Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve, the lines \(x = 1\) and \(x = 4\), and the \(x\)-axis. State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of the area of this region.
OCR MEI C2 2006 January Q5
5
  1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2006 January Q6
6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x + 9\). Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show that the curve has a stationary point of inflection when \(x = 3\).
OCR MEI C2 2006 January Q7
7 In Fig. 7, A and B are points on the circumference of a circle with centre O . Angle \(\mathrm { AOB } = 1.2\) radians. The arc length AB is 6 cm . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-3_371_723_1048_833} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the radius of the circle.
  2. Calculate the length of the chord AB .
OCR MEI C2 2006 January Q8
8 Find \(\int \left( x ^ { \frac { 1 } { 2 } } + \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x\).
OCR MEI C2 2006 January Q9
9 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-4_591_985_312_701} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} The graph of \(\log _ { 10 } y\) against \(x\) is a straight line as shown in Fig. 9 .
  1. Find the equation for \(\log _ { 10 } y\) in terms of \(x\).
  2. Find the equation for \(y\) in terms of \(x\). Section B (36 marks)
OCR MEI C2 2006 January Q10
10 The equation of a curve is \(y = 7 + 6 x - x ^ { 2 }\).
  1. Use calculus to find the coordinates of the turning point on this curve. Find also the coordinates of the points of intersection of this curve with the axes, and sketch the curve.
  2. Find \(\int _ { 1 } ^ { 5 } \left( 7 + 6 x - x ^ { 2 } \right) \mathrm { d } x\), showing your working.
  3. The curve and the line \(y = 12\) intersect at ( 1,12 ) and ( 5,12 ). Using your answer to part (ii), find the area of the finite region between the curve and the line \(y = 12\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-5_643_1034_331_513} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).
OCR MEI C2 2007 January Q1
1 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).