Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 Q3
  1. Express \(\mathrm { f } ( x )\) in factorised form.
  2. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
  3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
  4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\).
OCR MEI C2 Q12
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 Q6
  1. On the copy of Fig. 5, draw by eye a tangent to the curve at the point where \(x = 2\). Hence find an estimate of the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
  2. Calculate the \(y\)-values on the curve when \(x = 1.8\) and \(x = 2.2\). Hence calculate another approximation to the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
OCR MEI C2 Q11
  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 2007 January Q13
13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\).
Number of months after start-up \(( x )\)123456
Profit for this month \(( \pounds y )\)5008001200190030004800
The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.
  1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
  3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
  4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
  5. State one way in which this model is unrealistic.
OCR MEI C2 2009 January Q5
5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-3_979_1077_422_536} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} On the insert, draw the graph of
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 2009 January Q12
12 Answer part (ii) of this question on the insert provided. The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted.
Years after proposal accepted \(( t )\)12345
Cost \(( \pounds y\) million \()\)250300360440530
The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.
  1. Show that \(y = a b ^ { t }\) may be written as $$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
  3. Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
  4. According to this model, what was the estimated cost of the project when it was first accepted?
  5. Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.
OCR MEI C2 2010 January Q12
12 Answer part (ii) of this question on the insert provided. Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005.
Year195519651975198519952005
Population (millions)131161209277372492
Source: United Nations Such estimates are used to model future population growth and world needs of resources. One model is \(P = a 10 ^ { b t }\), where the population is \(P\) millions, \(t\) is the number of years after 1945 and \(a\) and \(b\) are constants.
  1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) is a straight line of gradient \(b\). State the intercept of this line on the vertical axis.
  2. On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of \(\log _ { 10 } P\) against \(t\). Draw, by eye, a line of best fit on your graph.
  3. Use your graph to find the equation for \(P\) in terms of \(t\).
  4. Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate.
OCR MEI C2 2006 June Q12
12 Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10 ^ { b t }\), where \(t\) is the time in years after 2000.
  1. Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis.
  2. The table gives the data for the population from 2001 to 2005.
    Year20012002200320042005
    \(t\)12345
    \(P\)79008800100001130012800
    Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(t\). Draw a line of best fit for the data.
  3. Use your graph to find the equation for \(P\) in terms of \(t\).
  4. Predict the population in 2008 according to this model.
OCR MEI C2 2008 June Q1
1 Express \(\frac { 7 \pi } { 6 }\) radians in degrees.
OCR MEI C2 2008 June Q3
3 State the transformation which maps the graph of \(y = x ^ { 2 } + 5\) onto the graph of \(y = 3 x ^ { 2 } + 15\).
OCR MEI C2 2008 June Q4
4 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 2008 June Q5
5 In Fig. 5, A and B are the points on the curve \(y = 2 ^ { x }\) with \(x\)-coordinates 3 and 3.1 respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-2_700_728_1197_705} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the gradient of the chord AB . Give your answer correct to 2 decimal places.
  2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2 ^ { x }\) at A .
OCR MEI C2 2008 June Q6
6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x }\). Find the equation of the curve, given that it passes through the point \(( 9,105 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-3_420_522_264_810} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} A sector of a circle of radius 6 cm has angle 1.6 radians, as shown in Fig. 7.
Find the area of the sector.
Hence find the area of the shaded segment.
OCR MEI C2 2008 June Q8
5 marks
8 The 11th term of an arithmetic progression is 1 . The sum of the first 10 terms is 120 . Find the 4th term. [5]
OCR MEI C2 2008 June Q9
9 Use logarithms to solve the equation \(5 ^ { x } = 235\), giving your answer correct to 2 decimal places.
OCR MEI C2 2008 June Q10
10 Showing your method, solve the equation \(2 \sin ^ { 2 } \theta = \cos \theta + 2\) for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR MEI C2 2008 June Q11
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-4_1022_942_356_603} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows a sketch of the cubic curve \(y = \mathrm { f } ( x )\). The values of \(x\) where it crosses the \(x\)-axis are - 5 , - 2 and 2 , and it crosses the \(y\)-axis at \(( 0 , - 20 )\).
  1. Express \(\mathrm { f } ( x )\) in factorised form.
  2. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
  3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
  4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a2188eac-08f7-4e75-a76d-fe35b13a2e5f-5_689_1006_269_568} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
  5. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough.
  6. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15\), for \(0 \leqslant x \leqslant 0.5\). Calculate \(\int _ { 0 } ^ { 0.5 } \left( 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15 \right) \mathrm { d } x\) and state what this represents.
    Hence find the volume of water in the trough as given by this model.
OCR MEI C2 2008 June Q13
13 The percentage of the adult population visiting the cinema in Great Britain has tended to increase since the 1980s. The table shows the results of surveys in various years.
Year\(1986 / 87\)\(1991 / 92\)\(1996 / 97\)\(1999 / 00\)\(2000 / 01\)\(2001 / 02\)
Percentage of the
adult population
visiting the cinema
314454565557
Source: Department of National Statistics, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
This growth may be modelled by an equation of the form $$P = a t ^ { b } ,$$ where \(P\) is the percentage of the adult population visiting the cinema, \(t\) is the number of years after the year 1985/86 and \(a\) and \(b\) are constants to be determined.
  1. Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(\log _ { 10 } t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. \section*{Answer part (ii) of this question on the insert provided.}
  2. Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(\log _ { 10 } t\). Draw by eye a line of best fit for the data.
  3. Use your graph to find the equation for \(P\) in terms of \(t\).
  4. Predict the percentage of the adult population visiting the cinema in the year 2007/2008 (i.e. when \(t = 22\) ), according to this model.
OCR MEI C2 2010 June Q1
1 You are given that $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\). Give your answers as fractions.
OCR MEI C2 2010 June Q2
2
  1. Evaluate \(\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }\).
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )\) where \(\mathrm { f } ( r )\) and \(a\) are to be determined.
OCR MEI C2 2010 June Q3
3
  1. Differentiate \(x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50\).
  2. Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50\).
OCR MEI C2 2010 June Q4
4 In this question, \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x\). Fig. 4 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e5ac28f3-d61a-4b40-8b47-28c930761a28-2_789_887_1427_628} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} On separate diagrams, sketch the curves \(y = \mathrm { f } ( 2 x )\) and \(y = 3 \mathrm { f } ( x )\), labelling the coordinates of their intersections with the axes and their turning points.
OCR MEI C2 2010 June Q5
5 Find \(\int _ { 2 } ^ { 5 } \left( 1 - \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x\).
OCR MEI C2 2010 June Q6
6 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.