Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 2015 June Q2
2 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).
OCR MEI C2 2015 June Q3
3 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression.
OCR MEI C2 2015 June Q4
4 A sector of a circle has angle 1.5 radians and area \(27 \mathrm {~cm} ^ { 2 }\). Find the perimeter of the sector.
OCR MEI C2 2015 June Q5
5 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.
OCR MEI C2 2015 June Q6
6
  1. On the same axes, sketch the curves \(y = 3 ^ { x }\) and \(y = 3 ^ { 2 x }\), identifying clearly which is which.
  2. Given that \(3 ^ { 2 x } = 729\), find in either order the values of \(3 ^ { x }\) and \(x\).
OCR MEI C2 2015 June Q7
7 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).
OCR MEI C2 2015 June Q8
8 Fig. 8 shows the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\). It is a straight line passing through the points \(( 2,8 )\) and \(( 0,2 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-2_460_634_1868_717} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Find the equation relating \(\log _ { 10 } y\) and \(\log _ { 10 } x\) and hence find the equation relating \(y\) and \(x\).
OCR MEI C2 2015 June Q9
9 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_253_1486_328_292} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing's cross-section. Calculate angle A and the area of the cross-section.
  2. Jean then modifies her design for the wing. Fig. 9.2 shows the new cross-section, with 1 unit for each of \(x\) and \(y\) representing one centimetre. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_431_1682_970_194} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Here are some of the coordinates that Jean used to draw the new cross-section.
    Upper surfaceLower surface
    \(x\)\(y\)\(x\)\(y\)
    0000
    41.454-0.85
    81.568-0.76
    121.2712-0.55
    161.0416-0.30
    200200
    Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section.
OCR MEI C2 2015 June Q10
10 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).
  1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
  2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve.
  3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\). Write down the coordinates of the minimum point of the transformed curve.
OCR MEI C2 2015 June Q11
11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.
  1. How many of Jill's descendants would there be in generation 8 ?
  2. How many of Jill's descendants would there be altogether in the first 15 generations?
  3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
  4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}
OCR MEI C2 2016 June Q1
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = 6 \sqrt { x }\).
  2. Find \(\int \frac { 12 } { x ^ { 2 } } \mathrm {~d} x\).
OCR MEI C2 2016 June Q2
2 A sequence is defined as follows.
\(u _ { 1 } = a\), where \(a > 0\)
To obtain \(u _ { r + 1 }\)
  • find the remainder when \(u _ { r }\) is divided by 3 ,
  • multiply the remainder by 5 ,
  • the result is \(u _ { r + 1 }\).
Find \(\sum _ { r = 2 } ^ { 4 } u _ { r }\) in each of the following cases.
  1. \(a = 5\)
  2. \(a = 6\)
OCR MEI C2 2016 June Q3
3 An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12 . Calculate the difference between the tenth terms of the AP and the GP. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-2_581_855_1644_587} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows triangle ABC , where \(\mathrm { AB } = 7.2 \mathrm {~cm} , \mathrm { AC } = 5.6 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 68 ^ { \circ }\).
Calculate the size of angle ACB .
OCR MEI C2 2016 June Q5
5
  1. Fig. 5 shows the graph of a sine function. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-3_534_1154_312_450} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} State the equation of this curve.
  2. Sketch the graph of \(y = \sin x - 3\) for \(0 ^ { \circ } \leqslant x \leqslant 450 ^ { \circ }\).
OCR MEI C2 2016 June Q6
6 A sector of a circle has radius \(r \mathrm {~cm}\) and sector angle \(\theta\) radians. It is divided into two regions, A and B . Region A is an isosceles triangle with the equal sides being of length \(a \mathrm {~cm}\), as shown in Fig. 6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-3_407_469_1343_612} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} \section*{Not to scale}
  1. Express the area of B in terms of \(a , r\) and \(\theta\).
  2. Given that \(r = 12\) and \(\theta = 0.8\), find the value of \(a\) for which the areas of A and B are equal. Give your answer correct to 3 significant figures.
OCR MEI C2 2016 June Q7
7
  1. Show that, when \(x\) is an acute angle, \(\tan x \sqrt { 1 - \sin ^ { 2 } x } = \sin x\).
  2. Solve \(4 \sin ^ { 2 } y = \sin y\) for \(0 ^ { \circ } \leqslant y \leqslant 360 ^ { \circ }\).
OCR MEI C2 2016 June Q8
8
  1. Simplify \(\log _ { a } 1 - \log _ { a } \left( a ^ { m } \right) ^ { 3 }\).
  2. Use logarithms to solve the equation \(3 ^ { 2 x + 1 } = 1000\). Give your answer correct to 3 significant figures.
OCR MEI C2 2016 June Q9
9 Fig. 9 shows the cross-section of a straight, horizontal tunnel. The \(x\)-axis from 0 to 6 represents the floor of the tunnel. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-4_668_734_456_662} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.
\(x\)0123456
\(y\)04.04.95.04.94.00
The length of the tunnel is 50 m .
  1. Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel.
  2. An engineer models the height of the roof of the tunnel using the curve \(y = \frac { 5 } { 81 } \left( 108 x - 54 x ^ { 2 } + 12 x ^ { 3 } - x ^ { 4 } \right)\). This curve is symmetrical about \(x = 3\).
    (A) Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel.
    (B) Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel.
OCR MEI C2 2016 June Q10
10
  1. Calculate the gradient of the chord of the curve \(y = x ^ { 2 } - 2 x\) joining the points at which the values of \(x\) are 5 and 5.1.
  2. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 2 x\), find and simplify \(\frac { \mathrm { f } ( 5 + h ) - \mathrm { f } ( 5 ) } { h }\).
  3. Use your result in part (ii) to find the gradient of the curve \(y = x ^ { 2 } - 2 x\) at the point where \(x = 5\), showing your reasoning.
  4. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 2 x\) at the point where \(x = 5\). Find the area of the triangle formed by this tangent and the coordinate axes.
OCR MEI C2 2016 June Q11
11 There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012-2013 flu epidemic in the UK were as follows.
Week12345678910
Number of flu viruses710243240386396234480
These data may be modelled by an equation of the form \(y = a \times 10 ^ { b t }\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.
  1. Explain why this model leads to a straight-line graph of \(\log _ { 10 } y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\).
  2. Complete the values of \(\log _ { 10 } y\) in the table, draw the graph of \(\log _ { 10 } y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. During the decline of the epidemic, an appropriate model was $$y = 921 \times 10 ^ { - 0.137 w }$$ where \(y\) is the number of flu viruses detected in week \(w\) of the decline.
  3. Use this to find the number of viruses detected in week 4 of the decline.
OCR MEI C2 2006 January Q11
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
  3. Find the equation of the tangent to the curve at the point \(( - 1,7 )\). Find also the coordinates of the point where this tangent crosses the curve again.
OCR MEI C2 2011 January Q11
  1. Use calculus to find \(\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x\) and state what this represents.
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\) is a decreasing function.
OCR MEI C2 2009 June Q10
  1. On the insert, complete the table and plot \(h\) against \(\log _ { 10 } t\), drawing by eye a line of best fit.
  2. Use your graph to find an equation for \(h\) in terms of \(\log _ { 10 } t\) for this model.
  3. Find the height of the tree at age 100 years, as predicted by this model.
  4. Find the age of the tree when it reaches a height of 29 m , according to this model.
  5. Comment on the suitability of the model when the tree is very young.
OCR MEI C2 2013 June Q5
  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 \(x = 2\).
    \(6 S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  3. Another geometric progression has first term \(2 a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\).
  4. A third geometric progression has first term \(a\) and common ratio \(r ^ { 2 }\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\).
OCR MEI C2 Q11
  1. The speed-time graph on the insert sheet provides the axes and the first two points plotted. Plot the remainder of these points and join them with a smooth curve. The area between this curve and the \(t\)-axis represents the distance travelled by the car in this time.
  2. Using the trapezium rule with 6 values of \(t\) estimate the area under the curve to give the distance travelled. Illustrate on your graph the area found.
  3. John's teacher suggests that the equation of the curve could be \(v = 6 t - \frac { 1 } { 2 } t ^ { 2 }\). Find, by calculus, the area between this curve and the \(t\) axis.
  4. Plot this curve on your graph. Comment on whether the estimates obtained in parts (ii) and (iii) are overestimates or underestimates. 12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure}
  5. Express \(y\) as a function of \(x\).
  6. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
  7. Prove that for the maximum value of \(A\), \(y = x\) exactly.
    \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
    Insert sheet for question 11}