Questions — OCR MEI C3 (366 questions)

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OCR MEI C3 2016 June Q1
1 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 1 + \cos \frac { 1 } { 2 } x \right) \mathrm { d } x\).
OCR MEI C3 2016 June Q2
2 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by \(\mathrm { f } ( x ) = \ln x\) and \(\mathrm { g } ( x ) = 2 + \mathrm { e } ^ { x }\), for \(x > 0\).
Find the exact value of \(x\), given that \(\mathrm { fg } ( x ) = 2 x\).
OCR MEI C3 2016 June Q3
3 Find \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\), giving your answer in an exact form.
OCR MEI C3 2016 June Q4
4 By sketching the graphs of \(y = | 2 x + 1 |\) and \(y = - x\) on the same axes, show that the equation \(| 2 x + 1 | = - x\) has two roots. Find these roots.
OCR MEI C3 2016 June Q5
5 The volume \(V \mathrm {~m} ^ { 3 }\) of a pile of grain of height \(h\) metres is modelled by the equation $$V = 4 \sqrt { h ^ { 3 } + 1 } - 4$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 2\). At a certain time, the height of the pile is 2 metres, and grain is being added so that the volume is increasing at a rate of \(0.4 \mathrm {~m} ^ { 3 }\) per minute.
  2. Find the rate at which the height is increasing at this time.
OCR MEI C3 2016 June Q6
6 Fig. 6 shows part of the curve \(\sin 2 y = x - 1\). P is the point with coordinates \(\left( 1.5 , \frac { 1 } { 12 } \pi \right)\) on the curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414a6b7f-cd96-4fa0-9521-ebe500bab375-2_458_691_1610_687} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). Hence find the exact gradient of the curve \(\sin 2 y = x - 1\) at the point P . The part of the curve shown is the image of the curve \(y = \arcsin x\) under a sequence of two geometrical transformations.
  2. Find \(y\) in terms of \(x\) for the curve \(\sin 2 y = x - 1\). Hence describe fully the sequence of transformations.
    \(7 \quad\) You are given that \(n\) is a positive integer.
    By expressing \(x ^ { 2 n } - 1\) as a product of two factors, prove that \(2 ^ { 2 n } - 1\) is divisible by 3 . Section B (36 marks)
OCR MEI C3 2016 June Q8
8 Fig. 8 shows the curve \(y = \frac { x } { \sqrt { x + 4 } }\) and the line \(x = 5\). The curve has an asymptote \(l\).
The tangent to the curve at the origin O crosses the line \(l\) at P and the line \(x = 5\) at Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414a6b7f-cd96-4fa0-9521-ebe500bab375-3_643_921_703_573} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that for this curve \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + 8 } { 2 ( x + 4 ) ^ { \frac { 3 } { 2 } } }\).
  2. Find the coordinates of the point P .
  3. Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line \(x = 5\).
OCR MEI C3 2016 June Q9
9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x }\) and \(k\) is a constant greater than 1 . The curve crosses the \(y\)-axis at P and has a turning point Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414a6b7f-cd96-4fa0-9521-ebe500bab375-4_783_951_392_557} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the \(y\)-coordinate of P in terms of \(k\).
  2. Show that the \(x\)-coordinate of Q is \(\frac { 1 } { 4 } \ln k\), and find the \(y\)-coordinate in its simplest form.
  3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \ln k\). Give your answer in the form \(a k + b\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } \left( x + \frac { 1 } { 4 } \ln k \right)\).
  4. (A) Show that \(\mathrm { g } ( x ) = \sqrt { k } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)\).
    (B) Hence show that \(\mathrm { g } ( x )\) is an even function.
    (C) Deduce, with reasons, a geometrical property of the curve \(y = \mathrm { f } ( x )\). \section*{END OF QUESTION PAPER}
OCR MEI C3 Q1
1 Solve the equation \(| 3 x + 2 | = 1\).
OCR MEI C3 Q7
7 Fig. 7 shows the curve defined implicitly by the equation $$y ^ { 2 } + y = x ^ { 3 } + 2 x ,$$ together with the line \(x = 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{185840c8-2799-44cd-a6d8-00d10c038c2c-03_465_378_534_808} \captionsetup{labelformat=empty} \caption{Not to scale}
\end{figure} Fig. 7 Find the coordinates of the points of intersection of the line and the curve.
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.
OCR MEI C3 Q8
8 Fig. 8 shows part of the curve \(y = x \sin 3 x\). It crosses the \(x\)-axis at P . The point on the curve with \(x\)-coordinate \(\frac { 1 } { 6 } \pi\) is Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{185840c8-2799-44cd-a6d8-00d10c038c2c-03_421_789_1748_610} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the \(x\)-coordinate of P .
  2. Show that Q lies on the line \(y = x\).
  3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
  4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).
OCR MEI C3 Q9
9 The function \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\) has domain \(- 3 \leqslant x \leqslant 3\).
Fig. 9 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{185840c8-2799-44cd-a6d8-00d10c038c2c-04_540_943_477_550} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Show algebraically that the function is even. State how this property relates to the shape of the curve.
  2. Find the gradient of the curve at the point \(\mathrm { P } ( 2 , \ln 5 )\).
  3. Explain why the function does not have an inverse for the domain \(- 3 \leqslant x \leqslant 3\). The domain of \(\mathrm { f } ( x )\) is now restricted to \(0 \leqslant x \leqslant 3\). The inverse of \(\mathrm { f } ( x )\) is the function \(\mathrm { g } ( x )\).
  4. Sketch the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) on the same axes. State the domain of the function \(\mathrm { g } ( x )\). Show that \(\mathrm { g } ( x ) = \sqrt { \mathrm { e } ^ { x } - 1 }\).
  5. Differentiate \(\mathrm { g } ( x )\). Hence verify that \(\mathrm { g } ^ { \prime } ( \ln 5 ) = 1 \frac { 1 } { 4 }\). Explain the connection between this result and your answer to part (ii). \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} MEI STRUCTURED MATHEMATICS \section*{4753/1} Methods for Advanced Mathematics (C3)
    Wednesday
OCR MEI C3 Q3
  1. \(\quad y = 2 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C3 Q9
(A) \(( x - y ) \left( x ^ { 2 } + x y + y ^ { 2 } \right) = x ^ { 3 } - y ^ { 3 }\),
(B) \(\left( x + \frac { 1 } { 2 } y \right) ^ { 2 } + \frac { 3 } { 4 } y ^ { 2 } = x ^ { 2 } + x y + y ^ { 2 }\).
(ii) Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x ^ { 3 } > y ^ { 3 }\).
(i) Verify the following statement: $$\text { ' } 2 ^ { p } - 1 \text { is a prime number for all prime numbers } p \text { less than } 11 \text { '. }$$ (ii) Calculate \(23 \times 89\), and hence disprove this statement:
' \(2 ^ { p } - 1\) is a prime number for all prime numbers \(p\) '.
OCR MEI C3 Q9
9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f8be5ab-d241-4027-af26-c49da9394adc-4_401_799_488_593} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. On the Insert sketch graphs of
    (A) \(y = 2 \mathrm { f } ( x )\),
    (B) \(y = \mathrm { f } ( - x )\),
    (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
  2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
  3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
  4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
    Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
  5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
  6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation.
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
  7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation.
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
  8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation.
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
  9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}
OCR MEI C3 Q9
9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.
  1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
  2. The data collected are as follows.
    Time \(( t\) years \()\)01234
    Number of bats, \(P\)100170300340360
    Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
    (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
  3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
  4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
  5. Sketch the graph of \(y = 2 \mathrm { f } ( x )\)
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
  6. Sketch the graph of \(y = \mathrm { f } ( 2 x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.}
  7. Plot \(\ln P\) against \(t\).
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
  8. Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph.
    \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}