Questions — OCR MEI (4456 questions)

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OCR MEI C2 2015 June Q9
11 marks Moderate -0.8
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
13 marks Standard +0.3
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
12 marks Standard +0.3
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 C3 2009 January Q1
3 marks Easy -1.8
1 Solve the inequality \(| x - 1 | < 3\).
OCR MEI C3 2009 January Q2
7 marks Moderate -0.3
2
  1. Differentiate \(x \cos 2 x\) with respect to \(x\).
  2. Integrate \(x \cos 2 x\) with respect to \(x\).
OCR MEI C3 2009 January Q3
3 marks Moderate -0.8
3 Given that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \ln ( x - 1 )\) and \(\mathrm { g } ( x ) = 1 + \mathrm { e } ^ { 2 x }\), show that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).
OCR MEI C3 2009 January Q4
5 marks Moderate -0.3
4 Find the exact value of \(\int _ { 0 } ^ { 2 } \sqrt { 1 + 4 x } \mathrm {~d} x\), showing your working.
OCR MEI C3 2009 January Q5
8 marks Moderate -0.8
5
  1. State the period of the function \(\mathrm { f } ( x ) = 1 + \cos 2 x\), where \(x\) is in degrees.
  2. State a sequence of two geometrical transformations which maps the curve \(y = \cos x\) onto the curve \(y = \mathrm { f } ( x )\).
  3. Sketch the graph of \(y = \mathrm { f } ( x )\) for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
OCR MEI C3 2009 January Q6
3 marks Moderate -0.8
6
  1. Disprove the following statement. $$\text { 'If } p > q \text {, then } \frac { 1 } { p } < \frac { 1 } { q } \text {. }$$
  2. State a condition on \(p\) and \(q\) so that the statement is true.
OCR MEI C3 2009 January Q7
7 marks Standard +0.3
7 The variables \(x\) and \(y\) satisfy the equation \(x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 5\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \left( \frac { y } { x } \right) ^ { \frac { 1 } { 3 } }\). Both \(x\) and \(y\) are functions of \(t\).
  2. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) when \(x = 1 , y = 8\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 6\). Section B (36 marks)
OCR MEI C3 2009 January Q8
18 marks Standard +0.3
8 Fig. 8 shows the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\). P is the point on this curve with \(x\)-coordinate 1 , and R is the point \(\left( 0 , - \frac { 7 } { 8 } \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{56672660-b7dc-4e10-8039-1c041e75b598-3_1022_995_479_575} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the gradient of PR.
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that PR is a tangent to the curve.
  3. Find the exact coordinates of the turning point Q .
  4. Differentiate \(x \ln x - x\). Hence, or otherwise, show that the area of the region enclosed by the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is \(\frac { 59 } { 24 } - \frac { 1 } { 4 } \ln 2\).
OCR MEI C3 2009 January Q9
18 marks Standard +0.3
9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 2 x - x ^ { 2 } } }\).
The curve has asymptotes \(x = 0\) and \(x = a\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{56672660-b7dc-4e10-8039-1c041e75b598-4_655_800_431_669} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find \(a\). Hence write down the domain of the function.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x - 1 } { \left( 2 x - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }\). Hence find the coordinates of the turning point of the curve, and write down the range of the function. The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
  3. (A) Show algebraically that \(\mathrm { g } ( x )\) is an even function.
    (B) Show that \(\mathrm { g } ( x - 1 ) = \mathrm { f } ( x )\).
    (C) Hence prove that the curve \(y = \mathrm { f } ( x )\) is symmetrical, and state its line of symmetry.
OCR MEI C3 2010 January Q1
4 marks Easy -1.2
1 Solve the equation \(\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } = 0\).
OCR MEI C3 2010 January Q2
6 marks Moderate -0.3
2 The temperature \(T\) in degrees Celsius of water in a glass \(t\) minutes after boiling is modelled by the equation \(T = 20 + b \mathrm { e } ^ { - k t }\), where \(b\) and \(k\) are constants. Initially the temperature is \(100 ^ { \circ } \mathrm { C }\), and after 5 minutes the temperature is \(60 ^ { \circ } \mathrm { C }\).
  1. Find \(b\) and \(k\).
  2. Find at what time the temperature reaches \(50 ^ { \circ } \mathrm { C }\).
OCR MEI C3 2010 January Q3
7 marks Moderate -0.3
3
  1. Given that \(y = \sqrt [ 3 ] { 1 + 3 x ^ { 2 } }\), use the chain rule to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
  2. Given that \(y ^ { 3 } = 1 + 3 x ^ { 2 }\), use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Show that this result is equivalent to the result in part (i).
OCR MEI C3 2010 January Q4
8 marks Moderate -0.8
4 Evaluate the following integrals, giving your answers in exact form.
  1. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x ^ { 2 } + 1 } \mathrm {~d} x\).
  2. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x + 1 } \mathrm {~d} x\).
OCR MEI C3 2010 January Q5
4 marks Moderate -0.8
5 The curves in parts (i) and (ii) have equations of the form \(y = a + b \sin c x\), where \(a , b\) and \(c\) are constants. For each curve, find the values of \(a , b\) and \(c\).
  1. \includegraphics[max width=\textwidth, alt={}, center]{3b3e20ee-457c-46be-b2e5-12573bee2fbf-2_455_679_1800_365}
  2. \includegraphics[max width=\textwidth, alt={}, center]{3b3e20ee-457c-46be-b2e5-12573bee2fbf-2_374_679_2311_365}
OCR MEI C3 2010 January Q6
4 marks Moderate -0.3
6 Write down the conditions for \(\mathrm { f } ( x )\) to be an odd function and for \(\mathrm { g } ( x )\) to be an even function.
Hence prove that, if \(\mathrm { f } ( x )\) is odd and \(\mathrm { g } ( x )\) is even, then the composite function \(\mathrm { gf } ( x )\) is even.
OCR MEI C3 2010 January Q7
3 marks Standard +0.3
7 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: let \(\arcsin x = \theta\).] Section B (36 marks)
OCR MEI C3 2010 January Q8
17 marks Standard +0.3
8 Fig. 8 shows part of the curve \(y = x \cos 3 x\).
The curve crosses the \(x\)-axis at \(\mathrm { O } , \mathrm { P }\) and Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b3e20ee-457c-46be-b2e5-12573bee2fbf-3_551_1189_925_479} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the exact coordinates of P and Q .
  2. Find the exact gradient of the curve at the point P . Show also that the turning points of the curve occur when \(x \tan 3 x = \frac { 1 } { 3 }\).
  3. Find the area of the region enclosed by the curve and the \(x\)-axis between O and P , giving your answer in exact form.
OCR MEI C3 2010 January Q9
19 marks Standard +0.3
9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 1 } { x ^ { 2 } + 1 }\) for the domain \(0 \leqslant x \leqslant 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b3e20ee-457c-46be-b2e5-12573bee2fbf-4_974_1211_358_466} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 6 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }\), and hence that \(\mathrm { f } ( x )\) is an increasing function for \(x > 0\).
  2. Find the range of \(\mathrm { f } ( x )\).
  3. Given that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 6 - 18 x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 3 } }\), find the maximum value of \(\mathrm { f } ^ { \prime } ( x )\). The function \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\).
  4. Write down the domain and range of \(\mathrm { g } ( x )\). Add a sketch of the curve \(y = \mathrm { g } ( x )\) to a copy of Fig. 9 .
  5. Show that \(\mathrm { g } ( x ) = \sqrt { \frac { x + 1 } { 2 - x } }\).
OCR MEI C3 2009 June Q1
3 marks Easy -1.2
1 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sin 3 x \mathrm {~d} x\).
OCR MEI C3 2009 June Q2
8 marks Moderate -0.8
2 A radioactive substance decays exponentially, so that its mass \(M\) grams can be modelled by the equation \(M = A \mathrm { e } ^ { - k t }\), where \(t\) is the time in years, and \(A\) and \(k\) are positive constants.
  1. An initial mass of 100 grams of the substance decays to 50 grams in 1500 years. Find \(A\) and \(k\).
  2. The substance becomes safe when \(99 \%\) of its initial mass has decayed. Find how long it will take before the substance becomes safe.
OCR MEI C3 2009 June Q3
3 marks Moderate -0.8
3 Sketch the curve \(y = 2 \arccos x\) for \(- 1 \leqslant x \leqslant 1\).
OCR MEI C3 2009 June Q4
3 marks Easy -1.3
4 Fig. 4 shows a sketch of the graph of \(y = 2 | x - 1 |\). It meets the \(x\) - and \(y\)-axes at ( \(a , 0\) ) and ( \(0 , b\) ) respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1167a0e5-48c8-48e0-b2d1-76a50bad03ad-2_478_556_1247_792} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find the values of \(a\) and \(b\).