Questions — OCR MEI (4333 questions)

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OCR MEI C3 2012 January Q2
4 marks Moderate -0.8
2 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined as follows. $$\begin{array} { l l } \mathrm { f } ( x ) = \ln x , & x > 0 \\ \mathrm {~g} ( x ) = 1 + x ^ { 2 } , & x \in \mathbb { R } \end{array}$$ Write down the functions \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\), and state whether these functions are odd, even or neither.
OCR MEI C3 2012 January Q3
5 marks Standard +0.3
3 Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos \frac { 1 } { 2 } x \mathrm {~d} x = \frac { \sqrt { 2 } } { 2 } \pi + 2 \sqrt { 2 } - 4\).
OCR MEI C3 2012 January Q5
6 marks Moderate -0.3
5 Each of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) below is obtained using a sequence of two transformations applied to the corresponding dashed graph. In each case, state suitable transformations, and hence find expressions for \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\).
  1. \includegraphics[max width=\textwidth, alt={}, center]{8b8958be-0ebc-4f72-ac3f-c16a8ec9e4ab-2_430_712_1366_680}
  2. \includegraphics[max width=\textwidth, alt={}, center]{8b8958be-0ebc-4f72-ac3f-c16a8ec9e4ab-2_394_608_1925_731}
OCR MEI C3 2012 January Q6
8 marks Moderate -0.3
6 Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius \(r\) metres of the oil slick \(t\) hours after the start of the leak is modelled by the equation $$r = 20 \left( 1 - \mathrm { e } ^ { - 0.2 t } \right) .$$
  1. Find the radius of the slick when \(t = 2\), and the rate at which the radius is increasing at this time.
  2. Find the rate at which the area of the slick is increasing when \(t = 2\).
OCR MEI C3 2012 January Q7
8 marks Standard +0.3
7 Fig. 7 shows the curve \(x ^ { 3 } + y ^ { 3 } = 3 x y\). The point P is a turning point of the curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b8958be-0ebc-4f72-ac3f-c16a8ec9e4ab-3_583_513_708_776} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x ^ { 2 } } { y ^ { 2 } - x }\).
  2. Hence find the exact \(x\)-coordinate of P . Section B (36 marks)
OCR MEI C3 2012 January Q8
18 marks Standard +0.3
8 Fig. 8 shows the curve \(y = \frac { x } { \sqrt { x - 2 } }\), together with the lines \(y = x\) and \(x = 11\). The curve meets these lines at P and Q respectively. R is the point \(( 11,11 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b8958be-0ebc-4f72-ac3f-c16a8ec9e4ab-4_609_736_440_667} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Verify that the \(x\)-coordinate of P is 3 .
  2. Show that, for the curve, \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x - 4 } { 2 ( x - 2 ) ^ { \frac { 3 } { 2 } } }\). Hence find the gradient of the curve at P . Use the result to show that the curve is not symmetrical about \(y = x\).
  3. Using the substitution \(u = x - 2\), show that \(\int _ { 3 } ^ { 11 } \frac { x } { \sqrt { x - 2 } } \mathrm {~d} x = 25 \frac { 1 } { 3 }\). Hence find the area of the region PQR bounded by the curve and the lines \(y = x\) and \(x = 11\).
OCR MEI C3 2012 January Q9
18 marks Challenging +1.2
9 Fig. 9 shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The function \(y = \mathrm { f } ( x )\) is given by $$f ( x ) = \ln \left( \frac { 2 x } { 1 + x } \right) , x > 0$$ The curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis at P , and the line \(x = 2\) at Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b8958be-0ebc-4f72-ac3f-c16a8ec9e4ab-5_552_636_470_715} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Verify that the \(x\)-coordinate of P is 1 . Find the exact \(y\)-coordinate of Q .
  2. Find the gradient of the curve at P. [Hint: use \(\ln \frac { a } { b } = \ln a - \ln b\).] The function \(\mathrm { g } ( x )\) is given by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } } { 2 - \mathrm { e } ^ { x } } , \quad x < \ln 2 .$$ The curve \(y = \mathrm { g } ( x )\) crosses the \(y\)-axis at the point R .
  3. Show that \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\). Write down the gradient of \(y = \mathrm { g } ( x )\) at R .
  4. Show, using the substitution \(u = 2 - \mathrm { e } ^ { x }\) or otherwise, that \(\int _ { 0 } ^ { \ln \frac { 4 } { 3 } } \mathrm {~g} ( x ) \mathrm { d } x = \ln \frac { 3 } { 2 }\). Using this result, show that the exact area of the shaded region shown in Fig. 9 is \(\ln \frac { 32 } { 27 }\).
    [0pt] [Hint: consider its reflection in \(y = x\).]
OCR MEI C3 2013 January Q1
6 marks Standard +0.3
1
  1. Given that \(y = \mathrm { e } ^ { - x } \sin 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence show that the curve \(y = \mathrm { e } ^ { - x } \sin 2 x\) has a stationary point when \(x = \frac { 1 } { 2 } \arctan 2\).
OCR MEI C3 2013 January Q3
2 marks Easy -1.2
3 Express \(1 < x < 3\) in the form \(| x - a | < b\), where \(a\) and \(b\) are to be determined.
OCR MEI C3 2013 January Q4
8 marks Standard +0.3
4 The temperature \(\theta ^ { \circ } \mathrm { C }\) of water in a container after \(t\) minutes is modelled by the equation $$\theta = a - b \mathrm { e } ^ { - k t } ,$$ where \(a , b\) and \(k\) are positive constants.
The initial and long-term temperatures of the water are \(15 ^ { \circ } \mathrm { C }\) and \(100 ^ { \circ } \mathrm { C }\) respectively. After 1 minute, the temperature is \(30 ^ { \circ } \mathrm { C }\).
  1. Find \(a , b\) and \(k\).
  2. Find how long it takes for the temperature to reach \(80 ^ { \circ } \mathrm { C }\).
OCR MEI C3 2013 January Q5
5 marks Moderate -0.3
5 The driving force \(F\) newtons and velocity \(v \mathrm {~km} \mathrm {~s} ^ { - 1 }\) of a car at time \(t\) seconds are related by the equation \(F = \frac { 25 } { v }\).
  1. Find \(\frac { \mathrm { d } F } { \mathrm {~d} v }\).
  2. Find \(\frac { \mathrm { d } F } { \mathrm {~d} t }\) when \(v = 50\) and \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 1.5\).
OCR MEI C3 2013 January Q6
5 marks Standard +0.3
6 Evaluate \(\int _ { 0 } ^ { 3 } x ( x + 1 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\), giving your answer as an exact fraction.
OCR MEI C3 2013 January Q7
4 marks Moderate -0.8
7
  1. Disprove the following statement: $$3 ^ { n } + 2 \text { is prime for all integers } n \geqslant 0 .$$
  2. Prove that no number of the form \(3 ^ { n }\) (where \(n\) is a positive integer) has 5 as its final digit.
OCR MEI C3 2013 January Q8
17 marks Standard +0.3
8 Fig. 8 shows parts of the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), where \(\mathrm { f } ( x ) = \tan x\) and \(\mathrm { g } ( x ) = 1 + \mathrm { f } \left( x - \frac { 1 } { 4 } \pi \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aad64998-748a-437a-8a26-6c5715c9366e-3_684_881_404_575} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Describe a sequence of two transformations which maps the curve \(y = \mathrm { f } ( x )\) to the curve \(y = \mathrm { g } ( x )\). It can be shown that \(\mathrm { g } ( x ) = \frac { 2 \sin x } { \sin x + \cos x }\).
  2. Show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { 2 } { ( \sin x + \cos x ) ^ { 2 } }\). Hence verify that the gradient of \(y = \mathrm { g } ( x )\) at the point \(\left( \frac { 1 } { 4 } \pi , 1 \right)\) is the same as that of \(y = \mathrm { f } ( x )\) at the origin.
  3. By writing \(\tan x = \frac { \sin x } { \cos x }\) and using the substitution \(u = \cos x\), show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( x ) \mathrm { d } x = \int _ { \frac { 1 } { \sqrt { 2 } } } ^ { 1 } \frac { 1 } { u } \mathrm {~d} u\). Evaluate this integral exactly.
  4. Hence find the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\), the \(x\)-axis and the lines \(x = \frac { 1 } { 4 } \pi\) and \(x = \frac { 1 } { 2 } \pi\).
OCR MEI C3 2013 January Q9
19 marks Standard +0.3
9 Fig. 9 shows the line \(y = x\) and the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - 1 \right)\). The line and the curve intersect at the origin and at the point \(\mathrm { P } ( a , a )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aad64998-748a-437a-8a26-6c5715c9366e-4_684_880_372_571} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Show that \(\mathrm { e } ^ { a } = 1 + 2 a\).
  2. Show that the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = a\) is \(\frac { 1 } { 2 } a\). Hence find, in terms of \(a\), the area enclosed by the curve and the line \(y = x\).
  3. Show that the inverse function of \(\mathrm { f } ( x )\) is \(\mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = \ln ( 1 + 2 x )\). Add a sketch of \(y = \mathrm { g } ( x )\) to the copy of Fig. 9.
  4. Find the derivatives of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\). Hence verify that \(\mathrm { g } ^ { \prime } ( a ) = \frac { 1 } { \mathrm { f } ^ { \prime } ( a ) }\). Give a geometrical interpretation of this result.
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 \(b\).
OCR MEI C3 2009 June Q5
7 marks Moderate -0.3
5 The equation of a curve is given by \(\mathrm { e } ^ { 2 y } = 1 + \sin x\).
  1. By differentiating implicitly, find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find an expression for \(y\) in terms of \(x\), and differentiate it to verify the result in part (i).
OCR MEI C3 2009 June Q6
5 marks Moderate -0.3
6 Given that \(\mathrm { f } ( x ) = \frac { x + 1 } { x - 1 }\), show that \(\mathrm { ff } ( x ) = x\).
Hence write down the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). What can you deduce about the symmetry of the curve \(y = \mathrm { f } ( x )\) ?
OCR MEI C3 2009 June Q7
7 marks Standard +0.3
7
  1. Show that
    (A) \(( x - y ) \left( x ^ { 2 } + x y + y ^ { 2 } \right) = x ^ { 3 } - y ^ { 3 }\),
    (B) \(\left( x + \frac { 1 } { 2 } y \right) ^ { 2 } + \frac { 3 } { 4 } y ^ { 2 } = x ^ { 2 } + x y + y ^ { 2 }\).
  2. Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x ^ { 3 } > y ^ { 3 }\). Section B (36 marks)
OCR MEI C3 2009 June Q8
18 marks Standard +0.3
8 Fig. 8 shows the line \(y = x\) and parts of the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), where $$\mathrm { f } ( x ) = \mathrm { e } ^ { x - 1 } , \quad \mathrm {~g} ( x ) = 1 + \ln x$$ The curves intersect the axes at the points A and B , as shown. The curves and the line \(y = x\) meet at the point C . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1167a0e5-48c8-48e0-b2d1-76a50bad03ad-3_807_897_1016_625} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the exact coordinates of A and B . Verify that the coordinates of C are \(( 1,1 )\).
  2. Prove algebraically that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).
  3. Evaluate \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in terms of e.
  4. Use integration by parts to find \(\int \ln x \mathrm {~d} x\). Hence show that \(\int _ { \mathrm { e } ^ { - 1 } } ^ { 1 } \mathrm {~g} ( x ) \mathrm { d } x = \frac { 1 } { \mathrm { e } }\).
  5. Find the area of the region enclosed by the lines OA and OB , and the arcs AC and BC .
OCR MEI C3 2009 June Q9
18 marks Moderate -0.3
9 Fig. 9 shows the curve \(y = \frac { x ^ { 2 } } { 3 x - 1 }\).
P is a turning point, and the curve has a vertical asymptote \(x = a\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1167a0e5-48c8-48e0-b2d1-76a50bad03ad-4_844_1486_447_331} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the value of \(a\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ( 3 x - 2 ) } { ( 3 x - 1 ) ^ { 2 } }\).
  3. Find the exact coordinates of the turning point P . Calculate the gradient of the curve when \(x = 0.6\) and \(x = 0.8\), and hence verify that P is a minimum point.
  4. Using the substitution \(u = 3 x - 1\), show that \(\int \frac { x ^ { 2 } } { 3 x - 1 } \mathrm {~d} x = \frac { 1 } { 27 } \int \left( u + 2 + \frac { 1 } { u } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = \frac { 2 } { 3 }\) and \(x = 1\).
OCR MEI C3 2011 June Q1
4 marks Moderate -0.5
1 Solve the equation \(| 2 x - 1 | = | x |\).