Questions — OCR MEI C3 (366 questions)

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OCR MEI C3 2010 January Q9
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 2011 January Q1
1 Given that \(y = \sqrt [ 3 ] { 1 + x ^ { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
OCR MEI C3 2011 January Q2
2 Solve the inequality \(| 2 x + 1 | \geqslant 4\).
OCR MEI C3 2011 January Q3
3 The area of a circular stain is growing at a rate of \(1 \mathrm {~mm} ^ { 2 }\) per second. Find the rate of increase of its radius at an instant when its radius is 2 mm .
OCR MEI C3 2011 January Q4
4 Use the triangle in Fig. 4 to prove that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = 1\). For what values of \(\theta\) is this proof valid? \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9857ad89-b315-4cd6-8d8d-26a509ca52a8-2_362_606_906_772} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
OCR MEI C3 2011 January Q5
5
  1. On a single set of axes, sketch the curves \(y = \mathrm { e } ^ { x } - 1\) and \(y = 2 \mathrm { e } ^ { - x }\).
  2. Find the exact coordinates of the point of intersection of these curves.
OCR MEI C3 2011 January Q7
7 Fig. 7 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 1 + 2 \arctan x , x \in \mathbb { R }\). The scales on the \(x\) - and \(y\)-axes are the same. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9857ad89-b315-4cd6-8d8d-26a509ca52a8-3_859_849_386_648} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the range of f , giving your answer in terms of \(\pi\).
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\), and add a sketch of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) to the copy of Fig. 7. Section B (36 Marks)
OCR MEI C3 2011 January Q8
8
  1. Use the substitution \(u = 1 + x\) to show that $$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { 1 + x } \mathrm {~d} x = \int _ { a } ^ { b } \left( u ^ { 2 } - 3 u + 3 - \frac { 1 } { u } \right) \mathrm { d } u$$ where \(a\) and \(b\) are to be found.
    Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { 1 + x } \mathrm {~d} x\), giving your answer in exact form. Fig. 8 shows the curve \(y = x ^ { 2 } \ln ( 1 + x )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9857ad89-b315-4cd6-8d8d-26a509ca52a8-4_830_809_902_667} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Verify that the origin is a stationary point of the curve.
  3. Using integration by parts, and the result of part (i), find the exact area enclosed by the curve \(y = x ^ { 2 } \ln ( 1 + x )\), the \(x\)-axis and the line \(x = 1\).
OCR MEI C3 2011 January Q9
9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \cos ^ { 2 } x } , - \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\), together with its asymptotes \(x = \frac { 1 } { 2 } \pi\) and \(x = - \frac { 1 } { 2 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9857ad89-b315-4cd6-8d8d-26a509ca52a8-5_917_1395_413_374} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Use the quotient rule to show that the derivative of \(\frac { \sin x } { \cos x }\) is \(\frac { 1 } { \cos ^ { 2 } x }\).
  2. Find the area bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \frac { 1 } { 2 } \mathrm { f } \left( x + \frac { 1 } { 4 } \pi \right)\).
  3. Verify that the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) cross at ( 0,1 ).
  4. State a sequence of two transformations such that the curve \(y = \mathrm { f } ( x )\) is mapped to the curve \(y = \mathrm { g } ( x )\). On the copy of Fig. 9, sketch the curve \(y = \mathrm { g } ( x )\), indicating clearly the coordinates of the minimum point and the equations of the asymptotes to the curve.
  5. Use your result from part (ii) to write down the area bounded by the curve \(y = \mathrm { g } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = - \frac { 1 } { 4 } \pi\). RECOGNISING ACHIEVEMENT
OCR MEI C3 2012 January Q1
1 Differentiate \(x ^ { 2 } \tan 2 x\).
OCR MEI C3 2012 January Q2
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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.