Questions — OCR MEI C3 (366 questions)

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OCR MEI C3 2013 June Q2
2
  1. Factorise fully \(n ^ { 3 } - n\).
  2. Hence prove that, if \(n\) is an integer, \(n ^ { 3 } - n\) is divisible by 6 .
OCR MEI C3 2013 June Q3
3 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 1 - 2 \sin x\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Fig. 3 shows the curve \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-3_732_807_349_612} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Write down the range of the function \(\mathrm { f } ( x )\).
  2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\).
  3. Find \(\mathrm { f } ^ { \prime } ( 0 )\). Hence write down the gradient of \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 1,0 )\).
OCR MEI C3 2013 June Q4
4 Water flows into a bowl at a constant rate of \(10 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) (see Fig. 4). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-3_422_385_1628_815} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} When the depth of water in the bowl is \(h \mathrm {~cm}\), the volume of water is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \pi h ^ { 2 }\). Find the rate at which the depth is increasing at the instant in time when the depth is 5 cm .
OCR MEI C3 2013 June Q5
5 Given that \(y = \ln \left( \sqrt { \frac { 2 x - 1 } { 2 x + 1 } } \right)\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 x - 1 } - \frac { 1 } { 2 x + 1 }\).
OCR MEI C3 2013 June Q6
6 Using a suitable substitution or otherwise, show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 x } { 3 + \cos 2 x } \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).
OCR MEI C3 2013 June Q7
7
  1. Show algebraically that the function \(\mathrm { f } ( x ) = \frac { 2 x } { 1 - x ^ { 2 } }\) is odd. Fig. 7 shows the curve \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 4\), together with the asymptote \(x = 1\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-4_730_817_431_607} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
  2. Use the copy of Fig. 7 to complete the curve for \(- 4 \leqslant x \leqslant 4\).
OCR MEI C3 2013 June Q8
8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 1 - x ) \mathrm { e } ^ { 2 x }\), with its turning point P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-5_716_810_404_609} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down the coordinates of the intercepts of \(y = \mathrm { f } ( x )\) with the \(x\) - and \(y\)-axes.
  2. Find the exact coordinates of the turning point P .
  3. Show that the exact area of the region enclosed by the curve and the \(x\) - and \(y\)-axes is \(\frac { 1 } { 4 } \left( e ^ { 2 } - 3 \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
  4. Express \(\mathrm { g } ( x )\) in terms of \(x\). Sketch the curve \(y = \mathrm { g } ( x )\) on the copy of Fig. 8, indicating the coordinates of its intercepts with the \(x\) - and \(y\)-axes and of its turning point.
  5. Write down the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\) and the \(x\)-and \(y\)-axes.
OCR MEI C3 2013 June Q9
9 Fig. 9 shows the curve with equation \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\). It has an asymptote \(x = a\) and turning point P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-6_752_867_356_584} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the value of \(a\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 3 x ^ { 2 } } { 3 y ^ { 2 } ( 2 x - 1 ) ^ { 2 } }\). Hence find the coordinates of the turning point P , giving the \(y\)-coordinate to 3 significant figures.
  3. Show that the substitution \(u = 2 x - 1\) transforms \(\int \frac { x } { \sqrt [ 3 ] { 2 x - 1 } } \mathrm {~d} x\) to \(\frac { 1 } { 4 } \int \left( u ^ { \frac { 2 } { 3 } } + u ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4.5\).
OCR MEI C3 2014 June Q1
1 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } ( 1 - \sin 3 x ) \mathrm { d } x\), giving your answer in exact form.
OCR MEI C3 2014 June Q3
3 Solve the equation \(| 3 - 2 x | = 4 | x |\).
OCR MEI C3 2014 June Q4
4 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = a + \cos b x , 0 \leqslant x \leqslant 2 \pi ,$$ and \(a\) and \(b\) are positive constants. The curve has stationary points at \(( 0,3 )\) and \(( 2 \pi , 1 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1a06289-d9e9-4f6b-ab58-70db1a4748ef-2_424_620_922_719} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Find \(a\) and \(b\).
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\), and state its domain and range.
OCR MEI C3 2014 June Q5
5 A spherical balloon of radius \(r \mathrm {~cm}\) has volume \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\). The balloon is inflated at a constant rate of \(10 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of \(r\) when \(r = 8\).
OCR MEI C3 2014 June Q6
6 The value \(\pounds V\) of a car \(t\) years after it is new is modelled by the equation \(V = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants which depend on the make and model of the car.
  1. Brian buys a new sports car. Its value is modelled by the equation $$V = 20000 \mathrm { e } ^ { - 0.2 t } .$$ Calculate how much value, to the nearest \(\pounds 100\), this car has lost after 1 year.
  2. At the same time as Brian buys his car, Kate buys a new hatchback for \(\pounds 15000\). Her car loses \(\pounds 2000\) of its value in the first year. Show that, for Kate's car, \(k = 0.143\) correct to 3 significant figures.
  3. Find how long it is before Brian's and Kate's cars have the same value.
OCR MEI C3 2014 June Q7
7 Either prove or disprove each of the following statements.
  1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.'
  2. 'If \(m\) and \(n\) are consecutive even numbers, then \(m n\) is divisible by 8 .'
OCR MEI C3 2014 June Q8
8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1a06289-d9e9-4f6b-ab58-70db1a4748ef-3_481_681_447_676} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show algebraically that \(\mathrm { f } ( x )\) is an odd function. Interpret this result geometrically.
  2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \left( 2 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }\). Hence find the exact gradient of the curve at the origin.
  3. Find the exact area of the region bounded by the curve, the \(x\)-axis and the line \(x = 1\).
  4. (A) Show that if \(y = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\), then \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\).
    (B) Differentiate \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\) implicitly to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y ^ { 3 } } { x ^ { 3 } }\). Explain why this expression cannot be used to find the gradient of the curve at the origin.
OCR MEI C3 2014 June Q9
9 Fig. 9 shows the curve \(y = x \mathrm { e } ^ { - 2 x }\) together with the straight line \(y = m x\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P . The dashed line is the tangent at P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1a06289-d9e9-4f6b-ab58-70db1a4748ef-4_424_972_383_559} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Show that the \(x\)-coordinate of P is \(- \frac { 1 } { 2 } \ln m\).
  2. Find, in terms of \(m\), the gradient of the tangent to the curve at P . You are given that OP and this tangent are equally inclined to the \(x\)-axis.
  3. Show that \(m = \mathrm { e } ^ { - 2 }\), and find the exact coordinates of P .
  4. Find the exact area of the shaded region between the line OP and the curve. \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {N } }\)}
OCR MEI C3 2015 June Q1
1 Fig. 1 shows part of the curve \(y = \mathrm { e } ^ { 2 x } \cos x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{955bebfb-04a3-4cd9-a33e-a8ba4b73e2ba-2_670_1029_404_504} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Find the coordinates of the turning point P .
OCR MEI C3 2015 June Q2
2 Find \(\int \sqrt [ 3 ] { 2 x - 1 } \mathrm {~d} x\).
OCR MEI C3 2015 June Q3
3 Find the exact value of \(\int _ { 1 } ^ { 2 } x ^ { 3 } \ln x \mathrm {~d} x\).
OCR MEI C3 2015 June Q4
4 Fig. 4 shows a cone with its axis vertical. The angle between the axis and the slant edge is \(45 ^ { \circ }\). Water is poured into the cone at a constant rate of \(5 \mathrm {~cm} ^ { 3 }\) per second. At time \(t\) seconds, the height of the water surface above the vertex O of the cone is \(h \mathrm {~cm}\), and the volume of water in the cone is \(V \mathrm {~cm} ^ { 3 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{955bebfb-04a3-4cd9-a33e-a8ba4b73e2ba-2_296_405_1804_831} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find \(V\) in terms of \(h\).
Hence find the rate at which the height of water is increasing when the height is 10 cm .
[0pt] [You are given that the volume \(V\) of a cone of height \(h\) and radius \(r\) is \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) ].
OCR MEI C3 2015 June Q5
5 A curve has implicit equation \(y ^ { 2 } + 2 x \ln y = x ^ { 2 }\).
Verify that the point \(( 1,1 )\) lies on the curve, and find the gradient of the curve at this point.
OCR MEI C3 2015 June Q6
6 Solve each of the following equations, giving your answers in exact form.
  1. \(6 \arcsin x - \pi = 0\).
  2. \(\arcsin x = \arccos x\).
OCR MEI C3 2015 June Q7
7
  1. The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \frac { 1 - x } { 1 + x } , x \neq - 1$$ Show that \(\mathrm { f } ( \mathrm { f } ( x ) ) = x\).
    Hence write down \(\mathrm { f } ^ { - 1 } ( x )\).
  2. The function \(\mathrm { g } ( x )\) is defined for all real \(x\) by $$\mathrm { g } ( x ) = \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } }$$ Prove that \(\mathrm { g } ( x )\) is even. Interpret this result in terms of the graph of \(y = \mathrm { g } ( x )\).
OCR MEI C3 2015 June Q8
8 Fig. 8 shows the line \(y = 1\) and the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x }\). The curve touches the \(x\)-axis at \(\mathrm { P } ( 2,0 )\) and has another turning point at the point Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{955bebfb-04a3-4cd9-a33e-a8ba4b73e2ba-4_960_1472_450_285} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 4 } { x ^ { 2 } }\), and find \(\mathrm { f } ^ { \prime \prime } ( x )\). Hence find the coordinates of Q and, using \(\mathrm { f } ^ { \prime \prime } ( x )\), verify that it is a maximum point.
  2. Verify that the line \(y = 1\) meets the curve \(y = \mathrm { f } ( x )\) at the points with \(x\)-coordinates 1 and 4 . Hence find the exact area of the shaded region enclosed by the line and the curve. The curve \(y = \mathrm { f } ( x )\) is now transformed by a translation with vector \(\binom { - 1 } { - 1 }\). The resulting curve has equation \(y = \mathrm { g } ( x )\).
  3. Show that \(\mathrm { g } ( x ) = \frac { x ^ { 2 } - 3 x } { x + 1 }\).
  4. Without further calculation, write down the value of \(\int _ { 0 } ^ { 3 } \mathrm {~g} ( x ) \mathrm { d } x\), justifying your answer.
OCR MEI C3 2015 June Q9
9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } - 1 , x \in \mathbb { R } .$$ The curve crosses the \(x\)-axis at O and P , and has a turning point at Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{955bebfb-04a3-4cd9-a33e-a8ba4b73e2ba-5_867_988_497_525} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the exact \(x\)-coordinate of P .
  2. Show that the \(x\)-coordinate of Q is \(\ln 2\) and find its \(y\)-coordinate.
  3. Find the exact area of the region enclosed by the curve and the \(x\)-axis. The domain of \(\mathrm { f } ( x )\) is now restricted to \(x \geqslant \ln 2\).
  4. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). Write down its domain and range, and sketch its graph on the copy of Fig. 9.