Questions — OCR MEI C3 (386 questions)

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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 \(b\).
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 2012 June Q2
3 marks Easy -1.2
2 Solve the inequality \(| 2 x + 1 | > 4\).
OCR MEI C3 2012 June Q3
4 marks Moderate -0.3
3 Find the gradient at the point \(( 0 , \ln 2 )\) on the curve with equation \(\mathrm { e } ^ { 2 y } = 5 - \mathrm { e } ^ { - x }\).
OCR MEI C3 2012 June Q4
6 marks Moderate -0.3
4 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7b77c646-2bc5-4166-b22e-3c1229abd722-2_476_572_861_751} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Find the value of \(a\).
  2. Write down the range of \(\mathrm { f } ( x )\).
  3. Sketch the curve \(y = \mathrm { f } \left( \frac { 1 } { 3 } x \right) - 1\).
OCR MEI C3 2012 June Q5
6 marks Moderate -0.8
5 A termites' nest has a population of \(P\) million. \(P\) is modelled by the equation \(P = 7 - 2 \mathrm { e } ^ { - k t }\), where \(t\) is in years, and \(k\) is a positive constant.
  1. Calculate the population when \(t = 0\), and the long-term population, given by this model.
  2. Given that the population when \(t = 1\) is estimated to be 5.5 million, calculate the value of \(k\).
OCR MEI C3 2012 June Q6
8 marks Standard +0.3
6 Fig. 6 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 2 \arcsin x , - 1 \leqslant x \leqslant 1\).
Fig. 6 also shows the curve \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\).
P is the point on the curve \(y = \mathrm { f } ( x )\) with \(x\)-coordinate \(\frac { 1 } { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7b77c646-2bc5-4166-b22e-3c1229abd722-3_711_693_466_685} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Find the \(y\)-coordinate of P , giving your answer in terms of \(\pi\). The point Q is the reflection of P in \(y = x\).
  2. Find \(\mathrm { g } ( x )\) and its derivative \(\mathrm { g } ^ { \prime } ( x )\). Hence determine the exact gradient of the curve \(y = \mathrm { g } ( x )\) at the point Q . Write down the exact gradient of \(y = \mathrm { f } ( x )\) at the point P .