Questions C3 (1200 questions)

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OCR MEI C3 Q5
5 Find \(\int _ { 2 } ^ { 3 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\), giving your answer to 1 decimal place.
OCR MEI C3 Q6
6 Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x )\) and hence or otherwise find the value of \(\int _ { 2 } ^ { 3 } \ln x \mathrm {~d} x\), giving your answer in the form \(\ln a + b\), where \(a\) and \(b\) are to be determined.
OCR MEI C3 Q7
7 Two quantities, \(x\) and \(\theta\), vary with time and are related by the equation \(x = 5 \sin \theta - 4 \cos \theta\).
  1. Find the value of \(x\) when \(\theta = \frac { \pi } { 2 }\).
  2. When \(\theta = \frac { \pi } { 2 }\), its rate of increase (in suitable units) is given by \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = 0.1\). Show that at that moment \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4\).
OCR MEI C3 Q8
8 You are given that \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 1 }\) for all real values of \(x\).
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }\).
  2. Hence show that there is a stationary value at \(\left( 1 , \frac { 1 } { 2 } \right)\) and find the coordinates of the other stationary point.
  3. The graph of the curve is shown in Fig. 8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-3_518_892_1612_705} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure} State whether the curve is odd or even and prove the result algebraically.
  4. Show that \(\int _ { 1 } ^ { 4 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x = \int _ { a } ^ { b } k \frac { 1 } { u + 1 } \mathrm {~d} u\), where the values of \(a , b\) and \(k\) are to be determined.
  5. Hence find the area of the shaded region in Fig. 8.
OCR MEI C3 Q9
9 The curve in Fig. 9.1 has equation \(\sqrt { x } + \sqrt { y } = 1\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_426_647_299_667} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Show that this is part, but not all of the curve \(y = 1 - 2 \sqrt { x } + x\). Sketch the full curve \(y = 1 - 2 \sqrt { x } + x\).
  2. Fig.9.2 shows a star shape made up of four parts, one of which is given in part (i) above. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_380_681_1197_651} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} For each of the sections of the shape labelled \(\mathrm { A } , \mathrm { B }\) and C , state the equation of the curve and the domain.
  3. The shape shown in Fig.9.2 is made into that in Fig. 10.3 by stretching the part of the figure for which \(y > 0\) by a scale factor of 2 . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_405_686_1996_605} \captionsetup{labelformat=empty} \caption{Fig. 9.3}
    \end{figure} Find the area of this shape.
OCR MEI C3 Q1
1 Prove that the product of any three consecutive integers is a multiple of 6 .
OCR MEI C3 Q2
2
  1. Sketch the graph of \(y = | 2 x - 3 |\).
  2. Hence, or otherwise, solve the inequality \(| 2 x - 3 | < 5\). Illustrate your answer on your graph.
OCR MEI C3 Q3
3 Differentiate the following functions.
  1. \(\quad y = \left( x ^ { 2 } + 3 \right) ^ { 5 }\)
  2. \(y = \frac { \sin 2 x } { x }\)
OCR MEI C3 Q4
4 A curve has equation \(y ^ { 2 } = 5 x - 4\).
Find the gradient of the curve at the points where \(x = 8\).
OCR MEI C3 Q5
5 Given that \(x\) and \(t\) are related by the formula \(x = x _ { 0 } \mathrm { e } ^ { - 3 t }\), show that \(t = \ln \left( \frac { a } { x } \right) ^ { b }\) where \(a\) and \(b\) are to be determined.
OCR MEI C3 Q6
6
  1. Find \(\int ( 2 x - 3 ) ^ { 7 } \mathrm {~d} x\).
  2. Use the substitution \(u = x ^ { 2 } + 1\), or otherwise, to find \(\int _ { 1 } ^ { 2 } x \left( x ^ { 2 } + 1 \right) ^ { 3 } \mathrm {~d} x\).
OCR MEI C3 Q7
7 The functions \(f , g\) and \(h\) are defined as follows. $$\mathrm { f } ( x ) = 2 x \quad \mathrm {~g} ( x ) = x ^ { 2 } \quad \mathrm {~h} ( x ) = x + 2$$ Find each of the following as functions of \(x\).
  1. \(\mathrm { f } ^ { 2 } ( x )\),
  2. \(\operatorname { fgh } ( x )\),
  3. \(\mathrm { h } ^ { - 1 } ( x )\).
OCR MEI C3 Q8
8 A curve has equation \(y = ( x + 2 ) \mathrm { e } ^ { - x }\).
  1. Find the coordinates of the points where the curve cuts the axes.
  2. Find the coordinates of the stationary point, S , on the curve.
  3. By evaluating \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at S , determine whether the stationary point is a maximum or a minimum.
  4. Sketch the curve in the domain \(- 3 < x < 3\).
  5. Find where the normal to the curve at the point \(( 0,2 )\) cuts the curve again.
  6. Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
OCR MEI C3 Q1
1 John asserts that the expression \(n ^ { 2 } + n + 11\) is prime for all positive integer values of \(n\). Show that John is wrong in his assertion.
OCR MEI C3 Q2
2
  1. Show that \(\mathrm { f } ( x ) = \left| x ^ { 3 } \right|\) is an even function.
  2. It is suggested that the function \(\mathrm { g } ( x ) = ( x - 1 ) ^ { 3 }\) is odd. Prove that this is false.
OCR MEI C3 Q4
4 The volume of a sphere, \(V \mathrm {~cm} ^ { 3 }\) is given by the formula \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) where \(r \mathrm {~cm}\) is the radius.
The radius of a sphere increases at a constant rate of 2 cm per second.
Find the rate of increase of \(V\) when \(r = 10 \mathrm {~cm}\).
OCR MEI C3 Q5
5 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } = 25\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { y }\).
  2. Hence find the equation of the normal to the circle at the point ( 3,4 ).
OCR MEI C3 Q6
6
  1. Find \(\int x \cos 2 x d x\).
  2. Using the substitution \(u = x ^ { 2 } + 1\), or otherwise, find the exact value of \(\int _ { 2 } ^ { 3 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x\).
OCR MEI C3 Q7
7 Fig. 7 shows the graphs of the curves \(y = \mathrm { e } ^ { - x }\) and \(y = \mathrm { e } ^ { - x } \sin x\) for \(0 \leq x \leq \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-3_407_793_1085_740} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The maximum point on \(y = \mathrm { e } ^ { - x } \sin x\) is at A , and the curves touch at B .
\(\mathrm { A } ^ { \prime }\) and \(\mathrm { B } ^ { \prime }\) are the points on the \(x\)-axis such that \(\mathrm { A } ^ { \prime } \mathrm { A }\) and \(\mathrm { B } ^ { \prime } \mathrm { B }\) are parallel to the \(y\)-axis.
Show that \(\mathrm { OA } ^ { \prime } = \mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }\).
OCR MEI C3 Q8
8 Fig. 8 shows part of the graph of the function \(y = 5 x ( 2 x - 1 ) ^ { 3 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-4_508_803_450_703} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the \(x\)-coordinate of S , the turning point of the curve.
  2. Find the area of the shaded region enclosed between the curve and the \(x\)-axis.
  3. Given that \(\mathrm { f } ( x ) = 5 x ( 2 x - 1 ) ^ { 3 }\), show that \(\mathrm { f } ( x + 0.5 ) = 40 x ^ { 3 } ( x + 0.5 )\).
  4. Find \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } 40 x ^ { 3 } ( x + 0.5 ) \mathrm { d } x\).
  5. Explain, with the aid of a sketch, the connection between your answer to parts (ii) and (iv).
OCR MEI C3 Q2
2
  1. Expand \(\left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 }\).
  2. Hence find \(\int \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 } \mathrm {~d} x\).
OCR MEI C3 Q3
3
  1. Sketch the graph of \(y = | 3 x - 6 |\).
  2. Solve the equation \(| 3 x - 6 | = x + 4\) and illustrate your answer on your graph.
    \(4 \quad\) Find \(\int x \sin 3 x \mathrm {~d} x\).
    \(5 \quad\) Make \(x\) the subject of \(t = \ln \sqrt { \frac { 5 } { ( x - 3 ) } }\).
OCR MEI C3 Q6
6 The function \(\mathrm { f } ( x )\) is defined as \(\mathrm { f } ( x ) = \frac { \ln x } { x }\). The graph of the function is shown in Fig. 6 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c7998a08-229a-40d2-ba34-b5f264139295-2_369_675_1930_689} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Give the coordinates of the point, P , where the curve crosses the \(x\)-axis.
  2. Use calculus to find the coordinates of the stationary point, Q , and show that it is a maximum.
OCR MEI C3 Q7
7 An oil slick is circular with radius \(r \mathrm {~km}\) and area \(A \mathrm {~km} ^ { 2 }\). The radius increases with time at a rate given by \(\frac { \mathrm { d } r } { \mathrm {~d} t } = 0.5\), in kilometres per hour.
  1. Show that \(\frac { \mathrm { dA } } { \mathrm { d } t } = \pi r\).
  2. Find the rate of increase of the area of the slick at a time when the radius is 6 km .
OCR MEI C3 Q8
8 Fig. 8 shows the graph of \(y = x \sqrt { 1 + x }\). The point P on the curve is on the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c7998a08-229a-40d2-ba34-b5f264139295-3_433_800_895_587} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down the coordinates of P .
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x + 2 } { 2 \sqrt { 1 + x } }\).
  3. Hence find the coordinates of the turning point on the curve. What can you say about the gradient of the curve at P ?
  4. By using a suitable substitution, show that \(\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u\). Evaluate this integral, giving your answer in an exact form.
    What does this value represent?
  5. Use your answer to part (ii) to differentiate \(y = x \sqrt { 1 + x } \sin 2 x\) with respect to \(x\).
    (You need not simplify your result.)