Questions — OCR MEI C3 (366 questions)

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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.)
OCR MEI C3 Q9
9 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = 2 x - 1$$ for all real values of \(x\).
  1. State the ranges of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\). Explain why \(\mathrm { f } ( x )\) has no inverse.
  2. Find an expression for the inverse function \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\). Sketch the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\) on the same axes.
  3. Find expressions for \(\operatorname { gf } ( x )\) and \(\operatorname { fg } ( x )\).
  4. Solve the equation \(\operatorname { gf } ( x ) = \mathrm { fg } ( x )\). Sketch the graphs of \(y = \operatorname { gf } ( x )\) and \(y = \operatorname { fg } ( x )\) on the same axes to illustrate your answer.
  5. Show that the equation \(\mathrm { f } ( x + a ) = \mathrm { g } ^ { 2 } ( x )\) has no solution if \(a > \frac { 1 } { 4 }\).
OCR MEI C3 Q1
1 Find \(\int \sqrt [ 3 ] { 2 x - 1 } \mathrm {~d} x\).
OCR MEI C3 Q2
2 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]{6ea594c5-52ba-4467-a098-cb66004b5a38-1_959_1469_748_317} \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 Q3
3 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } ( 1 - \sin 3 x ) \mathrm { d } x\), giving your answer in exact form.
OCR MEI C3 Q4
4 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]{6ea594c5-52ba-4467-a098-cb66004b5a38-2_431_977_728_602} \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.
OCR MEI C3 Q5
5 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 Q1
1 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]{75eebbfb-7bfa-4382-a6d7-1c5a7f3f419a-1_722_817_450_642} \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 \(\frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 } - 3 \right)\). 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.