Questions — OCR MEI (4301 questions)

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OCR MEI C3 Q3
19 marks Standard +0.3
3 A curve is defined by the equation \(y = 2 x \ln ( 1 + x )\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence verify that the origin is a stationary point of the curve.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that the origin is a minimum point.
  3. Using the substitution \(u = 1 + x\), show that \(\int \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x = \int \left( u - 2 + \frac { 1 } { u } \right) \mathrm { d } u\). Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x\), giving your answer in an exact form.
  4. Using integration by parts and your answer to part (iii), evaluate \(\int _ { 0 } ^ { 1 } 2 x \ln ( 1 + x ) \mathrm { d } x\).
OCR MEI C3 Q1
18 marks Standard +0.3
1 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt { 4 - x ^ { 2 } }\) for \(- 2 \leqslant x \leqslant 2\).
  1. Show that the curve \(y = \sqrt { 4 - x ^ { 2 } }\) is a semicircle of radius 2 , and explain why it is not the whole of this circle. Fig. 9 shows a point \(\mathrm { P } ( a , b )\) on the semicircle. The tangent at P is shown. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9e68f5e0-3394-4962-acd9-25bb31f09f2b-1_628_935_728_657} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure}
  2. (A) Use the gradient of OP to find the gradient of the tangent at P in terms of \(a\) and \(b\).
    (B) Differentiate \(\sqrt { 4 - x ^ { 2 } }\) and deduce the value of \(\mathrm { f } ^ { \prime } ( a )\).
    (C) Show that your answers to parts (A) and (B) are equivalent. The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } ( x - 2 )\), for \(0 \leqslant x \leqslant 4\).
  3. Describe a sequence of two transformations that would map the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { g } ( x )\). Hence sketch the curve \(y = \mathrm { g } ( x )\).
  4. Show that if \(y = \mathrm { g } ( x )\) then \(9 x ^ { 2 } + y ^ { 2 } = 36 x\).
OCR MEI C3 Q2
18 marks Standard +0.3
2 Fig. 7 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x \sqrt { 1 + x }\). The curve meets the \(x\)-axis at the origin and at the point P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9e68f5e0-3394-4962-acd9-25bb31f09f2b-2_487_875_487_624} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Verify that the point P has coordinates \(( - 1,0 )\). Hence state the domain of the function \(\mathrm { f } ( x )\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + 3 x } { 2 \sqrt { 1 + x } }\).
  3. Find the exact coordinates of the turning point of the curve. Hence write down the range of the function.
  4. Use the substitution \(u = 1 + x\) to show that $$\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( \begin{array} { l l } u ^ { \frac { 3 } { 2 } } & u ^ { \frac { 1 } { 2 } } \end{array} \right) \mathrm { d } u .$$ Hence find the area of the region enclosed by the curve and the \(x\)-axis.
OCR MEI C3 Q3
16 marks Standard +0.3
3 Fig. 7 shows the curve \(y = \frac { x ^ { 2 } } { 1 + 2 x ^ { 3 } }\). It is undefined at \(x = a\); the line \(x = a\) is a vertical asymptote. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9e68f5e0-3394-4962-acd9-25bb31f09f2b-3_654_1034_463_531} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the value of \(a\), giving your answer correct to 3 significant figures.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 2 x ^ { 4 } } { \left( 1 + 2 x ^ { 3 } \right) ^ { 2 } }\). Hence determine the coordinates of the turning points of the curve.
  3. Show that the area of the region between the curve and the \(x\)-axis from \(x = 0\) to \(x = 1\) is \(\frac { 1 } { 6 } \ln 3\).
OCR MEI C3 Q1
20 marks Standard +0.3
1 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aee8da6a-7d5c-442f-9729-55d81d9a606f-1_427_968_432_584} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the exact coordinates of P .
  2. Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
  3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  4. Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
  5. Find the gradient of the curve at the origin. Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
  6. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.
OCR MEI C3 Q2
17 marks Standard +0.3
2 Fig. 8 shows part of the curve \(y = x \sin 3 x\). It crosses the \(x\)-axis at P . The point on the curve with \(x\)-coordinate \(\frac { 1 } { 6 } \pi\) is Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aee8da6a-7d5c-442f-9729-55d81d9a606f-2_418_769_516_673} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the \(x\)-coordinate of P .
  2. Show that Q lies on the line \(y = x\).
  3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
  4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).
OCR MEI C3 Q3
19 marks Moderate -0.3
3 The function \(f ( x ) = \ln \left( t + x ^ { 2 } \right)\) has domain \(- 3 \leqslant x \leqslant 3\).
Fig. 9 shows the graph of \(y = f ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aee8da6a-7d5c-442f-9729-55d81d9a606f-3_510_895_523_604} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Show algebraically that the function is even. State how this property relates to the shape of the curve.
  2. Find the gradient of the curve at the point \(\mathrm { P } ( 2 , \ln 5 )\).
  3. Explain why the function does not have an inverse for the domain \(- 3 \leqslant x \leqslant 3\). The domain of \(f ( x )\) is now restricted to \(0 \leqslant x \leqslant 3\). The inverse of \(f ( x )\) is the function \(g ( x )\).
  4. Sketch the curves \(y = f ( x )\) and \(y = g ( x )\) on the same axes. State the domain of the function \(g ( x )\),
    Show that \(\mathrm { g } ( x ) = \sqrt { \mathrm { e } ^ { x } - 1 }\).
  5. Differentiate \(\mathrm { g } ( \mathrm { x } )\). Hence verify that \(\mathrm { g } ^ { \prime } ( \ln 5 ) = 1 \frac { 1 } { 4 }\). Explain the connection between this result and your answer to part (ii).
OCR MEI C3 Q1
5 marks Standard +0.3
1 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]{431d496a-a606-4b92-9f5c-e12b074a7ba9-1_295_403_542_871} \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 Q2
5 marks Standard +0.3
2 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 Q3
5 marks Moderate -0.3
3 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 Q4
5 marks Standard +0.3
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]{431d496a-a606-4b92-9f5c-e12b074a7ba9-2_414_379_485_838} \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 Q5
18 marks Moderate -0.3
5 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]{431d496a-a606-4b92-9f5c-e12b074a7ba9-3_921_1398_538_414} \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 \(x = - \frac { 1 } { 4 } \pi\).
OCR MEI C3 Q6
7 marks Moderate -0.3
6 When the gas in a balloon is kept at a constant temperature, the pressure \(P\) in atmospheres and the volume \(V \mathrm {~m} ^ { 3 }\) are related by the equation $$P = \frac { k } { V } ,$$ where \(k\) is a constant. [This is known as Boyle's Law.]
When the volume is \(100 \mathrm {~m} ^ { 3 }\), the pressure is 5 atmospheres, and the volume is increasing at a rate of \(10 \mathrm {~m} ^ { 3 }\) per second.
  1. Show that \(k = 500\).
  2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} V }\) in terms of \(V\).
  3. Find the rate at which the pressure is decreasing when \(V = 100\).
OCR MEI C3 Q7
7 marks Standard +0.3
7 Fig. 4 shows a cone. The angle between the axis and the slant edge is \(30 ^ { \circ }\). Water is poured into the cone at a constant rate of \(2 \mathrm {~cm} ^ { 3 }\) per second. At time \(t\) seconds, the radius of the water surface is \(r \mathrm {~cm}\) and the volume of water in the cone is \(V \mathrm {~cm} ^ { 3 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{431d496a-a606-4b92-9f5c-e12b074a7ba9-4_363_391_1447_887} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Write down the value of \(\frac { \mathrm { d } V } { \mathrm {~d} t }\).
  2. Show that \(V = \frac { \sqrt { 3 } } { 3 } \pi r ^ { 3 }\), and find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\).
    [0pt] [You may assume that the volume of a cone of height \(h\) and radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  3. Use the results of parts (i) and (ii) to find the value of \(\frac { \mathrm { d } r } { \mathrm {~d} t }\) when \(r = 2\).
OCR MEI C3 Q8
6 marks Moderate -0.8
8 Fig. 4 is a diagram of a garden pond. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{431d496a-a606-4b92-9f5c-e12b074a7ba9-5_295_742_410_693} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The volume \(V \mathrm {~m} ^ { 3 }\) of water in the pond when the depth is \(h\) metres is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 3 - h ) .$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} h }\). Water is poured into the pond at the rate of \(0.02 \mathrm {~m} ^ { 3 }\) per minute.
  2. Find the value of \(\frac { \mathrm { d } h } { \mathrm {~d} t }\) when \(h = 0.4\).
OCR MEI C3 Q1
4 marks Easy -1.2
1
  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]{8350e810-3ceb-4876-a7a8-249e17616057-1_718_813_567_644} \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 Q3
6 marks Standard +0.3
3 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]{8350e810-3ceb-4876-a7a8-249e17616057-2_433_716_569_710}

  2. \includegraphics[max width=\textwidth, alt={}, center]{8350e810-3ceb-4876-a7a8-249e17616057-2_396_612_1130_761}
OCR MEI C3 Q4
6 marks Standard +0.3
4 Fig. 4 shows the curve \(y = f ( x )\), where \(f ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8350e810-3ceb-4876-a7a8-249e17616057-3_480_573_410_785} \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 Q5
4 marks Moderate -0.8
5 You are given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are odd functions, defined for \(x \in \mathbb { R }\).
  1. Given that \(\mathrm { s } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )\), prove that \(\mathrm { s } ( x )\) is an odd function.
  2. Given that \(\mathrm { p } ( x ) = \mathrm { f } ( x ) \mathrm { g } ( x )\), determine whether \(\mathrm { p } ( x )\) is odd, even or neither.
OCR MEI C3 Q6
5 marks Moderate -0.8
6
  1. State the algebraic condition for the function \(\mathrm { f } ( x )\) to be an even function.
    What geometrical property does the graph of an even function have?
  2. State whether the following functions are odd, even or neither.
    (A) \(\mathrm { f } ( x ) = x ^ { 2 } - 3\)
    (B) \(\mathrm { g } ( x ) = \sin x + \cos x\)
    (C) \(\mathrm { h } ( x ) = \frac { 1 } { x + x ^ { 3 } }\)
OCR MEI C3 Q7
18 marks Standard +0.8
7 Fig. 8 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { - \frac { 1 } { 5 } x } \sin x\), for all \(x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8350e810-3ceb-4876-a7a8-249e17616057-4_645_1100_461_516} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Sketch the graphs of
    (A) \(y = \mathrm { f } ( 2 x )\),
    (B) \(y = \mathrm { f } ( x + \pi )\).
  2. Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\). Hence find the coordinates of P .
  3. Show that \(\mathrm { f } ( x + \pi ) = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \mathrm { f } ( x )\). Hence, using the substitution \(u = x - \pi\), show that $$\int _ { \pi } ^ { 2 \pi } \mathrm { f } ( x ) \mathrm { d } x = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \int _ { 0 } ^ { \pi } \mathrm { f } ( u ) \mathrm { d } u .$$ Interpret this result graphically. [You should not attempt to integrate \(\mathrm { f } ( x )\).]
OCR MEI C3 Q2
18 marks Standard +0.3
2 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]{27f6c723-b199-48f1-ab18-22cc0b4b017b-2_866_979_576_573} \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.
OCR MEI C3 Q3
8 marks Standard +0.3
3 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]{27f6c723-b199-48f1-ab18-22cc0b4b017b-3_864_844_465_688} \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.
OCR MEI C3 Q4
3 marks Easy -1.2
4 Given that \(\mathrm { f } ( x ) = 2 \ln x\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { x }\), find the composite function \(\mathrm { gf } ( x )\), expressing your answer as simply as possible.
OCR MEI C3 Q5
4 marks Standard +0.3
5 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.