Questions C3 (1200 questions)

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OCR MEI C3 2016 June Q1
1 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 1 + \cos \frac { 1 } { 2 } x \right) \mathrm { d } x\).
OCR MEI C3 2016 June Q2
2 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by \(\mathrm { f } ( x ) = \ln x\) and \(\mathrm { g } ( x ) = 2 + \mathrm { e } ^ { x }\), for \(x > 0\).
Find the exact value of \(x\), given that \(\mathrm { fg } ( x ) = 2 x\).
OCR MEI C3 2016 June Q3
3 Find \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\), giving your answer in an exact form.
OCR MEI C3 2016 June Q4
4 By sketching the graphs of \(y = | 2 x + 1 |\) and \(y = - x\) on the same axes, show that the equation \(| 2 x + 1 | = - x\) has two roots. Find these roots.
OCR MEI C3 2016 June Q5
5 The volume \(V \mathrm {~m} ^ { 3 }\) of a pile of grain of height \(h\) metres is modelled by the equation $$V = 4 \sqrt { h ^ { 3 } + 1 } - 4$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 2\). At a certain time, the height of the pile is 2 metres, and grain is being added so that the volume is increasing at a rate of \(0.4 \mathrm {~m} ^ { 3 }\) per minute.
  2. Find the rate at which the height is increasing at this time.
OCR MEI C3 2016 June Q6
6 Fig. 6 shows part of the curve \(\sin 2 y = x - 1\). P is the point with coordinates \(\left( 1.5 , \frac { 1 } { 12 } \pi \right)\) on the curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414a6b7f-cd96-4fa0-9521-ebe500bab375-2_458_691_1610_687} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). Hence find the exact gradient of the curve \(\sin 2 y = x - 1\) at the point P . The part of the curve shown is the image of the curve \(y = \arcsin x\) under a sequence of two geometrical transformations.
  2. Find \(y\) in terms of \(x\) for the curve \(\sin 2 y = x - 1\). Hence describe fully the sequence of transformations.
    \(7 \quad\) You are given that \(n\) is a positive integer.
    By expressing \(x ^ { 2 n } - 1\) as a product of two factors, prove that \(2 ^ { 2 n } - 1\) is divisible by 3 . Section B (36 marks)
OCR MEI C3 2016 June Q8
8 Fig. 8 shows the curve \(y = \frac { x } { \sqrt { x + 4 } }\) and the line \(x = 5\). The curve has an asymptote \(l\).
The tangent to the curve at the origin O crosses the line \(l\) at P and the line \(x = 5\) at Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414a6b7f-cd96-4fa0-9521-ebe500bab375-3_643_921_703_573} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that for this curve \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + 8 } { 2 ( x + 4 ) ^ { \frac { 3 } { 2 } } }\).
  2. Find the coordinates of the point P .
  3. Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line \(x = 5\).
OCR MEI C3 2016 June Q9
9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x }\) and \(k\) is a constant greater than 1 . The curve crosses the \(y\)-axis at P and has a turning point Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414a6b7f-cd96-4fa0-9521-ebe500bab375-4_783_951_392_557} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the \(y\)-coordinate of P in terms of \(k\).
  2. Show that the \(x\)-coordinate of Q is \(\frac { 1 } { 4 } \ln k\), and find the \(y\)-coordinate in its simplest form.
  3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \ln k\). Give your answer in the form \(a k + b\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } \left( x + \frac { 1 } { 4 } \ln k \right)\).
  4. (A) Show that \(\mathrm { g } ( x ) = \sqrt { k } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)\).
    (B) Hence show that \(\mathrm { g } ( x )\) is an even function.
    (C) Deduce, with reasons, a geometrical property of the curve \(y = \mathrm { f } ( x )\). \section*{END OF QUESTION PAPER}
OCR MEI C3 Q1
1 Solve the equation \(| 3 x + 2 | = 1\).
OCR MEI C3 Q7
7 Fig. 7 shows the curve defined implicitly by the equation $$y ^ { 2 } + y = x ^ { 3 } + 2 x ,$$ together with the line \(x = 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{185840c8-2799-44cd-a6d8-00d10c038c2c-03_465_378_534_808} \captionsetup{labelformat=empty} \caption{Not to scale}
\end{figure} Fig. 7 Find the coordinates of the points of intersection of the line and the curve.
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.
OCR MEI C3 Q8
8 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]{185840c8-2799-44cd-a6d8-00d10c038c2c-03_421_789_1748_610} \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 Q9
9 The function \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\) has domain \(- 3 \leqslant x \leqslant 3\).
Fig. 9 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{185840c8-2799-44cd-a6d8-00d10c038c2c-04_540_943_477_550} \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 \(\mathrm { f } ( x )\) is now restricted to \(0 \leqslant x \leqslant 3\). The inverse of \(\mathrm { f } ( x )\) is the function \(\mathrm { g } ( x )\).
  4. Sketch the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) on the same axes. State the domain of the function \(\mathrm { g } ( x )\). Show that \(\mathrm { g } ( x ) = \sqrt { \mathrm { e } ^ { x } - 1 }\).
  5. Differentiate \(\mathrm { g } ( x )\). Hence verify that \(\mathrm { g } ^ { \prime } ( \ln 5 ) = 1 \frac { 1 } { 4 }\). Explain the connection between this result and your answer to part (ii). \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} MEI STRUCTURED MATHEMATICS \section*{4753/1} Methods for Advanced Mathematics (C3)
    Wednesday
AQA C3 2007 January Q1
1 Use the mid-ordinate rule with four strips of equal width to find an estimate for \(\int _ { 1 } ^ { 5 } \frac { 1 } { 1 + \ln x } \mathrm {~d} x\), giving your answer to three significant figures.
(4 marks)
AQA C3 2007 January Q2
2 Describe a sequence of two geometrical transformations that maps the graph of \(y = \sec x\) onto the graph of \(y = 1 + \sec 3 x\).
AQA C3 2007 January Q3
3 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 - x ^ { 2 } , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 2 } { x + 1 } , & \text { for real values of } x , x \neq - 1 \end{array}$$
  1. Find the range of f.
  2. The inverse of g is \(\mathrm { g } ^ { - 1 }\).
    1. Find \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the range of \(\mathrm { g } ^ { - 1 }\).
  3. The composite function gf is denoted by h .
    1. Find \(\mathrm { h } ( x )\), simplifying your answer.
    2. State the greatest possible domain of h .
AQA C3 2007 January Q4
4
  1. Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
  2. Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
  3. The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694} The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
AQA C3 2007 January Q5
5
    1. Show that the equation $$2 \cot ^ { 2 } x + 5 \operatorname { cosec } x = 10$$ can be written in the form \(2 \operatorname { cosec } ^ { 2 } x + 5 \operatorname { cosec } x - 12 = 0\).
    2. Hence show that \(\sin x = - \frac { 1 } { 4 }\) or \(\sin x = \frac { 2 } { 3 }\).
  2. Hence, or otherwise, solve the equation $$2 \cot ^ { 2 } ( \theta - 0.1 ) + 5 \operatorname { cosec } ( \theta - 0.1 ) = 10$$ giving all values of \(\theta\) in radians to two decimal places in the interval \(- \pi < \theta < \pi\).
    (3 marks)
AQA C3 2007 January Q6
6
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
    1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
    2. \(y = x ^ { 2 } \tan x\).
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
    2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).
AQA C3 2007 January Q7
7
  1. Sketch the graph of \(y = | 2 x |\).
  2. On a separate diagram, sketch the graph of \(y = 4 - | 2 x |\), indicating the coordinates of the points where the graph crosses the coordinate axes.
  3. Solve \(4 - | 2 x | = x\).
  4. Hence, or otherwise, solve the inequality \(4 - | 2 x | > x\).
AQA C3 2007 January Q8
8 The diagram shows the curve \(y = \cos ^ { - 1 } x\) for \(- 1 \leqslant x \leqslant 1\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-4_492_698_1640_671}
  1. Write down the exact coordinates of the points \(A\) and \(B\).
  2. The equation \(\cos ^ { - 1 } x = 3 x + 1\) has only one root. Given that the root of this equation is \(\alpha\), show that \(0.1 \leqslant \alpha \leqslant 0.2\).
  3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 3 } \left( \cos ^ { - 1 } x _ { n } - 1 \right)\) with \(x _ { 1 } = 0.1\) to find the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to three decimal places.
AQA C3 2007 January Q9
9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}
    1. Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
      (2 marks)
    2. Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
    1. Write down the \(y\)-coordinate of \(A\).
    2. Show that \(x = \ln 2\) at \(B\).
  3. Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
  4. Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.
AQA C3 2008 January Q1
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
    1. \(y = \left( 2 x ^ { 2 } - 5 x + 1 \right) ^ { 20 }\);
    2. \(y = x \cos x\).
  2. Given that $$y = \frac { x ^ { 3 } } { x - 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k x ^ { 2 } ( x - 3 ) } { ( x - 2 ) ^ { 2 } }$$ where \(k\) is a positive integer.
AQA C3 2008 January Q2
2
  1. Solve the equation \(\cot x = 2\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
  2. Show that the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\) can be written as $$2 \cot ^ { 2 } x - 3 \cot x - 2 = 0$$
  3. Solve the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
AQA C3 2008 January Q3
3 The equation $$x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0$$ has a single root, \(\alpha\).
  1. Show that \(\alpha\) lies between - 0.33 and - 0.32 .
  2. Show that the equation \(x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0\) can be rearranged into the form $$x = \frac { 1 } { 3 } \left( x ^ { 4 } - 1 \right)$$
  3. Use the iteration \(x _ { n + 1 } = \frac { \left( x _ { n } ^ { 4 } - 1 \right) } { 3 }\) with \(x _ { 1 } = - 0.3\) to find \(x _ { 4 }\), giving your answer to three significant figures.
AQA C3 2008 January Q4
4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 3 } , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { x - 3 } , & \text { for real values of } x , x \neq 3 \end{array}$$
  1. State the range of f.
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 64\).
    1. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the range of \(\mathrm { g } ^ { - 1 }\).