Questions C3 (1200 questions)

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Edexcel C3 2009 June Q2
2. (a) Use the identity \(\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1\) to prove that \(\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1\).
(b) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta + 4 \sec \theta + \sec ^ { 2 } \theta = 2$$
Edexcel C3 2009 June Q3
  1. Rabbits were introduced onto an island. The number of rabbits, \(P , t\) years after they were introduced is modelled by the equation
$$P = 80 \mathrm { e } ^ { \frac { 1 } { 5 } t } , \quad t \in \mathbb { R } , t \geqslant 0$$
  1. Write down the number of rabbits that were introduced to the island.
  2. Find the number of years it would take for the number of rabbits to first exceed 1000.
  3. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\).
  4. Find \(P\) when \(\frac { \mathrm { d } P } { \mathrm {~d} t } = 50\).
Edexcel C3 2009 June Q4
4. (i) Differentiate with respect to \(x\)
  1. \(x ^ { 2 } \cos 3 x\)
  2. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\)
    (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C3 2009 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bcb0c693-66ae-4b97-99f8-b10fb9396886-07_721_1217_237_397} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The curve meets the coordinate axes at the points \(A ( 0,1 - k )\) and \(B \left( \frac { 1 } { 2 } \ln k , 0 \right)\), where \(k\) is a constant and \(k > 1\), as shown in Figure 2. On separate diagrams, sketch the curve with equation
  1. \(y = | f ( x ) |\),
  2. \(y = \mathrm { f } ^ { - 1 } ( x )\). Show on each sketch the coordinates, in terms of \(k\), of each point at which the curve meets or cuts the axes. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - k\),
  3. state the range of f ,
  4. find \(\mathrm { f } ^ { - 1 } ( x )\),
  5. write down the domain of \(\mathrm { f } ^ { - 1 }\).
Edexcel C3 2009 June Q6
  1. (a) Use the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\), to show that
$$\cos 2 A = 1 - 2 \sin ^ { 2 } A$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 3 \sin 2 x
& C _ { 2 } : \quad y = 4 \sin ^ { 2 } x - 2 \cos 2 x \end{aligned}$$ (b) Show that the \(x\)-coordinates of the points where \(C _ { 1 }\) and \(C _ { 2 }\) intersect satisfy the equation $$4 \cos 2 x + 3 \sin 2 x = 2$$ (c) Express \(4 \cos 2 x + 3 \sin 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places.
(d) Hence find, for \(0 \leqslant x < 180 ^ { \circ }\), all the solutions of $$4 \cos 2 x + 3 \sin 2 x = 2$$ giving your answers to 1 decimal place.
Edexcel C3 2009 June Q7
7. The function f is defined by $$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$
  1. Show that \(\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }\) The function g is defined by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
  2. Differentiate \(\mathrm { g } ( x )\) to show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }\)
  3. Find the exact values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) = 1\)
Edexcel C3 2009 June Q8
8. (a) Write down \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
(b) Find, for \(0 < x < \pi\), all the solutions of the equation $$\operatorname { cosec } x - 8 \cos x = 0$$ giving your answers to 2 decimal places.
Edexcel C3 2010 June Q1
  1. (a) Show that
$$\frac { \sin 2 \theta } { 1 + \cos 2 \theta } = \tan \theta$$ (b) Hence find, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), all the solutions of $$\frac { 2 \sin 2 \theta } { 1 + \cos 2 \theta } = 1$$ Give your answers to 1 decimal place.
Edexcel C3 2010 June Q2
2. A curve \(C\) has equation $$y = \frac { 3 } { ( 5 - 3 x ) ^ { 2 } } , \quad x \neq \frac { 5 } { 3 }$$ The point \(P\) on \(C\) has \(x\)-coordinate 2. Find an equation of the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C3 2010 June Q3
3. \(\mathrm { f } ( x ) = 4 \operatorname { cosec } x - 4 x + 1\), where \(x\) is in radians.
  1. Show that there is a root \(\alpha\) of \(\mathrm { f } ( x ) = 0\) in the interval [1.2,1.3].
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = \frac { 1 } { \sin x } + \frac { 1 } { 4 }$$
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { \sin x _ { n } } + \frac { 1 } { 4 } , \quad x _ { 0 } = 1.25$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  4. By considering the change of sign of \(\mathrm { f } ( x )\) in a suitable interval, verify that \(\alpha = 1.291\) correct to 3 decimal places.
Edexcel C3 2010 June Q4
4. The function \(f\) is defined by $$f : x \mapsto | 2 x - 5 | , \quad x \in \mathbb { R }$$
  1. Sketch the graph with equation \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
  2. Solve \(\mathrm { f } ( x ) = 15 + x\). The function \(g\) is defined by $$g : x \mapsto x ^ { 2 } - 4 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 5$$
  3. Find fg(2).
  4. Find the range of g.
Edexcel C3 2010 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52f73407-14c5-46e6-b911-aa096b9b5893-08_701_1125_246_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with the equation \(y = \left( 2 x ^ { 2 } - 5 x + 2 \right) \mathrm { e } ^ { - x }\).
  1. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
  2. Show that \(C\) crosses the \(x\)-axis at \(x = 2\) and find the \(x\)-coordinate of the other point where \(C\) crosses the \(x\)-axis.
  3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  4. Hence find the exact coordinates of the turning points of \(C\).
Edexcel C3 2010 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52f73407-14c5-46e6-b911-aa096b9b5893-10_781_858_239_575} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with the equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve has a turning point at \(A ( 3 , - 4 )\) and also passes through the point \(( 0,5 )\).
  1. Write down the coordinates of the point to which \(A\) is transformed on the curve with equation
    1. \(y = | \mathrm { f } ( x ) |\),
    2. \(y = 2 f \left( \frac { 1 } { 2 } x \right)\).
  2. Sketch the curve with equation $$y = \mathrm { f } ( | x | )$$ On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the \(y\)-axis. The curve with equation \(y = \mathrm { f } ( x )\) is a translation of the curve with equation \(y = x ^ { 2 }\).
  3. Find \(\mathrm { f } ( x )\).
  4. Explain why the function f does not have an inverse.
Edexcel C3 2010 June Q7
7. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 4 decimal places.
(b) (i) Find the maximum value of \(2 \sin \theta - 1.5 \cos \theta\).
(ii) Find the value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum occurs. Tom models the height of sea water, \(H\) metres, on a particular day by the equation $$H = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) hours is the number of hours after midday.
(c) Calculate the maximum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this maximum occurs.
(d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.
Edexcel C3 2010 June Q8
8. (a) Simplify fully $$\frac { 2 x ^ { 2 } + 9 x - 5 } { x ^ { 2 } + 2 x - 15 }$$ Given that $$\ln \left( 2 x ^ { 2 } + 9 x - 5 \right) = 1 + \ln \left( x ^ { 2 } + 2 x - 15 \right) , \quad x \neq - 5$$ (b) find \(x\) in terms of e.
Edexcel C3 2011 June Q1
Differentiate with respect to \(x\)
  1. \(\quad \ln \left( x ^ { 2 } + 3 x + 5 \right)\)
  2. \(\frac { \cos x } { x ^ { 2 } }\)
Edexcel C3 2011 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a0c2a69f-1196-4a07-a368-5dab3efaf316-04_460_725_260_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(R ( 4 , - 3 )\), as shown in Figure 1. Sketch, on separate diagrams, the graphs of
  1. \(y = 2 \mathrm { f } ( x + 4 )\),
  2. \(y = | \mathrm { f } ( - x ) |\). On each diagram, show the coordinates of the point corresponding to \(R\).
Edexcel C3 2011 June Q4
4. The function \(f\) is defined by $$\mathrm { f } : x \mapsto 4 - \ln ( x + 2 ) , \quad x \in \mathbb { R } , x \geqslant - 1$$
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Find the domain of \(\mathrm { f } ^ { - 1 }\). The function \(g\) is defined by $$\mathrm { g } : x \mapsto \mathrm { e } ^ { x ^ { 2 } } - 2 , \quad x \in \mathbb { R }$$
  3. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  4. Find the range of fg.
Edexcel C3 2011 June Q5
5. The mass, \(m\) grams, of a leaf \(t\) days after it has been picked from a tree is given by $$m = p \mathrm { e } ^ { - k t }$$ where \(k\) and \(p\) are positive constants.
When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.
  1. Write down the value of \(p\).
  2. Show that \(k = \frac { 1 } { 4 } \ln 3\).
  3. Find the value of \(t\) when \(\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3\).
Edexcel C3 2011 June Q6
6. (a) Prove that $$\frac { 1 } { \sin 2 \theta } - \frac { \cos 2 \theta } { \sin 2 \theta } = \tan \theta , \quad \theta \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise,
  1. show that \(\tan 15 ^ { \circ } = 2 - \sqrt { 3 }\),
  2. solve, for \(0 < x < 360 ^ { \circ }\), $$\operatorname { cosec } 4 x - \cot 4 x = 1$$
Edexcel C3 2011 June Q7
7. $$f ( x ) = \frac { 4 x - 5 } { ( 2 x + 1 ) ( x - 3 ) } - \frac { 2 x } { x ^ { 2 } - 9 } , \quad x \neq \pm 3 , x \neq - \frac { 1 } { 2 }$$
  1. Show that $$f ( x ) = \frac { 5 } { ( 2 x + 1 ) ( x + 3 ) }$$ The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The point \(P \left( - 1 , - \frac { 5 } { 2 } \right)\) lies on \(C\).
  2. Find an equation of the normal to \(C\) at \(P\).
Edexcel C3 2011 June Q8
  1. (a) Express \(2 \cos 3 x - 3 \sin 3 x\) in the form \(R \cos ( 3 x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your answers to 3 significant figures.
$$\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } \cos 3 x$$ (b) Show that \(\mathrm { f } ^ { \prime } ( x )\) can be written in the form $$\mathrm { f } ^ { \prime } ( x ) = R \mathrm { e } ^ { 2 x } \cos ( 3 x + \alpha )$$ where \(R\) and \(\alpha\) are the constants found in part (a).
(c) Hence, or otherwise, find the smallest positive value of \(x\) for which the curve with equation \(y = \mathrm { f } ( x )\) has a turning point.
Edexcel C3 2012 June Q1
  1. Express
$$\frac { 2 ( 3 x + 2 ) } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ as a single fraction in its simplest form.
Edexcel C3 2012 June Q2
2. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { } \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) , \quad x \neq - 3$$ The equation \(x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0\) has a single root which is between 1 and 2
  2. Use the iteration formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) , n \geqslant 0$$ with \(x _ { 0 } = 1\) to find, to 2 decimal places, the value of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). The root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
  3. By choosing a suitable interval, prove that \(\alpha = 1.272\) to 3 decimal places.
Edexcel C3 2012 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3fbdfb55-5dd5-44ab-b031-d39e64bdfc3b-04_538_953_251_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) which has equation $$y = \mathrm { e } ^ { x \sqrt { 3 } } \sin 3 x , \quad - \frac { \pi } { 3 } \leqslant x \leqslant \frac { \pi } { 3 }$$
  1. Find the \(x\) coordinate of the turning point \(P\) on \(C\), for which \(x > 0\) Give your answer as a multiple of \(\pi\).
  2. Find an equation of the normal to \(C\) at the point where \(x = 0\)