Questions — Edexcel (10514 questions)

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Edexcel C3 2010 June Q4
10 marks Moderate -0.8
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
12 marks Standard +0.3
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
10 marks Moderate -0.3
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
15 marks Standard +0.3
7.
  1. 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.
    1. Find the maximum value of \(2 \sin \theta - 1.5 \cos \theta\).
    2. 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.
  3. Calculate the maximum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this maximum occurs.
  4. 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
7 marks Moderate -0.8
8.
  1. Simplify fully $$\frac { 2 x ^ { 2 } + 9 x - 5 } { x ^ { 2 } + 2 x - 15 }$$ Given that $$\ln \left( 2 x ^ { 2 } + 9 x - 5 \right) = 1 + \ln \left( x ^ { 2 } + 2 x - 15 \right) , \quad x \neq - 5$$
  2. find \(x\) in terms of e.
Edexcel C3 2011 June Q1
5 marks Moderate -0.8
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
6 marks Moderate -0.3
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
8 marks Moderate -0.3
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
11 marks Moderate -0.3
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
12 marks Standard +0.3
6.
  1. Prove that $$\frac { 1 } { \sin 2 \theta } - \frac { \cos 2 \theta } { \sin 2 \theta } = \tan \theta , \quad \theta \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
  2. 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
13 marks Standard +0.3
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
12 marks Standard +0.3
  1. 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. $$\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } \cos 3 x$$
  2. 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).
  3. 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
4 marks Moderate -0.8
  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
9 marks Moderate -0.3
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
9 marks Standard +0.3
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 \(x = 0\)
Edexcel C3 2012 June Q4
7 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3fbdfb55-5dd5-44ab-b031-d39e64bdfc3b-06_560_1145_210_386} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows part of the curve with equation \(y = \mathrm { f } ( x )\) The curve passes through the points \(P ( - 1.5,0 )\) and \(Q ( 0,5 )\) as shown.
On separate diagrams, sketch the curve with equation
  1. \(y = | f ( x ) |\)
  2. \(y = \mathrm { f } ( | x | )\)
  3. \(y = 2 f ( 3 x )\) Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
Edexcel C3 2012 June Q5
9 marks Standard +0.3
  1. Express \(4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
  2. Hence show that $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta$$
  3. Hence or otherwise solve, for \(0 < \theta < \pi\), $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = 4$$ giving your answers in terms of \(\pi\).
Edexcel C3 2012 June Q6
14 marks Moderate -0.3
6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \mapsto \ln x , \quad x > 0 \end{aligned}$$
  1. State the range of f.
  2. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
  3. Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
  4. Find \(\mathrm { f } ^ { - 1 }\), the inverse function of f , stating its domain.
  5. On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.
Edexcel C3 2012 June Q7
11 marks Moderate -0.3
  1. Differentiate with respect to \(x\),
    1. \(x ^ { \frac { 1 } { 2 } } \ln ( 3 x )\)
    2. \(\frac { 1 - 10 x } { ( 2 x - 1 ) ^ { 5 } }\), giving your answer in its simplest form.
  2. Given that \(x = 3 \tan 2 y\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
Edexcel C3 2013 June Q1
4 marks Moderate -0.5
  1. Express
$$\frac { 3 x + 5 } { x ^ { 2 } + x - 12 } - \frac { 2 } { x - 3 }$$ as a single fraction in its simplest form.
Edexcel C3 2013 June Q2
5 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-03_499_1099_210_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x > 0\), where f is an increasing function of \(x\). The curve crosses the \(x\)-axis at the point \(( 1,0 )\) and the line \(x = 0\) is an asymptote to the curve. On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( 2 x ) , x > 0\)
  2. \(y = | \mathrm { f } ( x ) | , x > 0\) Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the \(x\)-axis.
Edexcel C3 2013 June Q3
10 marks Standard +0.3
3. $$f ( x ) = 7 \cos x + \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),
  1. find the exact value of \(R\) and the value of \(\alpha\) to one decimal place.
  2. Hence solve the equation $$7 \cos x + \sin x = 5$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
  3. State the values of \(k\) for which the equation $$7 \cos x + \sin x = k$$ has only one solution in the interval \(0 \leqslant x < 360 ^ { \circ }\)
Edexcel C3 2013 June Q4
11 marks Moderate -0.8
  1. The functions f and g are defined by
$$\begin{array} { l l } \mathrm { f } : x \mapsto 2 | x | + 3 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto 3 - 4 x , & x \in \mathbb { R } \end{array}$$
  1. State the range of f.
  2. Find \(\mathrm { fg } ( 1 )\).
  3. Find \(\mathrm { g } ^ { - 1 }\), the inverse function of g .
  4. Solve the equation $$\operatorname { gg } ( x ) + [ \mathrm { g } ( x ) ] ^ { 2 } = 0$$
Edexcel C3 2013 June Q5
10 marks Moderate -0.3
5.
  1. Differentiate $$\frac { \cos 2 x } { \sqrt { x } }$$ with respect to \(x\).
  2. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { 2 } 3 x \right)\) can be written in the form $$\mu \left( \tan 3 x + \tan ^ { 3 } 3 x \right)$$ where \(\mu\) is a constant.
  3. Given \(x = 2 \sin \left( \frac { y } { 3 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\), simplifying your answer.
Edexcel C3 2013 June Q6
9 marks Standard +0.3
  1. Use an appropriate double angle formula to show that $$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$ and state the value of the constant \(\lambda\).
  2. Solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$ You must show all your working. Give your answers in terms of \(\pi\).